Third Order Corrections to the Ground State Energy of a Bose Gas in the Gross–Pitaevskii Regime

Abstract

For a translation invariant system of N bosons in the Gross–Pitaevskii regime, we establish a precise bound for the ground state energy $E_N$. While the leading, order N, contribution to $E_N$ has been known since (Lieb et al. in Phys Rev A 61:043602, 2000; Lieb and Yngvason in Phys Rev Lett 80:2504–2507, 1998) and the second order corrections (of order one) have been first determined in Boccato et al. (Acta Math 222(2):219–335, 2019), our estimate also resolves the next term in the asymptotic expansion of $E_N$, which is of the order $(logN)/N$.

Introduction

In the Gross–Pitaevskii regime, we consider a gas of N bosons moving on the unit torus $\mathrm{\Lambda }\simeq {[0;1]}^3$, interacting through a repulsive potential with scattering length of the order 1/N. The Hamilton operator of such a system has the form

$$\begin{array}{c}\hfill H_N=\sum _{j=1}^N-{\mathrm{\Delta }}_{x_j}+\sum _{i<j}^N{N}^{2}V(N(x_i-x_j))\end{array}$$ 1.1

and acts, according to the bosonic statistics, on $L_s^2({\mathrm{\Lambda }}^N)$, the subspace of $L^2({\mathrm{\Lambda }}^N)$ consisting of functions that are symmetric w.r.t. permutations. Here, we are going to assume that $V\in L^3({\mathbb{R}}^3)$ is non-negative, compactly supported and spherically symmetric. We denote by $\mathfrak{a}$ its scattering length, which is defined through the solution f of the zero energy scattering equation

$$\begin{array}{c}\hfill \left[-,\mathrm{\Delta },+,\frac{1}{2},V\right]f=0\end{array}$$

with the boundary condition $f(x)\to 1$, as $|x|\to \infty$, by requiring that

$$\begin{array}{c}\hfill f(x)=1-\frac{\mathfrak{a}}{|x|}\end{array}$$

outside the support of V. By scaling, the scattering length of the potential $N^{2}V(N·)$ appearing in (1.1) is then given by $\mathfrak{a}/N$. Observe that, after rescaling $x\to Nx$, the Gross–Pitaevskii regime equivalently describes a gas of particles interacting through the fixed potential V, at density $\rho =N^{-2}$.

As first established in [29, 31], the ground state energy $E_N$ of (1.1) is given, to leading order, by

$$\begin{array}{c}\hfill E_N=4\pi \mathfrak{a}N+o(N)\end{array}$$

in the limit $N\to \infty$. In [28, 30, 34], it was also shown that the corresponding ground state vectors exhibit complete Bose–Einstein condensation; all particles, up to a fraction vanishing in the limit $N\to \infty$, can be described by the zero-momentum one-particle orbital ${\phi }_0\in L^2(\mathrm{\Lambda })$, defined by ${\phi }_0(x)=1$ for all $x\in \mathrm{\Lambda }$. In the last years, more precise bounds on the rate of condensation have been derived. For $V\in L^3({\mathbb{R}}^3)$, it was shown in [6] that, for any normalized sequence ${\psi }_N\in L_s^2({\mathrm{\Lambda }}^N)$ of approximate ground state vectors, satisfying

$$\begin{array}{c}\hfill ⟨{\psi }_N,H_N{\psi }_N⟩⩽E_N+K,\end{array}$$

the number of particles that are orthogonal to ${\phi }_0$ remains bounded by a constant proportional to K (but independent of N), in the limit $N\to \infty$ (a simplified proof of condensation has been recently proposed in [11], using the approach developed in [16]).

This optimal estimate on the rate of condensation was used in [6] as input for a rigorous version of Bogoliubov theory [8], showing that the ground state energy of (1.1) satisfies

$$\begin{array}{c}\hfill E_N=4\pi \mathfrak{a}(N-1)+e_{\mathrm{\Lambda }}{\mathfrak{a}}^2-\frac{1}{2}\sum _{p\in 2\pi {\mathbb{Z}}^3\backslash \{0\}}[p^2+8\pi \mathfrak{a}-\sqrt{{|p|}^4+16\pi \mathfrak{a}{p}^2}-\frac{{(8\pi \mathfrak{a})}^2}{2p^2}]+\mathcal{O}(N^{-1/4})\end{array}$$ 1.2

and that the spectrum of $H_N-E_N$ below a threshold $\zeta >0$ consists of eigenvalues having the form

$$\begin{array}{c}\hfill \sum _{p\in 2\pi {\mathbb{Z}}^3\backslash \{0\}}{n}_p\sqrt{{|p|}^4+16\pi \mathfrak{a}{p}^2}+\mathcal{O}((1+{\zeta }^3)N^{-1/4})\end{array}$$ 1.3

where $n_p\in \mathbb{N}$ for all $p\in 2\pi {\mathbb{Z}}^3\backslash \{0\}$. In (1.2), we defined

$$\begin{array}{c}\hfill e_{\mathrm{\Lambda }}=2-\underset{M\to \infty }{lim}\sum _{\begin{array}{c}p\in {\mathbb{Z}}^3\backslash \{0\}:\\ |p_1|,|p_2|,|p_3|⩽M\end{array}}\frac{cos|p|}{p^2}\end{array}$$ 1.4

The optimal bound on the condensation rate and the estimates (1.2), (1.3) on the low-energy spectrum of (1.1) have been later extended to Bose gases trapped by an external potentials in [14, 15, 33, 35], to bosons moving in a box with Neumann boundary conditions in [13], to systems interacting through a potential with scattering length of the order $N^{-1+\kappa }$, for sufficiently small $\kappa >0$, in [1, 12] and to Bose gases in the two-dimensional Gross–Pitaevskii regime in [18, 19]. Recently, an upper bound matching (1.2) was proven in [3], for particles interacting through a non-integrable, hard-sphere potential. New and simpler proofs of (1.2), (1.3) have been obtained in [25] and, very recently, in [16] (also beyond the Gross–Pitaevskii regime, for $\kappa >0$ small enough). Some rigorous bounds are also available at positive temperatures; to leading order, the free energy in the Gross–Pitaevskii regime was determined in [20], up to temperatures comparable with the critical temperature for condensation. Upper bounds for the free energy capturing also the next order corrections have been obtained in [7, 17].

Bogoliubov theory has been recently also used to determine equilibrium properties of Bose gases in the thermodynamic limit, where we consider N particles moving in the box ${[0;L]}^3$, with periodic boundary conditions, letting $N,L\to \infty$ keeping the density $\rho =N/L^3$ fixed. In [27], Lee–Huang–Yang derived a formula for the asymptotic behavior of the ground state energy per particle, in the dilute regime, to leading- and next-to-leading order. Their result was then improved by Wu in [38], by Hugenholtz-Pines in [26] and by Sawada in [37], who predicted that

$$\begin{array}{c}\hfill e(\rho )=\underset{\begin{array}{c}N,L\to \infty \\ N/L^3=\rho \end{array}}{lim}\frac{E_{N,L}}{N}=4\pi \mathfrak{a}\rho \left[1,+,\frac{128}{15\sqrt{\pi }},{(\rho {\mathfrak{a}}^3)}^{1/2},+,8,(,\frac{4}{3},\pi ,-,\sqrt{3},),\rho ,{\mathfrak{a}}^3,log,(12\pi \rho {\mathfrak{a}}^3),+,\cdots \right]\\ \hfill \end{array}$$ 1.5

up to lower order corrections, in the limit $\rho {\mathfrak{a}}^3\to 0$. The validity of the first term on the r.h.s. of (1.5) has been known since [21] (upper bound) and [31] (matching lower bound). As for the second term in (1.5) (the Lee–Huang–Yang correction), a lower bound was proven in [22] and, for more general interaction potentials (including hard-sphere interactions), in [23]. Recently, an optimal lower bound was also derived in [24] for the free energy at positive temperature (chosen so that the energy of thermal excitations is comparable to the Lee–Huang–Yang correction). As an upper bound, the first two terms in (1.5) were first established in [39]. A simpler proof, which applies to more general potentials (but not to hard-sphere interactions) was obtained in [4]. For the hard-sphere potential, on the other hand, the derivation of an upper bound matching (1.5) to second order is still an open problem. However, an upper bound establishing the validity of the first term on the r.h.s. of (1.5), with an error of the Lee–Huang–Yang order (but with the wrong constant) was recently proven in [2]. There is still no rigorous result about the third term on the r.h.s. of (1.5), neither as a lower nor as an upper bound to the ground state energy per particle.

In this paper, we improve (1.2), establishing the next contribution to the ground state energy in the Gross–Pitaevskii regime, which turns out to be of the order $(logN)/N$. The next theorem is our main result.

Theorem 1.1

Let $V\in L^3(\mathrm{\Lambda })$ be non-negative, spherically symmetric, and compactly supported. Let ${\mathrm{\Lambda }}_{+}^{\ast }=2\pi {\mathbb{Z}}^3\backslash \{0\}$. Then the ground state energy $E_N$ of the Hamiltonian $H_N$ from (1.1) satisfies

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill E_N=& 4\pi \mathfrak{a}(N-1)+e_{\mathrm{\Lambda }}{\mathfrak{a}}^2\hfill \\ \hfill & -\frac{1}{2}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}[p^2+8\pi \mathfrak{a}-\sqrt{{|p|}^4+16\pi \mathfrak{a}{p}^2}-\frac{{(8\pi \mathfrak{a})}^2}{2p^2}]\hfill \\ \hfill & -64\pi (\frac{4}{3}\pi -\sqrt{3}){\mathfrak{a}}^4\frac{(logN)}{N}+\mathcal{O}({(logN)}^{1/2}/N)\hfill \end{array}\end{array}$$ 1.6

as $N\to \infty$, with $e_{\mathrm{\Lambda }}$ defined in (1.4).

Remarks

1. Let ${\mathfrak{a}}_N=\mathfrak{a}/N$ be the scattering length of the potential in (1.1). With $\rho =N$, we observe that $\rho {\mathfrak{a}}_N^3={\mathfrak{a}}^3/N^2$. We conclude that third term on the r.h.s. of (1.6) is consistent with the prediction (1.5) for the third term in the asymptotic expansion of ground state energy per particle in the thermodynamic limit.

2. With our analysis, we could also improve the estimate (1.3) for the low-energy spectrum of $H_N-E_N$, showing that, below a threshold $\zeta >0$, it consists of eigenvalues having the form

   $$\begin{array}{c}\hfill \sum _{p\in 2\pi {\mathbb{Z}}^3\backslash \{0\}}{n}_p\sqrt{{|p|}^4+16\pi \mathfrak{a}{p}^2}+\mathcal{O}(C_{\zeta }{(logN)}^{1/2}/N)\end{array}$$

   for an appropriate constant $C_{\zeta }>0$, depending polynomially on $\zeta$.

3. Expansions of the ground state energy of Bose gases beyond second order have been previously obtained in the mean-field limit [10, 32, 36]. Moreover, for systems of N particles interacting through a potential of the form $N^{3\beta -1}V(N^{\beta }·)$, for a $\beta \in (0;1)$, the ground state energy was recently resolved to order $N^{-1+\beta }$ in [9].

4. At the expense of a slightly longer proof, with our techniques we could prove that the error term in (1.6) is $\mathcal{O}(N^{-1})$.

5. Heuristically, the appearance of a term of order $(logN)/N$ on the r.h.s. of (1.6) can be understood by perturbation theory. The bounds (1.2), (1.3) are proven in [6] showing that, after appropriate unitary transformations, $H_N$ can be approximated by a Fock space Hamiltonian of the form

   $$\begin{array}{c}\hfill H_{\text{quadr}}=E_N^{(0)}+\sum _{p\in 2\pi {\mathbb{Z}}^3\backslash \{0\}}\sqrt{{|p|}^4+16\pi \mathfrak{a}{p}^2}{a}_p^{\ast }{a}_p\end{array}$$

   quadratic in creation and annihilation operators. Here, $E_N^{(0)}$ denotes the approximation to the ground state energy of $H_N$ appearing on the r.h.s. of (1.2). The ground state of $H_{\text{quadr}}$ is the Fock space vacuum $\mathrm{\Omega }=\{1,0,0,\cdots \}$. The main correction to $H_{\text{quadr}}$ is given by cubic terms in creation and annihilation operators, whose expectation vanishes in the state $\mathrm{\Omega }$. By second order perturbation theory, we obtain therefore

   $$\begin{array}{c}\hfill E_N\simeq E_N^{(0)}+⟨\mathrm{\Omega },W{(H_{\text{quadr}}-E_N^{(0)})}^{-1}W\mathrm{\Omega }⟩\end{array}$$ 1.7

   where W has the form

   $$\begin{array}{c}\hfill W=\frac{1}{\sqrt{N}}\sum _{p,r\in 2\pi {\mathbb{Z}}^3\backslash \{0\}}{\phi }_{p,r}[a_{p+r}^{\ast }{a}_{-p}^{\ast }{a}_{-r}^{\ast }+\text{h.c.}]\end{array}$$

   for some appropriate coefficients ${\phi }_{p,r}$. The operator W includes all cubic terms which do not vanish when acting on $\mathrm{\Omega }$. Using the canonical commutation relations $[a_p,a_q^{\ast }]={\delta }_{\mathit{pq}}$, $[a_p,a_q]=0$, we find

   $$\begin{array}{c}\hfill E_N=E_N^{(0)}+\frac{1}{N}\sum _{p,r}\frac{{\phi }_{p,r}[{\phi }_{p,r}+{\phi }_{p+r,r}+{\phi }_{r,p}+{\phi }_{r,p+r}+{\phi }_{p+r,p}+{\phi }_{p,p+r}]}{{\epsilon }_p+{\epsilon }_r+{\epsilon }_{p+r}}\end{array}$$

   with the dispersion ${\epsilon }_p=\sqrt{{|p|}^4+16\pi \mathfrak{a}{p}^2}$. Determining the precise form of the coefficients ${\phi }_{p,r}$ is not trivial (it requires understanding precisely which corrections to the quadratic Hamiltonian $H_{\text{quadr}}$ are important); it turns out that ${\phi }_{p,r}$ scales as momentum to the power $-4$, for $|p|,|r|⩽CN$. Taking into account that ${\epsilon }_p\simeq p^2$, for large p, this produces a correction to $E_N^{(0)}$, exactly of the order $(logN)/N$. Although this argument could be probably also made into a rigorous proof, our approach is different, since it resolves the correct energy through two unitary conjugations, one with the exponential of a quadratic and, respectively, a cubic expression in (modified) creation and annihilation operators (our unitary conjugations implement the perturbative expansion leading to (1.7)).

Excitation Hamiltonians and Proof of Theorem 1.1

In order to determine the low-energy spectrum of the Hamilton operator (1.1), it is convenient, first of all, to factor out the Bose–Einstein condensate, focussing on its orthogonal excitations. To this end, we observe that an arbitrary wave function ${\psi }_N\in L_s^2({\mathrm{\Lambda }}^N)$ can be uniquely decomposed as

$$\begin{array}{c}\hfill {\psi }_N={\alpha }_0{\phi }_0^{\otimes N}+{\alpha }_1{\otimes }_s{\phi }_0^{\otimes (N-1)}+\cdots +{\alpha }_N\end{array}$$

with ${\alpha }_j\in L_{\perp }^2{(\mathrm{\Lambda })}^{{\otimes }_{s}j}$, where $L_{\perp }^2(\mathrm{\Lambda })$ denotes the orthogonal complement of the condensate wave function ${\phi }_0$, defined by ${\phi }_0(x)=1$ for all $x\in \mathrm{\Lambda }$ (and where ${\otimes }_s$ indicates the symmetric tensor product). This observation allows us to define a unitary operator $U_N:L_s^2({\mathrm{\Lambda }}^N)\to {\mathcal{F}}_{\perp }^{⩽N}$ mapping the original Hilbert space $L_s^2({\mathrm{\Lambda }}^N)$ into the truncated Fock space

$${\mathcal{F}}_{\perp }^{⩽N}=\underset{n=0}{\overset{N}{⨁}}{L}_{\perp {\phi }_0}^2{(\mathrm{\Lambda })}^{{\otimes }_{s}n},$$

setting $U_N{\psi }_N=\{{\alpha }_0,\cdots ,{\alpha }_N\}$. The map $U_N$ is characterized by its action on number of particle-preserving products of creation and annihilation operators, given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill U_N{a}_0^{\ast }{a}_0{U}_N^{\ast }=& N-{\mathcal{N}}_{+}\hfill \\ \hfill U_N{a}_p^{\ast }{a}_0{U}_N^{\ast }=& \sqrt{N}{b}_p^{\ast }\hfill \\ \hfill U_N{a}_0^{\ast }{a}_p{U}_N^{\ast }=& \sqrt{N}{b}_p\hfill \\ \hfill U_N{a}_p^{\ast }{a}_q{U}_N^{\ast }=& a_p^{\ast }{a}_q\hfill \end{array}\end{array}$$ 2.1

for momenta $p,q\in {\mathrm{\Lambda }}_{+}^{\ast }={\mathrm{\Lambda }}^{\ast }\backslash \{0\}$ (where ${\mathrm{\Lambda }}^{\ast }=2\pi {\mathbb{Z}}^3$ is the dual lattice to $\mathrm{\Lambda }$). Here $a_p^{\ast }=a({\phi }_p)$ and $a_p=a^{\ast }({\phi }_p)$ are creation and annihilation operators creating and, respectively, annihilating a particle with momentum $p\in {\mathrm{\Lambda }}^{\ast }$, described by the plane wave ${\phi }_p(x)=e^{-ip·x}$. Furthermore, ${\mathcal{N}}_{+}={\sum }_{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{a}_p^{\ast }{a}_p$ denotes the number of particles operator on ${\mathcal{F}}_{\perp }^{⩽N}$ and, for $p\in {\mathrm{\Lambda }}_{+}^{\ast }$, we introduced the modified creation and annihilation operators

$$\begin{array}{c}\hfill b_p:=\sqrt{\frac{N-{\mathcal{N}}_{+}}{N}}{a}_p,b_p^{\ast }:=a_p^{\ast }\sqrt{\frac{N-{\mathcal{N}}_{+}}{N}}.\end{array}$$

These operators act on ${\mathcal{F}}_{\perp }^{⩽N}$, they are bounded by the square root of ${\mathcal{N}}_{+}$, in the sense that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \Vert b_p\xi \Vert ⩽& \Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert \hfill \\ \hfill \Vert b_p^{\ast }\xi \Vert ⩽& \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$

and they satisfy the commutation relations

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [b_p,b_q^{\ast }]=& \left(1,-,\frac{{\mathcal{N}}_{+}}{N}\right){\delta }_{p,q}-\frac{1}{N}{a}_q^{\ast }{a}_p\hfill \\ \hfill [b_p,b_q]=& [b_p^{\ast },b_q^{\ast }]=0,\hfill \end{array}\end{array}$$ 2.2

and

$$\begin{array}{c}\hfill [b_p,a_q^{\ast }{a}_r]={\delta }_{p,q}{b}_r,[b_p^{\ast },a_q^{\ast }{a}_r]=-{\delta }_{p,r}{b}_q^{\ast }.\end{array}$$ 2.3

Rewriting the Hamilton operator (1.1) in momentum space, using the language of second quantization, we find

$$\begin{array}{c}\hfill H_N=\sum _{p\in {\mathrm{\Lambda }}^{\ast }}{p}^2{a}_p^{\ast }{a}_p+\frac{1}{2N}\sum _{p,q,r\in {\mathrm{\Lambda }}^{\ast }}\widehat{V}(r/N)a_{p+r}^{\ast }{a}_q^{\ast }{a}_{q+r}{a}_p.\end{array}$$ 2.4

This expression allows us to compute the excitation Hamiltonian ${\mathcal{L}}_N=U_N{H}_N{U}_N^{\ast }$, defined on the excitation space ${\mathcal{F}}_{\perp }^{⩽N}$, using the rules (2.1). We find

$$\begin{array}{c}\hfill {\mathcal{L}}_N:=U_N{H}_N{U}_N^{\ast }={\mathcal{L}}_N^{(0)}+\mathcal{K}+{\mathcal{L}}_N^{(2,V)}+{\mathcal{L}}_N^{(3)}+{\mathcal{V}}_N,\end{array}$$ 2.5

where we introduced the kinetic and potential energy operators

$$\begin{array}{c}\hfill \mathcal{K}=\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{a}_p^{\ast }{a}_p,{\mathcal{V}}_N=\frac{1}{2N}\sum _{\begin{array}{c}r\in {\mathrm{\Lambda }}^{\ast },p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r\ne -p,-q\end{array}}\widehat{V}(r/N)a_{p+r}^{\ast }{a}_q^{\ast }{a}_p{a}_{q+r}\end{array}$$ 2.6

and we set

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{L}}_N^{(0)}=& \frac{\widehat{V}(0)}{2}(N-1)\hfill \\ \hfill {\mathcal{L}}_N^{(2,V)}=& \sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}(p/N)\left(b_p^{\ast },b_p,-,\frac{1}{N},a_p^{\ast },a_p\right)\hfill \\ \hfill & +\frac{1}{2}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}(p/N)(b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p})-\frac{\widehat{V}(0)}{2N}{\mathcal{N}}_{+}({\mathcal{N}}_{+}-1)\hfill \\ \hfill {\mathcal{L}}_N^{(3)}=& \frac{1}{\sqrt{N}}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}\widehat{V}(p/N)\left(b_{p+q}^{\ast },a_{-p}^{\ast },a_q,+,a_q^{\ast },a_{-p},b_{p+q}\right).\hfill \end{array}\end{array}$$ 2.7

After conjugation with $U_N$, the vacuum vector $\mathrm{\Omega }\in {\mathcal{F}}_{\perp }^{⩽N}$ corresponds to the factorized wave function ${\phi }_0^{\otimes N}$, which is still very far, energetically, from the ground state of (2.4). In the next step, we are going to renormalize the excitation Hamiltonian (2.5), factoring out the microscopic correlation structure characterizing its low-energy states. To describe correlations, we fix $\ell >0$ and consider the ground state solution of the Neumann problem

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [-\mathrm{\Delta }+\frac{1}{2}V]f_{\ell }& ={\lambda }_{\ell }{f}_{\ell }(x)|x|⩽N\ell \hfill \\ \hfill {\partial }_r{f}_{\ell }(x)& =0|x|=N\ell \hfill \end{array}\end{array}$$

on the ball $|x|⩽N\ell$, normalized so that $f_{\ell }(x)=1$ for $|x|=N\ell$. We extend $f_{\ell }(x)=1$ for all $|x|>N\ell$ and we set $w_{\ell }(x)=1-f_{\ell }(x)$. We denote $f_{N,\ell }(x)=f_{\ell }(Nx)$ the solution of the rescaled Neumann problem

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [-\mathrm{\Delta }+\frac{N^2}{2}V(N·)]f_{N,\ell }& ={\lambda }_{\ell }{f}_{N,\ell }|x|⩽\ell \hfill \\ \hfill {\partial }_r{f}_{N,\ell }(x)& =0|x|=\ell \hfill \end{array}\end{array}$$ 2.8

on the ball $|x|⩽\ell$, with $f_{N,\ell }(x)=1$ for all $|x|⩾\ell$. As above, we set $w_{N,\ell }(x)=1-f_{N,\ell }(x)$. With a slight abuse of notation we use the same notation for $f_{N,\ell }$ and for its periodisation on the torus $\mathrm{\Lambda }$. The next Lemma, whose proof can be found in [5, Appendix B], collects important bounds for the functions $f_{\ell }$ and $w_{\ell }$, and for the eigenvalue ${\lambda }_{\ell }$.

Lemma 2.1

Let V be as in the assumptions of Theorem 1.1.

• (i)
  The eigenvalue ${\lambda }_{\ell }$ appearing in (2.8) satisfies

  $$\begin{array}{c}\hfill {\lambda }_{\ell }=\frac{3\mathfrak{a}}{{(N\ell )}^3}\left[1,+,\frac{9}{5},\frac{\mathfrak{a}}{N\ell },+,\mathcal{O},\left(\frac{{\mathfrak{a}}^2}{{(N\ell )}^2}\right)\right].\end{array}$$
• (ii)
  There exists a constant $C>0$ such that

  $$\begin{array}{c}\hfill \left|{\int }_{{\mathbb{R}}^3},V,(x),f_{\ell },(x),d,x,-,8,\pi ,\mathfrak{a},\left(1,+,\frac{3}{2},\frac{\mathfrak{a}}{N\ell }\right)\right|⩽\frac{C{\mathfrak{a}}^3}{{(N\ell )}^2}\end{array}$$ 2.9

  for $\ell \in (0,1/2)$.
• (iii)
  There exists $C>0$ with

  $$\begin{array}{c}\hfill w_{\ell }(x)⩽\frac{C}{|x|+1},|\mathrm{\nabla }{w}_{\ell }(x)|⩽\frac{C}{x^2+1}.\end{array}$$ 2.10

The Fourier coefficients of $f_{N,\ell }$ are given by

$$\begin{array}{c}\hfill {\widehat{f}}_{N,\ell }(p)={\int }_{\mathrm{\Lambda }}{f}_{N,\ell }(x)e^{-ip·x}dx={\delta }_{p,0}-N^{-3}{\widehat{w}}_{\ell }(p/N)\end{array}$$

where we defined

$$\begin{array}{c}\hfill {\widehat{w}}_{\ell }(q)={\int }_{{\mathbb{R}}^3}{w}_{N,\ell }(x)e^{-iq·x}dx\end{array}$$

for all $q\in {\mathbb{R}}^3$. For $p\in {\mathrm{\Lambda }}^{\ast }=2\pi {\mathbb{Z}}^3$, we consider the coefficients

$$\begin{array}{c}\hfill {\eta }_p=-N^{-2}{\widehat{w}}_{\ell }(p/N)\end{array}$$ 2.11

By (2.8), they satisfy the equation

$$\begin{array}{c}\hfill p^2{\eta }_p+\frac{1}{2}\widehat{V}(p/N)+\frac{1}{2N}\sum _{q\in {\mathrm{\Lambda }}^{\ast }}\widehat{V}((p-q)/N){\eta }_q=N^3{\lambda }_{\ell }{\widehat{\chi }}_{\ell }(p)+N^2{\lambda }_{\ell }\sum _{q\in {\mathrm{\Lambda }}^{\ast }}{\widehat{\chi }}_{\ell }(p-q){\eta }_q.\end{array}$$ 2.12

Here ${\chi }_{\ell }$ is the characteristic function of the ball of radius $\ell$, centered at the origin. Through the coefficients ${\eta }_p$ we define the antisymmetric operator

$$\begin{array}{c}\hfill B_{\eta }=\frac{1}{2}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{\eta }_p(b_p^{\ast }{b}_{-p}^{\ast }-b_p{b}_{-p}).\end{array}$$

Conjugating (2.5) with the generalized Bogoliubov transformation $e^{B_{\eta }}$, we define the renormalized excitation Hamiltonian ${\tilde{\mathcal{G}}}_N=e^{-B_{\eta }}{\mathcal{L}}_N{e}^{B_{\eta }}$. As shown in [5], ${\tilde{\mathcal{G}}}_N$ has the form

$$\begin{array}{c}\hfill {\tilde{\mathcal{G}}}_N\simeq C_{{\tilde{\mathcal{G}}}_N}+{\mathcal{Q}}_{{\tilde{\mathcal{G}}}_N}+{\mathcal{C}}_{{\tilde{\mathcal{G}}}_N}+{\mathcal{V}}_N\end{array}$$

up to small corrections. Here $C_{{\tilde{\mathcal{G}}}_N}$ is a constant, while ${\mathcal{Q}}_{\widetilde{{\mathcal{G}}_N}},{\mathcal{C}}_{{\tilde{\mathcal{G}}}_N}$ are quadratic and, respectively, cubic contributions in creation and annihilation operators. As discussed in [5], this form of the excitation Hamiltonian is still not enough to determine its spectrum (not even up to errors of order one, in N), because the cubic term ${\mathcal{C}}_{{\tilde{\mathcal{G}}}_N}$ is not negligible. A second renormalization, this time with a unitary transformation given by the exponential of a cubic expression in creation and annihilation operators, must be used to get rid of ${\mathcal{C}}_{{\tilde{\mathcal{G}}}_N}$. The resulting twice renormalized excitation Hamiltonian has the form

$$\begin{array}{c}\hfill {\tilde{\mathcal{J}}}_N\simeq C_{{\tilde{\mathcal{J}}}_N}+{\mathcal{Q}}_{{\tilde{\mathcal{J}}}_N}+{\mathcal{V}}_N\end{array}$$ 2.13

again up to small corrections. At this point the quadratic part of ${\tilde{\mathcal{J}}}_N$ has the form

$$\begin{array}{c}\hfill {\mathcal{Q}}_{{\tilde{\mathcal{J}}}_N}=\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}[{\tilde{F}}_p{b}_p^{\ast }{b}_p+\frac{1}{2}{\tilde{G}}_p(b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p})]\end{array}$$ 2.14

with the coefficients

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\tilde{F}}_p& =[p^2+(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p]cosh(2{\eta }_p)+(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_{p}sinh(2{\eta }_p)\hfill \\ \hfill {\tilde{G}}_p& =[p^2+(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p]sinh(2{\eta }_p)+(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_{p}cosh(2{\eta }_p).\hfill \end{array}\end{array}$$ 2.15

To compute the spectrum of (2.13), it is convenient to diagonalize (2.14), conjugating it with another generalized Bogoliubov transformation. As shown in [5, Lemma 5.1], the coefficients (2.15) satisfy the bounds

$$\begin{array}{c}\hfill \frac{p^2}{2}⩽{\tilde{F}}_p⩽C(1+p^2),|{\tilde{G}}_p|⩽\frac{C}{p^2},|{\tilde{G}}_p|⩽{\tilde{F}}_p\end{array}$$ 2.16

for all $p\in {\mathrm{\Lambda }}_{+}^{\ast }$. As a consequence, we can define coefficients ${\tau }_p$ requiring that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill tanh(2{\tau }_p)& =-\frac{{\tilde{G}}_p}{{\tilde{F}}_p}=-\frac{[p^2+(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p]sinh(2{\eta }_p)+(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_{p}cosh(2{\eta }_p)}{[p^2+(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p]cosh(2{\eta }_p)+(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_{p}sinh(2{\eta }_p)}.\hfill \end{array}\end{array}$$ 2.17

With this choice of the coefficients ${\tau }_p$, it was proven in [5] that

$${\tilde{\mathcal{M}}}_N=e^{-B_{\tau }}{\tilde{\mathcal{J}}}_N{e}^{B_{\tau }}\simeq C_{{\tilde{\mathcal{M}}}_N}+\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}\epsilon (p)a_p^{\ast }{a}_p+{\mathcal{V}}_N$$

for appropriate constant $C_{{\mathcal{M}}_N}$ and dispersion $\epsilon (p)$. It is then easy to determine the low-energy spectrum of ${\tilde{\mathcal{M}}}_N$ (using the positivity of the potential energy operator ${\mathcal{V}}_N$, and its smallness on states with few low-momentum excitations); see [5] for the details.

In the present work, to improve the energy resolution up to errors smaller than $(logN)/N$, we find it more convenient to combine $e^{B_{\eta }}$ and $e^{B_{\tau }}$ into a single generalized Bogoliubov transformation. To this end, we define the coefficients

$$\begin{array}{c}\hfill {\mu }_p={\eta }_p+{\tau }_p\end{array}$$ 2.18

for all $p\in {\mathrm{\Lambda }}^{\ast }=2\pi {\mathbb{Z}}^3$, with ${\eta }_p$ as in (2.11) and ${\tau }_p$ as introduced in (2.17). In the next lemma, we collect important properties of the coefficients ${\eta }_p$, ${\tau }_p$, ${\mu }_p$, and

$$\begin{array}{c}\hfill {\gamma }_p=cosh{\mu }_p,{\sigma }_p=sinh{\mu }_p.\end{array}$$ 2.19

We will systematically use such properties in estimates throughout the paper.

Lemma 2.2

Let V be as in the assumptions of Theorem 1.1.

• (i)
  There exists a constant $C>0$ such that

  $$\begin{array}{c}\hfill |{\eta }_p{|⩽C|p|}^{-2}.\end{array}$$

  Moreover,

  $$\begin{array}{c}\hfill |{\eta }_0{|⩽C,\Vert \eta \Vert }_2⩽C,\sum _{p\in {\mathrm{\Lambda }}^{\ast }}{p}^2|{\eta }_p{|}^2⩽CN,{\Vert \eta \Vert }_q⩽Cmax\{1,N^{3-2q}\}\end{array}$$ 2.20

  for all $q⩾1$.
• (ii)
  There exists a constant $C>0$ such that

  $$\begin{array}{c}\hfill |{\tau }_p{|⩽C|p|}^{-4},\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{\tau }_p^2+\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{\tau }_p^2<C.\end{array}$$ 2.21
• (iii)
  As a consequence of (i) and (ii) and of the definitions of ${\mu }_p,{\gamma }_p,{\sigma }_p$, we have

  $$\begin{array}{c}\hfill |{\mu }_p|+|{\sigma }_p{|⩽C|p|}^{-2},|{\gamma }_p|⩽C\end{array}$$ 2.22

  and

  $$\begin{array}{c}\hfill {\Vert \mu \Vert }_2+{\Vert \sigma \Vert }_2⩽C,\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2({\mu }_p^2+{\sigma }_p^2)⩽CN\end{array}$$ 2.23

  for a suitable $C>0$.

Proof

We only prove the last bound in (2.20) since the other estimates in (i) are shown in [5, Appendix B], the estimates in (ii) follow from the definition of ${\tau }_p$ together with (2.12) and (2.16), and (iii) is an immediate consequence of (i) and (ii). We have

$$\begin{array}{cc}\hfill {\Vert \eta \Vert }_q^q& =\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}|{\eta }_p{|}^q=\sum _{p:|p|⩽N}{|p|}^{-2q}+\sum _{p:|p|>N}{|{\eta }_p|}^q\hfill \\ \hfill & ⩽Cmax\{N^{3-2q},1\}+\sum _{p:|p|>N}\frac{|(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p{|}^q}{{|p|}^{2q}}+C\sum _{p:|p|>N}\frac{|({\widehat{\chi }}_{\ell }\ast {\widehat{f}}_{N,\ell }{)}_p{|}^q}{{|p|}^{2q}}\hfill \end{array}$$

With Hölder’s inequality, we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \sum _{p:|p|>N}\frac{|(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p{|}^q}{{|p|}^{2q}}& ⩽\Vert \widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{\Vert }_2^q\Vert \frac{\chi (|.|>N)}{{|.|}^{2q}}{\Vert }_{2/(2-q)}\hfill \\ \hfill & ⩽C{N}^{3q/2}{N}^{-(7q-6)/2}⩽C{N}^{3-2q}.\hfill \end{array}\end{array}$$

$\square$

Using the coefficients ${\mu }_p$ we define the antisymmetric operator

$$\begin{array}{c}\hfill B_{\mu }=B_{\eta }+B_{\tau }=\frac{1}{2}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{\mu }_p(b_p^{\ast }{b}_{-p}^{\ast }-\text{h.c.}).\end{array}$$ 2.24

With the corresponding generalized Bogoliubov transformation $e^{B_{\mu }}$, we define the renormalized excitation Hamiltonian

$$\begin{array}{c}\hfill {\mathcal{G}}_N=e^{-B_{\mu }}{\mathcal{L}}_N{e}^{B_{\mu }}.\end{array}$$ 2.25

The advantage that we have, when working with ${\mathcal{G}}_N$ rather than with ${\mathcal{L}}_N$, is that, after removing the microscopic correlation structure through $e^{-B_{\mu }}$, low-energy states of ${\mathcal{G}}_N$ have only few excitations, with bounded energy. More precisely, we obtain the following a-priori estimates on products of the energy operator ${\mathcal{H}}_N=\mathcal{K}+{\mathcal{V}}_N$ with arbitrary powers of the number of particles operator ${\mathcal{N}}_{+}$.

Proposition 2.3

Let ${\psi }_N\in L_s^2({\mathrm{\Lambda }}^N)$ be a normalized sequence of approximate ground state vectors of the Hamilton operator (1.1), satisfying ${\psi }_N=\chi (H_N⩽E_N+K){\psi }_N$, where $E_N$ is the ground state energy of (1.1) and $K>0$ is fixed. Let ${\xi }_N=e^{-B_{\mu }}{U}_N{\psi }_N$ be the corresponding normalized sequence of approximate ground state vectors of the renormalized excitation Hamiltonian (2.25). Let $k\in \mathbb{N}$. Then, there exists $C>0$ (depending on K and k) such that

$$\begin{array}{c}\hfill ⟨{\xi }_N,({\mathcal{H}}_N+1){({\mathcal{N}}_{+}+1)}^k{\xi }_N⟩⩽C\end{array}$$ 2.26

for all $N\in \mathbb{N}$.

A-priori bounds of the form (2.26) have been established in [5, Prop. 4.1], for a sequence ${\xi }_N^{\prime }=e^{-B_{\eta }}{U}_N{\psi }_N$, defined in terms of the generalized Bogoliubov transformation generated by $B_{\eta }$, rather than in terms of that generated by $B_{\mu }$. From (2.24), the difference $B_{\mu }-B_{\eta }=B_{\tau }$ is associated with the kernel $\tau$ exhibiting, by (2.21), fast decay in momentum space. For this reason, the estimate in Prop. 2.3 follows from the bounds for ${\xi }_N^{\prime }$ in [5, Prop. 4.1]. This is shown in Appendix B.

In the next theorem, whose proof is deferred to Sect. 3, we determine the operator ${\mathcal{G}}_N$, up to very small errors (which can be controlled through the a-priori estimates in Prop. 2.3).

Theorem 2.4

Let $V\in L^3({\mathbb{R}}^3)$ be non-negative, compactly supported, and spherically symmetric. Let ${\mathcal{G}}_N$ be defined as in (2.25) with parameter $\ell \in (0,\frac{1}{2})$ small enough. For $p\in {\mathrm{\Lambda }}_{+}^{\ast }$, let ${\gamma }_p=cosh{\mu }_p$ and ${\sigma }_p=sinh{\mu }_p$. Moreover, define the constant

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_{{\mathcal{G}}_N}=& \frac{(N-1)}{2}\widehat{V}(0)+\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}[p^2{\sigma }_p^2+\widehat{V}(p/N)({\sigma }_p{\gamma }_p+{\sigma }_p^2)]\hfill \\ \hfill & +\frac{1}{2N}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}((p-q)/N){\sigma }_q{\gamma }_q{\sigma }_p{\gamma }_p+\frac{1}{N}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}[p^2{\eta }_p^2+\frac{1}{2N}(\widehat{V}(·/N)\ast \eta {)}_p{\eta }_p]\hfill \\ \hfill & -\frac{1}{N}\sum _{p\in {\mathrm{\Lambda }}^{\ast },q\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}(p/N){\eta }_p{\sigma }_q^2\hfill \end{array}\end{array}$$ 2.27

and the cubic operator

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathcal{C}}_{{\mathcal{G}}_N}=\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}\widehat{V}(p/N)b_{p+q}^{\ast }{b}_{-p}^{\ast }({\gamma }_q{b}_q+{\sigma }_q{b}_{-q}^{\ast })+\mathrm{h}.\mathrm{c}..\end{array}\end{array}$$ 2.28

Furthermore, let ${\mathcal{T}}_{{\mathcal{G}}_N}={\mathcal{T}}_N^{(2)}+{\mathcal{T}}_N^{(4)}$, with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{T}}_N^{(2)}=& \frac{1}{2N^2}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}(\widehat{V}(·/N)\ast \eta {)}_p(b_p{b}_{-p}+b_p^{\ast }{b}_{-p}^{\ast })(1+{2\Vert \sigma \Vert }_2^2)\hfill \\ \hfill & +\frac{1}{2N}\sum _{q\in {\mathrm{\Lambda }}_{+}^{\ast }}(2{\sigma }_q^2+\frac{{\gamma }_q{\sigma }_q}{{\mu }_q}-1)\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{\eta }_p(b_p{b}_{-p}+b_p^{\ast }{b}_{-p}^{\ast })\hfill \\ \hfill {\mathcal{T}}_{{\mathcal{G}}_N}^{(4)}=& \frac{1}{2N}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast },r\in {\mathrm{\Lambda }}^{\ast }}\widehat{V}(r/N){\sigma }_p{\sigma }_{q+r}{b}_{p+r}^{\ast }{b}_q^{\ast }{b}_{-p}^{\ast }{b}_{-q-r}^{\ast }+\text{h.c.}\hfill \\ \hfill & +\frac{1}{N^2}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}(\widehat{V}(·/N)\ast \eta {)}_p{\gamma }_q{\sigma }_q{b}_p^{\ast }{b}_{-p}^{\ast }{b}_q^{\ast }{b}_{-q}^{\ast }+\mathrm{h}.\mathrm{c}.\hfill \\ \hfill & -\frac{1}{2N}\sum _{p\in {\mathrm{\Lambda }}^{\ast },q\in {\mathrm{\Lambda }}_{+}^{\ast }}(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p({\gamma }_q{\sigma }_q-\frac{{\eta }_q}{2}-\frac{{\sigma }_q^2}{2{\mu }_q})b_p^{\ast }{b}_{-p}^{\ast }{b}_q^{\ast }{b}_{-q}^{\ast }+\mathrm{h}.\mathrm{c}..\hfill \end{array}\end{array}$$ 2.29

Then

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{G}}_N=& C_{{\mathcal{G}}_N}+\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}\sqrt{{|p|}^4+2{(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell })}_p{p}^2}{a}_p^{\ast }{a}_p\hfill \\ \hfill & -\frac{1}{N}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}(\widehat{V}(q/N)+\widehat{V}((q+p)/N)){\eta }_q[({\gamma }_p^2+{\sigma }_p^2)b_p^{\ast }{b}_p+{\gamma }_p{\sigma }_p(b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p})]\hfill \\ \hfill & +{\mathcal{C}}_{{\mathcal{G}}_N}+{\mathcal{V}}_N+{\mathcal{T}}_{{\mathcal{G}}_N}+{\mathcal{E}}_{{\mathcal{G}}_N}\hfill \end{array}\end{array}$$ 2.30

where, for every $0<\epsilon <1$, we have

$$\begin{array}{c}\hfill \pm {\mathcal{E}}_{{\mathcal{G}}_N}⩽\epsilon {\mathcal{N}}_{+}+\frac{C}{\epsilon N}({\mathcal{H}}_N+{\mathcal{N}}_{+}^3+1)({\mathcal{N}}_{+}+1)\end{array}$$ 2.31

with ${\mathcal{H}}_N=\mathcal{K}+{\mathcal{V}}_N$.

Remark

In the representation (2.30) of the renormalized excitation Hamiltonian, we distinguish three types of error terms (terms which will not contribute to the energy of the Hamiltonian, up to order $(logN)/N$). First of all, in ${\mathcal{E}}_{{\mathcal{G}}_N}$ we absorb several contributions that are controlled by the second term on the r.h.s. of (2.31). With the a-priori bounds in Prop. 2.3, these terms are small, of order $N^{-1}$, in the limit $N\to \infty$. Other contributions to the error ${\mathcal{E}}_{{\mathcal{G}}_N}$ are bounded by $\epsilon {\mathcal{N}}_{+}$, for an arbitrary small $\epsilon >0$. Since the cubic conjugation only increases ${\mathcal{N}}_{+}$ by $\mathcal{O}(N^{-1})$, we will control these terms using a little bit of kinetic energy. Terms in ${\mathcal{T}}_{{\mathcal{G}}_N}$, on the other hand, could only be controlled, at this point, by

$$|⟨\xi ,{\mathcal{T}}_{{\mathcal{G}}_N}\xi ⟩|⩽C{N}^{-1/2}\Vert {\mathcal{K}}^{1/2}\xi \Vert \Vert ({\mathcal{N}}_{+}+1)\xi \Vert$$

or by

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{T}}_{{\mathcal{G}}_N}\xi ⟩|⩽C{N}^{-1/2}\Vert {\mathcal{K}}^{1/2}{\mathcal{N}}_{+}^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert .\end{array}$$

Notice, in the two bounds, the presence of the operator $({\mathcal{N}}_{+}+1)$, instead of just ${\mathcal{N}}_{+}$, due to the fact that contributions in ${\mathcal{T}}_{{\mathcal{G}}_N}$ only contain creation or annihilation operators, but never both. Since moreover the cubic conjugation changes the expectation of $\mathcal{K}$ by order one, this estimate does not yet allow us conclude that ${\mathcal{T}}_{{\mathcal{G}}_N}$ is negligible (instead, we first have to apply the cubic conjugation to ${\mathcal{T}}_{{\mathcal{G}}_N}$ and only afterwards we will be able to show that it can be dropped).

To get rid of the cubic term ${\mathcal{C}}_{{\mathcal{G}}_N}$ in (2.30), we conjugate ${\mathcal{G}}_N$ with a second unitary transformation, given by the exponential of a cubic expression in (modified) creation and annihilation operators. We define

$$\begin{array}{c}\hfill A=\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}{b}_{r+v}^{\ast }{b}_{-r}^{\ast }({\eta }_r{\gamma }_v{b}_v+{\nu }_{r,v}{b}_{-v}^{\ast })-\mathrm{h}.\mathrm{c}.=A_{\gamma }+A_{\nu },\end{array}$$ 2.32

where

$$\begin{array}{c}\hfill {\nu }_{r,v}=\frac{2r^2{\eta }_r{\sigma }_v}{{|r+v|}^2+r^2+v^2},\end{array}$$ 2.33

and $\eta$, $\gamma$, and $\sigma$ were defined in (2.11) and in (2.19) (recall from Lemma 2.2 that $|{\eta }_r|,|{\sigma }_r{|⩽C|r|}^{-2}$, while $|{\gamma }_r|⩽C$, for an appropriate constant $C>0$). The choice of the coefficients in A guarantees that the commutator $[{\mathcal{H}}_N,A]$ produces a contribution cancelling the cubic term ${\mathcal{C}}_{{\mathcal{G}}_N}$ in (2.28). Compared with the cubic phase used in [5] and in later works, where the coefficient in front of the operator $b_{r+v}^{\ast }{b}_{-r}^{\ast }{b}_{-v}^{\ast }$ was simply given by ${\eta }_r{\sigma }_v$, we modify here the choice of ${\nu }_{r,v}$ to eliminate certain terms arising from the commutator $[\mathcal{K},A]$ which would not be negligible at the level of accuracy required to show Theorem 1.1; see the remark after Lemma 4.1 (notice that in [5], the operator A was defined summing only over momenta r, v with $|r|\gg |v|$; in this region, ${\nu }_{r,v}\simeq {\eta }_r{\sigma }_v$).

For our analysis, it is very important to control the growth of number and energy of excitations, w.r.t. conjugation by $e^A$.

Proposition 2.5

Let A be defined as in (2.32), ${\mathcal{N}}_{+}$ be the number of particles operator on ${\mathcal{F}}_{\perp }^{⩽N}$ and ${\mathcal{H}}_N=\mathcal{K}+{\mathcal{V}}_N$ as be defined as in Theorem 2.4. Then, for any $k\in \mathbb{N}$, for any $s\in [0;1]$, there exists $C>0$ (depending on k) such that

$$\begin{array}{c}\hfill e^{-sA}{({\mathcal{N}}_{+}+1)}^k{e}^{\mathit{sA}}⩽C{({\mathcal{N}}_{+}+1)}^k.\end{array}$$ 2.34

Furthermore, there is $C>0$ such that

$$\begin{array}{c}\hfill ⟨\xi ,e^{-sA}{\mathcal{N}}_{+}{e}^{\mathit{sA}}\xi ⟩⩽⟨\xi ,{\mathcal{N}}_{+}\xi ⟩+\frac{C}{N}⟨\xi ,{({\mathcal{N}}_{+}+1)}^2\xi ⟩.\end{array}$$ 2.35

Moreover, for every $k\in \mathbb{N}$ and every $s\in [0;1]$, there exists $C>0$ such that

$$\begin{array}{c}\hfill \begin{array}{c}\hfill e^{-sA}({\mathcal{H}}_N+1){({\mathcal{N}}_{+}+1)}^k{e}^{\mathit{sA}}⩽C({\mathcal{H}}_N+1){({\mathcal{N}}_{+}+1)}^k+C{({\mathcal{N}}_{+}+1)}^{k+2}.\end{array}\end{array}$$ 2.36

The proof of Proposition 2.5 will be given in Sect. 4 (there, the choice of the coefficients (2.33) will become clear).

With A defined as in (2.32), we introduce the cubically renormalized excitation Hamiltonian

$$\begin{array}{c}\hfill {\mathcal{J}}_N=e^{-A}{\mathcal{G}}_N{e}^A.\end{array}$$ 2.37

In the next theorem, we describe the operator ${\mathcal{J}}_N$.

Theorem 2.6

Let $V\in L^3({\mathbb{R}}^3)$ be non-negative, compactly supported, and spherically symmetric. Let ${\mathcal{J}}_N$ be defined in (2.37). Let

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_{{\mathcal{J}}_N}=& 4\pi \mathfrak{a}(N-1)+e_{\mathrm{\Lambda }}{\mathfrak{a}}^2-\frac{1}{2}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}[p^2+8\pi \mathfrak{a}-\sqrt{{|p|}^4+16\pi \mathfrak{a}{p}^2}-\frac{{(8\pi \mathfrak{a})}^2}{2p^2}]\hfill \\ & +\frac{1}{N}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}[(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p+(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_{p+q}]\hfill \\ & \times {\eta }_p{\eta }_q\frac{2{\eta }_{q+p}{(q+p)}^2-2{\eta }_q(p·q)}{p^2+q^2+{(p+q)}^2}\hfill \end{array}\end{array}$$ 2.38

with $e_{\mathrm{\Lambda }}$ defined in (1.4). Then we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{J}}_N=& C_{{\mathcal{J}}_N}+\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}\sqrt{{|p|}^4+16\pi \mathfrak{a}{p}^2}{a}_p^{\ast }{a}_p+{\mathcal{V}}_N+{\mathcal{E}}_{{\mathcal{J}}_N}\hfill \end{array}\end{array}$$ 2.39

where, for any $\epsilon >0$ ($\epsilon$ can also depend on N, provided $\epsilon >C(logN)/N$),

$$\begin{array}{c}\hfill \pm {\mathcal{E}}_{{\mathcal{J}}_N}⩽\epsilon \mathcal{K}+\frac{C}{N}[{(logN)}^{1/2}+{\epsilon }^{-1}]({\mathcal{H}}_N+1){({\mathcal{N}}_{+}+1)}^4\end{array}$$

The proof of Theorem 2.6 will be given below, in Sect. 5.

We can now apply Theorem 2.6, to show our main result.

Proof of Theorem 1.1

We claim that the ground state energy $E_N$ of (1.1) is such that

$$\begin{array}{c}\hfill |E_N-C_{{\mathcal{J}}_N}|⩽\frac{C{(logN)}^{1/2}}{N}.\end{array}$$

Upper bound. From (2.39), taking $\epsilon >0$ a constant, we obtain

$$\begin{array}{c}\hfill {\mathcal{J}}_N⩽C_{{\mathcal{J}}_N}+C\mathcal{K}+{\mathcal{V}}_N+\frac{C{(logN)}^{1/2}}{N}({\mathcal{H}}_N+1){({\mathcal{N}}_{+}+1)}^4.\end{array}$$

We conclude, taking expectation in the vacuum, that

$$\begin{array}{c}\hfill E_N⩽⟨\mathrm{\Omega },{\mathcal{J}}_N\mathrm{\Omega }⟩⩽C_{{\mathcal{J}}_N}+\frac{C{(logN)}^{1/2}}{N}.\end{array}$$

Lower bound. From (2.39), taking $\epsilon ={(logN)}^{-1/2}$, and using the positivity of ${\mathcal{V}}_N$, we find

$$\begin{array}{c}\hfill {\mathcal{J}}_N⩾C_{{\mathcal{J}}_N}+[1-\frac{1}{{(logN)}^{1/2}}]\mathcal{K}-\frac{C{(logN)}^{1/2}}{N}({\mathcal{H}}_N+1){({\mathcal{N}}_{+}+1)}^4.\end{array}$$

Let ${\theta }_N\in {\mathcal{F}}_{\perp }^{⩽N}$ denote a normalized ground state vector for ${\mathcal{J}}_N$. Then ${\xi }_N=e^A{\theta }_N$ is a normalized ground state vector of the excitation Hamiltonian ${\mathcal{G}}_N$ defined in (2.25). Combining Prop. 2.3 with Prop. 2.5, we conclude that

$$\begin{array}{c}\hfill ⟨{\theta }_N,({\mathcal{H}}_N+1){({\mathcal{N}}_{+}+1)}^4{\theta }_N⟩⩽C⟨{\xi }_N,({\mathcal{H}}_N+1){({\mathcal{N}}_{+}+1)}^5{\xi }_N⟩⩽C\end{array}$$

and therefore that

$$\begin{array}{c}\hfill E_N=⟨{\theta }_N,{\mathcal{J}}_N{\theta }_N⟩⩾C_{{\mathcal{J}}_N}-\frac{C{(logN)}^{1/2}}{N}.\end{array}$$

To conclude the proof of Theorem 1.1, we still need to evaluate the constant $C_{{\mathcal{J}}_N}$. To this end, we write (2.38) as

$$\begin{array}{c}\hfill C_{{\mathcal{J}}_N}=4\pi \mathfrak{a}(N-1)+e_{\mathrm{\Lambda }}{\mathfrak{a}}^2-\frac{1}{2}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}[p^2+8\pi \mathfrak{a}-\sqrt{{|p|}^4+16\pi \mathfrak{a}{p}^2}-\frac{{(8\pi \mathfrak{a})}^2}{2p^2}]+{\tilde{C}}_{{\mathcal{J}}_N}\end{array}$$

with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\tilde{C}}_{{\mathcal{J}}_N}& =\frac{1}{N}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}[(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_r+(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_{r+v}]\hfill \\ \hfill & \times {\eta }_r{\eta }_v\frac{2{\eta }_{r+v}{(r+v)}^2-2{\eta }_v(r·v)}{r^2+v^2+{(r+v)}^2}\hfill \\ \hfill \end{array}\end{array}$$ 2.40

We first show that

$$\begin{array}{c}\hfill {\tilde{C}}_{{\mathcal{J}}_N}=\frac{1024{\pi }^4{\mathfrak{a}}^4}{N}\sum _{r,v\in {\mathrm{\Lambda }}_{+}^{\ast }:|r|,|v|⩽N}\frac{r·v-v^2}{r^2+v^2+r·v}\frac{1}{r^2{v}^4}+\mathcal{O}(N^{-1}).\end{array}$$

To reach this goal, we apply the scattering equation (2.12) to the second line of (2.40). Noticing that the contribution arising from the r.h.s. of (2.12) is negligible (the r.h.s decays faster, it makes the term of order $N^{-1}$), and combining with symmetry we arrive at

$$\begin{array}{c}\hfill \begin{array}{cc}& {\tilde{C}}_{{\mathcal{J}}_N}\hfill \\ & =\frac{1}{8N}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}\frac{(r·v)-v^2}{r^2+v^2+r·v}\frac{1}{r^2{v}^4}(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_v^2(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_r(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_{r+v}\hfill \\ & -\frac{1}{8N}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}\frac{1}{r^2+v^2+r·v}\frac{1}{r^2{v}^2}(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_{r+v}^2(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_r(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_v\hfill \\ & +\frac{1}{8N}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}\frac{(r·v)}{r^2+v^2+r·v}\frac{1}{r^2{v}^4}(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_r^2(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_v^2+\mathcal{O}(N^{-1}).\hfill \end{array}\end{array}$$ 2.41

In the next step, we restrict all sums to $|r|,|v|⩽N$. For the second term on the r.h.s. of the last equation, it is easy to check that the corresponding error is negligible, of order $N^{-1}$. In fact,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |\frac{1}{N}& \sum _{r,v:|r|>N}\frac{1}{r^2+v^2+r·v}\frac{1}{r^2{v}^2}(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_{r+v}^2(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_r(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_v|\hfill \\ \hfill ⩽& \frac{C}{N}\sum _{r,v:|r|>N}\frac{1}{(r^2+v^2)v^2{r}^2}|(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_r|⩽\frac{C}{N}\sum _{|r|>N}\frac{1}{{|r|}^3}|(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_r|⩽\frac{C}{N}\hfill \end{array}\end{array}$$ 2.42

using the bound $\Vert \widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{\Vert }_2={\Vert N^{3}V(N·)f_{N,\ell }\Vert }_2⩽C{N}^{3/2}$. Similarly, one can also bound the contribution to this term arising from the region $|v|>N$. Let us now consider the last term on the r.h.s. of (2.41). Observing that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |& \frac{1}{N}\sum _{r,v:|v|>N}\frac{(r·v)}{r^2+v^2+r·v}\frac{1}{r^2{v}^4}(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_r^2(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_v^2|\hfill \\ \hfill & ⩽\frac{C}{N^2}\sum _{r,v:|v|>N}\frac{1}{r^2+v^2}\frac{1}{|r|v^2}|(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_r{|}^2⩽\frac{C}{N^2}\sum _{r\in {\mathrm{\Lambda }}_{+}^{\ast }}\frac{1}{r^2}|(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_r{|}^2⩽\frac{C}{N},\hfill \end{array}\end{array}$$

we can restrict the sum to $|v|⩽N$. To restrict it also to $|r|⩽N$, we estimate, using a change of variable $v\to -v$,

$$\begin{array}{cc}\hfill |\frac{1}{N}& \sum _{|v|⩽N,|r|>N}\frac{(r·v)}{r^2+v^2+r·v}\frac{1}{r^2{v}^4}(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_r^2(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_v^2|\hfill \\ \hfill =& \frac{1}{2}\sum _{|v|⩽N,|r|>N}\frac{{(r·v)}^2}{(r^2+v^2+r·v)(r^2+v^2-r·v)}\frac{1}{r^2{v}^4}(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_r^2(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_v^2\hfill \\ \hfill ⩽& C\sum _{|v|⩽N,|r|>N}\frac{1}{(r^2+v^2)v^2}\frac{1}{r^2}(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_r^2(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_v^2\hfill \\ \hfill ⩽& \frac{1}{N}\sum _{|r|>N}\frac{1}{{|r|}^3}(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_r^2⩽\frac{C}{N}.\hfill \end{array}$$ 2.43

The term in the first line of (2.41) can be handled similarly. In fact, we control the contribution proportional to $-v^2$ as in (2.42). As for the contribution proportional to $(r·v)$, we proceed analogously to (2.43). There is here an additional term arising from the change of variable $v\to -v$, due to the potential ${(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell })}_{r+v}$, which can be bounded using that

$$\begin{array}{c}\hfill |{(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell })}_{r+v}-{(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell })}_{r-v}|⩽C|v|/N.\end{array}$$ 2.44

Finally, we replace all renormalized potentials ${(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell })}_p$ with factors of ${(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell })}_0$, and then, using (2.9), by factors of $8\pi \mathfrak{a}$. To bound the corresponding errors, we rely again on estimates of the form (2.44). Considering an example among the terms arising from the contribution on the second line of (2.41), we can bound

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |\frac{1}{N}\sum _{|r|,|v|⩽N}& \frac{1}{r^2+v^2+r·v}\frac{1}{r^2{v}^2}(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_{r+v}^2(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_v\hfill \\ \hfill & \times [(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_r-(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_0]|\hfill \\ \hfill & ⩽\frac{C}{N^2}\sum _{|r|,|v|⩽N}\frac{1}{r^2+v^2}\frac{1}{|r|v^2}⩽\frac{C}{N^2}\sum _{|r|,|v|⩽N}\frac{1}{{|r|}^{5/2}{|v|}^{5/2}}⩽\frac{C}{N}.\hfill \end{array}\end{array}$$

To estimate the errors arising from the third line of (2.41), on the other hand, we proceed similarly to (2.43), performing a change of variable $v\to -v$, to bound

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |\frac{1}{N}& \sum _{|r|,|v|⩽N}\frac{(r·v)}{r^2+v^2+r·v}\frac{1}{r^2{v}^4}(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_v[(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_r-(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_0]|\hfill \\ \hfill & =|\frac{1}{2N}\sum _{|r|,|v|⩽N}\frac{{(r·v)}^2}{(r^2+v^2+r·v)(r^2+v^2-r·v)}\frac{1}{r^2{v}^4}(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_v\hfill \\ \hfill & \times [(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_r-(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_0]|\hfill \\ \hfill & ⩽\frac{C}{N^2}\sum _{|r|,|v|⩽N}\frac{|r|}{v^2{(r^2+v^2)}^2}⩽\frac{C}{N^2}\sum _{|r|,|v|⩽N}\frac{1}{{|r|}^{5/2}{|v|}^{5/2}}⩽\frac{C}{N}.\hfill \end{array}\end{array}$$

Also the errors from the first line of (2.41) can be bounded analogously (also here the change of variables $v\to -v$ needed to handle terms proportional to $(r·v)$ will produce additional contributions, containing an additional difference ${(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell })}_{r+v}-{(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell })}_{r-v}$, which can be estimated by |v|/N and can be handled similarly as above). We conclude that

$$\begin{array}{c}\hfill {\tilde{C}}_{{\mathcal{J}}_N}=\frac{1024{\pi }^4{\mathfrak{a}}^4}{N}\sum _{\begin{array}{c}r,v\in 2\pi {\mathbb{Z}}^3\backslash \{0\}\\ |r|,|v|⩽N\end{array}}\frac{r·v-v^2}{r^2+v^2+r·v}\frac{1}{r^2{|v|}^4}+\mathcal{O}(N^{-1}).\end{array}$$ 2.45

At this point, we can approximate the sum with an integral. Consider first the contribution proportional to $-v^2$. For $\tilde{r}=({\tilde{r}}_1,{\tilde{r}}_2,{\tilde{r}}_3),\tilde{v}=({\tilde{v}}_1,{\tilde{v}}_2,{\tilde{v}}_3)\in 2\pi {\mathbb{Z}}^3$, with $2\pi ⩽|\tilde{r}|,|\tilde{v}|⩽N$ and $r\in B_{\tilde{r}}=[{\tilde{r}}_1-\pi ;{\tilde{r}}_1+\pi ]\times [{\tilde{r}}_2-\pi ,{\tilde{r}}_2+\pi ]\times [{\tilde{r}}_3-\pi ;{\tilde{r}}_3+\pi ]$, $v\in B_{\tilde{v}}=[{\tilde{v}}_1-\pi ;{\tilde{v}}_1+\pi ]\times [{\tilde{v}}_2-\pi ;{\tilde{v}}_2+\pi ]\times [{\tilde{v}}_3-\pi ;{\tilde{v}}_3+\pi ]$, we find

$$\begin{array}{c}\hfill |\frac{1}{r^2+v^2+r·v}\frac{1}{r^2{v}^2}-\frac{1}{{\tilde{r}}^2+{\tilde{v}}^2+\tilde{r}·\tilde{v}}\frac{1}{{\tilde{r}}^2{\tilde{v}}^2}|⩽C\frac{1}{r^2+v^2}\frac{1}{r^2{v}^2}{(|r|}^{-1}+{|v|}^{-1}).\end{array}$$

Setting

$$\begin{array}{c}\hfill U=\bigcup _{\begin{array}{c}\tilde{r},\tilde{v}\in 2\pi {\mathbb{Z}}^3:\\ 2\pi ⩽|\tilde{r}|,|\tilde{v}|⩽N\end{array}}{B}_{\tilde{r}}\times B_{\tilde{v}}\end{array}$$

this implies that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |\sum _{\begin{array}{c}r,v\in 2\pi {\mathbb{Z}}^3\\ 2\pi ⩽|r|,|v|⩽N\end{array}}\frac{1}{r^2+v^2+r·v}\frac{1}{r^2{v}^2}-& \frac{1}{{(2\pi )}^6}{\int }_U\frac{1}{r^2+v^2+r·v}\frac{1}{r^2{v}^2}drdv|\hfill \\ \hfill & ⩽C{\int }_{|r|,|v|⩾\pi }\frac{1}{r^2+v^2}\frac{1}{r^2{|v|}^3}drdv⩽C.\hfill \end{array}\end{array}$$

Observing that

$$\begin{array}{c}\hfill \{(r,v)\in {\mathbb{R}}^6:2\pi ⩽|r|,|v|⩽N-C\}\subset U\subset \{(r,v)\in {\mathbb{R}}^6:\pi ⩽|r|,|v|⩽N+C\},\end{array}$$

and estimating

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\int }_{\{\pi ⩽|r|⩽2\pi \}\cup \{N-C⩽|r|⩽N+C\}}dr{\int }_{\pi ⩽|v|⩽N+C}dv\frac{1}{r^2+v^2+r·v}\frac{1}{r^2{v}^2}⩽C\end{array}\end{array}$$

we conclude that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |\sum _{\begin{array}{c}r,v\in 2\pi {\mathbb{Z}}^3\\ 2\pi ⩽|r|,|v|⩽N\end{array}}\frac{1}{r^2+v^2+r·v}\frac{1}{r^2{v}^2}-& \frac{1}{{(2\pi )}^6}{\int }_{2\pi ⩽|r|,|v|⩽N}\frac{1}{r^2+v^2+r·v}\frac{1}{r^2{v}^2}drdv|⩽C.\hfill \end{array}\end{array}$$

Using the identity

$$\begin{array}{c}\hfill \sum _{|r|,|v|⩽N}\frac{r·v}{r^2+v^2+r·v}\frac{1}{r^2{|v|}^4}=-\sum _{|r|,|v|⩽N}\frac{{(r·v)}^2}{[r^2+v^2+r·v][r^2+v^2-r·v]}\frac{1}{r^2{|v|}^4}\end{array}$$

we can obtain a similar bound also for the contribution associated with the factor $r·v$ appearing on the r.h.s. of (2.45). Thus

$$\begin{array}{c}\hfill {\tilde{C}}_{{\mathcal{J}}_N}=\frac{1024{\pi }^4{\mathfrak{a}}^4}{N}\frac{1}{{(2\pi )}^6}{\int }_{2\pi ⩽|r|,|v|⩽N}\frac{r·v-v^2}{r^2+v^2+r·v}\frac{1}{r^2{|v|}^4}drdv+\mathcal{O}(N^{-1}).\end{array}$$

By explicit computation (for fixed r, we first integrate over v using spherical coordinates $(|v|,\theta ,\phi )$, with $r·v=|r||v|cos\theta$; the result of the v integral is a radial function of r, which can be integrated using spherical coordinates for r), we find

$$\begin{array}{c}\hfill {\tilde{C}}_{{\mathcal{J}}_N}=-64\pi (\frac{4}{3}\pi -\sqrt{3}){\mathfrak{a}}^4\frac{(logN)}{N}+\mathcal{O}(N^{-1}),\end{array}$$

which concludes the proof of (1.6). $\square$

Quadratic Renormalization: Proof of Theorem 2.4

In order to show Theorem 2.4 we will rely on bounds controlling the growth of the number of excitations and of their energy w.r.t. the action of the generalized Bogoliubov transformation $e^{B_{\mu }}$. Recall that, from Lemma (2.23), ${\Vert \mu \Vert }_2⩽C$ uniformly in N. As shown for example in [13, Lemma 3.1], this implies that for every $j⩾0$ there exists $C>0$ such that

$$\begin{array}{c}\hfill e^{-B_{\mu }}{({\mathcal{N}}_{+}+1)}^j{e}^{B_{\mu }}⩽C{({\mathcal{N}}_{+}+1)}^j.\end{array}$$ 3.1

From (2.20), we also obtain rough, non-uniform estimates on the growth of the energy. The proof of the following lemma can be found in [5, Lemma 7.1] (the coefficients ${\mu }_p$ and ${\eta }_p$ satisfy the same bounds).

Lemma 3.1

Let $\mathcal{K},{\mathcal{V}}_N$ be defined as in (2.6). Under the same assumptions as in Theorem 2.4, for every $j\in \mathbb{N}$ there exists $C>0$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^{-B_{\mu }}\mathcal{K}{({\mathcal{N}}_{+}+1)}^j{e}^{B_{\mu }}⩽& C\mathcal{K}{({\mathcal{N}}_{+}+1)}^j+CN{({\mathcal{N}}_{+}+1)}^{j+1}\hfill \\ \hfill e^{-B_{\mu }}{\mathcal{V}}_N{({\mathcal{N}}_{+}+1)}^j{e}^{B_{\mu }}⩽& C{\mathcal{V}}_N{({\mathcal{N}}_{+}+1)}^j+CN{({\mathcal{N}}_{+}+1)}^j.\hfill \end{array}\end{array}$$ 3.2

To prove Theorem 2.4, we will also need to compute the action of the generalized Bogoliubov transformation $e^{B_{\mu }}$ on creation and annihilation operators more precisely. To this end, we introduce the notation ${\gamma }_p=cosh{\mu }_p$, ${\sigma }_p=sinh{\mu }_p$ and we define operators $d_p,d_p^{\ast }$, for $p\in {\mathrm{\Lambda }}_{+}^{\ast }$, through the identities

$$\begin{array}{c}\hfill e^{-B_{\mu }}{b}_p{e}^{B_{\mu }}={\gamma }_p{b}_p+{\sigma }_p{b}_{-p}^{\ast }+d_p,e^{-B_{\mu }}{b}_p^{\ast }{e}^{B_{\mu }}={\gamma }_p{b}_p^{\ast }+{\sigma }_p{b}_{-p}+d_p^{\ast }.\end{array}$$ 3.3

In position space, we similarly introduce operator-valued distributions ${\check{d}}_x,{\check{d}}_x^{\ast }$, requiring that

$$\begin{array}{c}\hfill e^{-B_{\mu }}{\check{b}}_x{e}^{B_{\mu }}=b({\check{\gamma }}_x)+b^{\ast }({\check{\sigma }}_x)+{\check{d}}_x,e^{-B_{\mu }}{\check{b}}_x^{\ast }{e}^{B_{\mu }}=b{({\check{\gamma }}_x)}^{\ast }+b({\check{\sigma }}_y)+{\check{d}}_x^{\ast }.\end{array}$$

Here ${\check{\sigma }}_x(y)=\check{\sigma }(y;x)={\sum }_{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{\sigma }_{p}exp(ip·(x-y))$ and similarly for the distribution ${\check{\gamma }}_x$.

On states with few excitations, ie ${\mathcal{N}}_{+}\ll N$, the operators $b^{\ast },b$ are close to the standard creation and annihilation operators $a^{\ast },a$. Thus, we expect $B_{\mu }$ to act almost as a Bogoliubov transformation; equivalently, we expect $d^{\ast },d$ to be small. To prove bounds on the fields $d^{\ast },d$, we will use the integral representation proven in the following lemma.

Lemma 3.2

For $s\in [0,1]$, let ${\gamma }_p^{(s)}=cosh(s{\mu }_p)$ and ${\sigma }_p^{(s)}=sinh(s{\mu }_p)$. Then

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill d_p=& -\frac{1}{N}{\int }_0^{1}ds{e}^{-(1-s)B_{\mu }}[{\gamma }_p^{(s)}({\mu }_p{\mathcal{N}}_{+}{b}_{-p}^{\ast }+\sum _{q\in {\mathrm{\Lambda }}_{+}^{\ast }}{\mu }_q{b}_q^{\ast }{a}_{-q}^{\ast }{a}_p)\hfill \\ \hfill & +{\sigma }_p^{(s)}({\mu }_p{\mathcal{N}}_{+}{b}_p+\sum _{q\in {\mathrm{\Lambda }}_{+}^{\ast }}{\mu }_q{a}_{-p}^{\ast }{a}_{-q}{b}_q)]e^{(1-s)B_{\mu }}.\hfill \end{array}\end{array}$$ 3.4

Proof

The commutators

$$\begin{array}{c}\hfill \begin{array}{c}\hfill [b_p,B]={\mu }_p(1-\frac{{\mathcal{N}}_{+}}{N})b_{-p}^{\ast }-\frac{1}{N}\sum _{q\in {\mathrm{\Lambda }}_{+}^{\ast }}{\mu }_q{b}_q^{\ast }{a}_{-q}^{\ast }{a}_p\\ \hfill [b_{-p}^{\ast },B]={\mu }_p{b}_p(1-\frac{{\mathcal{N}}_{+}}{N})-\frac{1}{N}\sum _{q\in {\mathrm{\Lambda }}_{+}^{\ast }}{\mu }_q{a}_{-p}^{\ast }{a}_{-q}{b}_q\end{array}\end{array}$$

imply the identity

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{d}{\mathit{ds}}(e^{s{B}_{\mu }}({\gamma }_p^{(s)}{b}_p+{\sigma }_p^{(s)}{b}_{-p}^{\ast })e^{-s{B}_{\mu }})=& \frac{1}{N}{e}^{s{B}_{\mu }}[{\gamma }_p^{(s)}({\mu }_p{\mathcal{N}}_{+}{b}_{-p}^{\ast }+\sum _{q\in {\mathrm{\Lambda }}_{+}^{\ast }}{\mu }_q{b}_q^{\ast }{a}_{-q}^{\ast }{a}_p)\hfill \\ \hfill & +{\sigma }_p^{(s)}({\mu }_p{b}_p{\mathcal{N}}_{+}+\sum _{q\in {\mathrm{\Lambda }}_{+}^{\ast }}{\mu }_q{a}_{-p}^{\ast }{a}_{-q}{b}_q)]e^{-s{B}_{\mu }}.\hfill \end{array}\end{array}$$

Integrating both sides from $s=0$ to $s=1$ and comparing with (3.3) concludes the proof. $\square$

In the next lemma, we collect bounds for the operators $d^{\ast },d$ that will be used throughout the proof of Theorem 2.4. Similar estimates have been shown in [5], but only under the assumption that the ${\ell }^2$-norm of the coefficients in the Bogoliubov transformation is small enough (which is not satisfied here, since we included $\tau$ in (2.18)). With the help of the representation (3.4), we relax this assumption.

Lemma 3.3

Let $\mu \in {\ell }^2({\mathrm{\Lambda }}_{+}^{\ast })$, and let $n\in \mathbb{N}$. Then there exists $C>0$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \Vert {({\mathcal{N}}_{+}+1)}^{n/2}{d}_p\xi \Vert ⩽& \frac{C}{N}[|{\mu }_p|\Vert {({\mathcal{N}}_{+}+1)}^{(n+3)/2}\xi \Vert +\Vert b_p{({\mathcal{N}}_{+}+1)}^{(n+2)/2}\xi \Vert ]\hfill \\ \hfill \Vert {({\mathcal{N}}_{+}+1)}^{n/2}{d}_p^{\ast }\xi \Vert ⩽& \frac{C}{N}\Vert {({\mathcal{N}}_{+}+1)}^{(n+3)/2}\xi \Vert \hfill \\ \hfill \Vert {({\mathcal{N}}_{+}+1)}^{n/2}{a}_q{d}_p\xi \Vert ⩽& \frac{C}{N}[|{\mu }_q|\Vert {({\mathcal{N}}_{+}+1)}^{(n+3)/2}{a}_p\xi \Vert +|{\mu }_p|\Vert {({\mathcal{N}}_{+}+1)}^{(n+3)/2}{a}_q\xi \Vert \hfill \\ \hfill & +{\delta }_{p,-q}|{\mu }_q|\Vert {({\mathcal{N}}_{+}+1)}^{(n+3)/2}\xi \Vert +\Vert {({\mathcal{N}}_{+}+1)}^{(n+2)/2}{a}_p{a}_q\xi \Vert \hfill \\ \hfill & +|{\mu }_p||{\mu }_q|\Vert {({\mathcal{N}}_{+}+1)}^{(n+4)/2}\xi \Vert ]\hfill \\ \hfill \Vert {({\mathcal{N}}_{+}+1)}^{n/2}{a}_q{d}_p^{\ast }\xi \Vert ⩽& \frac{C}{N}[|{\mu }_q|\Vert {({\mathcal{N}}_{+}+1)}^{(n+4)/2}\xi \Vert +\Vert {({\mathcal{N}}_{+}+1)}^{(n+3)/2}{a}_q\xi \Vert \hfill \\ \hfill & +{\delta }_{p,-q}\Vert {({\mathcal{N}}_{+}+1)}^{(n+2)/2}\xi \Vert ],\hfill \end{array}\end{array}$$ 3.5

for all $p\in {\mathrm{\Lambda }}_{+}^{\ast }$ and $\xi \in {\mathcal{F}}_{\perp }^{⩽N}$. Moreover, as distributions,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \Vert {({\mathcal{N}}_{+}+1)}^{n/2}{\check{d}}_x\xi \Vert ⩽& \frac{C}{N}[\Vert {({\mathcal{N}}_{+}+1)}^{(n+3)/2}\xi \Vert +\Vert {\check{a}}_x{({\mathcal{N}}_{+}+1)}^{(n+2)/2}\xi \Vert ]\hfill \\ \hfill \Vert {({\mathcal{N}}_{+}+1)}^{n/2}{\check{a}}_y{\check{d}}_x\xi \Vert ⩽& \frac{C}{N}[\Vert {\check{a}}_x{({\mathcal{N}}_{+}+1)}^{(n+1)/2}\xi \Vert +(1+|\check{\mu }(x-y)|)\Vert {({\mathcal{N}}_{+}+1)}^{(n+2)/2}\xi \Vert \hfill \\ \hfill & +\Vert {\check{a}}_y{({\mathcal{N}}_{+}+1)}^{(n+3)/2}\xi \Vert +\Vert {\check{a}}_x{\check{a}}_y{({\mathcal{N}}_{+}+1)}^{(n+2)/2}\xi \Vert ]\hfill \\ \hfill \Vert {({\mathcal{N}}_{+}+1)}^{n/2}{\check{d}}_x{\check{d}}_y\xi \Vert ⩽& \frac{C}{N^2}[\Vert {({\mathcal{N}}_{+}+1)}^{(n+6)/2}\xi \Vert +|\check{\mu }(x-y)|\Vert {({\mathcal{N}}_{+}+1)}^{(n+4)/2}\xi \Vert \hfill \\ \hfill & +\Vert {\check{a}}_x{({\mathcal{N}}_{+}+1)}^{(n+5)/2}\xi \Vert +\Vert {\check{a}}_y{({\mathcal{N}}_{+}+1)}^{(n+5)/2}\xi \Vert \hfill \\ \hfill & +\Vert {\check{a}}_x{\check{a}}_y{({\mathcal{N}}_{+}+1)}^{(n+4)/2}\xi \Vert ].\hfill \end{array}\end{array}$$ 3.6

A proof of Lemma 3.3 is given in Appendix A. The main message from Lemma 3.3 is the fact that d-operators, defined through (3.3), are small (and therefore the action of $e^{-B_{\mu }}$ is close to the action of a Bogoliubov transformation) on states with few excitations. For $d_p$ (but not for $d_p^{\ast }$), these bounds also allow us to extract some decay for large momenta $p\in {\mathrm{\Lambda }}_{+}^{\ast }$.

We proceed now with the computation of the renormalized excitation Hamiltonian ${\mathcal{G}}_N$, which will lead to the proof of Theorem 2.4. From (2.5), we find

$$\begin{array}{c}\hfill {\mathcal{G}}_N=\frac{\widehat{V}(0)}{2}(N-1)+{\mathcal{G}}_N^{(2,\mathcal{K})}+{\mathcal{G}}_N^{(2,V)}+{\mathcal{G}}_N^{(3)}+{\mathcal{G}}_N^{(4)}\end{array}$$

with

$$\begin{array}{cc}\hfill {\mathcal{G}}_N^{(2,\mathcal{K})}& =e^{-B_{\mu }}\mathcal{K}{e}^{B_{\mu }},{\mathcal{G}}_N^{(2,V)}=e^{-B_{\mu }}{\mathcal{L}}_N^{(2,V)}{e}^{B_{\mu }},\hfill \\ \hfill {\mathcal{G}}_N^{(3)}& =e^{-B_{\mu }}{\mathcal{L}}_N^{(3)}{e}^{B_{\mu }},{\mathcal{G}}_N^{(4)}=e^{-B_{\mu }}{\mathcal{V}}_N{e}^{B_{\mu }}.\hfill \end{array}$$

The form of the operators ${\mathcal{G}}_N^{(2,K)},{\mathcal{G}}_N^{(2,V)},{\mathcal{G}}_N^{(3)},{\mathcal{G}}_N^{(4)}$ will be determined in Props. 3.4–3.7, up to negligible errors. To this end, we will argue similarly as in the proof of [5, Proposition 3.2]; however, to resolve the energy to lower order, we will need to keep several additional terms, which did not play an important role in [5].

First of all, we establish the form of the operator ${\mathcal{G}}_N^{(2,K)}=e^{-B_{\mu }}\mathcal{K}{e}^{B_{\mu }}$.

Proposition 3.4

Under the same assumptions of Theorem 2.4 we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{G}}_N^{(2,K)}=& \mathcal{K}+\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2[{\sigma }_p^2+2{\sigma }_p^2{b}_p^{\ast }{b}_p+{\gamma }_p{\sigma }_p(b_p{b}_{-p}+b_p^{\ast }{b}_{-p}^{\ast })]+\frac{1}{N}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{\eta }_p^2(1-{\mathcal{N}}_{+})\hfill \\ \hfill & +\frac{1}{N}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{\eta }_p^2[{\sigma }_q^2+({\gamma }_q^2+{\sigma }_q^2)b_q^{\ast }{b}_q+{\gamma }_q{\sigma }_q(b_q{b}_{-q}+b_q^{\ast }{b}_{-q}^{\ast })]\hfill \\ \hfill & +\frac{1}{2N}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{\eta }_p(b_p{b}_{-p}+b_p^{\ast }{b}_{-p}^{\ast })[{\sigma }_q^2-\frac{{\gamma }_q{\sigma }_q}{{\eta }_q}+1]\hfill \\ \hfill & +\frac{1}{2N}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{\eta }_p{b}_p^{\ast }{b}_{-p}^{\ast }{b}_q^{\ast }{b}_{-q}^{\ast }[{\gamma }_q{\sigma }_q-\frac{{\sigma }_q^2}{{\eta }_q}]+\mathrm{h}.\mathrm{c}.\hfill \\ \hfill & +\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{\eta }_p\left(b_{-p},d_p,+,d_p^{\ast },b_{-p}^{\ast }\right)+{\mathcal{E}}_{{\mathcal{G}}_N^K}\hfill \end{array}\end{array}$$ 3.7

with

$$\begin{array}{c}\hfill \pm {\mathcal{E}}_{{\mathcal{G}}_N^K}⩽\epsilon {\mathcal{N}}_{+}+\frac{C}{\epsilon N}(\mathcal{K}+{\mathcal{N}}_{+}^2+1)({\mathcal{N}}_{+}+1).\end{array}$$ 3.8

Remark

Compared with the corresponding result in [5], we additionally need to keep track of the terms appearing in the third and fourth line of (3.7). These terms will be included in the operator ${\mathcal{T}}_{{\mathcal{G}}_N}$ defined in (2.29) and eventually will be proven to be negligible after cubic conjugation.

Proof of Proposition 3.4

Writing

$$\begin{array}{c}\hfill \mathcal{K}=\frac{N-1}{N}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{b}_p^{\ast }{b}_p+\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{b}_p^{\ast }{b}_p\frac{{\mathcal{N}}_{+}}{N}+\mathcal{K}\frac{{({\mathcal{N}}_{+}-1)}^2}{N^2},\end{array}$$

and using (3.2) to get rid of the last term, we find

$$\begin{array}{c}\hfill e^{-B_{\mu }}\mathcal{K}{e}^{B_{\mu }}=\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{e}^{-B_{\mu }}{b}_p^{\ast }{b}_p{e}^{B_{\mu }}+\frac{1}{N}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{e}^{-B_{\mu }}{b}_p^{\ast }{b}_q^{\ast }{b}_p{b}_q{e}^{B_{\mu }}+{\mathcal{E}}_{\mathcal{K},1}\end{array}$$ 3.9

with

$$\begin{array}{c}\hfill \pm {\mathcal{E}}_{\mathcal{K},1}⩽C{N}^{-1}(\mathcal{K}+{\mathcal{N}}_{+}^2+1)({\mathcal{N}}_{+}+1).\end{array}$$

We decompose the first term on the r.h.s. of (3.9) as

$$\begin{array}{c}\hfill \sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{e}^{-B_{\mu }}{b}_p^{\ast }{b}_p{e}^{B_{\mu }}=E_1+E_2+E_3,\end{array}$$

with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill E_1=& \sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2({\gamma }_p{b}_{-p}^{\ast }+{\sigma }_p{b}_{-p})({\gamma }_p{b}_p+{\sigma }_p{b}_{-p}^{\ast })\hfill \\ \hfill E_2=& \sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2({\gamma }_p{b}_p^{\ast }+{\sigma }_p{b}_{-p})d_p+\mathrm{h}.\mathrm{c}.\hfill \\ \hfill E_3=& \sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{d}_p^{\ast }{d}_p.\hfill \end{array}\end{array}$$

With the commutation relations (2.2) (and using $|{\mu }_p-{\eta }_p|=|{\tau }_p{|⩽C/|p|}^4$) we obtain

$$\begin{array}{c}\hfill E_1=\mathcal{K}+\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2({\sigma }_p^2-{\eta }_p^2\frac{{\mathcal{N}}_{+}}{N})+\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}(p^2{\gamma }_p{\sigma }_p(b_p{b}_{-p}+b_p^{\ast }{b}_{-p}^{\ast })+2p^2{\sigma }_p^2{b}_p^{\ast }{b}_p)+{\mathcal{E}}_{\mathcal{K},2},\end{array}$$ 3.10

with $\pm {\mathcal{E}}_{\mathcal{K},2}⩽C{N}^{-1}\mathcal{K}({\mathcal{N}}_{+}+1)$. Applying (3.5), we find (see [5, Equation above (7.16)])

$$\begin{array}{c}\hfill \pm E_3⩽C{N}^{-1}(\mathcal{K}+{\mathcal{N}}_{+}^2+1)({\mathcal{N}}_{+}+1).\end{array}$$ 3.11

As for $E_2$, using (3.5), $|{\gamma }_p{-1|⩽C/|p|}^4$, $|{\sigma }_p-{\mu }_p{|⩽C/|p|}^6$ and again $|{\mu }_p-{\eta }_p|=|{\tau }_p{|⩽C/|p|}^4$, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill E_2=& \sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{\eta }_p(b_{-p}{d}_p+\mathrm{h}.\mathrm{c}.)+\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2(b_p^{\ast }{d}_p+\mathrm{h}.\mathrm{c}.)+{\mathcal{E}}_{\mathcal{K},3},\hfill \end{array}\end{array}$$

with $\pm {\mathcal{E}}_{\mathcal{K},3}⩽C{N}^{-1}{({\mathcal{N}}_{+}+1)}^2$. The first term contributes to the r.h.s. of (3.7). As for the second term, we expand $d_p$ using (3.4). In the resulting expression, we observe that the coefficients ${\gamma }_p^{(s)}$ and ${\sigma }_p^{(s)}$ can be replaced by 1 and $s{\mu }_p$, respectively, up to negligible errors. As an example, consider

$$\begin{array}{cc}\hfill \pm \frac{1}{N}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}& p^2{\mu }_q{\int }_0^{1}ds({\gamma }_p^{(s)}-1)(\langle \xi ,b_p^{\ast }{e}^{(s-1)B_{\mu }}{b}_q^{\ast }{a}_{-q}^{\ast }{a}_p{e}^{-(s-1)B_{\mu }}\xi \rangle +\mathrm{h}.\mathrm{c}.)\hfill \\ \hfill ⩽& \frac{C}{N}{\int }_0^{1}ds\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}\frac{1}{{|q|}^2}\Vert b_q{e}^{-(s-1)B_{\mu }}{b}_p\xi \Vert \Vert a_p{\mathcal{N}}_{+}^{1/2}{e}^{-(s-1)B_{\mu }}\xi \Vert \hfill \\ \hfill ⩽& C{N}^{-1}⟨\xi ,{({\mathcal{N}}_{+}+1)}^3\xi ⟩,\hfill \end{array}$$ 3.12

We obtain (replacing $a,a^{\ast }$ with $b,b^{\ast }$ only produces small corrections)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}& p^2(b_p^{\ast }{d}_p+\mathrm{h}.\mathrm{c}.)\hfill \\ \hfill =& -\frac{1}{N}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{\mu }_p{\int }_0^{1}ds{b}_p^{\ast }{e}^{(s-1)B_{\mu }}{b}_{-p}^{\ast }(\sum _{q\in {\mathrm{\Lambda }}_{+}^{\ast }}{b}_q^{\ast }{b}_q+1)e^{-(s-1)B_{\mu }}+\mathrm{h}.\mathrm{c}.\hfill \\ \hfill & -\frac{1}{N}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{\mu }_q{\int }_0^{1}ds{b}_p^{\ast }{e}^{(s-1)B_{\mu }}{b}_q^{\ast }{b}_{-q}^{\ast }{b}_p{e}^{-(s-1)B_{\mu }}+\mathrm{h}.\mathrm{c}.\hfill \\ \hfill & -\frac{1}{N}\sum _{p,q,\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{\mu }_p{\mu }_q{\int }_0^{1}dss{b}_p^{\ast }{e}^{(s-1)B_{\mu }}{b}_{-p}^{\ast }{b}_q{b}_{-q}{e}^{-(s-1)B_{\mu }}+\mathrm{h}.\mathrm{c}.+{\mathcal{E}}_{\mathcal{K},4}\hfill \end{array}\end{array}$$

with $\pm {\mathcal{E}}_{\mathcal{K},4}⩽C{N}^{-1}{({\mathcal{N}}_{+}+1)}^3$. With (3.3) we observe that $e^{(s-1)B_{\mu }}{b}_{-p}^{\ast }{e}^{-(s-1)B_{\mu }}\simeq b_{-p}^{\ast }$ (in the first and third line) and that $e^{(s-1)B_{\mu }}{b}_{-p}^{\ast }{e}^{-(s-1)B_{\mu }}\simeq {\mu }_p{b}_p$ (in second line), up to contributions that can be bounded similarly as in (3.12). Setting $t=1-s$, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}& p^2(b_p^{\ast }{d}_p+\mathrm{h}.\mathrm{c}.)\hfill \\ \hfill =& -\frac{1}{N}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{\mu }_p{b}_p^{\ast }{b}_{-p}^{\ast }{\int }_0^{1}dt{e}^{-t{B}_{\mu }}(b_q^{\ast }{b}_q+t{\mu }_q{b}_q^{\ast }{b}_{-q}^{\ast }+(1-t){\mu }_q{b}_q{b}_{-q})e^{t{B}_{\mu }}+\mathrm{h}.\mathrm{c}.\hfill \\ \hfill & -\frac{1}{N}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{\mu }_p(b_p^{\ast }{b}_{-p}^{\ast }+\mathrm{h}.\mathrm{c}.)+{\mathcal{E}}_{\mathcal{K},5}\hfill \end{array}\end{array}$$

with $\pm {\mathcal{E}}_{\mathcal{K},5}⩽C{N}^{-1}{({\mathcal{N}}_{+}+1)}^3$. Applying again (3.3), noticing that contributions involving $d,d^{\ast }$ operators are negligible as a consequence of (3.5), rearranging terms in normal order, computing the integrals over t explicitly and using that $|{\mu }_p-{\eta }_p{|⩽C/|p|}^4$, we arrive at

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill E_2=& \sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2({\eta }_p{b}_{-p}{d}_p+\mathrm{h}.\mathrm{c}.)\hfill \\ \hfill & -\frac{1}{N}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{\eta }_p{b}_p^{\ast }{b}_{-p}^{\ast }[({\sigma }_q^2+\frac{{\gamma }_q{\sigma }_q}{{\mu }_q})b_q^{\ast }{b}_q+\frac{1}{2}(\frac{{\sigma }_q^2}{{\mu }_q}+{\gamma }_q{\sigma }_q)(b_q{b}_{-q}+b_q^{\ast }{b}_{-q}^{\ast })]\hfill \\ \hfill & -\frac{1}{N}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{\eta }_p(b_p^{\ast }{b}_{-p}^{\ast }+\mathrm{h}.\mathrm{c}.)[1+\frac{1}{2}\sum _{q\in {\mathrm{\Lambda }}_{+}^{\ast }}({\sigma }_q^2+\frac{{\gamma }_q{\sigma }_q}{{\mu }_q}-1)]+{\mathcal{E}}_{\mathcal{K},6},\hfill \end{array}\end{array}$$ 3.13

where $\pm {\mathcal{E}}_{\mathcal{K},6}⩽C{N}^{-1}{({\mathcal{N}}_{+}+1)}^3$.

Finally, we consider the second term on the r.h.s. of (3.9). Again we apply (3.3) and we observe that all contributions involving the operators $d,d^{\ast }$ are negligible, by (3.5). Furthermore, we notice that the coefficients ${\gamma }_p$ and ${\sigma }_p$ can be replaced everywhere by one and, respectively, ${\eta }_p$, up to a negligible error (using again $|{\mu }_p-{\eta }_p{|⩽C/|p|}^4$). Finally, we remark that all terms proportional to $b_p^{\ast }{b}_p$ are also irrelevant (because the sum over p can be controlled by $\mathcal{K}$). Collecting all terms proportional to $b_p^{\ast }{b}_{-p}^{\ast },b_p{b}_{-p}$ (and all commutators arising from normal ordering), we arrive at

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{1}{N}& \sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{e}^{-B_{\mu }}{b}_p^{\ast }{b}_q^{\ast }{b}_q{b}_p{e}^{B_{\mu }}\hfill \\ \hfill =& \frac{1}{N}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{\eta }_p^2(1+\sum _{q\in {\mathrm{\Lambda }}_{+}^{\ast }}{\sigma }_q^2)+\frac{1}{N}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{\eta }_p^2[({\gamma }_q^2+{\sigma }_q^2)b_q^{\ast }{b}_q+{\gamma }_q{\sigma }_q(b_q{b}_{-q}+b_q^{\ast }{b}_{-q}^{\ast })]\hfill \\ \hfill & +\frac{1}{N}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{\eta }_p{b}_p^{\ast }{b}_{-p}^{\ast }(({\gamma }_q^2+{\sigma }_q^2)b_q^{\ast }{b}_q+{\gamma }_q{\sigma }_q(b_q{b}_{-q}+b_q^{\ast }{b}_{-q}^{\ast }))+\mathrm{h}.\mathrm{c}.\hfill \\ \hfill & +\frac{1}{N}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{\eta }_p{b}_p^{\ast }{b}_{-p}^{\ast }(1+\sum _{q\in {\mathrm{\Lambda }}_{+}^{\ast }}{\sigma }_q^2)+{\mathcal{E}}_{\mathcal{K},7},\hfill \end{array}\end{array}$$ 3.14

with $\pm {\mathcal{E}}_{\mathcal{K},7}⩽C{N}^{-1}(\mathcal{K}+{\mathcal{N}}_{+}^2+1)({\mathcal{N}}_{+}+1)$. Lastly, we notice that, in (3.13) and in (3.14), the quartic terms which contain both creation and annihilation operators can be treated as errors bounded by $\epsilon {\mathcal{N}}_{+}+C{\epsilon }^{-1}{N}^{-1}\mathcal{K}{({\mathcal{N}}_{+}+1)}^2$. As an example, consider

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \pm \frac{1}{N}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}& p^2{\eta }_p{\gamma }_q{\sigma }_q⟨\xi ,b_p^{\ast }{b}_{-p}^{\ast }{b}_q{b}_{-q}\xi ⟩+\mathrm{h}.\mathrm{c}.\hfill \\ \hfill ⩽& \frac{C}{N}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2|{\eta }_p||{\sigma }_q|\Vert b_q^{\ast }{b}_{-p}{b}_p\xi \Vert \Vert b_{-q}\xi \Vert ⩽\epsilon ⟨\xi ,{\mathcal{N}}_{+}\xi ⟩+\frac{C}{\epsilon N}⟨\xi ,\mathcal{K}{({\mathcal{N}}_{+}+1)}^2\xi ⟩.\hfill \end{array}\end{array}$$ 3.15

Combining this with (3.10), (3.11), (3.13), we obtain (3.7). $\square$

Next, we consider the operator ${\mathcal{G}}_N^{(2,V)}=e^{-B_{\mu }}{\mathcal{L}}_N^{(2,V)}{e}^{B_{\mu }}$.

Proposition 3.5

Under the same assumptions of Theorem 2.4 we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{G}}_N^{(2,V)}=& \sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}(p/N)\left({\sigma }_p^2,+,{\gamma }_p,{\sigma }_p,-,{\eta }_p,\frac{{\mathcal{N}}_{+}}{N}\right)\hfill \\ \hfill & +\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}(p/N){({\gamma }_p+{\sigma }_p)}^2\left(b_p^{\ast },b_p,+,\frac{1}{2},(b_p{b}_{-p}+b_p^{\ast }{b}_{-p}^{\ast })\right)\hfill \\ \hfill & +\frac{1}{2}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}(p/N)[b_{-p}{d}_p+d_p(b_{-p}+{\eta }_p{b}_p^{\ast })+\mathrm{h}.\mathrm{c}.]+{\mathcal{E}}_{{\mathcal{G}}_N^V},\hfill \end{array}\end{array}$$ 3.16

with

$$\begin{array}{c}\hfill \pm {\mathcal{E}}_{{\mathcal{G}}_N^V}⩽C{N}^{-1}(\mathcal{K}+{\mathcal{N}}_{+}^3+1)({\mathcal{N}}_{+}+1).\end{array}$$

Proof of Proposition 3.4

With (2.7), we write

$$\begin{array}{c}\hfill {\mathcal{G}}_N^{(2,V)}=e^{-B_{\mu }}{\mathcal{L}}_N^{(2,V)}{e}^{B_{\mu }}=F_1+F_2+F_3+F_4\end{array}$$

with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill F_1=& \sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}(p/N)e^{-B_{\mu }}{b}_p^{\ast }{b}_p{e}^{B_{\mu }}\hfill \\ \hfill F_2=& -\frac{1}{N}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}(p/N)e^{-B_{\mu }}{a}_p^{\ast }{a}_p{e}^{B_{\mu }}\hfill \\ \hfill F_3=& \frac{1}{2}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}(p/N)e^{-B_{\mu }}(b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p})e^{B_{\mu }}\hfill \\ \hfill F_4=& \frac{\widehat{V}(0)}{2N}{e}^{-B_{\mu }}{\mathcal{N}}_{+}({\mathcal{N}}_{+}-1)e^{B_{\mu }}.\hfill \end{array}\end{array}$$

Clearly $\pm (F_2+F_4)⩽C{N}^{-1}{({\mathcal{N}}_{+}+1)}^2$. Proceeding as in [5, Proposition 7.2] (using (3.3) and the bounds (3.5)) and normal ordering using (2.2), we find

$$\begin{array}{c}\hfill F_1=\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}(p/N)(({\gamma }_p^2+{\sigma }_p^2)b_p^{\ast }{b}_p+{\gamma }_p{\sigma }_p(b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p}))+{\mathcal{E}}_{V,1},\end{array}$$

with $\pm {\mathcal{E}}_{V,1}⩽C{N}^{-1}{({\mathcal{N}}_{+}+1)}^2$. As for $F_3$, we apply (3.3) to decompose

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill F_3=& \frac{1}{2}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}(p/N)(({\gamma }_p^2+{\sigma }_p^2)(b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p})+2{\gamma }_p{\sigma }_p(2b_p^{\ast }{b}_p+1)-2{\eta }_p\frac{{\mathcal{N}}_{+}}{N})\hfill \\ \hfill & +\frac{1}{2}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}(p/N)[b_{-p}{d}_p+d_p(b_{-p}+{\eta }_p{b}_p^{\ast })]+\mathrm{h}.\mathrm{c}.+{\mathcal{E}}_{V,2},\hfill \end{array}\end{array}$$

with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{E}}_{V,2}=& \frac{1}{2}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}(p/N)[2({\eta }_p-{\gamma }_p{\sigma }_p)\frac{{\mathcal{N}}_{+}}{N}+({\gamma }_p-1)(b_{-p}{d}_p+d_p{b}_{-p})\hfill \\ \hfill & +{\sigma }_p{b}_p^{\ast }{d}_p+({\sigma }_p-{\eta }_p)d_p{b}_{-p}^{\ast }+d_p^{\ast }{d}_{-p}^{\ast }+d_{-p}{d}_p].\hfill \end{array}\end{array}$$ 3.17

Using (3.5), we bound the last two terms of ${\mathcal{E}}_{V,2}$ by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \pm \frac{1}{2}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}& \widehat{V}(p/N)(⟨\xi ,d_p^{\ast }{d}_{-p}^{\ast }\xi ⟩+\mathrm{h}.\mathrm{c}.)\hfill \\ \hfill ⩽& C\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}|\widehat{V}(p/N)|\Vert {({\mathcal{N}}_{+}+1)}^{1/2}{d}_{-p}^{\ast }\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{-1/2}{d}_p\xi \Vert \hfill \\ \hfill ⩽& \frac{C}{N^2}\Vert {({\mathcal{N}}_{+}+1)}^2\xi \Vert (\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}\frac{1}{{|p|}^2}|\widehat{V}(p/N)|{)}^{1/2}\hfill \\ \hfill & \times (\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{|{\mu }_p|}^2\Vert ({\mathcal{N}}_{+}+1)\xi {\Vert }^2+\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2\Vert b_p{({\mathcal{N}}_{+}+1)}^{1/2}\xi {\Vert }^2{)}^{1/2}\hfill \\ \hfill ⩽& C{N}^{-1}⟨\xi ,(\mathcal{K}+{\mathcal{N}}_{+}^3+1)({\mathcal{N}}_{+}+1)\xi ⟩.\hfill \end{array}\end{array}$$

Using (3.5), the rest of the terms of ${\mathcal{E}}_{V,2}$ can be shown to be negligible as well. Arranging the main contributions to $F_1$ and $F_3$ in normal order, we obtain (3.16), up to another negligible remainder. $\square$

We now discuss the cubic term ${\mathcal{G}}_N^{(3)}=e^{-B_{\mu }}{\mathcal{L}}_N^{(3)}{e}^{B_{\mu }}$.

Proposition 3.6

Under the same assumptions of Theorem 2.4 we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{G}}_N^{(3)}=& {\mathcal{C}}_{{\mathcal{G}}_N}+{\mathcal{E}}_{{\mathcal{G}}_N^3}\hfill \end{array}\end{array}$$ 3.18

where ${\mathcal{C}}_{{\mathcal{G}}_N}$ is defined as in (2.28), and

$$\begin{array}{c}\hfill \pm {\mathcal{E}}_{{\mathcal{G}}_N^3}⩽\epsilon {\mathcal{N}}_{+}+\frac{C}{\epsilon N}({\mathcal{H}}_N+{\mathcal{N}}_{+}^2+1)({\mathcal{N}}_{+}+1).\end{array}$$

Proof

From (2.7), we find

$$\begin{array}{c}\hfill {\mathcal{G}}_N^{(3)}=\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}\widehat{V}(p/N)e^{-B_{\mu }}(b_{p+q}^{\ast }{b}_{-p}^{\ast }{b}_q+\mathrm{h}.\mathrm{c}.)e^{B_{\mu }}+{\mathcal{E}}_{3,1},\end{array}$$

with

$$\begin{array}{c}\hfill {\mathcal{E}}_{3,1}=\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}\widehat{V}(p/N)e^{-B_{\mu }}{b}_{p+q}^{\ast }{a}_{-p}^{\ast }\frac{{\mathcal{N}}_{+}}{N}{a}_q{e}^{B_{\mu }}+\mathrm{h}.\mathrm{c}..\end{array}$$ 3.19

Let us first focus on the main term, we will show later that ${\mathcal{E}}_{3,1}$ can be absorbed in the error ${\mathcal{E}}_{{\mathcal{G}}_N^3}$. With (3.3), we decompose

$$\begin{array}{c}\hfill \frac{1}{\sqrt{N}}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}\widehat{V}(p/N)e^{-B_{\mu }}{b}_{p+q}^{\ast }{b}_{-p}^{\ast }{b}_q{e}^{B_{\mu }}=M_0+M_1+M_2+M_3,\end{array}$$

where, for $j=0,\cdots ,3$, $M_j$ collects contributions with j factors $d,d^{\ast }$, ie.

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill M_0=& \frac{1}{\sqrt{N}}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}\widehat{V}(p/N)({\gamma }_{p+q}{b}_{p+q}^{\ast }+{\sigma }_{p+q}{b}_{-p-q})({\gamma }_p{b}_{-p}^{\ast }+{\sigma }_p{b}_p)({\gamma }_q{b}_q+{\sigma }_q{b}_{-q}^{\ast })\hfill \end{array}\end{array}$$

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill M_1=& \frac{1}{\sqrt{N}}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}\widehat{V}(p/N)\{({\gamma }_{p+q}{b}_{p+q}^{\ast }+{\sigma }_{p+q}{b}_{-p-q})({\gamma }_p{b}_{-p}^{\ast }+{\sigma }_p{b}_p)d_q\hfill \\ & +({\gamma }_{p+q}{b}_{p+q}^{\ast }+{\sigma }_{p+q}{b}_{-p-q})d_{-p}^{\ast }({\gamma }_q{b}_q+{\sigma }_q{b}_{-q}^{\ast })\hfill \\ & +d_{p+q}^{\ast }({\gamma }_p{b}_{-p}^{\ast }+{\sigma }_p{b}_p)({\gamma }_q{b}_q+{\sigma }_q{b}_{-q}^{\ast })\}\hfill \\ \hfill M_2=& \frac{1}{\sqrt{N}}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}\widehat{V}(p/N)\{({\gamma }_{p+q}{b}_{p+q}^{\ast }+{\sigma }_{p+q}{b}_{-p-q}^{\ast })d_{-p}^{\ast }{d}_q\hfill \\ & +d_{p+q}^{\ast }({\gamma }_p{b}_{-p}^{\ast }+{\sigma }_p{b}_p)d_q+d_{p+q}^{\ast }{d}_{-p}^{\ast }({\gamma }_q{b}_q+{\sigma }_q{b}_{-q}^{\ast })\}\hfill \\ \hfill M_3=& \frac{1}{\sqrt{N}}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}\widehat{V}(p/N)d_{p+q}^{\ast }{d}_{-p}^{\ast }{d}_q.\hfill \end{array}\end{array}$$

Let us first consider $M_0$. Rearranging terms in normal order and noticing that all contributions arising from commutators are negligible (because, due to translation invariance, labels of creation operators cannot coincide with labels of annihilation operators without violating the condition $p,q,p+q\ne 0$), we find

$$\begin{array}{cc}\hfill M_0=& {\mathcal{C}}_{{\mathcal{G}}_N}+\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}\widehat{V}(p/N)({\gamma }_{p+q}{\gamma }_p-1)b_{p+q}^{\ast }{b}_{-p}^{\ast }({\gamma }_q{b}_q+{\sigma }_q{b}_{-q}^{\ast })+\mathrm{h}.\mathrm{c}.\hfill \\ & +\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}\widehat{V}(p/N){\gamma }_{p+q}{\sigma }_p{b}_{p+q}^{\ast }({\gamma }_q{b}_q+{\sigma }_q{b}_{-q}^{\ast })b_p+\mathrm{h}.\mathrm{c}.\hfill \\ & +\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}\widehat{V}(p/N){\sigma }_{p+q}{\gamma }_p{b}_{-p}^{\ast }({\gamma }_q{b}_q+{\sigma }_q{b}_{-q}^{\ast })b_{-p-q}+\mathrm{h}.\mathrm{c}.\hfill \\ & +\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}\widehat{V}(p/N){\sigma }_{p+q}{\sigma }_p({\gamma }_q{b}_q+{\sigma }_q{b}_{-q}^{\ast })b_p{b}_{-p-q}+\mathrm{h}.\mathrm{c}.+{\mathcal{E}}_{3,2}\hfill \end{array}$$ 3.20

with $\pm {\mathcal{E}}_{3,2}⩽C{N}^{-3/2}{({\mathcal{N}}_{+}+1)}^2$. Except for ${\mathcal{C}}_{{\mathcal{G}}_N}$, all the terms in $M_0$ can also be treated as error as they are bounded by $\epsilon {\mathcal{N}}_{+}+C{\epsilon }^{-1}{N}^{-1}{({\mathcal{N}}_{+}+1)}^2$. In fact, considering for example one of the contributions in the last line of (3.20), we have

$$\begin{array}{cc}\hfill |\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}& \widehat{V}(p/N){\sigma }_{p+q}{\sigma }_p{\gamma }_q⟨\xi ,b_q{b}_p{b}_{-p-q}\xi ⟩|\hfill \\ \hfill ⩽& \frac{C}{\sqrt{N}}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}|{\eta }_p||{\eta }_{p+q}|\Vert b_q{b}_p{b}_{-p-q}{({\mathcal{N}}_{+}+1)}^{-1}\xi \Vert \Vert ({\mathcal{N}}_{+}+1)\xi \Vert \hfill \\ \hfill ⩽& \frac{C}{\sqrt{N}}[\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}\Vert b_q{b}_p{({\mathcal{N}}_{+}+1)}^{-1/2}{\xi \Vert }^2{]}^{1/2}[\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}|{\eta }_{p+q}{|}^2{|{\eta }_p|}^2{]}^{1/2}\hfill \\ \hfill ⩽& C{N}^{-1/2}\Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert \Vert ({\mathcal{N}}_{+}+1)\xi \Vert .\hfill \end{array}$$

Next, we show that the terms $M_1$, $M_2$, and $M_3$ are negligible. First, let us consider $M_1$. Terms with at least one $\gamma$-coefficient can be estimated by Cauchy-Schwarz, using (3.5). For example,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \pm & \frac{1}{\sqrt{N}}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}\widehat{V}(p/N){\gamma }_{p+q}{\sigma }_q(⟨\xi ,b_{p+q}^{\ast }{d}_{-p}^{\ast }{b}_{-q}^{\ast }\xi ⟩+\mathrm{h}.\mathrm{c}.)\hfill \\ \hfill ⩽& \frac{C}{\sqrt{N}}[\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}\frac{|\widehat{V}(p/N)|}{{|p+q|}^2}{\sigma }_q^2\Vert {({\mathcal{N}}_{+}+1)}^{3/2}{\xi \Vert }^2{]}^{\frac{1}{2}}[\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}{|p+q|}^2{\Vert {({\mathcal{N}}_{+}+1)}^{-1}{d}_{-p}{b}_{p+q}\xi \Vert }^2{]}^{\frac{1}{2}}\hfill \\ \hfill ⩽& C{N}^{-1}⟨\xi ,(\mathcal{K}+{\mathcal{N}}_{+}^2+1)({\mathcal{N}}_{+}+1)\xi ⟩.\hfill \end{array}\end{array}$$

The term proportional to $b_{-p-q}{b}_p{d}_q$ can be handled similarly, estimating $\Vert b_{-p-q}^{\ast }({\mathcal{N}}_{+}+1)\xi \Vert ⩽\Vert {({\mathcal{N}}_{+}+1)}^{3/2}\xi \Vert$ and using the factor ${\sigma }_{p+q}$ to sum over q. The terms proportional to $b_{-p-q}{d}_{-p}^{\ast }{b}_{-q}^{\ast }$ or $d_{p+q}^{\ast }{b}_p{b}_{-q}^{\ast }$ are slightly more challenging, because we prefer to avoid commutators between b and d operators. Still, using (3.5) (and the smallness of $[b_{-p},b_{-p-q}^{\ast }]$, for $q\ne 0$) we can estimate

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \pm & \frac{1}{\sqrt{N}}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}\widehat{V}(p/N){\sigma }_{p+q}{\sigma }_q(⟨\xi ,b_{-p-q}{d}_{-p}^{\ast }{b}_{-q}^{\ast }\xi ⟩+\mathrm{h}.\mathrm{c}.)\hfill \\ \hfill ⩽& \frac{C\Vert {({\mathcal{N}}_{+}+1)}^{3/2}\xi \Vert }{N^{3/2}}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}|\widehat{V}(p/N)||{\sigma }_{p+q}||{\sigma }_q|[|{\eta }_p|\Vert {({\mathcal{N}}_{+}+1)}^{1/2}{b}_{-p-q}^{\ast }\xi \Vert +\Vert b_{-p}{b}_{-p-q}^{\ast }\xi \Vert ]\hfill \\ \hfill ⩽& C{N}^{-1}⟨\xi ,(\mathcal{K}+{\mathcal{N}}_{+}^2+1)({\mathcal{N}}_{+}+1)\xi ⟩\hfill \end{array}\end{array}$$

and similarly for the term proportional to $d_{p+q}^{\ast }{b}_p{b}_{-q}^{\ast }$. Thus

$$\begin{array}{c}\hfill \pm (M_1+\mathrm{h}.\mathrm{c}.)⩽C{N}^{-1}(\mathcal{K}+{\mathcal{N}}_{+}^2+1)({\mathcal{N}}_{+}+1).\end{array}$$

As for $M_2$, it follows from [5, Eq. (7.32)] that

$$\begin{array}{c}\hfill \pm (M_2+\mathrm{h}.\mathrm{c}.)⩽C{N}^{-1}({\mathcal{V}}_N+{\mathcal{N}}_{+}+1)({\mathcal{N}}_{+}+1).\end{array}$$

To bound $M_3$, we switch to position space. With (3.6) and using (2.20) to show $\Vert \check{\eta }{\Vert }_{\infty }⩽CN$, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,(M_3+\mathrm{h}.\mathrm{c}.)\xi ⟩|⩽& \int dxdy{N}^{5/2}V(N(x-y))\Vert {({\mathcal{N}}_{+}+1)}^{-1}{\check{d}}_x{\check{d}}_y\xi \Vert \Vert ({\mathcal{N}}_{+}+1){\check{d}}_x\xi \Vert \hfill \\ \hfill ⩽& \frac{C}{N^3}\int dxdy{N}^{5/2}V(N(x-y))(\Vert {({\mathcal{N}}_{+}+1)}^{5/2}\xi \Vert +\Vert {\check{a}}_x{({\mathcal{N}}_{+}+1)}^2\xi \Vert )\hfill \\ \hfill & \times (N\Vert ({\mathcal{N}}_{+}+1)\xi \Vert +\Vert {\check{a}}_x{({\mathcal{N}}_{+}+1)}^{3/2}\xi \Vert +\Vert {\check{a}}_x{\check{a}}_y({\mathcal{N}}_{+}+1)\xi \Vert )\hfill \\ \hfill ⩽& C{N}^{-3/2}⟨\xi ,({\mathcal{V}}_N+{\mathcal{N}}_{+}^2+1)({\mathcal{N}}_{+}+1)\xi ⟩.\hfill \end{array}\end{array}$$

Finally, let us get back to the term ${\mathcal{E}}_{3,1}$, from (3.19). Using Lemma 3.1, we can write

$$\begin{array}{c}\hfill {\mathcal{E}}_{3,1}=\frac{1}{N^{3/2}}\sum _{\begin{array}{c}p,q,r\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}\widehat{V}(p/N)e^{-B_{\mu }}{b}_{p+q}^{\ast }{b}_{-p}^{\ast }{e}^{B_{\mu }}{e}^{-B_{\mu }}{\mathcal{N}}_{+}{e}^{B_{\mu }}{e}^{-B_{\mu }}{b}_q{e}^{B_{\mu }}+{\mathcal{E}}_{3,3}\end{array}$$

where $\pm {\mathcal{E}}_{3,3}⩽C{N}^{-1}({\mathcal{H}}_N+1)({\mathcal{N}}_{+}+1)$. Now, we expand $e^{-B_{\mu }}{b}_{p+q}^{\ast }{b}_{-p}^{\ast }{e}^{B_{\mu }}$ and, on the other side, $e^{-B_{\mu }}{b}_q{e}^{B_{\mu }}$ using (3.3). The resulting terms can be controlled as above; by (3.1), the additional factor $e^{-B_{\mu }}{\mathcal{N}}_{+}{e}^{B_{\mu }}$ does not affect the estimates (when applying Cauchy-Schwarz, it is however important not to act with the kinetic energy operator $\mathcal{K}$ on $e^{-B_{\mu }}{\mathcal{N}}_{+}{e}^{B_{\mu }}$). At the end, the main contribution has the form ${\mathcal{C}}_{{\mathcal{G}}_N}{e}^{-B_{\mu }}{\mathcal{N}}_{+}{e}^{B_{\mu }}/N$ (with ${\mathcal{C}}_{{\mathcal{G}}_N}$ as in (2.28)) and can be bounded, using repeatedly (3.1), by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \pm \frac{1}{N}\langle \xi ,& {\mathcal{C}}_{{\mathcal{G}}_N}{e}^{-B_{\mu }}{\mathcal{N}}_{+}{e}^{B_{\mu }}\xi \rangle \hfill \\ \hfill ⩽& \frac{1}{N^{3/2}}(\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}\frac{|\widehat{V}(p/N)|}{p^2}[\Vert b_q{e}^{-B_{\mu }}{\mathcal{N}}_{+}{e}^{B_{\mu }}{\xi \Vert }^2+{\sigma }_q^2{\Vert {\mathcal{N}}_{+}^{1/2}{e}^{-B_{\mu }}{\mathcal{N}}_{+}{e}^{B_{\mu }}\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill & \times (\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}{p}^2{\Vert b_{-p}{b}_{p+q}\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill ⩽& C{N}^{-1}⟨\xi ,(\mathcal{K}+{\mathcal{N}}_{+}^2+1)({\mathcal{N}}_{+}+1)\xi ⟩.\hfill \end{array}\end{array}$$

We conclude that $\pm {\mathcal{E}}_{3,1}⩽C{N}^{-1}(\mathcal{K}+{\mathcal{N}}_{+}^2+1)({\mathcal{N}}_{+}+1)$. Together with (3.20) and with the bounds for $M_1,M_2,M_3$, this concludes the proof of the proposition. $\square$

Finally, we consider the action of $B_{\mu }$ on the operator ${\mathcal{V}}_N$, defined in (2.6).

Proposition 3.7

Under the same assumptions of Theorem 2.4 we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{G}}_N^{(4)}=& {\mathcal{V}}_N+\frac{1}{2N}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}((p-q)/N){\sigma }_p{\gamma }_p{\sigma }_q{\gamma }_q(1+\frac{1}{N}-2\frac{{\mathcal{N}}_{+}}{N})\hfill \\ \hfill & +\frac{1}{2N}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}((p-q)/N){\eta }_q[{\gamma }_p^2{b}_p^{\ast }{b}_{-p}^{\ast }(1+\frac{1}{N}-\frac{{\mathcal{N}}_{+}}{N})+2{\gamma }_p{\sigma }_p{b}_p^{\ast }{b}_p+{\sigma }_p^2{b}_p{b}_{-p}\hfill \\ \hfill & +b_{-p}{d}_p+d_p(b_{-p}+{\eta }_p{b}_p^{\ast })+\text{h.c.}]\hfill \\ \hfill & +\frac{1}{N^2}\sum _{p,q,u\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}((p-q)/N){\eta }_p{\eta }_q[({\gamma }_u^2+{\sigma }_u^2)b_u^{\ast }{b}_u+{\gamma }_u{\sigma }_u(b_u{b}_{-u}+b_u^{\ast }{b}_{-u}^{\ast })+{\sigma }_u^2]\hfill \\ \hfill & +\frac{1}{2N}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast },r\in {\mathrm{\Lambda }}^{\ast }}\widehat{V}(r/N){\gamma }_{p+r}{\gamma }_q{\sigma }_p{\sigma }_{q+r}{b}_{p+r}^{\ast }{b}_q^{\ast }{b}_{-p}^{\ast }{b}_{-q-r}^{\ast }+\text{h.c.}\hfill \\ \hfill & +\frac{1}{N^2}\sum _{p,q,u\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}((p-q)/N){\eta }_q{b}_p^{\ast }{b}_{-p}^{\ast }[({\gamma }_u{\sigma }_u{b}_u^{\ast }{b}_{-u}^{\ast })+{\sigma }_u^2]+\mathrm{h}.\mathrm{c}.\hfill \\ \hfill & +{\mathcal{E}}_{{\mathcal{G}}_N^4}\hfill \end{array}\end{array}$$ 3.21

with

$$\begin{array}{c}\hfill \pm {\mathcal{E}}_{{\mathcal{G}}_N^4}⩽\epsilon {\mathcal{N}}_{+}+\frac{C}{\epsilon N}({\mathcal{H}}_N+{\mathcal{N}}_{+}^2+1)({\mathcal{N}}_{+}+1).\end{array}$$

Remark

Compared with the corresponding result in [5], here we also keep track of the terms appearing in the fifth and sixth line of (3.21). These terms will be included in the operator ${\mathcal{T}}_{{\mathcal{G}}_N}$ defined in (2.29) and will be shown to be negligible after cubic conjugation.

Proof

Proceeding as in the proof of [5, Lemma 7.4], we find (with the notation ${\sum }_{p,q,r}^{\ast }={\sum }_{p,q\in {\mathrm{\Lambda }}_{+}^{\ast },r\in {\mathrm{\Lambda }}^{\ast }:r\ne -p,-q}$)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{G}}_N^{(4)}=& \frac{N+1}{2N^2}\sum ^{\ast }\widehat{V}(r/N)e^{-B_{\mu }}{b}_{p+r}^{\ast }{b}_q^{\ast }{b}_p{b}_{q+r}{e}^{B_{\mu }}\hfill \\ \hfill & +\frac{1}{N^2}\sum ^{\ast }\widehat{V}(r/N)e^{-B_{\mu }}{b}_{p+r}^{\ast }{b}_q^{\ast }{b}_u^{\ast }{b}_u{b}_p{b}_{q+r}{e}^{B_{\mu }}+{\mathcal{E}}_{4,1},\hfill \end{array}\end{array}$$ 3.22

with $\pm {\mathcal{E}}_{4,1}⩽C{N}^{-1}({\mathcal{V}}_N+{\mathcal{N}}_{+}+1)({\mathcal{N}}_{+}+1)$. Again, following [5, Lemma 7.4], we can write the first term on the r.h.s. as

$$\begin{array}{c}\hfill \frac{N+1}{2N^2}\sum ^{\ast }\widehat{V}(r/N)e^{-B_{\mu }}{b}_{p+r}^{\ast }{b}_q^{\ast }{b}_p{b}_{q+r}{e}^{B_{\mu }}=V_0+V_1+{\mathcal{E}}_{4,2}\end{array}$$ 3.23

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill V_0=& \frac{N+1}{2N^2}\sum ^{\ast }\widehat{V}(r/N)[{\gamma }_{p+r}{\gamma }_q{b}_{p+r}^{\ast }{b}_q^{\ast }+{\gamma }_{p+r}{\sigma }_q{b}_{p+r}^{\ast }{b}_{-q}+{\sigma }_{p+r}{\sigma }_q{b}_{-p-r}{b}_{-q}\hfill \\ \hfill & +{\sigma }_{p+r}{\gamma }_q(b_q^{\ast }{b}_{-p-r}-N^{-1}{a}_q^{\ast }{a}_{-p-r})]\hfill \\ \hfill & \times [{\sigma }_p{\sigma }_{q+r}{b}_{-p}^{\ast }{b}_{-q-r}^{\ast }+{\sigma }_p{\gamma }_{q+r}{b}_{-p}^{\ast }{b}_{q+r}+{\gamma }_p{\gamma }_{q+r}{b}_p{b}_{q+r}\hfill \\ \hfill & +{\gamma }_p{\sigma }_{q+r}(b_{-q-r}^{\ast }{b}_p-N^{-1}{a}_{-q-r}^{\ast }{a}_p)]\hfill \\ \hfill & +\frac{N+1}{2N^2}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}((p-q)/N){\gamma }_q{\sigma }_q[({\gamma }_p^2{b}_p^{\ast }{b}_{-p}^{\ast }+2{\gamma }_p{\sigma }_p{b}_p^{\ast }{b}_p-N^{-1}{\gamma }_p{\sigma }_p{a}_p^{\ast }{a}_p\hfill \\ \hfill & +{\sigma }_p^2{b}_p{b}_{-p})(1-\frac{{\mathcal{N}}_{+}}{N})+\mathrm{h}.\mathrm{c}.]\hfill \\ \hfill & +\frac{N+1}{2N^2}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}((p-q)/N){\gamma }_p{\sigma }_p{\gamma }_q{\sigma }_q(1-\frac{{\mathcal{N}}_{+}}{N}{)}^2,\hfill \\ \hfill V_1=& \frac{1}{2N}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}(\frac{p-q}{N}){\gamma }_q{\sigma }_q[d_p({\gamma }_p{b}_{-p}+{\sigma }_p{b}_p^{\ast })+({\gamma }_p{b}_p+{\sigma }_p{b}_{-p}^{\ast })d_{-p}]+\mathrm{h}.\mathrm{c}.\hfill \end{array}\end{array}$$

and $\pm {\mathcal{E}}_{4,2}⩽C{N}^{-1}({\mathcal{V}}_N+{\mathcal{N}}_{+}+1)({\mathcal{N}}_{+}+1)$ (this error includes also the terms $V_{12},V_{13}$ in the proof of [5, Lemma 7.4]). Considering separately quartic, quadratic and constant contributions to $V_0$, we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill V_0=& {\mathcal{V}}_N+\frac{1}{2N}\sum _{r\in {\mathrm{\Lambda }}^{\ast },p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}(r/N){\gamma }_{p+r}{\gamma }_q{\sigma }_p{\sigma }_{q+r}{b}_{p+r}^{\ast }{b}_q^{\ast }{b}_{-p}^{\ast }{b}_{-q-r}^{\ast }+\text{h.c.}\hfill \\ \hfill & +\frac{1}{N}\sum _{r\in {\mathrm{\Lambda }}^{\ast },p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}(r/N){\gamma }_{p+r}{\gamma }_q{\sigma }_p{\gamma }_{q+r}{b}_{p+r}^{\ast }{b}_q^{\ast }{b}_{-p}^{\ast }{b}_{q+r}+\text{h.c.}\hfill \\ \hfill & +\frac{1}{N}\sum _{r\in {\mathrm{\Lambda }}^{\ast },p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}(r/N)({\gamma }_{p+r}{\gamma }_q-1)b_{p+r}^{\ast }{b}_q^{\ast }{b}_p{b}_{q+r}+\text{h.c.}\hfill \\ \hfill & +\frac{1}{2N}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\end{array}}\widehat{V}((p-q)/N){\gamma }_q{\sigma }_q\hfill \\ \hfill & \times [{\gamma }_p^2{b}_p^{\ast }{b}_{-p}^{\ast }\left(1,+,\frac{1}{N},-,\frac{{\mathcal{N}}_{+}}{N}\right)+2{\gamma }_p{\sigma }_p{b}_p^{\ast }{b}_p+{\sigma }_p^2{b}_p{b}_{-p}+\text{h.c.}]\hfill \\ \hfill & +\frac{1}{2N}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\end{array}}\widehat{V}((p-q)/N){\sigma }_q{\gamma }_q{\sigma }_p{\gamma }_p\left(1,+,\frac{1}{N},-,2,\frac{{\mathcal{N}}_{+}}{N}\right)+{\mathcal{E}}_{4,3}\hfill \end{array}\end{array}$$ 3.24

where $\pm {\mathcal{E}}_{4,3}⩽C{N}^{-1}({\mathcal{H}}_N+{\mathcal{N}}_{+}+1)({\mathcal{N}}_{+}+1)$. Additionally, except for ${\mathcal{V}}_N$, the quartic terms containing both creation and annihilation operators can be considered as errors and are bounded by $\epsilon {\mathcal{N}}_{+}+C{\epsilon }^{-1}{N}^{-1}\mathcal{K}{({\mathcal{N}}_{+}+1)}^2$ as in (3.15). For the term $V_1$, proceeding as for (3.17), we get

$$\begin{array}{c}\hfill V_1=\frac{1}{2N}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}((p-q)/N){\sigma }_q{\gamma }_q[(b_{-p}{d}_p+d_p(b_{-p}+{\eta }_p{b}_p^{\ast })]+\text{h.c.}\end{array}$$ 3.25

Let us now consider the second term on the r.h.s. of (3.22). Proceeding as in the proof of [5, Lemma 7.4], we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{1}{N^2}\sum ^{\ast }& \widehat{V}(r/N)e^{-B_{\mu }}{b}_{p+r}^{\ast }{b}_q^{\ast }{b}_u^{\ast }{b}_u{b}_p{b}_{q+r}{e}^{B_{\mu }}\hfill \\ \hfill =& \frac{1}{N^2}\sum _{p,q,u\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}((p-q)/N){\sigma }_q{\gamma }_q{\sigma }_p{\gamma }_p[({\gamma }_u^2+{\sigma }_u^2)b_u^{\ast }{b}_u+{\gamma }_u{\sigma }_u(b_u{b}_{-u}+b_u^{\ast }{b}_{-u}^{\ast })+{\sigma }_u^2]\hfill \\ \hfill & +W_1+{\mathcal{E}}_{4,4}\hfill \end{array}\end{array}$$ 3.26

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill W_1=& \frac{1}{N^2}\sum _{p,q,u\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}((p-q)/N){\sigma }_q{\gamma }_q\hfill \\ \hfill & \times [{\gamma }_p^2{b}_p^{\ast }{b}_{-p}^{\ast }+2{\gamma }_p{\sigma }_p{b}_p^{\ast }{b}_p-N^{-1}{\gamma }_p{\sigma }_p{a}_p^{\ast }{a}_p+{\sigma }_p^2{b}_p{b}_{-p}+{\gamma }_p{b}_{-p}^{\ast }{d}_p^{\ast }+{\sigma }_p{b}_p{d}_p^{\ast }\hfill \\ \hfill & +{\gamma }_p{d}_{-p}^{\ast }{b}_p^{\ast }+{\sigma }_p{d}_p^{\ast }{b}_p+d_p^{\ast }{d}_{-p}^{\ast }](e^{-B}{b}_u^{\ast }{b}_u{e}^B)(1-{\mathcal{N}}_{+}/N)+\text{h.c.}+{\mathcal{E}}_{4,4}\hfill \end{array}\end{array}$$

and $\pm {\mathcal{E}}_{4,4}⩽C{N}^{-1}({\mathcal{V}}_N+{\mathcal{N}}_{+}+1)({\mathcal{N}}_{+}+1)$ (the first line on the r.h.s. of (3.26) corresponds to the term $W_0$ in the proof of [5, Lemma 7.4]; the term $W_2$ is absorbed here into the error ${\mathcal{E}}_{4,4}$). We decompose

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill W_1=& \frac{1}{N^2}\sum _{p,q,u\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}((p-q)/N){\sigma }_q{\gamma }_q{\gamma }_p^2{b}_p^{\ast }{b}_{-p}^{\ast }{e}^{-B_{\mu }}{b}_u^{\ast }{b}_u{e}^{B_{\mu }}+\text{h.c.}+W_{11}\hfill \\ \hfill =& \frac{1}{N^2}\sum _{p,q,u\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}((p-q)/N){\sigma }_q{\gamma }_q{\gamma }_p^2{b}_p^{\ast }{b}_{-p}^{\ast }({\gamma }_u{b}_u^{\ast }+{\sigma }_u{b}_{-u})({\gamma }_u{b}_u+{\sigma }_u{b}_{-u}^{\ast })+\text{h.c.}\hfill \\ \hfill & +W_{11}+W_{12}.\hfill \end{array}\end{array}$$ 3.27

Furthermore, we can write

$$\begin{array}{cc}\hfill W_{11}=& \frac{1}{N^2}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}((p-q)/N){\sigma }_q{\gamma }_q(2{\gamma }_p{\sigma }_p{b}_p^{\ast }{b}_p-N^{-1}{\gamma }_p{\sigma }_p{a}_p^{\ast }{a}_p+{\sigma }_p^2{b}_p{b}_{-p})\hfill \\ \hfill & \times e^{-B_{\mu }}{\mathcal{N}}_{+}(1-({\mathcal{N}}_{+}-1)/N)e^{B_{\mu }}(1-{\mathcal{N}}_{+}/N)+\text{h.c.}\hfill \\ \hfill & -\frac{1}{N^3}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}((p-q)/N){\sigma }_q{\gamma }_q{\gamma }_p^2{b}_p^{\ast }{b}_{-p}^{\ast }{e}^{-B_{\mu }}{\mathcal{N}}_{+}(1-({\mathcal{N}}_{+}-1)/N)e^{B_{\mu }}{\mathcal{N}}_{+}+\text{h.c.}\hfill \\ \hfill & +\frac{1}{N^2}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}((p-q)/N){\sigma }_q{\gamma }_q({\gamma }_p{b}_{-p}^{\ast }{d}_p^{\ast }+{\sigma }_p{b}_p{d}_p^{\ast }+{\gamma }_p{d}_{-p}^{\ast }{b}_p^{\ast }+{\sigma }_p{d}_p^{\ast }{b}_p+d_{p+q}^{\ast }{d}_q^{\ast })\hfill \\ \hfill & \times e^{-B_{\mu }}{\mathcal{N}}_{+}(1-({\mathcal{N}}_{+}-1)/N)e^{B_{\mu }}(1-{\mathcal{N}}_{+}/N)+\text{h.c.}\hfill \\ \hfill =& W_{111}+W_{112}+W_{113}.\hfill \end{array}$$

With (3.1), we can bound

$$\begin{array}{cc}\hfill |⟨\xi ,W_{111}\xi ⟩|⩽& \frac{C}{N^2}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}|\widehat{V}((p-q)/N)||{\mu }_q|\Vert b_p{({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \hfill \\ \hfill & \times \Vert b_p{({\mathcal{N}}_{+}+1)}^{-1/2}{e}^{-B_{\eta }}{\mathcal{N}}_{+}{e}^{B_{\eta }}(1-{\mathcal{N}}_{+}/N)\xi \Vert \hfill \\ \hfill & +\frac{C}{N^2}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}|\widehat{V}((p-q)/N)||{\mu }_q||{\sigma }_p{|}^2{\Vert ({\mathcal{N}}_{+}+1)\xi \Vert }^2\hfill \\ \hfill ⩽& \frac{C}{N}⟨\xi ,{({\mathcal{N}}_{+}+1)}^2\xi ⟩.\hfill \end{array}$$

We control $W_{112}$ in position space, again with the help of (3.1). We find

$$\pm W_{112}⩽C{N}^{-3/2}({\mathcal{V}}_N+{\mathcal{N}}_{+}+1)({\mathcal{N}}_{+}+1).$$

As for $W_{113}$, we partially switch to position space and use Eq. (3.5), (3.6), the bound $\Vert \check{\sigma }\ast \check{\gamma }{\Vert }_{\infty }⩽CN$ and the inequality

$$\frac{C}{N^2}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}|\widehat{V}((p-q)/N)||{\mu }_p||{\mu }_q|⩽C<\infty$$

to estimate

$$\begin{array}{cc}\hfill |⟨\xi ,W_{113}\xi ⟩|⩽& \frac{C}{N^3}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}|\widehat{V}((p-q)/N)||{\mu }_q|[|{\mu }_p|\Vert {({\mathcal{N}}_{+}+1)}^{3/2}{b}_p\xi \Vert +\Vert b_p{b}_{-p}({\mathcal{N}}_{+}+1)\xi \Vert \hfill \\ \hfill & +|{\mu }_p{|}^2\Vert {({\mathcal{N}}_{+}+1)}^{3/2}\xi \Vert \Vert +|{\sigma }_p|(\Vert b_p^{\ast }{b}_p({\mathcal{N}}_{+}+1)\xi \Vert \hfill \\ \hfill & +\Vert ({\mathcal{N}}_{+}+1)\xi \Vert )]\Vert e^{-B}{\mathcal{N}}_{+}{e}^B\xi \Vert \hfill \\ \hfill & +\frac{C}{N^2}\int dxdy{N}^{3}V(N(x-y))|(\check{\sigma }\ast \check{\gamma })(x-y)|\hfill \\ \hfill & \times [\Vert {\check{b}}_x{\check{d}}_y\xi \Vert +\Vert r_x\Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\check{d}}_y\xi \Vert +\Vert {\check{d}}_x{\check{d}}_y\xi \Vert ]\Vert {\mathcal{N}}_{+}{e}^B\xi \Vert \hfill \\ \hfill ⩽& \frac{C}{N}⟨\xi ,(\mathcal{K}+{\mathcal{N}}_{+}+1)({\mathcal{N}}_{+}+1)\xi ⟩\hfill \\ \hfill & +\frac{C}{N^2}\int dxdy{N}^{3}V(N(x-y))[\Vert {\check{a}}_x{({\mathcal{N}}_{+}+1)}^{3/2}\xi \Vert +\Vert {\check{a}}_y{({\mathcal{N}}_{+}+1)}^{3/2}\xi \Vert \hfill \\ \hfill & +(N+|\check{\mu }(x-y)|)\Vert ({\mathcal{N}}_{+}+1)\xi \Vert +\Vert {\check{a}}_y{\check{a}}_x({\mathcal{N}}_{+}+1)\xi \Vert ]\Vert {\mathcal{N}}_{+}{e}^B\xi \Vert \hfill \\ \hfill ⩽& \frac{C}{N}⟨\xi ,(\mathcal{K}+{\mathcal{V}}_N+{\mathcal{N}}_{+}+1)({\mathcal{N}}_{+}+1)\xi ⟩.\hfill \\ \hfill \end{array}$$

As for the term $W_{12}$ on the r.h.s. of (3.27), we find, with (3.5),

$$\begin{array}{cc}\hfill |& ⟨\xi ,W_{12}\xi ⟩|\hfill \\ \hfill ⩽& \frac{C}{N^2}\sum _{p,q,u\in {\mathrm{\Lambda }}_{+}^{\ast }}|\widehat{V}((p-q)/N)||{\mu }_q|\hfill \\ \hfill & \times [\Vert {({\mathcal{N}}_{+}+1)}^{-\frac{3}{2}}{d}_u{b}_p{b}_{-p}\Vert \{\Vert b_u{({\mathcal{N}}_{+}+1)}^{\frac{3}{2}}\xi \Vert +|{\sigma }_u|\Vert {({\mathcal{N}}_{+}+1)}^2\xi \Vert +\Vert {({\mathcal{N}}_{+}+1)}^{\frac{3}{2}}{d}_u\xi \Vert \}\hfill \\ \hfill & +\{\Vert b_u{b}_p{b}_{-p}{({\mathcal{N}}_{+}+1)}^{-1/2}\xi \Vert +|{\sigma }_u|\Vert b_p{b}_{-p}\xi \Vert \}\Vert {({\mathcal{N}}_{+}+1)}^{1/2}{d}_u\xi \Vert ]\hfill \\ \hfill ⩽& \frac{C}{N^3}\sum _{p,q,u\in {\mathrm{\Lambda }}_{+}^{\ast }}|\widehat{V}((p-q)/N)||{\mu }_q|\hfill \\ \hfill & \times [|{\mu }_u{|}^2\Vert b_p{b}_{-p}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^2\xi \Vert +|{\mu }_u|\Vert b_p{b}_{-p}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{3/2}{b}_u\xi \Vert \hfill \\ \hfill & +\Vert b_u{b}_p{b}_{-p}{({\mathcal{N}}_{+}+1)}^{-\frac{1}{2}}\xi \Vert \Vert b_u{({\mathcal{N}}_{+}+1)}^{\frac{3}{2}}\xi \Vert +|{\mu }_u|\Vert b_u{b}_p{b}_{-p}{({\mathcal{N}}_{+}+1)}^{-\frac{1}{2}}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^2\xi \Vert ]\hfill \\ \hfill ⩽& \frac{C}{N^{3/2}}\Vert {\mathcal{K}}^{1/2}{({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^2\xi \Vert ⩽\frac{C}{N}⟨\xi ,(\mathcal{K}+{\mathcal{N}}_{+}^2+1)({\mathcal{N}}_{+}+1)\xi ⟩.\hfill \end{array}$$

Combining (3.22), (3.23), (3.24),(3.25), (3.26), (3.27) with the bounds for $W_{111},W_{112},W_{113},W_{12}$ and using $|{\gamma }_q{\sigma }_q-{\mu }_q{|⩽C|q|}^{-6}$, we obtain (3.21). $\square$

We are now ready to conclude the proof of Theorem 2.4.

Proof of Theorem 2.4

Collecting all terms linear in the $d,d^{\ast }$ operators from (3.7), (3.16) and from (3.21), we define

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{D}}_N=& \sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}[p^2{\eta }_p+\frac{1}{2}\widehat{V}(p/N)+\frac{1}{2N}(\widehat{V}(·/N)\ast \eta {)}_p]b_p{d}_{-p}+\text{h.c.}\hfill \\ \hfill & +\frac{1}{2}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p[d_p(b_{-p}+{\eta }_p{b}_p^{\ast })]+\mathrm{h}.\mathrm{c}..\hfill \end{array}\end{array}$$

With the scattering equation (2.12) and (3.5), we obtain

$$\begin{array}{c}\hfill {\mathcal{D}}_N=\frac{1}{2}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell })}_p[d_p(b_{-p}+{\eta }_p{b}_p^{\ast })]+\text{h.c.}+{\mathcal{E}}_1\end{array}$$

where $\pm {\mathcal{E}}_1⩽N^{-1}(\mathcal{K}+{\mathcal{N}}_{+}+1)({\mathcal{N}}_{+}+1)$. Handling the contribution proportional to ${\eta }_p$ as in [5, Section 7.5] (where the contribution is labeled $D_{33}$), and using (3.4) to expand the rest of the term, we find

$$\begin{array}{cc}\hfill {\mathcal{D}}_N=& -\frac{1}{2N}\sum _{\begin{array}{c}p\in {\mathrm{\Lambda }}^{\ast }\\ q\in {\mathrm{\Lambda }}_{+}^{\ast }\end{array}}(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p{\eta }_p[{\gamma }_q{\sigma }_q(b_q{b}_{-q}+b_q^{\ast }{b}_{-q}^{\ast })+({\sigma }_q^2+{\gamma }_q^2-1)b_q^{\ast }{b}_q+{\sigma }_q^2]\hfill \\ \hfill & -\frac{1}{2N}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p{\eta }_q\hfill \\ \hfill & \times {\int }_0^{1}ds{\gamma }_p^{(s)}{e}^{(-1+s)B_{\mu }}{b}_{-q}^{\ast }{a}_q^{\ast }{a}_p{e}^{(1-s)B_{\mu }}{b}_{-p}+\mathrm{h}.\mathrm{c}.+{\mathcal{E}}_2\hfill \end{array}$$

where $\pm {\mathcal{E}}_2⩽C{N}^{-1}{({\mathcal{N}}_{+}+1)}^2$. Next, we compute the action of $(1-s)B_{\mu }$ on $b_q^{\ast }{b}_{-q}^{\ast }{b}_p$; with (3.3), we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{D}}_N=& -\frac{1}{2N}\sum _{\begin{array}{c}p\in {\mathrm{\Lambda }}^{\ast }\\ q\in {\mathrm{\Lambda }}_{+}^{\ast }\end{array}}(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p{\eta }_p[{\gamma }_q{\sigma }_q(b_q{b}_{-q}+b_q^{\ast }{b}_{-q}^{\ast })+({\sigma }_q^2+{\gamma }_q^2-1)b_q^{\ast }{b}_q+{\sigma }_q^2]\hfill \\ & -\frac{1}{4N}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p{b}_p^{\ast }{b}_{-p}^{\ast }\hfill \\ & \times [(-{\mu }_q+{\gamma }_q{\sigma }_q)b_q^{\ast }{b}_{-q}^{\ast }+({\mu }_q+{\gamma }_q{\sigma }_q)b_q{b}_{-q}+2{\sigma }_q^2{b}_q^{\ast }{b}_q+{\sigma }_q^2]+\mathrm{h}.\mathrm{c}.\hfill \\ & +{\mathcal{E}}_3,\hfill \end{array}\end{array}$$

with $\pm {\mathcal{E}}_3⩽C{N}^{-1}(\mathcal{K}+{\mathcal{N}}_{+}^2+1)({\mathcal{N}}_{+}+1)$. In a similar way to (3.15), one can show that the contribution of the quartic terms which contain both creation and annihilation operators can be treated as errors bounded by $\epsilon {\mathcal{N}}_{+}+C{\epsilon }^{-1}{N}^{-1}\mathcal{K}{({\mathcal{N}}_{+}+1)}^2$. Combining this with all other contributions in (3.7), (3.16), (3.18) and (3.21), we obtain

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathcal{G}}_N=C_{{\mathcal{G}}_N}+\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}[{\mathrm{\Phi }}_p{b}_p^{\ast }{b}_p+\frac{1}{2}{\mathrm{\Gamma }}_p(b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p})]+{\mathcal{C}}_{{\mathcal{G}}_N}+{\mathcal{V}}_N+{\mathcal{T}}_{{\mathcal{G}}_N}+{\mathcal{E}}_{{\mathcal{G}}_N}\end{array}\end{array}$$ 3.28

where $C_{{\mathcal{G}}_N},{\mathcal{C}}_{{\mathcal{G}}_N},{\mathcal{T}}_{{\mathcal{G}}_N}$ are defined as in (2.27), (2.28), (2.29), and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathrm{\Phi }}_p=& p^2({\gamma }_p^2+{\sigma }_p^2)+\widehat{V}(p/N){({\gamma }_p+{\sigma }_p)}^2+\frac{2{\gamma }_p{\sigma }_p}{N}(\widehat{V}(·/N)\ast \eta {)}_p-\frac{({\gamma }_p^2+{\sigma }_p^2)}{N}\sum _{q\in {\mathrm{\Lambda }}^{\ast }}\widehat{V}(q/N){\eta }_q\hfill \\ \hfill {\mathrm{\Gamma }}_p=& 2p^2{\gamma }_p{\sigma }_p+\widehat{V}(p/N){({\gamma }_p+{\sigma }_p)}^2+\frac{({\gamma }_p^2+{\sigma }_p^2)}{N}(\widehat{V}(·/N){\ast \eta )}_p-\frac{2{\gamma }_p{\sigma }_p}{N}\sum _{q\in {\mathrm{\Lambda }}^{\ast }}\widehat{V}(q/N){\eta }_q.\hfill \end{array}\end{array}$$ 3.29

Moreover,

$$\begin{array}{c}\hfill \pm {\mathcal{E}}_{{\mathcal{G}}_N}⩽\epsilon {\mathcal{N}}_{+}+\frac{C}{\epsilon N}({\mathcal{H}}_N+{\mathcal{N}}_{+}^3+1)({\mathcal{N}}_{+}+1).\end{array}$$ 3.30

To conclude the proof of Theorem 2.4, we consider the quadratic term on the r.h.s. of (3.28). Adding and subtracting the contributions that will arise from the cubic conjugation in Theorem 2.6, we rewrite the coefficients in (3.29) as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathrm{\Phi }}_p& =F_p-\frac{({\gamma }_p^2+{\sigma }_p^2)}{N}\sum _{q\in {\mathrm{\Lambda }}_{+}^{\ast }}(\widehat{V}(q/N)+\widehat{V}((q+p)/N)){\eta }_q\hfill \\ \hfill {\mathrm{\Gamma }}_p& =G_p-\frac{2{\gamma }_p{\sigma }_p}{N}\sum _{q\in {\mathrm{\Lambda }}_{+}^{\ast }}(\widehat{V}(q/N)+\widehat{V}((q+p)/N)){\eta }_q\hfill \end{array}\end{array}$$

where

$$\begin{array}{cc}\hfill F_p& =p^2({\gamma }_p^2+{\sigma }_p^2)+(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p{({\gamma }_p+{\sigma }_p)}^2\hfill \\ \hfill G_p& =2p^2{\gamma }_p{\sigma }_p+(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p{({\gamma }_p+{\sigma }_p)}^2.\hfill \end{array}$$

Recalling ${\gamma }_p=cosh{\mu }_p=cosh({\eta }_p+{\tau }_p)$, ${\sigma }_p=sinh{\mu }_p=sinh({\eta }_p+{\tau }_p)$ and the definition (2.17) of the coefficients ${\tau }_p$, we obtain,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill G_p& =[p^2+(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p]sinh(2{\tau }_p+2{\eta }_p)+(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_{p}cosh(2{\tau }_p+2{\eta }_p)\hfill \\ \hfill & =[p^2+(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p]cosh(2{\tau }_p)(sinh(2{\eta }_p)+tanh(2{\tau }_p)cosh(2{\eta }_p))\hfill \\ \hfill & +(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_{p}cosh(2{\tau }_p)(cosh(2{\eta }_p)+tanh(2{\tau }_p)sinh(2{\eta }_p))\hfill \end{array}\end{array}$$

which implies that $G_p=0$, and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill F_p& =[p^2+(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p]cosh(2{\tau }_p+2{\eta }_p)+(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_{p}sinh(2{\tau }_p+2{\eta }_p)\hfill \\ \hfill & =[p^2+(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p]cosh(2{\tau }_p)(cosh(2{\eta }_p)+tanh(2{\tau }_p)sinh(2{\eta }_p))\hfill \\ \hfill & +(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_{p}cosh(2{\tau }_p)(sinh(2{\eta }_p)+tanh(2{\tau }_p)cosh(2{\eta }_p))\hfill \end{array}\end{array}$$ 3.31

leading to $F_p=\sqrt{{|p|}^4+2{(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell })}_p{p}^2}$. This concludes the proof of Theorem 2.4. $\square$

Proof of Proposition 2.5

The goal of this section is to show Prop. 2.5, controlling the growth of the number of excitations and of their energy with respect to cubic conjugation. To reach this goal, we are going to estimate the commutators of ${\mathcal{N}}_{+}$ and of the Hamiltonian ${\mathcal{H}}_N=\mathcal{K}+{\mathcal{V}}_N$ with the cubic operator A, as defined in (2.32). Since in the proof of Theorem 2.6 we will need to compute $e^{-A}{\mathcal{H}}_N{e}^A$, the next proposition contains precise estimates, which are not really needed in the proof of Prop. 2.5.

Lemma 4.1

We have

$$\begin{array}{c}\hfill [\mathcal{K},A]={[\mathcal{K},A]}_1+{[\mathcal{K},A]}_2+\mathrm{h}.\mathrm{c}.\end{array}$$ 4.1

where

$$\begin{array}{cc}\hfill {[\mathcal{K},A]}_1=& \frac{2}{\sqrt{N}}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}{r}^2{\eta }_r{b}_{r+v}^{\ast }{b}_{-r}^{\ast }({\gamma }_v{b}_v+{\sigma }_v{b}_{-v}^{\ast })\hfill \end{array}$$ 4.2

$$\begin{array}{cc}\hfill _2^{}=& \frac{2}{\sqrt{N}}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}(r·v){\eta }_r{\gamma }_v{b}_{r+v}^{\ast }{b}_{-r}^{\ast }{b}_v.\hfill \end{array}$$ 4.3

Moreover,

$$\begin{array}{c}\hfill [{\mathcal{V}}_N,A]={[{\mathcal{V}}_N,A]}_1+{[{\mathcal{V}}_N,A]}_2+\text{h.c.}+{\mathcal{E}}_{[{\mathcal{V}}_N,A]}\end{array}$$ 4.4

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [{\mathcal{V}}_N{,A]}_1=& \frac{1}{N^{3/2}}\sum _{\begin{array}{c}r,v,s\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v,r+v+s\ne 0\end{array}}\widehat{V}((r-s)/N){\eta }_s{b}_{r+v}^{\ast }{b}_{-r}^{\ast }({\gamma }_v{b}_v+{\sigma }_v{b}_{-v}^{\ast })\hfill \\ \hfill {[{\mathcal{V}}_N,A]}_2=& \frac{1}{N^{3/2}}\sum _{\begin{array}{c}r,v,s\in {\mathrm{\Lambda }}_{+}^{\ast }\\ s+v,r+v\ne 0\end{array}}\widehat{V}((r-s)/N){\eta }_s\frac{2{\eta }_{s+v}{s}^2-2{\sigma }_v(v·s)}{s^2+v^2+{|s+v|}^2}{b}_{r+v}^{\ast }{b}_{-r}^{\ast }{b}_{-v}^{\ast }\hfill \end{array}\end{array}$$ 4.5

and

$$\begin{array}{c}\hfill \begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_{[{\mathcal{V}}_N,A]}\xi ⟩|⩽C{N}^{-1}⟨\xi ,({\mathcal{H}}_N+{\mathcal{N}}_{+}^2+1)({\mathcal{N}}_{+}^2+1)\xi ⟩.\end{array}\end{array}$$ 4.6

Furthermore, we have

$$\begin{array}{c}\hfill |⟨{\xi }_1,[\mathcal{K}+{\mathcal{V}}_N,A]{\xi }_2⟩|⩽C⟨{\xi }_1,({\mathcal{H}}_N+{\mathcal{N}}_{+}^2+1){\xi }_1⟩+⟨{\xi }_2,({\mathcal{H}}_N+{\mathcal{N}}_{+}^2+1){\xi }_2⟩\end{array}$$ 4.7

for any ${\xi }_1,{\xi }_2\in {\mathcal{F}}_{+}^{⩽N}$.

Remark

The coefficients ${\nu }_{r,v}$ entering the cubic operator A are defined in (2.33) exactly so that the contribution to the commutator $[\mathcal{K},A]$ proportional to $b_{r+v}^{\ast }{b}_{-r}^{\ast }{b}_{-v}^{\ast }$ only enters in the term containing $r^2$ in (4.2) (and it is absent from the term containing $r·v$ in (4.3)).

To prove Lemma 4.1, we will use the following auxiliary lemma.

Lemma 4.2

For $r,v\in {\mathrm{\Lambda }}_{+}^{\ast }$, we define the coefficients

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\alpha }_{r,v}:=& \frac{1}{N^{1/2}}\sum _{\begin{array}{c}s\in {\mathrm{\Lambda }}_{+}^{\ast }\\ s+v\ne 0\end{array}}\widehat{V}((r-s)/N){\eta }_s[\frac{2{\eta }_{s+v}{s}^2}{s^2+v^2+{|s+v|}^2}-\frac{2{\sigma }_v(v·s)}{s^2+v^2+{|s+v|}^2}].\hfill \end{array}\end{array}$$ 4.8

Then, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \sum _{v\in {\mathrm{\Lambda }}_{+}^{\ast }}\underset{r\in {\mathrm{\Lambda }}_{+}^{\ast }}{sup}{|{\alpha }_{r,v}|}^2+\frac{1}{N}& \sum _{v\in {\mathrm{\Lambda }}_{+}^{\ast }}\left[\underset{z\in {\mathrm{\Lambda }}^{\ast }}{sup},\sum _{r\in {\mathrm{\Lambda }}_{+}^{\ast }},\frac{|{\alpha }_{r,v}{|}^2}{{(r+z)}^2}\right]+N^{1/2}\underset{r\in {\mathrm{\Lambda }}_{+}^{\ast }}{sup}|{\alpha }_{r,v}||v|⩽C.\hfill \end{array}\end{array}$$ 4.9

Proof

We split ${\alpha }_{r,v}={\alpha }_{r,v}^{(1)}+{\alpha }_{r,v}^{(2)}$, with ${\alpha }_{r,v}^{(1)},{\alpha }_{r,v}^{(2)}$ indicating the contribution of the first, respectively, the second term in the square brackets. We will show (4.9) separately, for ${\alpha }_{r,v}^{(1)}$ and ${\alpha }_{r,v}^{(2)}$. We have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \sum _v\underset{r}{sup}{|{\alpha }_{r,v}^{(1)}|}^2& ⩽\frac{C}{N}\sum _{s,s^{\prime },v}|{\eta }_s||{\eta }_{s+v}||{\eta }_{s^{\prime }+v}||{\eta }_{s^{\prime }}|\hfill \\ \hfill & ⩽\frac{C}{N}\sum _{v,s}\frac{|{\eta }_s||{\eta }_{s+v}|}{|v|}⩽\frac{C}{N}\sum _{|v|⩽N}\frac{1}{{|v|}^2}+\frac{C}{N^2}\sum _{s,v;|v|>N}|{\eta }_s||{\eta }_{s+v}|⩽C,\hfill \end{array}\end{array}$$

Similarly, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{1}{N}\sum _v\left[\underset{z\in {\mathrm{\Lambda }}^{\ast }}{sup},\sum _r,\frac{|{\alpha }_{r,v}^{(1)}{|}^2}{{(r+z)}^2}\right]& ⩽\frac{C}{N^2}\sum _v\underset{z\in {\mathrm{\Lambda }}^{\ast }}{sup}\sum _{s,s^{\prime },r}\frac{|\widehat{V}((r-s)/N)|}{{(r+z)}^2}|{\eta }_s||{\eta }_{s+v}||{\eta }_{s^{\prime }+v}||{\eta }_{s^{\prime }}|\hfill \\ \hfill & ⩽\frac{C}{N}\sum _{s,s^{\prime },v}|{\eta }_s||{\eta }_{s+v}||{\eta }_{s^{\prime }+v}||{\eta }_{s^{\prime }}|⩽C,\hfill \end{array}\end{array}$$

using the bound $\Vert \widehat{V}{(·/N)\Vert }_2⩽C{N}^{3/2}$, in the region $|r+z|>N$. The bound for the third summand on the left hand side of (4.9) follows immediately from ${\sum }_s|{\eta }_s||{\eta }_{s+v}{|⩽|v|}^{-1}$.

To handle ${\alpha }_{r,v}^{(2)}$, we decompose ${\alpha }_{r,v}^{(2)}={\alpha }_{r,v}^{(2,>)}+{\alpha }_{r,v}^{(2,<)}$ with

$$\begin{array}{cc}\hfill {\alpha }_{r,v}^{(2,>)}& =-\frac{1}{N^{1/2}}\sum _{\begin{array}{c}s\in {\mathrm{\Lambda }}_{+}^{\ast },|s|>N\\ s+v\ne 0\end{array}}\widehat{V}((r-s)/N){\eta }_s\frac{2{\sigma }_v(v·s)}{s^2+v^2+{|s+v|}^2},\hfill \\ \hfill {\alpha }_{r,v}^{(2,<)}& =-\frac{1}{N^{1/2}}\sum _{\begin{array}{c}s\in {\mathrm{\Lambda }}_{+}^{\ast },|s|⩽N\\ s+v\ne 0\end{array}}\widehat{V}((r-s)/N){\eta }_s\frac{2{\sigma }_v(v·s)}{s^2+v^2+{|s+v|}^2}\hfill \end{array}$$

Estimating $|{\alpha }_{r,v}^{(2,>)}|⩽|v||{\sigma }_v|/\sqrt{N}$, it is easy to check that (4.9) holds true, if we replace ${\alpha }_{r,v}$ with ${\alpha }_{r,v}^{(2,>)}$. Let us now consider the contribution of ${\alpha }_{r,v}^{(2,<)}$. Through a change of variable $s\to -s$ we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\alpha }_{r,v}^{(2,<)}:=& \frac{1}{N^{1/2}}\sum _{|s|⩽N}{\eta }_s{\sigma }_v(s·v)\left[\frac{\widehat{V}((r+s)/N)}{s^2+v^2+{(s-v)}^2},-,\frac{\widehat{V}((r-s)/N)}{s^2+v^2+{(s+v)}^2}\right]\hfill \\ \hfill =& \frac{1}{N^{1/2}}\sum _{|s|⩽N}[\widehat{V}((r+s)/N)-\widehat{V}((r-s)/N)]\frac{{\eta }_s{\sigma }_v(s·v)}{s^2+v^2+{(s+v)}^2}\hfill \\ \hfill & +\frac{1}{N^{1/2}}\sum _{|s|⩽N}\widehat{V}((r+s)/N)\frac{4{\eta }_s{\sigma }_v{(s·v)}^2}{(s^2+v^2+{(s+v)}^2)(s^2+v^2+{(s-v)}^2)}\hfill \\ \hfill =& {\alpha }_{r,v}^{(2,<,1)}+{\alpha }_{r,v}^{(2,<,2)}.\hfill \end{array}\end{array}$$ 4.10

The contribution of ${\alpha }_{r,v}^{(2,<,2)}$ to (4.9) can be bounded easily, as we did with ${\alpha }_{r,v}^{(1)}$. Let us focus on the contribution of ${\alpha }_{r,v}^{(2,<,1)}$. From the Lipschitz continuity of $\widehat{V}$, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \sum _v\underset{r}{sup}{|{\alpha }_{r,v}^{(2,<,1)}|}^2& ⩽\frac{C}{N^3}\sum _v|{\sigma }_v{|}^2{v}^2(\sum _{|s|⩽N}|{\eta }_s|{)}^2⩽C.\hfill \end{array}\end{array}$$ 4.11

Similarly, we can also estimate the last term on the l.h.s. of (4.9). As for the second term in (4.9), it can be bounded by (4.11), if we restrict the sum over $r\in {\mathrm{\Lambda }}_{+}^{\ast }$ to momenta with $|r+z|⩽N$. For $|r+z|>N$, on the other hand, we switch to position space and get

$$\begin{array}{cc}\hfill \frac{1}{N}& \sum _v\left[\underset{z}{sup},\sum _r,\frac{|{\alpha }_{r,v}^{(2,<,1)}{|}^2}{{(r+z)}^2}\right]⩽\frac{C}{N^2}\sum _v{|{\sigma }_v|}^2\underset{z}{sup}{\left(\sum _{\begin{array}{c}|r+z|>N\end{array}},\frac{1}{{(r+z)}^4}\right)}^{1/2}\hfill \\ \hfill & \times (\sum _{\begin{array}{c}r\\ |s_1|,|s_2|,\\ |s_3|,|s_4|⩽N\end{array}}\prod _{i=1,2,3,4}[\widehat{V}((r+s_i)/N)-\widehat{V}((r-s_i)/N)]\frac{{\eta }_{s_i}(s_i·v)}{s_i^2+v^2+{(s_i+v)}^2}{)}^{1/2}\hfill \\ \hfill & ⩽\frac{C}{N}\sum _v|{\sigma }_v{|}^2{v}^2({\int }_{x_1,x_2,x_2\in \mathrm{\Lambda }}d{x}_{1}d{x}_{2}d{x}_3|V(x_1)||V(x_2)||V(x_3)||V(x_1+x_2+x_3)|\hfill \\ \hfill & \times \sum _{\begin{array}{c}|s_1|,|s_2|,\\ |s_3|,|s_4|⩽N\end{array}}\prod _{i=1,2,3,4}\frac{|{\eta }_{s_i}|}{N}{)}^{1/2}\hfill \\ \hfill & ⩽{C\Vert V·(V\ast V\ast V)\Vert }_1⩽C\hfill \end{array}$$ 4.12

where we used the bound $|e^{i(s·x)/N}-e^{-i(s·x)/N}|⩽C|s|/N$ for $x\in \mathrm{\Lambda }$. $\square$

Now we are ready to prove Lemma 4.1.

Proof of Lemma 4.1

With the commutation relations (2.2) and the definition (2.32), we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [\mathcal{K},A]=& \frac{1}{\sqrt{N}}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}{\eta }_r{\gamma }_v({(r+v)}^2+r^2-v^2)b_{r+v}^{\ast }{b}_{-r}^{\ast }{b}_v\hfill \\ \hfill & +\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}{\nu }_{r,v}({(r+v)}^2+r^2+v^2)b_{r+v}^{\ast }{b}_{-r}^{\ast }{b}_{-v}^{\ast }+\text{h.c.}\hfill \end{array}\end{array}$$

Recalling (2.33), we immediately obtain (4.1).

A slightly longer (but still straightforward) computation shows that (4.4) holds true, with

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathcal{E}}_{[{\mathcal{V}}_N,A]}\hfill \\ \hfill =& \frac{1}{N^{3/2}}\sum _{\begin{array}{c}r,v,s\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v+s\ne 0\end{array}}\widehat{V}((r-s)/N)[\frac{2s^2{\eta }_s({\sigma }_{s+v}-{\eta }_{s+v})}{s^2+v^2+{(s+v)}^2}+\frac{2v^2[{\eta }_v({\sigma }_s-{\eta }_s)-{\eta }_s({\sigma }_v-{\eta }_v)]}{s^2+v^2+{(s+v)}^2}]\hfill \\ \hfill & \times b_{r+v}^{\ast }{b}_{-r}^{\ast }{b}_{-v}^{\ast }+\mathrm{h}.\mathrm{c}.\hfill \\ \hfill & +\frac{1}{N^{3/2}}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },p,r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ v-u,r+v-u,p+u\ne 0\end{array}}\widehat{V}(u/N){\eta }_r({\gamma }_v-{\gamma }_{v-u})b_{r+v-u}^{\ast }{b}_{-r}^{\ast }{a}_{p+u}^{\ast }{a}_p{b}_v+\mathrm{h}.\mathrm{c}.\hfill \\ \hfill & +\frac{1}{N^{3/2}}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },p,r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v,r+u,p+u\ne 0\end{array}}\widehat{V}(u/N){\eta }_r{\gamma }_v{b}_{r+v}^{\ast }{b}_{-r-u}^{\ast }{a}_{p+u}^{\ast }{a}_p{b}_v+\mathrm{h}.\mathrm{c}.\hfill \\ \hfill & +\frac{1}{N^{3/2}}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },p,r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ v-u\ne 0\\ r+v-u,p+u\ne 0\end{array}}\widehat{V}(u/N)[{\nu }_{r,v}+{\nu }_{r,v-u}+{\nu }_{r-u,v}]b_{r+v-u}^{\ast }{b}_{-r}^{\ast }{b}_{-v}^{\ast }{a}_{p+u}^{\ast }{a}_p+\mathrm{h}.\mathrm{c}.\hfill \end{array}\end{array}$$ 4.13

Using that $|{\sigma }_s-{\eta }_s|⩽C|{\tau }_s|$, we can bound the first term by

$$\begin{array}{c}\hfill |\frac{1}{N^{3/2}}\sum _{\begin{array}{c}r,v,s\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v+s\ne 0\end{array}}\widehat{V}((r+s)/N)[\frac{2s^2{\eta }_s({\sigma }_{s+v}-{\eta }_{s+v})}{s^2+v^2+{(s+v)}^2}+\frac{2v^2[{\eta }_v({\sigma }_s-{\eta }_s)-{\eta }_s({\sigma }_v-{\eta }_v)]}{s^2+v^2+{(s+v)}^2}]\\ \hfill \times ⟨\xi ,b_{r+v}^{\ast }{b}_{-r}^{\ast }{b}_{-v}^{\ast }\xi ⟩|⩽\frac{C}{N}\Vert {\mathcal{K}}^{1/2}\xi \Vert \Vert ({\mathcal{N}}_{+}+1)\xi \Vert .\end{array}$$

As for the second term on the r.h.s. of (4.13), we observe that $|{\gamma }_v-{\gamma }_{v-u}|⩽C({\eta }_v^2+{\eta }_{v-u}^2)$. We obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& |\frac{1}{N^{3/2}}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },p,r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ v-u,r+v-u,p+u\ne 0\end{array}}\widehat{V}(u/N){\eta }_r({\gamma }_v-{\gamma }_{v-u})⟨\xi ,b_{r+v-u}^{\ast }{b}_{-r}^{\ast }{a}_{p+u}^{\ast }{a}_p{b}_v\xi ⟩|\hfill \\ & ⩽\frac{C}{N}\Vert {\mathcal{K}}^{1/2}{({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{3/2}\xi \Vert .\hfill \end{array}\end{array}$$

To bound the third term, we switch to position space. We find

$$\begin{array}{cc}\hfill |\frac{1}{N^{3/2}}& \sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },p,r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v,r+u,p+u\ne 0\end{array}}\widehat{V}(u/N){\eta }_r{\gamma }_v⟨\xi ,b_{r+v}^{\ast }{b}_{-r-u}^{\ast }{a}_{p+u}^{\ast }{a}_p{b}_v\xi ⟩|\hfill \\ \hfill ⩽& \frac{1}{N^{1/2}}\int dxdydz{N}^{2}V(N(x-z))|\check{\eta }(y-z)||⟨\xi ,{\check{b}}_y^{\ast }{\check{b}}_z^{\ast }{\check{a}}_x^{\ast }{\check{a}}_{x}b({\check{\gamma }}_y)\xi ⟩|\hfill \\ \hfill ⩽& \frac{C}{N}[\int dxdydz{N}^{2}V(N(x-z)){\Vert {\check{b}}_y{\check{b}}_z{\check{a}}_x\xi \Vert }^2{]}^{1/2}\hfill \\ & \times [\int dxdydz{N}^{3}V(N(x-z))|\check{\eta }{(y-z)|}^2{\Vert {\check{a}}_{x}b({\check{\gamma }}_y)\xi \Vert }^2{]}^{1/2}\hfill \\ \hfill ⩽& \frac{C}{N}\Vert {\mathcal{V}}_N^{1/2}{({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{K}}^{1/2}{({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \hfill \end{array}$$

where, in the last step, we used $|\check{\eta }(y-z)|⩽C/|y-z|$ (see (2.11) and (2.10)) and Hardy’s inequality (and the estimate $\Vert {\check{a}}_{x}b({\check{\gamma }}_y)\xi \Vert ⩽\Vert {\check{a}}_x{\check{a}}_y\xi \Vert +\Vert {\check{a}}_x{\mathcal{N}}_{+}^{1/2}\xi \Vert$). Finally, let us consider the last term on the r.h.s. of (4.13). We only show how to bound the contribution proportional to ${\nu }_{r,v}$, the others can be estimated similarly. With $|{\nu }_{r,v}{|⩽C|r|}^{-2}{|v|}^{-2}$, we obtain

$$\begin{array}{cc}& |\frac{1}{N^{3/2}}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },p,r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ v-u\ne 0\\ r+v-u,p+u\ne 0\end{array}}\widehat{V}(u/N){\nu }_{r,v}⟨\xi ,b_{r+v-u}^{\ast }{b}_{-r}^{\ast }{b}_{-v}^{\ast }{a}_{p+u}^{\ast }{a}_p\xi ⟩|\hfill \\ & ⩽\frac{C}{N^{3/2}}[\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },p,r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ v-u\ne 0\\ r+v-u,p+u\ne 0\end{array}}\frac{|\widehat{V}(u/N)|}{{|p+u|}^2}\frac{1}{{|r|}^4{|v|}^4}{\Vert ({\mathcal{N}}_{+}+1)a_p\xi \Vert }^2{]}^{1/2}\hfill \\ & \times [\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },p,r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ v-u\ne 0\\ r+v-u,p+u\ne 0\end{array}}{|p+u|}^2{\Vert {({\mathcal{N}}_{+}+1)}^{-1}{a}_{p+u}{a}_{-v}{a}_{-r}{a}_{r+v-u}\xi \Vert }^2{]}^{1/2}\hfill \\ & ⩽\frac{C}{N}\Vert {\mathcal{K}}^{1/2}{({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{3/2}\xi \Vert .\hfill \end{array}$$

This concludes the proof of (4.6).

Proceeding as we did above, it is also simple to verify that the error term ${\mathcal{E}}_{[{\mathcal{V}}_N,A]}$ satisfy (4.7). To conclude the proof of the bound (4.7), we observe, first of all, that

$$|⟨{\xi }_1,{[\mathcal{K},A]}_1{\xi }_2⟩|⩽C\Vert {\mathcal{K}}^{1/2}{\xi }_1\Vert \Vert ({\mathcal{N}}_{+}+1){\xi }_2\Vert$$

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨{\xi }_1,& {[\mathcal{K},A]}_2{\xi }_2⟩|\hfill \\ \hfill ⩽& \frac{C}{N^{1/2}}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}|r||v||{\eta }_r|\Vert b_{r+v}{b}_{-r}{\xi }_1\Vert \Vert b_v{\xi }_2\Vert \hfill \\ \hfill ⩽& \frac{C}{N^{1/2}}[\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}{|r|}^2\Vert b_{r+v}{b}_{-r}{\xi }_1{\Vert }^2{]}^{1/2}[\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}|{\eta }_r{|}^2{|v|}^2{\Vert b_v{\xi }_2\Vert }^2{]}^{1/2}\hfill \\ \hfill ⩽& \frac{C}{N^{1/2}}\Vert {\mathcal{K}}^{1/2}{({\mathcal{N}}_{+}+1)}^{1/2}{\xi }_1\Vert \Vert {\mathcal{K}}^{1/2}{\xi }_2\Vert .\hfill \end{array}\end{array}$$ 4.14

We rewrite the term ${[{\mathcal{V}}_N,A]}_1$ as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [{\mathcal{V}}_N,A{]}_1=& \frac{1}{N^{3/2}}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}(\widehat{V}\left(·,/,N\right)\ast \eta )(r)b_{r+v}^{\ast }{b}_{-r}^{\ast }({\gamma }_v{b}_v+{\sigma }_v{b}_{-v}^{\ast })\hfill \\ & -\frac{1}{N^{3/2}}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}\widehat{V}(r/N){\eta }_0{b}_{r+v}^{\ast }{b}_{-r}^{\ast }({\gamma }_v{b}_v+{\sigma }_v{b}_{-v}^{\ast })\hfill \end{array}\end{array}$$

Using (2.11), the decomposition ${\eta }_p=-N{\delta }_{p,0}+N{\widehat{f}}_{N,\ell }(p)$, and the fact that $|{\eta }_0|⩽C$ (see (2.20)), we can bound

$$|⟨{\xi }_1,{[{\mathcal{V}}_N,A]}_1{\xi }_2⟩|⩽C\Vert {\mathcal{K}}^{1/2}{\xi }_1\Vert \Vert ({\mathcal{N}}_{+}+1){\xi }_2\Vert .$$

With (4.9), we bound ${[{\mathcal{V}}_N,A]}_2$ as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨{\xi }_1,{[{\mathcal{V}}_N,A]}_2{\xi }_2⟩|⩽& \frac{C}{N}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}|{\alpha }_{r,v}|\Vert {({\mathcal{N}}_{+}+1)}^{-1}{b}_{r+v}{b}_{-r}{b}_{-v}{\xi }_1\Vert \Vert ({\mathcal{N}}_{+}+1){\xi }_2\Vert \hfill \\ \hfill ⩽& \frac{C}{N^{1/2}}\Vert ({\mathcal{N}}_{+}+1){\xi }_2\Vert [\frac{1}{N}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}\frac{|{\alpha }_{r,v}{|}^2}{{|r|}^2}{]}^{1/2}\hfill \\ \hfill & \times [\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}{|r|}^2{\Vert b_{r+v}{b}_{-r}{b}_{-v}{({\mathcal{N}}_{+}+1)}^{-1}{\xi }_1\Vert }^2{]}^{1/2}\hfill \\ \hfill ⩽& \frac{C}{N^{1/2}}\Vert {\mathcal{K}}^{1/2}{\xi }_1\Vert \Vert ({\mathcal{N}}_{+}+1){\xi }_2\Vert .\hfill \end{array}\end{array}$$ 4.15

$\square$

We can now prove Proposition 2.5

Proof of Prop. 2.5

The proof of the bound (2.34) follows similarly as in [5, Prop. 4.2]. To show (2.35) we consider the function

$$g_{\xi }(s)=⟨\xi ,e^{-sA}{\mathcal{N}}_{+}{e}^{\mathit{sA}}\xi ⟩$$

for $s\in [0,1]$, and its derivative

$$g_{\xi }^{\prime }(s)=⟨\xi ,e^{-sA}[{\mathcal{N}}_{+},A]e^{\mathit{sA}}\xi ⟩$$

Since

$$\begin{array}{cc}\hfill [{\mathcal{N}}_{+},A]=& \frac{1}{\sqrt{N}}\sum _{r,v\in {\mathrm{\Lambda }}_{+}^{\ast }}{b}_{r+v}^{\ast }{b}_{-r}^{\ast }({\eta }_r{\gamma }_v{b}_v+3{\nu }_{r,v}{b}_{-v}^{\ast })+\text{h.c.}\hfill \end{array}$$

and using that $|{\nu }_{r,v}{|⩽C|r|}^{-2}{|v|}^{-2}$, we immediately find

$$\begin{array}{cc}\hfill |g_{\xi }^{\prime }(s)|⩽& \frac{C}{\sqrt{N}}\sum _{r,v\in {\mathrm{\Lambda }}_{+}^{\ast }}(|{\eta }_r|\Vert b_{r+v}{b}_{-r}{e}^{\mathit{sA}}\xi \Vert \Vert b_v{e}^{\mathit{sA}}\xi \Vert \hfill \\ \hfill & +|{\nu }_{r,v}|\Vert {({\mathcal{N}}_{+}+1)}^{-1}{b}_{r+v}{b}_{-r}{b}_{-v}{e}^{\mathit{sA}}\xi \Vert \Vert ({\mathcal{N}}_{+}+1)e^{\mathit{sA}}\xi \Vert )\hfill \\ \hfill ⩽& \frac{C}{\sqrt{N}}\Vert ({\mathcal{N}}_{+}+1)e^{\mathit{sA}}\xi \Vert \Vert {\mathcal{N}}_{+}^{1/2}{e}^{\mathit{sA}}\xi \Vert \hfill \\ \hfill ⩽& ⟨\xi ,e^{-sA}{\mathcal{N}}_{+}{e}^{\mathit{sA}}\xi ⟩+\frac{C}{N}⟨\xi ,e^{-sA}{({\mathcal{N}}_{+}+1)}^2{e}^{\mathit{sA}}\xi ⟩.\hfill \end{array}$$

Using Gronwall’s lemma, and Eq. (2.34), we have

$$g_{\xi }(s)⩽⟨\xi ,{\mathcal{N}}_{+}\xi ⟩+\frac{C}{N}⟨\xi ,{({\mathcal{N}}_{+}+1)}^2\xi ⟩,$$

which concludes the proof of (2.35). To show (2.36), we define

$$\begin{array}{c}\hfill {\phi }_{\xi }(s):=⟨\xi ,e^{-sA}{({\mathcal{N}}_{+}+1)}^k({\mathcal{H}}_N+1)e^{\mathit{sA}}\xi ⟩\end{array}$$

for $s\in [0,1]$. Differentiating with respect to s, we have

$$\begin{array}{cc}\hfill {\phi }_{\xi }^{\prime }(s)=& ⟨\xi ,e^{-sA}[{({\mathcal{N}}_{+}+1)}^k({\mathcal{H}}_N+1),A]e^{\mathit{sA}}\xi ⟩\hfill \\ \hfill =& ⟨\xi ,e^{-sA}{({\mathcal{N}}_{+}+1)}^k[\mathcal{K}+{\mathcal{V}}_N,A]e^{\mathit{sA}}\xi ⟩+⟨\xi ,e^{-sA}[{({\mathcal{N}}_{+}+1)}^k,A]({\mathcal{H}}_N+1)e^{\mathit{sA}}\xi ⟩\hfill \\ \hfill =& P_1+P_2.\hfill \end{array}$$

We consider first $P_1$. From (4.7), we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,e^{-sA}{({\mathcal{N}}_{+}+1)}^k[\mathcal{K}+{\mathcal{V}}_N,A]e^{\mathit{sA}}\xi ⟩|⩽& C⟨\xi ,e^{-sA}{({\mathcal{N}}_{+}+1)}^k({\mathcal{H}}_N+1)e^{\mathit{sA}}\xi ⟩\hfill \\ \hfill & +C⟨\xi ,{({\mathcal{N}}_{+}+1)}^{k+2}\xi ⟩.\hfill \end{array}\end{array}$$

As for the term $P_2$, we observe that the proof of [5, Prop. 4.4] (restricted to $k=1$) can be easily extended to general $k\in \mathbb{N}$. We conclude that

$$|P_2|⩽C⟨\xi ,e^{-sA}({\mathcal{H}}_N+1)({\mathcal{N}}_{+}+1)e^{\mathit{sA}}\xi ⟩+C⟨\xi ,{({\mathcal{N}}_{+}+1)}^{k+2}\xi ⟩.$$

Putting together the estimates for $P_1$ and $P_2$, we obtain

$${\phi }_{\xi }^{\prime }(s)⩽C{\phi }_{\xi }(s)+C⟨\xi ,{({\mathcal{N}}_{+}+1)}^{k+2}\xi ⟩$$

for any $\xi \in {\mathcal{F}}_{+}^{⩽N}$ and for some constant $C>0$. With Gronwall’s lemma we get the desired result. $\square$

Cubic Renormalization: Proof of Theorem 2.6

The starting point for proving Theorem 2.6 is the representation (2.30) for the excitation Hamiltonian ${\mathcal{G}}_N$. To determine the structure of ${\mathcal{J}}_N=e^{-A}{\mathcal{G}}_N{e}^A$, we will separately apply the cubic conjugation to the different summands in ${\mathcal{G}}_N$.

Control of quadratic terms

To conjugate quadratic terms in ${\mathcal{G}}_N$ (excluding the kinetic energy), we will make use of the following lemma.

Lemma 5.1

Let A be defined as in (2.32). For $p\in {\mathrm{\Lambda }}_{+}^{\ast }$, let $w_p\in \mathbb{R}$. Set

$$\begin{array}{c}\hfill \mathcal{W}=\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{w}_p{a}_p^{\ast }{a}_p\end{array}$$

Then

$$\begin{array}{c}\hfill |⟨\xi ,[\mathcal{W},A]\xi ⟩|⩽\frac{{C\Vert w\Vert }_{\infty }}{\sqrt{N}}\Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert \Vert ({\mathcal{N}}_{+}+1)\xi \Vert \end{array}$$ 5.1

Moreover,

$$\begin{array}{c}\hfill |⟨\xi ,[\mathcal{W},A]\xi ⟩|⩽\frac{C{sup}_{p\in {\mathrm{\Lambda }}_{+}^{\ast }}|w_p|/|p|}{\sqrt{N}}\Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert \Vert {\mathcal{K}}^{1/2}{({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \end{array}$$ 5.2

Suppose now $O_p\in \mathbb{C}$ for all $p\in {\mathrm{\Lambda }}_{+}^{\ast }$ and set

$$\begin{array}{c}\hfill \mathcal{O}=\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{O}_p{b}_p^{\ast }{b}_{-p}^{\ast }.\end{array}$$

Then

$$\begin{array}{c}\hfill |⟨\xi ,[\mathcal{O},A]\xi ⟩|⩽\frac{{C\Vert O\Vert }_2}{\sqrt{N}}\Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert \Vert ({\mathcal{N}}_{+}+1)\xi \Vert \end{array}$$ 5.3

and

$$\begin{array}{c}\hfill |⟨\xi ,[\mathcal{O},A]\xi ⟩|⩽\frac{C}{\sqrt{N}}{\left[{\Vert O\Vert }_{\infty },+,\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }},\frac{|O_p{|}^2}{p^2}\right]}^{1/2}\Vert {\mathcal{K}}^{1/2}\xi \Vert \Vert ({\mathcal{N}}_{+}+1)\xi \Vert .\end{array}$$ 5.4

Proof

From the identity

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [\mathcal{W},A]=& \frac{1}{\sqrt{N}}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}(w_{r+v}+w_r-w_v){\eta }_r{\gamma }_v{b}_{r+v}^{\ast }{b}_{-r}^{\ast }{b}_v+\mathrm{h}.\mathrm{c}.\hfill \\ & +\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}(w_{r+v}+w_r+w_v){\nu }_{r,v}{b}_{r+v}^{\ast }{b}_{-r}^{\ast }{b}_{-v}^{\ast }+\mathrm{h}.\mathrm{c}..\hfill \end{array}\end{array}$$

we find (5.1) and (5.2), using that $\gamma \in {\ell }^{\infty }$ and that $\eta ,\nu \in {\ell }^2$ ($\nu$ is square integrable in both its variables), uniformly in N. To prove (5.3), (5.4), we proceed as in the proof of [5, Prop. 8.2]. From the commutation relations (2.2), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [\mathcal{O}+{\mathcal{O}}^{\ast },A]=& \frac{2}{\sqrt{N}}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}{O}_{r+v}{b}_{-r-v}^{\ast }({\eta }_r{\gamma }_v{b}_v^{\ast }+{\nu }_{r,v}{b}_{-v})b_r+\mathrm{h}.\mathrm{c}.\hfill \\ & +\frac{2}{\sqrt{N}}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}{O}_r{b}_r^{\ast }({\eta }_r{\gamma }_v{b}_v^{\ast }+{\nu }_{r,v}{b}_{-v})b_{r+v}+\mathrm{h}.\mathrm{c}.\hfill \\ & +\frac{2}{\sqrt{N}}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}{O}_v{\nu }_{r,v}{b}_v^{\ast }{b}_{r+v}{b}_{-r}+\mathrm{h}.\mathrm{c}.\hfill \\ & -\frac{2}{\sqrt{N}}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}{O}_v{\eta }_r{\gamma }_v{b}_{r+v}^{\ast }{b}_{-r}^{\ast }{b}_{-v}^{\ast }+\mathrm{h}.\mathrm{c}.+\mathcal{E},\hfill \end{array}\end{array}$$ 5.5

where the error term $\mathcal{E}$ collects contributions due to the fact that the modified creation and annihilation operators $b,b^{\ast }$ do not exactly satisfy canonical commutation relations. We have

$$\begin{array}{c}\hfill |⟨\xi ,\mathcal{E}\xi ⟩|⩽\frac{{C\Vert O\Vert }_2}{N}\Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert \Vert ({\mathcal{N}}_{+}+1)\xi \Vert \end{array}$$ 5.6

and also

$$\begin{array}{c}\hfill |⟨\xi ,\mathcal{E}\xi ⟩|⩽\frac{C}{N}\left[{\Vert O\Vert }_{\infty },+,\sum _p,\frac{|O_p{|}^2}{p^2}\right]\Vert {\mathcal{K}}^{1/2}\xi \Vert \Vert ({\mathcal{N}}_{+}+1)\xi \Vert .\end{array}$$ 5.7

Bounding the explicit terms on the r.h.s. of (5.5) one by one, we conclude the proof of the lemma. As an example, consider the first term on the first line. We can estimate

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |\frac{2}{\sqrt{N}}\sum _{r,v}{O}_{r+v}{\eta }_r{\gamma }_v⟨\xi ,b_{-r-v}^{\ast }{b}_v^{\ast }{b}_r\xi ⟩|& ⩽\frac{C}{\sqrt{N}}\sum _{r,v}|O_{r+v}|\Vert b_{-r-v}{b}_v\xi \Vert \Vert b_r\xi \Vert \hfill \\ \hfill & ⩽\frac{C}{\sqrt{N}}{\Vert O\Vert }_2\Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert \Vert {\mathcal{N}}_{+}\xi \Vert \hfill \end{array}\end{array}$$

or, if $O\notin {\ell }^2$,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |\frac{2}{\sqrt{N}}\sum _{r,v}{O}_{r+v}{\eta }_r{\gamma }_v⟨& \xi ,b_{-r-v}^{\ast }{b}_v^{\ast }{b}_r\xi ⟩|\hfill \\ \hfill ⩽& \frac{C}{\sqrt{N}}\sum _{r,v}\frac{|O_{r+v}|}{|r+v|}|r+v|\Vert b_{-r-v}{b}_v\xi \Vert \Vert b_r\xi \Vert \hfill \\ \hfill ⩽& \frac{C}{\sqrt{N}}{\left[\sum _r,\frac{|O_r{|}^2}{r^2}\right]}^{1/2}\Vert {\mathcal{K}}^{1/2}\xi \Vert \Vert {\mathcal{N}}_{+}\xi \Vert .\hfill \end{array}\end{array}$$

In contrast to the proof of (5.1) and (5.2) (where we only used the ${\ell }^{\infty }$ norm of w and of $w_{·}/|·|$), here we need the decay of the observable O (because ${\eta }_r$ and $\Vert b_r\xi \Vert$ both decay in the same variable r and in one of the factors arising from the Cauchy-Schwarz inequality only the observable $O_{r+v}$ provides decays in v). The last term on the r.h.s. of (5.5) can be bounded similarly. The other terms can be estimated using the ${\ell }^{\infty }$ norm of O and the fact that $\gamma \in {\ell }^{\infty }$, $\eta ,\nu \in {\ell }^2$, uniformly in N. $\square$

Applying Lemma 5.1, we obtain the following proposition.

Proposition 5.2

Recall the operator ${\mathcal{T}}_N^{(2)}$, defined in (2.29). Furthermore, from (2.30), we consider the operators

$$\begin{array}{cc}\hfill D& =\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}[\sqrt{{|p|}^4+2{(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell })}_p{p}^2}-p^2]a_p^{\ast }{a}_p\hfill \\ \hfill E& =-\frac{1}{N}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}(\widehat{V}(q/N)+\widehat{V}((q+p)/N)){\eta }_q[({\gamma }_p^2+{\sigma }_p^2)b_p^{\ast }{b}_p+{\gamma }_p{\sigma }_p(b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p})]\hfill \end{array}$$

Then, for every $0<\epsilon <1$, we have

$$\begin{array}{c}\hfill \pm e^{-A}{\mathcal{T}}_N^{(2)}{e}^A⩽\epsilon \mathcal{K}+\frac{C}{\epsilon N}({\mathcal{H}}_N+{\mathcal{N}}_{+}^2+1)\end{array}$$ 5.8

and also

$$\begin{array}{c}\hfill e^{-A}D{e}^A=D+{\mathcal{E}}_D,e^{-A}E{e}^A=E+{\mathcal{E}}_E\end{array}$$

where

$$\begin{array}{c}\hfill \pm {\mathcal{E}}_D,{\mathcal{E}}_E⩽\epsilon {\mathcal{N}}_{+}+\frac{C}{\epsilon N}{({\mathcal{N}}_{+}+1)}^2\end{array}$$

Proof

First of all, we observe that

$$\begin{array}{c}\hfill |\sqrt{{|p|}^4+2{(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell })}_p{p}^2}-p^2|⩽C\end{array}$$

and also that

$$\begin{array}{cc}\hfill |\frac{1}{N}\sum _q(\widehat{V}(q/N)+\widehat{V}((q+p)/N)){\eta }_q({\gamma }_p^2+{\sigma }_p^2)|& ⩽C\hfill \\ \hfill |\frac{1}{N}\sum _q(\widehat{V}(q/N)+\widehat{V}((q+p)/N)){\eta }_q{\gamma }_p{\sigma }_p|& ⩽{C|p|}^{-2}\hfill \end{array}$$

for all $p\in {\mathrm{\Lambda }}_{+}^{\ast }$. Thus, we can apply Lemma 5.1 (and in particular (5.1) and (5.3)) to estimate the commutators [D, A], [E, A]. Writing

$$\begin{array}{c}\hfill {\mathcal{E}}_D={\int }_0^{1}ds{e}^{-sA}[D,A]e^{\mathit{sA}}\end{array}$$

we can therefore bound

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_D\xi ⟩|⩽\frac{C}{\sqrt{N}}{\int }_0^{1}ds\Vert {\mathcal{N}}_{+}^{1/2}{e}^{-sA}\xi \Vert \Vert ({\mathcal{N}}_{+}+1)e^{-sA}\xi \Vert .\end{array}$$

From Prop. 2.5, we conclude that

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_D\xi ⟩|⩽\frac{C}{\sqrt{N}}\Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert \Vert ({\mathcal{N}}_{+}+1)\xi \Vert +\frac{C}{N}{\Vert ({\mathcal{N}}_{+}+1)\xi \Vert }^2.\end{array}$$

Similarly, we can also bound ${\delta }_E$.

To show (5.8), we rewrite

$$\begin{array}{c}\hfill {\mathcal{T}}_N^{(2)}=\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{O}_p(b_p{b}_{-p}+b_{-p}^{\ast }{b}_p^{\ast })\end{array}$$

with

$$\begin{array}{c}\hfill O_p=\frac{(1+2\Vert \sigma {\Vert }_2^2)}{N^2}{(\widehat{V}(·/N)\ast \eta )}_p+\frac{1}{2N}\sum _{q\in {\mathrm{\Lambda }}_{+}^{\ast }}(2{\sigma }_q^2+\frac{{\gamma }_q{\sigma }_q}{{\mu }_q}-1)p^2{\eta }_p\end{array}$$

and we observe that $|O_p|⩽C/N$ for all $p\in {\mathrm{\Lambda }}_{+}^{\ast }$ and that

$$\begin{array}{c}\hfill \sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}\frac{|O_p{|}^2}{p^2}⩽\frac{C}{N}.\end{array}$$ 5.9

This implies that

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{T}}_N^{(2)}\xi ⟩|⩽{\left[\sum _p,\frac{|O_p{|}^2}{p^2}\right]}^{1/2}\Vert {\mathcal{K}}^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert ⩽\frac{C}{\sqrt{N}}\Vert {\mathcal{K}}^{1/2}\xi \Vert \Vert ({\mathcal{N}}_{+}+1)\xi \Vert \end{array}$$

and, by Lemma 5.1, also that

$$\begin{array}{c}\hfill |⟨\xi ,e^{-A}[{\mathcal{T}}_N^{(2)},A]e^A\xi ⟩|⩽\frac{C}{N}\Vert {\mathcal{K}}^{1/2}{e}^A\xi \Vert \Vert ({\mathcal{N}}_{+}+1)e^A\xi \Vert .\end{array}$$

Writing

$$\begin{array}{c}\hfill e^{-A}{\mathcal{T}}_N^{(2)}{e}^A={\mathcal{T}}_{{\mathcal{G}}_N}^{(2)}+{\int }_0^{1}ds{e}^{-sA}[{\mathcal{T}}_N^{(2)},A]e^{\mathit{sA}}\end{array}$$

and applying also Prop. 2.5, we arrive at (5.8). $\square$

Control of quartic terms

Next, we control the quartic error term ${\mathcal{T}}_{{\mathcal{G}}_N}^{(4)}$, defined in (2.29). To this end, we will make use of the following lemma.

Lemma 5.3

Let A be defined as in (2.32). We consider a quartic operator of the form

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{D}=& \frac{1}{N}\sum _{\begin{array}{c}r,p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+p+q\ne 0\end{array}}{D}_{r,p,q}{b}_{r+p}^{\ast }{b}_q^{\ast }{b}_{-p}^{\ast }{b}_{-r-q}^{\ast }+\mathrm{h}.\mathrm{c}.\hfill \end{array}\end{array}$$

where we assume that, for $r,p,q\in {\mathrm{\Lambda }}_{+}^{\ast }$, the coefficients $D_{r,p,q}$ are so that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill min\{\sum _{p,q}\underset{r}{sup}{|D_{r,p,q}|}^2+\frac{1}{N}& \sum _{p,q}\left[\underset{z}{sup},\sum _r,\frac{|D_{r,p,q}{|}^2}{{(r+z)}^2}\right]+\frac{1}{N^2}\sum _{p,q}{\left[\underset{z}{sup},\sum _r,\frac{|D_{r,p,q}|}{{(r+z)}^2}\right]}^2,\hfill \\ \hfill (r\leftrightarrow p),(r\leftrightarrow q)\}⩽C\end{array}\end{array}$$ 5.10

Here, $(r\leftrightarrow p)$ means the same quantity, but inverting the role of the momenta r and p (ie. in the first term, we sum over r, q and we take the supremum over p, and similarly for the other terms). This assumption reflects the fact that (in applications) one of the momenta r, p, q is the argument of the potential $\widehat{V}(./N)$ while the dependence on the other two momenta is square-summable. Then there exists $C>0$ such that

$$\begin{array}{c}\hfill |⟨\xi ,[\mathcal{D},A]\xi ⟩|⩽\frac{C}{\sqrt{N}}\Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert \Vert ({\mathcal{N}}_{+}+1)\xi \Vert +\frac{C}{N}⟨\xi ,(\mathcal{K}+1){({\mathcal{N}}_{+}+1)}^3\xi ⟩\end{array}$$

for all $\xi \in {\mathcal{F}}_{\perp }^{⩽N}$.

Proof

We decompose

$$\begin{array}{cc}\hfill [D,A]=& \frac{1}{N^{3/2}}\sum _{r,p,q,s,w}{D}_{r,p,q}{\eta }_s{\gamma }_w{b}_{s+w}^{\ast }{b}_{-s}^{\ast }[b_{r+p}^{\ast }{b}_q^{\ast }{b}_{-p}^{\ast }{b}_{-r-q}^{\ast },b_w]+\mathrm{h}.\mathrm{c}.\hfill \\ \hfill & -\frac{1}{N^{3/2}}\sum _{r,p,q,s,w}{D}_{r,p,q}{\eta }_s{\gamma }_w{b}_w^{\ast }[b_{r+p}^{\ast }{b}_q^{\ast }{b}_{-p}^{\ast }{b}_{-r-q}^{\ast },b_{-s}{b}_{s+w}]+\mathrm{h}.\mathrm{c}.\hfill \\ \hfill & -\frac{1}{N^{3/2}}\sum _{r,p,q,s,w}{D}_{r,p,q}{\nu }_{s,w}[b_{r+p}^{\ast }{b}_q^{\ast }{b}_{-p}^{\ast }{b}_{-r-q}^{\ast },b_{-w}{b}_{-s}{b}_{s+w}]+\mathrm{h}.\mathrm{c}.\hfill \\ \hfill =:& (\text{I}+\text{II}+\text{III})+\mathrm{h}.\mathrm{c}.\hfill \end{array}$$

After a long but straightforward computation based on the commutation relations (2.2), we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \text{I}=& -\frac{1}{N^{3/2}}\sum _{r,p,q,s}{D}_{r,p,q}^{\text{I}}{\gamma }_{r+p}{\eta }_s{b}_{s+p+r}^{\ast }{b}_q^{\ast }{b}_{-s}^{\ast }{b}_{-p}^{\ast }{b}_{-r-q}^{\ast }+{\mathcal{E}}_1\hfill \\ \hfill \text{II}=& \frac{1}{N^{3/2}}\sum _{r,p,q,w}{D}_{r,p,q}^{\text{II},1}({\eta }_q+{\eta }_{q-w}){\gamma }_w{b}_w^{\ast }{b}_{p+r}^{\ast }{b}_{-r-q}^{\ast }{b}_{-p}^{\ast }{b}_{w-q}\hfill \\ \hfill & +\frac{1}{N^{3/2}}\sum _{r,p,q}{D}_{r,p,q}^{\text{II},2}({\eta }_p+{\eta }_{p+r}){\gamma }_r{b}_r^{\ast }{b}_q^{\ast }{b}_{-r-q}^{\ast }+{\mathcal{E}}_2\hfill \end{array}\end{array}$$ 5.11

with the coefficients

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D_{r,p,q}^{\text{I}}& =D_{r,p,q}+D_{-r,q+r,p+r}+D_{r,-p-r,q}+D_{-r,-q,-p},\hfill \\ \hfill D_{r,p,q}^{\text{II},1}& =D_{r,p,q}+D_{-r,q+r,p+r}+D_{r,-q,-p}+D_{-r,-q,-p}+D_{r,p,-q-r},\hfill \\ \hfill D_{r,p,q}^{\text{II},2}& =D_{r,p,q}+D_{r+p+q,-q,-p}+D_{p-q,r+q,q}+D_{q-p,p,r+p}+D_{-r,-q,-p}+D_{-r-p-q,p,q}\hfill \end{array}\end{array}$$ 5.12

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \text{III}=& \frac{1}{N^{3/2}}\sum _{p,q,r,s}{D}_{r,p,q}^{\text{III},1}({\nu }_{s,-q}+{\nu }_{s,q-s}+{\nu }_{q,-s})b_{r+p}^{\ast }{b}_{-p}^{\ast }{b}_{-r-q}^{\ast }{b}_{-s}{b}_{s-q}\hfill \\ \hfill & +\frac{1}{N^{3/2}}\sum _{r,p,q}{D}_{r,p,q}^{\text{III},2}{\nu }_{p+r,-q-r}{b}_{-p}^{\ast }{b}_q^{\ast }{b}_{q-p}\hfill \\ \hfill & +\frac{1}{N^{3/2}}\sum _{p,q,r}{D}_{r,p,q}^{\text{III},3}({\nu }_{p-q,q+r}+{\nu }_{q+r,p-q})b_q^{\ast }{b}_{-p}^{\ast }{b}_{q-p}\hfill \\ \hfill & +\frac{1}{N^{3/2}}\sum _{r,p,q}{D}_{r,p,q}^{\text{III},4}({\nu }_{p-q,q+r}+{\nu }_{q+r,p-q})b_q^{\ast }{b}_{-p}^{\ast }{b}_{q-p}+{\mathcal{E}}_3\hfill \end{array}\end{array}$$ 5.13

with

$$\begin{array}{cc}\hfill D_{r,p,q}^{\text{III},1}=& D_{r,p,q}+D_{-r,q+r,p+r}+D_{-r,-q,-p}+D_{r,p,-q-r}\hfill \\ \hfill D_{r,p,q}^{\text{III},2}=& D_{r,p,q}+D_{r,p,-q-r}+D_{r,-p-r,-q-r}+D_{r,-p-r,q}+D_{-r,-q,p+r}\hfill \\ \hfill & +D_{-r,q+r,p+r}+D_{-r,-q,-p}+D_{-r,q+r,-p}+D_{p-q,r+q,q}+D_{p-q,-r-p,-p}\hfill \\ \hfill & +D_{q-p,p,p+r}+D_{q-p,-q,-q-r}\hfill \\ \hfill D_{r,p,q}^{\text{III},3}=& D_{r,p,q}+D_{r,p,-q-r}+D_{p-q,q+r,q}+D_{p-q,-p-r,q}+D_{r,-p-r,-q-r}+D_{r,-p-r,q}\hfill \\ \hfill D_{r,p,q}^{\text{III},4}=& D_{r,p,q}+D_{r,p,-q-r}+D_{q-p,p,p+r}+D_{q-p,p,-q-r}+D_{r,-p-r,-q-r}+D_{r,-p-r,q}.\hfill \end{array}$$

In (5.11), (5.13), the error terms ${\mathcal{E}}_1,{\mathcal{E}}_2,{\mathcal{E}}_3$ are produced by the fact that the commutation relations (2.2) are not precisely canonical and can be controlled by

$$\begin{array}{c}\hfill \pm {\mathcal{E}}_1,\pm {\mathcal{E}}_2,\pm {\mathcal{E}}_3⩽\frac{C}{N}(\mathcal{K}+1){({\mathcal{N}}_{+}+1)}^3.\end{array}$$

To bound the first term in (5.11), we can use Cauchy-Schwarz. Let us consider for example the contribution ${\text{I}}_1$, arising from the first term in the identity (5.12) for $D_{r,p,q}^{\text{I}}$. We find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\text{I}}_1\xi ⟩|& ⩽\frac{C}{N^{3/2}}\Vert ({\mathcal{N}}_{+}+1)\xi \Vert {\left[\sum _{r,p,q,s},\frac{|D_{r,p,q}{|}^2}{{(r+q)}^2},{\eta }_s^2\right]}^{1/2}\hfill \\ \hfill & \times {\left[\sum _{r,p,q,s},{(q+r)}^2,,{\Vert b_q{b}_{-s}{b}_{-p}{b}_{-r-q}{({\mathcal{N}}_{+}+1)}^{-1/2}\xi \Vert }^2\right]}^{1/2}\hfill \\ \hfill & ⩽\frac{C}{N^{3/2}}{\left[\sum _{p,q},\underset{z}{sup},\sum _r,\frac{|D_{r,p,q}{|}^2}{{(r+z)}^2}\right]}^{1/2}\Vert ({\mathcal{N}}_{+}+1)\xi \Vert \Vert {\mathcal{K}}^{1/2}({\mathcal{N}}_{+}+1)\xi \Vert .\hfill \end{array}\end{array}$$

With different choices of the weight in the Cauchy-Schwarz inequality, we can exchange the role of the labels r, p, q. Proceeding analogously for the terms arising from the other contributions to $D_{r,p,q}^{\text{I}}$, we arrive at

$$\begin{array}{cc}\hfill |⟨\xi ,\text{I}\xi ⟩|& ⩽\frac{C}{N^{3/2}}min\{\sum _{p,q}\underset{z}{sup}\sum _r\frac{|D_{r,p,q}{|}^2}{{(r+z)}^2},\sum _{r,q}\underset{z}{sup}\sum _p\frac{|D_{r,p,q}{|}^2}{{(p+z)}^2},\sum _{r,p}\underset{z}{sup}\sum _q\frac{|D_{r,p,q}{|}^2}{{(q+z)}^2}{\}}^{1/2}\hfill \\ \hfill & \times ⟨\xi ,(\mathcal{K}+1){({\mathcal{N}}_{+}+1)}^2\xi ⟩\hfill \\ \hfill & ⩽\frac{C}{N}⟨\xi ,(\mathcal{K}+1){({\mathcal{N}}_{+}+1)}^2\xi ⟩\hfill \end{array}$$

where we applied the assumption (5.10). Also the other quintic terms, in (5.11) and in (5.13), can be handled similarly. Let us now consider the cubic terms. The contribution to the cubic term on the last line of (5.11) arising from the first term in the expression for $D_{r,p,}^{\text{II},2}$ in (5.12) can be bounded by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\text{II}}_2\xi ⟩|& ⩽\frac{C\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }{N^{3/2}}{\left[\sum _{r,p,q},\frac{|D_{r,p,q}{|}^2}{r^2}\right]}^{1/2}{\left[\sum _{r,,p,q},({\eta }_p^2+{\eta }_{p+r}^2),r^2,{\Vert b_r{b}_q\xi \Vert }^2\right]}^{1/2}\hfill \\ \hfill & ⩽\frac{C}{N^{3/2}}{\left[\sum _{r,p,q},\frac{|D_{r,p,q}{|}^2}{r^2}\right]}^{1/2}\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{K}}^{1/2}{({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$

Analogously, the estimate also holds if we replace $r^{-2}$ with $q^{-2}$. But we cannot replace $r^{-2}$ with $p^{-2}$. So, we proceed slightly differently, using $|{\eta }_p|⩽C/p^2$ to bound

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\text{II}}_2\xi ⟩|& ⩽\frac{C\Vert ({\mathcal{N}}_{+}+1)\xi \Vert }{N^{3/2}}{\left[\sum _{r,q},{\left(\underset{z}{sup},\sum _p,\frac{|D_{r,p,q}|}{{(p+z)}^2}\right)}^2\right]}^{1/2}{\left[\sum _{r,q},{\Vert b_r{b}_q{({\mathcal{N}}_{+}+1)}^{-1/2}\xi \Vert }^2\right]}^{1/2}\hfill \\ \hfill & ⩽\frac{C}{N^{3/2}}{\left[\sum _{r,q},{\left(\underset{z}{sup},\sum _p,\frac{|D_{r,p,q}|}{{(p+z)}^2}\right)}^2\right]}^{1/2}\Vert ({\mathcal{N}}_{+}+1)\xi \Vert \Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$

From the assumption (5.10), we conclude that

$$\begin{array}{c}\hfill |⟨\xi ,{\text{II}}_2\xi ⟩|⩽\frac{C}{N}⟨\xi ,(\mathcal{K}+1)({\mathcal{N}}_{+}+1)\xi ⟩+\frac{C}{\sqrt{N}}\Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert \Vert ({\mathcal{N}}_{+}+1)\xi \Vert .\end{array}$$

Also the other contributions to the cubic term in (5.11) can be handled similarly. The second term ${\text{III}}_2$ on the r.h.s. of (5.13) can be estimated by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\text{III}}_2\xi ⟩|& ⩽\frac{C}{N^{3/2}}\sum _{r,p,q}|D_{r,p,q}^{\text{III},2}|\frac{|{\eta }_{p+r}|}{{|q+r|}^2}\Vert b_{-p}{b}_q\xi \Vert \Vert b_{q-p}\xi \Vert \hfill \\ \hfill & ⩽\frac{C}{N^{3/2}}min\{\sum _q\underset{r,p}{sup}|D_{r,p,q}^{\text{III},2}{|}^2,\sum _p\underset{r,q}{sup}|D_{r,p,q}^{\text{III},2}{|}^2{\}}^{1/2}\Vert {\mathcal{N}}_{+}\xi \Vert \Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert \hfill \\ \hfill & ⩽\frac{C}{N}\Vert {\mathcal{N}}_{+}\xi \Vert \Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$

As for the last two terms on the r.h.s. of (5.13), they can be controlled similarly as ${\text{II}}_2$. We skip the details. $\square$

We can now control the conjugation of the quartic component of ${\mathcal{T}}_{{\mathcal{G}}_N}$.

Proposition 5.4

Let ${\mathcal{T}}_{{\mathcal{G}}_N}^{(4)}$ be defined as in (2.29) and A as in (2.32). Then, for every $\epsilon >0$ small enough, we have

$$\begin{array}{c}\hfill \pm e^{-A}{\mathcal{T}}_N^{(4)}{e}^A⩽\epsilon \mathcal{K}+\frac{C}{\epsilon N}({\mathcal{H}}_N+1){({\mathcal{N}}_{+}+1)}^4\end{array}$$

Proof

We write

$$\begin{array}{c}\hfill ⟨\xi ,e^{-A}{\mathcal{T}}_N^{(4)}{e}^A\xi ⟩=⟨\xi ,{\mathcal{T}}_N^{(4)}\xi ⟩+{\int }_0^{1}ds⟨\xi ,e^{-sA}[{\mathcal{T}}_N^{(4)},A]e^{\mathit{sA}}\xi ⟩.\end{array}$$ 5.14

Next, we observe that

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{T}}_N^{(4)}\xi ⟩|⩽\frac{C}{\sqrt{N}}\Vert {\mathcal{K}}^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{3/2}\xi \Vert .\end{array}$$

In fact, the contribution of the first term in the definition (2.29) of ${\mathcal{T}}_N^{(4)}$ is bounded by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |\frac{1}{2N}& \sum _{p,q,r}\widehat{V}(r/N){\sigma }_p{\sigma }_{q+r}⟨\xi ,b_{p+r}^{\ast }{b}_q^{\ast }{b}_{-p}^{\ast }{b}_{-q-r}^{\ast }\xi ⟩|\hfill \\ \hfill ⩽& \frac{C}{N}\Vert {({\mathcal{N}}_{+}+1)}^{3/2}\xi \Vert {\left[\sum _{p,q,r},{(p+r)}^2,{\Vert b_{p+r}{b}_q{b}_{-p}{b}_{-q-r}{({\mathcal{N}}_{+}+1)}^{-3/2}\xi \Vert }^2\right]}^{1/2}\hfill \\ \hfill & \times {\left[\sum _{p,q,r},\frac{|\widehat{V}{(r/N)|}^2}{{(r+p)}^2},{\sigma }_p^2,{\sigma }_{q+r}^2\right]}^{1/2}\hfill \\ \hfill ⩽& \frac{C}{\sqrt{N}}\Vert {\mathcal{K}}^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{3/2}\xi \Vert .\hfill \end{array}\end{array}$$

Also the other contributions to ${\mathcal{T}}_{{\mathcal{G}}_N}^{(4)}$ can be bounded similarly. As for the second term on the r.h.s. of (5.14), we apply Lemma 5.3. To this end, we observe that all terms in ${\mathcal{T}}_{{\mathcal{G}}_N}^{(4)}$ satisfy the assumption (5.10) (with $D_{r,p,q}=\widehat{V}(r/N){\sigma }_p{\sigma }_{q+r}$ for the first term, $D_{r,p,q}=N^{-1}{(\widehat{V}(./N)\ast \eta )}_p{\gamma }_q{\sigma }_q{\delta }_{r,0}$ for the second term, $D_{r,p,q}={(\widehat{V}(./N)\ast {\widehat{f}}_{N,\ell })}_p({\gamma }_q{\sigma }_q-{\eta }_q/2-{\sigma }_q^2/(2{\mu }_q))$ for the third term). We obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,e^{-sA}& [{\mathcal{T}}_N^{(4)},A]e^{\mathit{sA}}\xi ⟩|\hfill \\ \hfill ⩽& \frac{C}{N}⟨\xi ,e^{-sA}(\mathcal{K}+1){({\mathcal{N}}_{+}+1)}^3{e}^{\mathit{sA}}\xi ⟩+\frac{C}{\sqrt{N}}\Vert {\mathcal{N}}_{+}^{1/2}{e}^{\mathit{sA}}\xi \Vert \Vert ({\mathcal{N}}_{+}+1)e^{\mathit{sA}}\xi \Vert \hfill \\ \hfill ⩽& \frac{C}{N}⟨\xi ,({\mathcal{H}}_N+1){({\mathcal{N}}_{+}+1)}^4\xi ⟩+\frac{C}{\sqrt{N}}\Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert \Vert ({\mathcal{N}}_{+}+1)\xi \Vert \hfill \end{array}\end{array}$$

where in the last step we also used Prop. 2.5. $\square$

Control of cubic term

In this section, we study the conjugation of the cubic term ${\mathcal{C}}_{{\mathcal{G}}_N}$. We will use the next lemma.

Lemma 5.5

Let ${\mathcal{C}}_{{\mathcal{G}}_N}$ be defined as in (2.28) and A as in (2.32). Then, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [{\mathcal{C}}_{{\mathcal{G}}_N},A]=& \frac{2}{N}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }:p+q\ne 0}(\widehat{V}(p/N)+\widehat{V}((p+q)/N)){\eta }_p\hfill \\ & \times [{\sigma }_q^2+({\gamma }_q^2+{\sigma }_q^2)b_q^{\ast }{b}_q+{\gamma }_q{\sigma }_q(b_q{b}_{-q}+b_q^{\ast }{b}_{-q}^{\ast })]\hfill \\ & +\frac{2}{N}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}[\widehat{V}(p/N)+\widehat{V}((p+q)/N)]{\sigma }_q{\eta }_p\frac{2{\eta }_{q+p}{(q+p)}^2-2{\sigma }_q(p·q)}{p^2+q^2+{(p+q)}^2}\hfill \\ & +\mathcal{B}+\mathcal{D}+{\mathcal{E}}_{[\mathcal{C},A]},\hfill \end{array}\end{array}$$ 5.15

where

$$\begin{array}{c}\hfill \mathcal{B}=\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{O}_p(b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p})\end{array}$$ 5.16

is a quadratic operator, with coefficients satisfying

$$\begin{array}{c}\hfill \sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}\frac{|O_p{|}^2}{p^2}⩽\frac{C}{N},{\Vert O\Vert }_{\infty }⩽\frac{C}{N}\end{array}$$ 5.17

and where

$$\begin{array}{c}\hfill \mathcal{D}=\frac{1}{N}\sum _{\begin{array}{c}r,p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+p+q\ne 0\end{array}}{D}_{r,p,q}{b}_{r+p}^{\ast }{b}_q^{\ast }{b}_{-p}^{\ast }{b}_{-r-q}^{\ast }+\mathrm{h}.\mathrm{c}.\end{array}$$ 5.18

is a quartic operator with coefficients $D_{r,p,q}$ satisfying (5.10). Furthermore, we have

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_{[\mathcal{C},A]}\xi ⟩|⩽\frac{C}{\sqrt{N}}\Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert \Vert {\mathcal{K}}^{1/2}({\mathcal{N}}_{+}+1)\xi \Vert +\frac{C}{N}⟨\xi ,(\mathcal{K}+1){({\mathcal{N}}_{+}+1)}^2\xi ⟩\end{array}$$ 5.19

for every $\xi \in {\mathcal{F}}_{\perp }^{⩽N}$.

Proof

From (2.28), (2.32), we can decompose

$$\begin{array}{c}\hfill [{\mathcal{C}}_{{\mathcal{G}}_N},A]=\sum _{i=1}^4{\mathrm{\Pi }}_i\end{array}$$

with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathrm{\Pi }}_1=& \frac{1}{\sqrt{N}}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}\widehat{V}(p/N){\gamma }_q[b_{p+q}^{\ast }{b}_{-p}^{\ast }{b}_q,A_{\nu }]+\mathrm{h}.\mathrm{c}.\hfill \\ \hfill {\mathrm{\Pi }}_2=& \frac{1}{\sqrt{N}}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}\widehat{V}(p/N){\gamma }_q[b_{p+q}^{\ast }{b}_{-p}^{\ast }{b}_q,A_{\gamma }]+\mathrm{h}.\mathrm{c}.\hfill \\ \hfill {\mathrm{\Pi }}_3=& \frac{1}{\sqrt{N}}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}\widehat{V}(p/N){\sigma }_q[b_{p+q}^{\ast }{b}_{-p}^{\ast }{b}_{-q}^{\ast },A_{\nu }]+\mathrm{h}.\mathrm{c}.\hfill \\ \hfill {\mathrm{\Pi }}_4=& \frac{1}{\sqrt{N}}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}\widehat{V}(p/N){\sigma }_q[b_{p+q}^{\ast }{b}_{-p}^{\ast }{b}_{-q}^{\ast },A_{\gamma }]+\mathrm{h}.\mathrm{c}..\hfill \end{array}\end{array}$$

We analyze the four terms separately. From (2.2), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathrm{\Pi }}_1=& \frac{1}{N}\sum _{\begin{array}{c}p,q,r\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q,r+q\ne 0\end{array}}\widehat{V}(p/N){\gamma }_q({\nu }_{r,-q-r}+{\nu }_{q,r}+{\nu }_{r,q})b_{-p-q}^{\ast }{b}_p^{\ast }{b}_{r+q}^{\ast }{b}_{-r}^{\ast }+\mathrm{h}.\mathrm{c}.\hfill \\ & +\frac{1}{N}\sum _{\begin{array}{c}p,q,r\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q,r+p\ne 0\end{array}}[\widehat{V}(p/N)+\widehat{V}((p+q)/N)]{\gamma }_q({\nu }_{r,-p-r}+{\nu }_{p,r}+{\nu }_{r,p})\hfill \\ & \times b_{p+q}^{\ast }{b}_{p+r}{b}_q{b}_{-r}+\mathrm{h}.\mathrm{c}.\hfill \\ & +\frac{2}{N}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}[\widehat{V}(p/N)+\widehat{V}((p+q)/N)]{\gamma }_q({\nu }_{p,q}+{\nu }_{q,p}+{\nu }_{p,-p-q})b_q^{\ast }{b}_{-q}^{\ast }\hfill \\ & +{\tilde{\mathcal{E}}}_{{\mathrm{\Pi }}_1}\hfill \end{array}\end{array}$$ 5.20

where the error ${\tilde{\mathcal{E}}}_{{\mathrm{\Pi }}_1}$ (and, similarly, the errors ${\tilde{\mathcal{E}}}_{{\mathrm{\Pi }}_j}$, $j=2,3,4$, below) is due to the fact that the commutation relations (2.2) are not precisely canonical and can be estimated by

$$\begin{array}{c}\hfill \pm {\tilde{\mathcal{E}}}_{{\mathrm{\Pi }}_1}⩽\frac{C}{N}(\mathcal{K}+1){({\mathcal{N}}_{+}+1)}^2.\end{array}$$

Using the fact that $|{\nu }_{p,q}{|⩽C|p|}^{-2}{|q|}^{-2}$, the second term on the r.h.s. of (5.20) can be bounded by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |\frac{1}{N}& \sum _{\begin{array}{c}p,q,r\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q,r+p\ne 0\end{array}}[\widehat{V}(p/N)+\widehat{V}((p+q)/N)]{\gamma }_q({\nu }_{r,-p-r}+{\nu }_{p,r}+{\nu }_{r,p})⟨\xi ,b_{p+q}^{\ast }{b}_{p+r}{b}_q{b}_{-r}\xi ⟩|\hfill \\ \hfill ⩽& \frac{C}{N}{\left[\sum _{p,q,r},{|r|}^{-4},{(|p|}^{-4},{+|p+r|}^{-4},)\Vert ,b_{p+q},{({\mathcal{N}}_{+}+1)}^{1/2},{\xi \Vert }^2\right]}^{1/2}\hfill \\ \hfill & {\left[\sum _{p,q,r},{\Vert b_{p+r}{b}_q{b}_{-r}{({\mathcal{N}}_{+}+1)}^{-1/2}\xi \Vert }^2\right]}^{1/2}\hfill \\ \hfill ⩽& \frac{C}{N}⟨\xi ,{({\mathcal{N}}_{+}+1)}^2\xi ⟩.\hfill \end{array}\end{array}$$ 5.21

As for the third term on the r.h.s. of (5.20), we notice that $|{\nu }_{p,-p-q}{|⩽C|p|}^{-2}{|p+q|}^{-2}$ implies that the quadratic term proportional to ${\nu }_{p,-p-q}$ is bounded by $C{N}^{-1}\mathcal{K}{\mathcal{N}}_{+}$. To handle the other contributions, we write

$$\begin{array}{c}\hfill {\nu }_{p,q}+{\nu }_{q,p}={\eta }_p{\sigma }_q-\frac{2q^2}{p^2+q^2+{|p+q|}^2}[{\eta }_p({\sigma }_q-{\eta }_q)-{\eta }_q({\sigma }_p-{\eta }_p)]-{\eta }_p{\sigma }_q\frac{2p·q}{p^2+q^2+{|p+q|}^2}.\end{array}$$ 5.22

The contributions proportional to ${\eta }_q({\sigma }_p-{\eta }_p)$ and ${\eta }_p({\sigma }_q-{\eta }_q)$ are bounded by $C{N}^{-1}({\mathcal{N}}_{+}+1)$, since $|{\sigma }_p-{\eta }_p{|⩽C|p|}^{-4}$. To bound the contribution of the last term, we distinguish $|p|>N$ and $|p|⩽N$. For $|p|>N$, we estimate

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |\frac{2}{N}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ |p|>N\end{array}}& (\widehat{V}(p/N)+\widehat{V}((p+q)/N)){\gamma }_q{\eta }_p{\sigma }_q\frac{2p·q}{p^2+q^2+{|p+q|}^2}⟨\xi ,b_q^{\ast }{b}_{-q}^{\ast }\xi ⟩|\hfill \\ \hfill ⩽& \frac{C}{N}\sum _{\begin{array}{c}q\in {\mathrm{\Lambda }}_{+}^{\ast }\end{array}}|q||{\sigma }_q|\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert b_q\xi \Vert ⩽\frac{C}{N}⟨\xi ,(\mathcal{K}+1)\xi ⟩\hfill \end{array}\end{array}$$ 5.23

since ${\sum }_{|p|>N}|{\eta }_p{||p|}^{-1}⩽C$. For $|p|⩽N$, we proceed with a change of variable $p\to -p$ and obtain

$$\begin{array}{cc}\hfill \frac{2}{N}& \sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast },|p|⩽N}{\eta }_p{\sigma }_q(p·q)[\frac{\widehat{V}(p/N)+\widehat{V}((p-q)/N)}{p^2+q^2+{|p-q|}^2}-\frac{\widehat{V}(p/N)+\widehat{V}((p+q)/N)}{p^2+q^2+{|p+q|}^2}]\hfill \\ \hfill & \times {\gamma }_q⟨\xi ,(b_q^{\ast }{b}_{-q}^{\ast }+\text{h.c.})\xi ⟩\hfill \\ \hfill =& \frac{2}{N}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ |p|⩽N\end{array}}[\widehat{V}((p-q)/N)-\widehat{V}((p+q)/N)]\frac{{\eta }_p{\sigma }_q(p·q)}{p^2+q^2+{|p+q|}^2}{\gamma }_q⟨\xi ,(b_q^{\ast }{b}_{-q}^{\ast }+\mathrm{h}.\mathrm{c}.)\xi ⟩\hfill \\ \hfill & +\frac{8}{N}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ |p|⩽N\end{array}}\frac{[\widehat{V}(p/N)+\widehat{V}((p+q)/N)]{\eta }_p{\sigma }_q{(p·q)}^2}{(p^2+q^2{+|p+q|}^2)(p^2+q^2{+|p-q|}^2)}{\gamma }_q⟨\xi ,(b_q^{\ast }{b}_{-q}^{\ast }+\mathrm{h}.\mathrm{c}.)\xi ⟩.\hfill \end{array}$$

Using Lipschitz contiuity of $\widehat{V}$ in the first term on the r.h.s and the bound ${\sum }_p{|p+q|}^{-2}{|p|}^{-2}⩽C{|q|}^{-1}$ in the second term, we obtain

$$\begin{array}{cc}\hfill |\frac{2}{N}& \sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast },|p|⩽N}{\eta }_p{\sigma }_q(p·q)[\frac{\widehat{V}(p/N)+\widehat{V}((p-q)/N)}{p^2+q^2+{|p-q|}^2}-\frac{\widehat{V}(p/N)+\widehat{V}((p+q)/N)}{p^2+q^2+{|p+q|}^2}]\hfill \\ \hfill & \times {\gamma }_q⟨\xi ,(b_q^{\ast }{b}_{-q}^{\ast }+\text{h.c.})\xi ⟩|\hfill \\ \hfill & ⩽\frac{C}{N}\sum _{q\in {\mathrm{\Lambda }}_{+}^{\ast }}{\sigma }_q|q|⟨\xi ,(b_q^{\ast }{b}_{-q}^{\ast }+\text{h.c.})\xi ⟩⩽\frac{C}{N}⟨\xi ,(\mathcal{K}+1)\xi ⟩\hfill \end{array}$$

proceeding similarly as in (5.23).

Noticing that the coefficients $D_{q,r,p}^{(1)}=\widehat{V}(p/N){\gamma }_q({\nu }_{r,-q-r}+{\nu }_{q,r}+{\nu }_{r,q})$ satisfy the assumption (5.10), we denote by ${\mathcal{D}}^{(1)}$ the quartic operator on the first line on the r.h.s. of (5.20); it will be absorbed in (5.18). We conclude that

$$\begin{array}{c}\hfill {\mathrm{\Pi }}_1=\frac{1}{N}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}[\widehat{V}(p/N)+\widehat{V}((p+q)/N)]{\eta }_p{\gamma }_q{\sigma }_q{b}_q^{\ast }{b}_{-q}^{\ast }+\mathrm{h}.\mathrm{c}.+{\mathcal{D}}^{(1)}+{\mathcal{E}}_{{\mathrm{\Pi }}_1},\end{array}$$ 5.24

where

$$\begin{array}{c}\hfill \pm {\mathcal{E}}_{{\mathrm{\Pi }}_1}⩽\frac{C}{N}(\mathcal{K}+1){({\mathcal{N}}_{+}+1)}^2.\end{array}$$

Let us now consider ${\mathrm{\Pi }}_2$. From (2.2), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathrm{\Pi }}_2=& \frac{1}{N}\sum _{\begin{array}{c}p,r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v,p-v\ne 0\end{array}}\widehat{V}(p/N)({\eta }_v+{\eta }_{r+v}){\gamma }_r{\gamma }_v{b}_{r+v}^{\ast }{b}_{p-v}^{\ast }{b}_{-p}^{\ast }{b}_r+\mathrm{h}.\mathrm{c}.\hfill \\ \hfill & -\frac{1}{N}\sum _{\begin{array}{c}q,r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v,q+v\ne 0\end{array}}[\widehat{V}(v/N)+\widehat{V}((v+q)/N)]{\gamma }_q{\gamma }_v{\eta }_r{b}_{-r}^{\ast }{b}_{r+v}^{\ast }{b}_{-v-q}^{\ast }{b}_{-q}+\mathrm{h}.\mathrm{c}.\hfill \\ \hfill & +\frac{1}{N}\sum _{\begin{array}{c}q,r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v,q+v\ne 0\end{array}}[\widehat{V}(v/N)+\widehat{V}((v+q)/N)]({\eta }_v+{\eta }_{v+r}){\gamma }_r{\gamma }_q{b}_{r+v}^{\ast }{b}_q^{\ast }{b}_r{b}_{q+v}+\mathrm{h}.\mathrm{c}.\hfill \\ \hfill & -\frac{1}{N}\sum _{\begin{array}{c}p,r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v,p+v\ne 0\end{array}}\widehat{V}(p/N){\eta }_r{\gamma }_v^2{b}_{r+v}^{\ast }{b}_{-r}^{\ast }{b}_{p+v}{b}_{-p}+\mathrm{h}.\mathrm{c}.\hfill \\ \hfill & +\frac{2}{N}\sum _{\begin{array}{c}q,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ q+v\ne 0\end{array}}[\widehat{V}(v/N)+\widehat{V}((v+q)/N)]{\eta }_v{\gamma }_q^2{b}_q^{\ast }{b}_q+{\tilde{\mathcal{E}}}_{{\mathrm{\Pi }}_2}\hfill \end{array}\end{array}$$ 5.25

with $\pm {\tilde{\mathcal{E}}}_{{\mathrm{\Pi }}_2}⩽C{N}^{-1}(\mathcal{K}+1){({\mathcal{N}}_{+}+1)}^2$. The first term on the r.h.s. can be controlled, using Cauchy-Schwarz, by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |\frac{1}{N}\sum _{p,r,v}& \widehat{V}(p/N)({\eta }_v+{\eta }_{r+v}){\gamma }_r{\gamma }_v⟨\xi ,b_{r+v}^{\ast }{b}_{p-v}^{\ast }{b}_{-p}^{\ast }{b}_r\xi ⟩|\hfill \\ \hfill ⩽& \frac{C}{N}{\left[\sum _{p,r,v},p^2,{\Vert b_{r+v}{b}_{p-v}{b}_{-p}\xi \Vert }^2\right]}^{1/2}{\left[\sum _{p,r,v},\frac{|\widehat{V}{(p/N)|}^2}{p^2},({\eta }_v^2+{\eta }_{r+v}^2),{\Vert b_r\xi \Vert }^2\right]}^{1/2}\hfill \\ \hfill ⩽& \frac{C}{\sqrt{N}}\Vert {\mathcal{K}}^{1/2}{\mathcal{N}}_{+}\xi \Vert \Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert \hfill \end{array}\end{array}$$ 5.26

The other quartic terms on the r.h.s. of (5.25) can be bounded similarly. We conclude that

$$\begin{array}{c}\hfill {\mathrm{\Pi }}_2=\frac{2}{N}\sum _{\begin{array}{c}q,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ q+v\ne 0\end{array}}[\widehat{V}(v/N)+\widehat{V}((v+q)/N)]{\eta }_v{\gamma }_q^2{b}_q^{\ast }{b}_q+{\mathcal{E}}_{{\mathrm{\Pi }}_2}\end{array}$$ 5.27

where

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_{{\mathrm{\Pi }}_2}\xi ⟩|⩽\frac{C}{\sqrt{N}}\Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert \Vert {\mathcal{K}}^{1/2}{\mathcal{N}}_{+}\xi \Vert +\frac{C}{N}⟨\xi ,(\mathcal{K}+1){({\mathcal{N}}_{+}+1)}^2\xi ⟩\end{array}$$

for all $\xi \in {\mathcal{F}}_{\perp }^{⩽N}$.

Next, we consider ${\mathrm{\Pi }}_3$. With (2.2), we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathrm{\Pi }}_3=& \frac{1}{N}\sum _{\begin{array}{c}p,q,r\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q,q+r\ne 0\end{array}}\widehat{V}(p/N){\sigma }_q({\nu }_{r,-q-r}+{\nu }_{q,r}+{\nu }_{r,q})b_{p+q}^{\ast }{b}_{-p}^{\ast }{b}_{-r}{b}_{r+q}+\mathrm{h}.\mathrm{c}.\hfill \\ & +\frac{1}{N}\sum _{\begin{array}{c}p,q,r\ne 0\\ p+q,p+r\ne 0\end{array}}[\widehat{V}(p/N)+\widehat{V}((p+q)/N)]{\sigma }_q({\nu }_{r,-p-r}+{\nu }_{p,q}+{\nu }_{r,p})\hfill \\ & \times b_{p+q}^{\ast }{b}_{-q}^{\ast }{b}_{-r}{b}_{r+p}+\mathrm{h}.\mathrm{c}.\hfill \\ & +\frac{2}{N}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}[\widehat{V}(p/N)+\widehat{V}((p+q)/N)]{\sigma }_q\hfill \\ & \times ({\nu }_{-p-q,q}+{\nu }_{q,-p-q}+{\nu }_{p,-p-q}+{\nu }_{-p-q,p}+{\nu }_{p,q}+{\nu }_{q,p})b_{-p}^{\ast }{b}_{-p}\hfill \\ & +\frac{2}{N}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}[\widehat{V}(p/N)+\widehat{V}((p+q)/N)]{\sigma }_q({\nu }_{-p-q,p}+{\nu }_{p,q}+{\nu }_{q,p})b_{-q}^{\ast }{b}_{-q}\hfill \\ & +\frac{2}{N}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}[\widehat{V}(p/N)+\widehat{V}((p+q)/N)]{\sigma }_q({\nu }_{-p-q,p}+{\nu }_{p,q}+{\nu }_{q,p})+{\tilde{\mathcal{E}}}_{{\mathrm{\Pi }}_3}\hfill \end{array}\end{array}$$ 5.28

where $\pm {\tilde{\mathcal{E}}}_{{\mathrm{\Pi }}_3}⩽C{N}^{-1}(\mathcal{K}+1){({\mathcal{N}}_{+}+1)}^2$. The first term on the r.h.s. can be handled similarly to (5.26). The second term can be bounded analogously to (5.21). Let us now consider the quadratic terms.

The term proportional to $b_{-p}^{\ast }{b}_{-p}$ is controlled by $C{N}^{-1}({\mathcal{N}}_{+}+1)$, since the sum over q can be bounded using $|{\sigma }_q{|⩽C|q|}^{-2}$, $|{\nu }_{r,s}{|⩽C|r|}^{-2}{|s|}^{-2}$. In the next quadratic term, proportional to $b_{-q}^{\ast }{b}_{-q}$, the bound $|{\nu }_{-p-q,p}{|⩽C|p|}^{-2}{|p+q|}^{-2}$ allows us to estimate the corresponding contribution by $C{N}^{-1}({\mathcal{N}}_{+}+1)$. To handle the contributions proportional to ${\nu }_{p,q}$, ${\nu }_{q,p}$, we recall (5.22). The contribution of all terms on the r.h.s. of (5.22), with the exception of the term proportional to ${\eta }_p{\sigma }_q$, can be estimated by $C{N}^{-1}logN{\mathcal{N}}_{+}$ and can therefore be neglected.

As for the constant term on the r.h.s. of (5.28), we write

$$\begin{array}{cc}\hfill {\sigma }_q({\nu }_{-p-q,p}+{\nu }_{p,q}+{\nu }_{q,p})=& {\eta }_p{\sigma }_q^2+{\sigma }_q{\eta }_p\frac{2{\eta }_{p+q}{(p+q)}^2-2{\sigma }_q(p·q)}{p^2+q^2+{(p+q)}^2}\hfill \\ \hfill & +\frac{2q^2{\sigma }_q}{p^2+q^2+{(p+q)}^2}[{\eta }_q({\sigma }_p-{\eta }_p)+{\eta }_p({\eta }_q-{\sigma }_q)]\hfill \\ \hfill & +\frac{2{(p+q)}^2{\sigma }_q}{p^2+q^2+{(p+q)}^2}{\eta }_{p+q}({\sigma }_p-{\eta }_p)\hfill \end{array}$$

and we notice that the contribution associated with the last two lines is of the order $N^{-1}$ (because these terms are all summable over p, q). We conclude that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathrm{\Pi }}_3=& \frac{2}{N}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}[\widehat{V}(p/N)+\widehat{V}((p+q)/N)]{\eta }_p{\sigma }_q^2\hfill \\ & +\frac{2}{N}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}[\widehat{V}(p/N)+\widehat{V}((p+q)/N)]{\sigma }_q{\eta }_p\frac{2{\eta }_{q+p}{(q+p)}^2-2{\sigma }_q(p·q)}{p^2+q^2+{(p+q)}^2}\hfill \\ & +\frac{2}{N}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}[\widehat{V}(v/N)+\widehat{V}((v+q)/N)]{\eta }_v{\sigma }_q^2{b}_q^{\ast }{b}_q+{\mathcal{E}}_{{\mathrm{\Pi }}_3}\hfill \end{array}\end{array}$$ 5.29

where

$$|⟨\xi ,{\mathcal{E}}_{{\mathrm{\Pi }}_3}\xi ⟩|⩽\frac{C}{\sqrt{N}}\Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert \Vert {\mathcal{K}}^{1/2}({\mathcal{N}}_{+}+1)\xi \Vert +\frac{C}{N}⟨\xi ,(\mathcal{K}+1){({\mathcal{N}}_{+}+1)}^2\xi ⟩$$

for all $\xi \in {\mathcal{F}}_{\perp }^{⩽N}$.

Finally, we consider ${\mathrm{\Pi }}_4$. Again with (2.2), we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathrm{\Pi }}_4=& -\frac{1}{N}\sum _{\begin{array}{c}p,r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v,p+v\ne 0\end{array}}\widehat{V}(p/N){\eta }_r{\gamma }_v{\sigma }_v{b}_{r+v}^{\ast }{b}_{-r}^{\ast }{b}_{p-v}^{\ast }{b}_{-p}^{\ast }+\mathrm{h}.\mathrm{c}.\hfill \\ & -\frac{1}{N}\sum _{\begin{array}{c}p,r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v,q+v\ne 0\end{array}}[\widehat{V}(v/N)+\widehat{V}((v+p)/N)]{\eta }_r{\gamma }_v{\sigma }_p{b}_{r+v}^{\ast }{b}_{-r}^{\ast }{b}_p^{\ast }{b}_{-p-v}^{\ast }+\mathrm{h}.\mathrm{c}.\hfill \\ & +\frac{1}{N}\sum _{\begin{array}{c}p,r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v,p+v\ne 0\end{array}}\widehat{V}(p/N)({\eta }_v+{\eta }_{r+v}){\gamma }_r{\sigma }_v{b}_{r+v}^{\ast }{b}_{p+v}{b}_{-p}{b}_r+\mathrm{h}.\mathrm{c}.\hfill \\ & +\frac{1}{N}\sum _{\begin{array}{c}p,r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v,p+v\ne 0\end{array}}[\widehat{V}(v/N)+\widehat{V}((v+p)/N)]({\eta }_v+{\eta }_{r+v}){\sigma }_p{\gamma }_r{b}_{r+v}^{\ast }{b}_{p+v}{b}_{-p}{b}_r+\mathrm{h}.\mathrm{c}.\hfill \\ & +\frac{1}{N}\sum _{\begin{array}{c}q,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ q+v\ne 0\end{array}}[\widehat{V}(v/N)+\widehat{V}((v+q)/N)]{\eta }_v{\gamma }_q{\sigma }_q{b}_q^{\ast }{b}_{-q}^{\ast }+\mathrm{h}.\mathrm{c}.\hfill \\ & +\frac{1}{N}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}[\widehat{V}(r/N)+\widehat{V}((r+v)/N)]({\eta }_{r+v}+{\eta }_v){\gamma }_r{\sigma }_v{b}_r{b}_{-r}+\mathrm{h}.\mathrm{c}.+{\tilde{\mathcal{E}}}_{{\mathrm{\Pi }}_4}\hfill \end{array}\end{array}$$ 5.30

with $\pm {\tilde{\mathcal{E}}}_{{\mathrm{\Pi }}_4}⩽C{N}^{-1}(\mathcal{K}+1){({\mathcal{N}}_{+}+1)}^2$. Since the coefficients $D_{v,r,p}^{(2)}=-\widehat{V}(p/N){\eta }_r{\gamma }_v{\sigma }_v$ and $D_{v,r,p}^{(3)}=[\widehat{V}(v/N)+\widehat{V}((v+p)/N)]{\eta }_r{\gamma }_v{\sigma }_p$ satisfy the assumption (5.10), we denote by ${\mathcal{D}}^{(2)}$ and ${\mathcal{D}}^{(3)}$ the quartic operators on the first and second line of (5.30); we will absorb them in (5.18). The other quartic contributions to ${\mathrm{\Pi }}_4$ can be bounded similarly to (5.21) (term on the fourth line) and (5.26) (term on the third line). As for the quadratic terms, the contribution proportional to ${\eta }_{r+v}$ in the last line is small, since

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \pm \frac{1}{N}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}& [\widehat{V}(r/N)+\widehat{V}((r+v)/N)]{\eta }_{r+v}{\gamma }_r{\sigma }_v(⟨\xi ,b_r{b}_{-r}\xi ⟩+\mathrm{h}.\mathrm{c}.)\hfill \\ \hfill ⩽& \frac{C}{N}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}\frac{1}{{|r+v|}^2{|v|}^2}\Vert b_r\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \hfill \\ \hfill ⩽& \frac{C}{N}\sum _{r\in {\mathrm{\Lambda }}_{+}^{\ast }}\frac{1}{r}\Vert b_r\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert ⩽\frac{C}{N}⟨\xi ,(\mathcal{K}+{\mathcal{N}}_{+}+1)\xi ⟩.\hfill \end{array}\end{array}$$

Similarly, in the contribution proportional to ${\eta }_v$ in the last line, we can replace ${\gamma }_r$ by 1, up to an error bounded by $N^{-1}({\mathcal{N}}_{+}+1)$. Observing that the coefficients

$$\begin{array}{c}\hfill O_r=\frac{1}{N}\sum _{v\in {\mathrm{\Lambda }}_{+}^{\ast }:r+v\ne 0}[\widehat{V}(r/N)+\widehat{V}((r+v)/N)]{\eta }_v{\sigma }_v\end{array}$$

satisfy (5.17), we conclude that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathrm{\Pi }}_4=& \frac{1}{N}\sum _{\begin{array}{c}q,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ q+v\ne 0\end{array}}[\widehat{V}(v/N)+\widehat{V}((v+q)/N)]{\eta }_v{\gamma }_q{\sigma }_q{b}_q^{\ast }{b}_{-q}^{\ast }+\mathrm{h}.\mathrm{c}.\hfill \\ & +\frac{1}{N}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}[\widehat{V}(r/N)+\widehat{V}((r+v)/N)]{\eta }_v{\sigma }_v{b}_r^{\ast }{b}_{-r}^{\ast }+\mathrm{h}.\mathrm{c}.\hfill \\ & +\mathcal{B}+{\mathcal{D}}^{(2)}+{\mathcal{D}}^{(3)}+{\mathcal{E}}_{{\mathrm{\Pi }}_4}\hfill \end{array}\end{array}$$ 5.31

where

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_{{\mathrm{\Pi }}_4}\xi ⟩|⩽\frac{C}{\sqrt{N}}\Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert \Vert {\mathcal{K}}^{1/2}({\mathcal{N}}_{+}+1)\xi \Vert +\frac{C}{N}⟨\xi ,(\mathcal{K}+1){({\mathcal{N}}_{+}+1)}^2\xi ⟩\end{array}$$

for all $\xi \in {\mathcal{F}}_{\perp }^{⩽N}$, where $\mathcal{B}$ has the form (5.16), with coefficients satisfying the corresponding conditions (5.17), and where ${\mathcal{D}}^{(2)},{\mathcal{D}}^{(3)}$ have the form (5.18), with coefficients satisfying (5.10). Combining (5.24), (5.27), (5.29), (5.31), we obtain (5.15). $\square$

Making use of the last lemma, we can compute $e^{-A}{\mathcal{C}}_{{\mathcal{G}}_N}{e}^A$, up to small errors.

Proposition 5.6

Let ${\mathcal{C}}_{{\mathcal{G}}_N}$ be defined as in (2.28) and A as in (2.32). Under the same assumptions as Theorem 2.6, we have

$$\begin{array}{c}\hfill \begin{array}{cc}& e^{-A}{\mathcal{C}}_{{\mathcal{G}}_N}{e}^A\hfill \\ & ={\mathcal{C}}_{{\mathcal{G}}_N}\hfill \\ & +\frac{2}{N}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}(\widehat{V}(p/N)+\widehat{V}((p+q)/N)){\eta }_p[{\sigma }_q^2+({\gamma }_q^2+{\sigma }_q^2)b_q^{\ast }{b}_q+{\gamma }_q{\sigma }_q(b_q{b}_{-q}+b_q^{\ast }{b}_{-q}^{\ast })]\hfill \\ & +\frac{2}{N}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}[\widehat{V}(p/N)+\widehat{V}((p+q)/N)]{\sigma }_q{\eta }_p\frac{2{\eta }_{q+p}{(q+p)}^2-2{\sigma }_q(p·q)}{p^2+q^2+{(p+q)}^2}+{\mathcal{E}}_{{{\mathcal{C}}_{\mathcal{G}}}_N},\hfill \end{array}\end{array}$$ 5.32

where, for every $\epsilon >0$ small enough, we have

$$\begin{array}{c}\hfill \pm {\mathcal{E}}_{{\mathcal{C}}_{{\mathcal{G}}_N}}⩽\epsilon \mathcal{K}+\frac{C}{\epsilon N}({\mathcal{H}}_N+1){({\mathcal{N}}_{+}+1)}^4\end{array}$$

for all $\xi \in {\mathcal{F}}_{\perp }^{⩽N}$.

Proof

We write

$$\begin{array}{c}\hfill e^{-A}{\mathcal{C}}_{{\mathcal{G}}_N}{e}^A={\mathcal{C}}_{{\mathcal{G}}_N}+{\int }_0^{1}ds{e}^{-sA}[{\mathcal{C}}_{{\mathcal{G}}_N},A]e^{\mathit{sA}}.\end{array}$$ 5.33

From Lemma 5.5, we recall the identity

$$\begin{array}{cc}\hfill [{\mathcal{C}}_{{\mathcal{G}}_N},A]=& \frac{2}{N}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }:p+q\ne 0}(\widehat{V}(p/N)+\widehat{V}((p+q)/N)){\eta }_p\hfill \\ & \times [{\sigma }_q^2+({\gamma }_q^2+{\sigma }_q^2)b_q^{\ast }{b}_q+{\gamma }_q{\sigma }_q(b_q{b}_{-q}+b_q^{\ast }{b}_{-q}^{\ast })]\hfill \\ & +\frac{2}{N}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}[\widehat{V}(p/N)+\widehat{V}((p+q)/N)]{\sigma }_q{\eta }_p\frac{2{\eta }_{q+p}{(q+p)}^2-2{\sigma }_q(p·q)}{p^2+q^2+{(p+q)}^2}\hfill \\ & +\mathcal{B}+\mathcal{D}+{\mathcal{E}}_{[\mathcal{C},A]},\hfill \end{array}$$

where $\mathcal{B}$ is a quadratic operator having the form (5.16), $\mathcal{D}$ is a quartic operator like (5.18) and ${\mathcal{E}}_{[\mathcal{C},A]}$ satisfies the bounds (5.19). From Prop. 5.2, we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{2}{N}\sum _{p,q}(\widehat{V}& (p/N)+\widehat{V}((p+q)/N)){\eta }_p{e}^{-sA}[({\gamma }_q^2+{\sigma }_q^2)b_q^{\ast }{b}_q+{\gamma }_q{\sigma }_q(b_q{b}_{-q}+b_q^{\ast }{b}_{-q}^{\ast })]e^{\mathit{sA}}\hfill \\ \hfill & =\frac{2}{N}\sum _{p,q}(\widehat{V}(p/N)+\widehat{V}((p+q)/N))[({\gamma }_q^2+{\sigma }_q^2)b_q^{\ast }{b}_q+{\gamma }_q{\sigma }_q(b_q{b}_{-q}+b_q^{\ast }{b}_{-q}^{\ast })]+{\mathcal{E}}_1\hfill \end{array}\end{array}$$

where, for any $\epsilon >0$ small enough,

$$\begin{array}{c}\hfill \pm {\mathcal{E}}_1⩽\epsilon {\mathcal{N}}_{+}+\frac{C}{\epsilon N}{({\mathcal{N}}_{+}+1)}^2.\end{array}$$

Using (5.17) and proceeding as in the proof of (5.8) (applying Lemma 5.1 and using that the coefficients $O_p$ in (5.16) satisfy (5.17)), we obtain that, for any $\epsilon >0$,

$$\begin{array}{c}\hfill \pm e^{-sA}\mathcal{B}{e}^{\mathit{sA}}⩽\epsilon \mathcal{K}+\frac{C}{\epsilon N}({\mathcal{H}}_N+{\mathcal{N}}_{+}^2+1).\end{array}$$

Moreover, proceeding as in Prop. 5.4 (we can apply Lemma 5.3, because the coefficients $D_{r,p,q}$ in (5.18) satisfy (5.10)), we obtain that, for any $\epsilon >0$,

$$\begin{array}{c}\hfill \pm e^{-sA}\mathcal{D}{e}^{\mathit{sA}}⩽\epsilon \mathcal{K}+\frac{C}{\epsilon N}({\mathcal{H}}_N+1){({\mathcal{N}}_{+}+1)}^4.\end{array}$$

From Prop. 2.5, we also get

$$\begin{array}{c}\hfill |⟨\xi ,e^{-sA}{\mathcal{E}}_{[\mathcal{C},A]}{e}^{\mathit{sA}}\xi ⟩|⩽\frac{C}{\sqrt{N}}\Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert \Vert {({\mathcal{H}}_N+1)}^{1/2}{({\mathcal{N}}_{+}+1)}^{3/2}\xi \Vert +\frac{C}{N}⟨\xi ,({\mathcal{H}}_N+1){({\mathcal{N}}_{+}+1)}^3\xi ⟩.\end{array}$$

Inserting in (5.33), we obtain (5.32). $\square$

Control of ${\mathcal{H}}_N=\mathcal{K}+{\mathcal{V}}_N$

Finally, we conjugate the Hamiltonian ${\mathcal{H}}_N$. Besides Lemma 4.1, we will also need estimates for the the commutators of the terms ${[\mathcal{K},A]}_2,{[{\mathcal{V}}_N,A]}_2$, defined in (4.3) and, respectively, (4.5), with A.

Lemma 5.7

Let A be defined as in (2.32) and ${[\mathcal{K},A]}_2$, ${[{\mathcal{V}}_N,A]}_2$ be defined as in Lemma 4.1. Then

$$\begin{array}{c}\hfill \pm [{[\mathcal{K},A]}_2,A]+\mathrm{h}.\mathrm{c}.⩽\frac{C{(logN)}^{1/2}}{N}(\mathcal{K}+1){({\mathcal{N}}_{+}+1)}^2.\end{array}$$ 5.34

Moreover,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [{[{\mathcal{V}}_N,A]}_2,A]+\mathrm{h}.\mathrm{c}.=& \frac{2}{N^2}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}[(\widehat{V}(·/N)\ast \eta {)}_p+(\widehat{V}(·/N)\ast \eta {)}_{p+q}]\hfill \\ & \times {\eta }_p{\sigma }_q\frac{2{\eta }_{q+p}{(q+p)}^2-2{\sigma }_q(p·q)}{p^2+q^2+{(p+q)}^2}+{\mathcal{E}}_{[{[{\mathcal{V}}_N,A]}_2,A]}\hfill \end{array}\end{array}$$ 5.35

with

$$\begin{array}{c}\hfill \pm {\mathcal{E}}_{[{[{\mathcal{V}}_N,A]}_2,A]}⩽\frac{C}{N}(\mathcal{K}+1){({\mathcal{N}}_{+}+1)}^2.\end{array}$$

Proof

We start by proving (5.34). Recalling the definitions (2.32) and (4.3) of A and ${[\mathcal{K},A]}_2$, we can split

$$\begin{array}{c}\hfill [{[\mathcal{K},A]}_2,A]+\text{h.c.}=\sum _{j=1}^4{T}_j\end{array}$$

with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill T_1=& \frac{2}{N}\sum _{\begin{array}{c}r,v,p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v,p+q\ne 0\end{array}}r·v{\eta }_r{\gamma }_v{\eta }_p{\gamma }_q[b_{r+v}^{\ast }{b}_{-r}^{\ast }{b}_v,b_{p+q}^{\ast }{b}_{-p}^{\ast }{b}_q]+\mathrm{h}.\mathrm{c}.\hfill \\ \hfill T_2=& \frac{2}{N}\sum _{\begin{array}{c}r,v,p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v,p+q\ne 0\end{array}}r·v{\eta }_r{\gamma }_v{\nu }_{p,q}[b_{r+v}^{\ast }{b}_{-r}^{\ast }{b}_v,b_{p+q}^{\ast }{b}_{-p}^{\ast }{b}_{-q}^{\ast }]+\mathrm{h}.\mathrm{c}.\hfill \\ \hfill T_3=& \frac{2}{N}\sum _{\begin{array}{c}r,v,p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v,p+q\ne 0\end{array}}r·v{\eta }_r{\gamma }_v{\eta }_p{\gamma }_q[b_v^{\ast }{b}_{r+v}{b}_{-r},b_{p+q}^{\ast }{b}_{-p}^{\ast }{b}_q]+\mathrm{h}.\mathrm{c}.\hfill \\ \hfill T_4=& \frac{2}{N}\sum _{\begin{array}{c}r,v,p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v,p+q\ne 0\end{array}}r·v{\eta }_r{\gamma }_v{\nu }_{p,q}[b_v^{\ast }{b}_{r+v}{b}_{-r},b_{p+q}^{\ast }{b}_{-p}^{\ast }{b}_{-q}^{\ast }]+\mathrm{h}.\mathrm{c}.\hfill \\ \hfill \end{array}\end{array}$$

Up to terms of lower order (due to the fact that the operators $b,b^{\ast }$ do not exactly satisfy canonical commutation relations), the operators $T_1,T_2$ only contain quartic contributions, satisfying (5.34). As an example, consider the term

$$\begin{array}{c}\hfill T_{1,1}=\frac{2}{N}\sum _{r,v,p}r·v{\eta }_r{\gamma }_v{\eta }_v{\gamma }_{v-p}{b}_{r+v}^{\ast }{b}_{-r}^{\ast }{b}_{-p}^{\ast }{b}_{v-p}+\mathrm{h}.\mathrm{c}.\end{array}$$

contributing to $T_1$, which can be bounded by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,T_{1,1}\xi ⟩|& ⩽\frac{C}{N}[\sum _{r,v,p}{(r+v)}^2{\Vert b_{r+v}{b}_{-r}{b}_{-p}{({\mathcal{N}}_{+}+1)}^{-1/2}\xi \Vert }^2{]}^{1/2}\hfill \\ \hfill & \times [\sum _{r,v,p}\frac{{(r·v)}^2}{{(r+v)}^2}{\eta }_r^2{\eta }_v^2{\Vert b_{v-p}{({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2{]}^{1/2}\hfill \\ \hfill & ⩽\frac{C{(logN)}^{1/2}}{N}\Vert {\mathcal{K}}^{1/2}{({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert ({\mathcal{N}}_{+}+1)\xi \Vert \hfill \\ \hfill & ⩽\frac{C{(logN)}^{1/2}}{N}⟨\xi ,(\mathcal{K}+1)({\mathcal{N}}_{+}+1)\xi ⟩\hfill \end{array}\end{array}$$ 5.36

where we estimated

$$\begin{array}{c}\hfill \sum _{r,v}\frac{{(r·v)}^2}{{(r+v)}^2}{\eta }_r^2{\eta }_v^2⩽C\sum _{r,v}\frac{1}{{(r+v)}^2}\frac{1}{v^2}|{\eta }_r|⩽C\sum _r\frac{1}{r}|{\eta }_r|⩽ClogN\end{array}$$

as can be proven separating $|r|⩽N$ and $|r|>N$ (in the second region, we can apply the last bound in (2.20)). All other contributions to $T_1,T_2$ can be handled similarly. Let us focus on the other terms. With (2.2), we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill T_3=& \frac{4}{N}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}r·v{\eta }_r{\eta }_{r+v}{\gamma }_v^2{b}_v^{\ast }{b}_v\hfill \\ & +\frac{2}{N}\sum _{\begin{array}{c}r,v,p\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v,r+v-p\ne 0\end{array}}r·v{\eta }_r({\eta }_{r+v-p}+{\eta }_{r+v}){\gamma }_v{\gamma }_p{b}_v^{\ast }{b}_{p-r-v}^{\ast }{b}_{-r}{b}_p+\mathrm{h}.\mathrm{c}.\hfill \\ & +\frac{2}{N}\sum _{\begin{array}{c}r,v,p\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v,r-p\ne 0\end{array}}r·v{\eta }_r({\eta }_{r+p}+{\eta }_r){\gamma }_v{\gamma }_p{b}_v^{\ast }{b}_{p+r}^{\ast }{b}_{r+v}{b}_p+\mathrm{h}.\mathrm{c}.\hfill \\ & -\frac{2}{N}\sum _{\begin{array}{c}r,v,p\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v,p+r\ne 0\end{array}}r·v{\eta }_r{\eta }_p{\gamma }_v^2{b}_{p+q}^{\ast }{b}_{-p}^{\ast }{b}_{r+v}{b}_{-r}+\mathrm{h}.\mathrm{c}.+{\tilde{\mathcal{E}}}_{T_3}\hfill \end{array}\end{array}$$ 5.37

with

$$\begin{array}{c}\hfill \pm {\tilde{\mathcal{E}}}_{T_3}⩽\frac{C}{N}(\mathcal{K}+1){({\mathcal{N}}_{+}+1)}^2.\end{array}$$ 5.38

Here, and in the rest of this proof, we will denote by ${\tilde{\mathcal{E}}}_{T_j}$ contributions due to the fact that the commutation relations (2.2) are not exactly canonical. All these contributions satisfy an estimate like (5.38) and are therefore negligible. All quartic terms on the r.h.s. of (5.37) are small. As an example, we can bound

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |\frac{2}{N}\sum _{r,v,p}& r·v{\eta }_r^2{\gamma }_v{\gamma }_p⟨\xi ,b_v^{\ast }{b}_{p+r}^{\ast }{b}_{r+v}{b}_p\xi ⟩|\hfill \\ \hfill ⩽& \frac{C}{N}(\sum _{r,v,p}\frac{r^2{v}^2}{{|r+v|}^2}{\eta }_r^2\Vert b_v{b}_{p+r}\xi \Vert {)}^{1/2}(\sum _{r,v,p}{\eta }_r^2{|r+v|}^2{\Vert b_{r+v}{b}_p\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill ⩽& \frac{C}{N}⟨\xi ,(\mathcal{K}+1)({\mathcal{N}}_{+}+1)\xi ⟩.\hfill \end{array}\end{array}$$ 5.39

All other quartic terms are bounded similarly. To estimate the quadratic term in the first line of (5.37), we observe that, with the change of variables $-r-v=r^{\prime }$ and $v=v^{\prime }$,

$$\begin{array}{c}\hfill \frac{4}{N}\sum _{r,v}r·v{\eta }_r{\eta }_{r+v}{\gamma }_v^2{b}_v^{\ast }{b}_v=-\frac{2}{N}\sum _{r,v}{v}^2{\eta }_r{\eta }_{r+v}{\gamma }_v^2{b}_v^{\ast }{b}_v,\end{array}$$ 5.40

which is clearly controlled by $\mathcal{K}/N$. Thus,

$$\begin{array}{c}\hfill |⟨\xi ,T_3\xi ⟩|⩽\frac{C{(logN)}^{1/2}}{N}\langle \xi ,(\mathcal{K}+1){({\mathcal{N}}_{+}+1)}^2\xi \rangle .\end{array}$$

Next, let us consider the term $T_4$. With (2.2), we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill T_4=& \frac{2}{N}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}r·v{\eta }_r{\gamma }_v({\nu }_{-r-v,r}+{\nu }_{r,-r-v}+{\nu }_{-r-v,v}+{\nu }_{v,-r-v}+{\nu }_{r,v}+{\nu }_{v,r})b_v^{\ast }{b}_{-v}^{\ast }+\mathrm{h}.\mathrm{c}.\hfill \\ & +\frac{2}{N}\sum _{\begin{array}{c}r,v,p\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v,r+p\ne 0\end{array}}r·v{\eta }_r{\gamma }_v({\nu }_{p,-r-p}+{\nu }_{r,p}+{\nu }_{p,r})b_v^{\ast }{b}_{r+p}^{\ast }{b}_{-p}^{\ast }{b}_{r+v}+\mathrm{h}.\mathrm{c}.\hfill \\ & +\frac{2}{N}\sum _{\begin{array}{c}r,v,p\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v,p-r-v\ne 0\end{array}}r·v{\eta }_r{\gamma }_v({\nu }_{p,r+v-p}+{\nu }_{-r-v,p}+{\nu }_{p,-r-v})b_v^{\ast }{b}_{-p}^{\ast }{b}_{p-r-v}^{\ast }{b}_{-r}+\mathrm{h}.\mathrm{c}.\hfill \\ & +{\tilde{\mathcal{E}}}_{T_4},\hfill \end{array}\end{array}$$

with $\pm {\tilde{\mathcal{E}}}_{T_4}⩽C{N}^{-1}(\mathcal{K}+1){({\mathcal{N}}_{+}+1)}^2$. All quartic terms can be bounded similarly as (5.39). For the first two quadratic terms on the first line (the ones proportional to ${\nu }_{-r-v,r}$ and ${\nu }_{r,-r-v}$), we notice that, for any $\epsilon >0$, ${\sum }_r{|r|}^{-3}{|r+v|}^{-2}⩽C{|v|}^{-2+\epsilon }$ and therefore, choosing $\epsilon <1/2$,

$$\begin{array}{c}\hfill \begin{array}{cc}& |\frac{2}{N}\sum _{r,v}r·v{\eta }_r{\gamma }_v({\nu }_{-r-v,r}+{\nu }_{r,-r-v})⟨\xi ,b_v^{\ast }{b}_{-v}^{\ast }\xi ⟩|\hfill \\ \hfill & ⩽\frac{C}{N}\sum _v{|v|}^{-1+\epsilon }\Vert b_v\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert ⩽\frac{C}{N}⟨\xi ,(\mathcal{K}+1)\xi ⟩.\hfill \end{array}\end{array}$$ 5.41

Using Eq. (5.22), we rewrite the rest of the quadratic terms as

$$\begin{array}{c}\hfill \begin{array}{cc}& \frac{2}{N}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}r·v{\eta }_r{\gamma }_v({\nu }_{-r-v,v}+{\nu }_{v,-r-v}+{\nu }_{r,v}+{\nu }_{v,r})b_v^{\ast }{b}_{-v}^{\ast }+\mathrm{h}.\mathrm{c}.\hfill \\ \hfill =& \frac{2}{N}\sum _{\begin{array}{c}r,v\end{array}}r·v{\eta }_r{\gamma }_v({\eta }_r+{\eta }_{r+v}){\sigma }_v{b}_v^{\ast }{b}_{-v}^{\ast }\hfill \\ & -\frac{4}{N}\sum _{\begin{array}{c}r,v\end{array}}\frac{{\eta }_r^2{\sigma }_v{(r·v)}^2}{r^2+v^2+{|r+v|}^2}{\gamma }_v{b}_v^{\ast }{b}_{-v}^{\ast }+\frac{4}{N}\sum _{\begin{array}{c}r,v\end{array}}\frac{{\eta }_r{\eta }_{r+v}{\sigma }_v((r+v)·v)(r·v)}{r^2+v^2+{|r+v|}^2}{\gamma }_v{b}_v^{\ast }{b}_{-v}^{\ast }\hfill \\ & -\frac{4}{N}\sum _{\begin{array}{c}r,v\end{array}}\frac{(r·v)v^2}{r^2+v^2+{|r+v|}^2}{\eta }_r(({\eta }_r+{\eta }_{r+v})({\sigma }_v-{\eta }_v)\hfill \\ & +{\eta }_v({\sigma }_{r+v}-{\eta }_{r+v})+{\eta }_v({\sigma }_r-{\eta }_r)){\gamma }_v{b}_v^{\ast }{b}_{-v}^{\ast }+\mathrm{h}.\mathrm{c}.\hfill \end{array}\end{array}$$

The terms in the last three lines can all be bounded, first summing over r and then proceeding similarly as in (5.41), by $C{N}^{-1}(\mathcal{K}+1)$. As for the first term on the r.h.s., the contribution proportional to $r·v{\eta }_r^2{\gamma }_v{\sigma }_v$ vanishes, as can be seen replacing $r\to -r$. Moreover, switching $r\to -r-v$,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{1}{N}\sum _{r,v}r·v{\eta }_r{\eta }_{r+v}{\gamma }_v{\sigma }_v(b_v^{\ast }{b}_{-v}^{\ast }+\text{h.c.})& =-\frac{1}{N}\sum _{r,v}(r+v)·v{\eta }_r{\eta }_{r+v}{\gamma }_v{\sigma }_v(b_v^{\ast }{b}_{-v}^{\ast }+\text{h.c.})\hfill \\ \hfill & =-\frac{1}{2N}\sum _{r,v}{v}^2{\eta }_r{\eta }_{r+v}{\gamma }_v{\sigma }_v(b_v^{\ast }{b}_{-v}^{\ast }+\text{h.c.})\hfill \end{array}\end{array}$$

With ${\sum }_r|{\eta }_r||{\eta }_{r+v}{|⩽C|v|}^{-1}$, we can proceed as in (5.41) to prove that also this contribution is bounded by $C{N}^{-1}(\mathcal{K}+1)$. We conclude that

$$\begin{array}{c}\hfill \pm T_4⩽\frac{C}{N}(\mathcal{K}+1){({\mathcal{N}}_{+}+1)}^2.\end{array}$$

Next, we show (5.35). We write

$$[{[{\mathcal{V}}_N,A]}_2,A]+\text{h.c.}=\sum _{j=1}^3{S}_j$$

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill S_1=& \frac{1}{N^{3/2}}\sum _{\begin{array}{c}r,v,p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v,p+q\ne 0\end{array}}{\alpha }_{r,v}{\eta }_p{\gamma }_q[b_{r+v}^{\ast }{b}_{-r}^{\ast }{b}_{-v}^{\ast },b_{p+q}^{\ast }{b}_{-p}^{\ast }{b}_q]+\mathrm{h}.\mathrm{c}.\hfill \\ \hfill S_2=& \frac{1}{N^{3/2}}\sum _{\begin{array}{c}r,v,p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v,p+q\ne 0\end{array}}{\alpha }_{r,v}{\eta }_p{\gamma }_q[b_{r+v}{b}_{-r}{b}_{-v},b_{p+q}^{\ast }{b}_{-p}^{\ast }{b}_q]+\mathrm{h}.\mathrm{c}.\hfill \\ \hfill S_3=& \frac{1}{N^{3/2}}\sum _{\begin{array}{c}r,v,p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v,p+q\ne 0\end{array}}{\alpha }_{r,v}{\nu }_{p,q}[b_{r+v}{b}_{-r}{b}_{-v},b_{p+q}^{\ast }{b}_{-p}^{\ast }{b}_{-q}^{\ast }]+\mathrm{h}.\mathrm{c}.,\hfill \end{array}\end{array}$$

and the coefficient ${\alpha }_{r,v}$ is defined as in (4.8). With (2.2), we find

$$\begin{array}{cc}\hfill S_1=& -\frac{1}{N^{3/2}}\sum _{\begin{array}{c}r,v,p\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v,p+r+v\ne 0\end{array}}{\eta }_p{\gamma }_{r+v}({\alpha }_{r,v}+{\alpha }_{-r-v,v}+{\alpha }_{r,-r-v})b_{p+r+v}^{\ast }{b}_{-p}^{\ast }{b}_{-r}^{\ast }{b}_{-v}^{\ast }+\text{h.c.}+{\tilde{\mathcal{E}}}_{S_1}\hfill \end{array}$$

where $\pm {\tilde{\mathcal{E}}}_{S_1}⩽C{N}^{-1}(\mathcal{K}+1){({\mathcal{N}}_{+}+1)}^2$. Let us bound, using Eq. (4.9), the contribution proportional to ${\alpha }_{r,v}$; the other terms can be controlled analogously.

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |\frac{1}{N^{3/2}}& \sum _{\begin{array}{c}r,v,p\end{array}}{\eta }_p{\gamma }_{r+v}{\alpha }_{r,v}⟨\xi ,b_{p+r+v}^{\ast }{b}_{-p}^{\ast }{b}_{-r}^{\ast }{b}_{-v}^{\ast }\xi ⟩|\hfill \\ \hfill & ⩽\frac{C}{N^{3/2}}(\sum _{\begin{array}{c}r,v,p\end{array}}|{\eta }_p{|}^2\frac{|{\alpha }_{r,v}{|}^2}{r^2}{)}^{1/2}(\sum _{\begin{array}{c}r,v,p\end{array}}{r}^2\Vert b_p{b}_r{b}_v{({\mathcal{N}}_{+}+1)}^{-1/2}{\xi \Vert }^2{)}^{1/2}\Vert ({\mathcal{N}}_{+}+1)\xi \Vert \hfill \\ \hfill & ⩽\frac{C}{N}⟨\xi ,(\mathcal{K}+1)({\mathcal{N}}_{+}+1)\xi ⟩.\hfill \end{array}\end{array}$$ 5.42

We conclude that $\pm S_1⩽C{N}^{-1}(\mathcal{K}+1){({\mathcal{N}}_{+}+1)}^2$.

Next, we consider the term $S_2$. Using (2.2), we find

$$\begin{array}{cc}\hfill S_2=& \frac{1}{N^{3/2}}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}({\eta }_{r+v}+{\eta }_v){\gamma }_r({\alpha }_{r,v}+{\alpha }_{-r-v,v}+{\alpha }_{v,r})b_r^{\ast }{b}_{-r}^{\ast }+\text{h.c.}\hfill \\ \hfill & +\frac{1}{N^{3/2}}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}({\eta }_p+{\eta }_v){\gamma }_{v+p}({\alpha }_{r,v}+{\alpha }_{v,r}+{\alpha }_{r,-r-v})b_{-p}^{\ast }{b}_{-r}{b}_{r+v}{b}_{-v-p}+\text{h.c.}\hfill \\ \hfill & +{\tilde{\mathcal{E}}}_{S_2},\hfill \end{array}$$

where $\pm {\tilde{\mathcal{E}}}_{S_2}⩽C{N}^{-1}(\mathcal{K}+1){({\mathcal{N}}_{+}+1)}^2$. The quartic terms can be controlled in a similar way to $S_1$, using the second bound in (4.9). Also the first two quadratic contributions, proportional to ${\alpha }_{r,v}$ and ${\alpha }_{-r-v,v}$, can be estimated using the second bound in (4.9). As for the last quadratic term, the one proportional to ${\alpha }_{v,r}$, it can be handled using the third bound of (4.9). In fact,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |\frac{1}{N^{3/2}}& \sum _{r,v}({\eta }_{r+v}+{\eta }_v){\gamma }_v{\alpha }_{v,r}⟨\xi ,b_r^{\ast }{b}_{-r}^{\ast }\xi ⟩|\hfill \\ \hfill & ⩽\frac{C}{N^2}\sum _{r,v}\frac{|{\eta }_{r+v}|+|{\eta }_v|}{|r|}\Vert b_r\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert ⩽\frac{C}{N}⟨\xi ,(\mathcal{K}+1)\xi ⟩.\hfill \end{array}\end{array}$$ 5.43

We conclude that $\pm S_2⩽C{N}^{-1}(\mathcal{K}+1){({\mathcal{N}}_{+}+1)}^2$.

Finally, we consider $S_3$. We decompose $S_3=S_3^{(0)}+S_3^{(2)}+S_3^{(4)}+{\tilde{\mathcal{E}}}_{S_3}$, where $\pm {\tilde{\mathcal{E}}}_{S_3}⩽C{N}^{-1}(\mathcal{K}+1){({\mathcal{N}}_{+}+1)}^2$,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill S_3^{(0)}=& \frac{2}{N^{3/2}}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}{\alpha }_{r,v}({\nu }_{r,v}+{\nu }_{r,-r-v}+{\nu }_{v,r}+{\nu }_{v,-r-v}+{\nu }_{-r-v,r}+{\nu }_{-r-v,v}),\hfill \\ \hfill S_3^{(2)}=& \frac{2}{N^{3/2}}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}({\alpha }_{r,v}+2{\alpha }_{v,r})({\nu }_{r,v}+{\nu }_{r,-r-v}+{\nu }_{v,r}+{\nu }_{v,-r-v}+{\nu }_{-r-v,r}+{\nu }_{-r-v,v})b_r^{\ast }{b}_r\hfill \end{array}\end{array}$$

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill S_3^{(4)}=& \frac{1}{N^{3/2}}\sum _{\begin{array}{c}r,v,p\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v,r+p\ne 0\end{array}}({\alpha }_{r,v}+{\alpha }_{v,r}+{\alpha }_{-r-v,v})({\nu }_{p,-r-p}+{\nu }_{p,r}+{\nu }_{r,p})b_{p+r}^{\ast }{b}_{-p}^{\ast }{b}_{r+v}{b}_{-v}+\text{h.c.}\hfill \end{array}\end{array}$$

Using (4.9) (in particular, the bound for the first two terms on the l.h.s. of (4.9)), as well as $|{\nu }_{r,p}|⩽C|{\eta }_r{||p|}^{-2}$, $S_3^{(4)}$ can be estimated by $C{N}^{-1}(\mathcal{K}+1)({\mathcal{N}}_{+}+1)$.

Also the quadratic terms are negligible. Indeed, the terms proportional to ${\alpha }_{r,v}$ are easily bounded by $N^{-3/2}({\mathcal{N}}_{+}+1)$, using the first estimate in (4.9). The quadratic terms proportional to ${\alpha }_{v,r}$, on the other hand, can be controlled using the third bound in (4.9) and $|{\nu }_{r,p}|⩽C|{\eta }_r||{\sigma }_p|$, similarly to (5.43). Finally, we consider the constant term in $S_3^{(0)}$. From the definition (2.33) of ${\nu }_{r,v}$, we arrive at

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill S_3^{(0)}=& \frac{2}{N^{3/2}}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}{\alpha }_{r,v}{\sigma }_v({\eta }_r+{\eta }_{r+v})\hfill \\ & +\frac{2}{N^{3/2}}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}{\alpha }_{r,v}[\frac{{\sigma }_v(r·v)({\eta }_{r+v}-{\eta }_r)}{r^2+v^2+{(r+v)}^2}+\frac{v^2{\eta }_v{\sigma }_{r+v}}{r^2+v^2+{(r+v)}^2}]\hfill \\ & +\frac{2}{N^{3/2}}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}{\alpha }_{r,v}\frac{v^2}{r^2+v^2+{(r+v)}^2}[{\eta }_v({\sigma }_r-{\eta }_r)-{\eta }_r({\sigma }_v-{\eta }_v)]\hfill \\ & +\frac{2}{N^{3/2}}\sum _{\begin{array}{c}r,v\in {\mathrm{\Lambda }}_{+}^{\ast }\\ r+v\ne 0\end{array}}{\alpha }_{r,v}({\nu }_{r,-r-v}+{\nu }_{-r-v,r})\hfill \end{array}\end{array}$$ 5.44

With (4.9), $|{\nu }_{r,p}|⩽C|{\eta }_r||{\sigma }_p|$ and $|{\sigma }_r-{\eta }_r|⩽C|{\tau }_r|$, it is easy to show that the terms in the third and fourth lines are small, bounded by $C{N}^{-1}$. The terms on the second line are also negligible, since

$$\begin{array}{cc}& |\frac{2}{N^{3/2}}\sum _{r,v}{\alpha }_{r,v}\frac{{\sigma }_v({\eta }_{r+v}-{\eta }_r)(r·v)+v^2{\eta }_v{\sigma }_{r+v}}{r^2+v^2+{(r+v)}^2}|\hfill \\ \hfill & ⩽\frac{C}{N^2}\sum _{r,v}(\frac{|{\eta }_{r+v}||{\sigma }_v|}{|r|}+\frac{|{\eta }_r||r|}{{(r+v)}^2{v}^2}+\frac{|{\eta }_v||v|}{{(r+v)}^2{r}^2})⩽\frac{C}{N}.\hfill \end{array}$$

Recalling the definition (4.8) of ${\alpha }_{r,v}$ to rewrite the first term on the r.h.s. of (5.44), we obtain (5.35). $\square$

Combining Lemma 4.1 with Lemma 5.7, we can now compute the action of A on the Hamiltonian ${\mathcal{H}}_N$.

Proposition 5.8

Let A be defined as in (2.32), and ${\mathcal{H}}_N=\mathcal{K}+{\mathcal{V}}_N$. Let ${\mathcal{C}}_{{\mathcal{G}}_N}$ be defined in (2.28), ${[\mathcal{K},A]}_2$ and ${[{\mathcal{V}}_N,A]}_2$ be defined in Lemma 4.1. Then, under the same assumptions as in Theorem 2.6, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^{-A}{\mathcal{H}}_N{e}^A=& {\mathcal{H}}_N-{\mathcal{C}}_{{\mathcal{G}}_N}\hfill \\ & -\frac{1}{N}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }:p+q\ne 0}(\widehat{V}(p/N)+\widehat{V}((p+q)/N)){\eta }_p\hfill \\ & \times [{\sigma }_q^2+({\gamma }_q^2+{\sigma }_q^2)b_q^{\ast }{b}_q+{\gamma }_q{\sigma }_q(b_q{b}_{-q}+b_q^{\ast }{b}_{-q}^{\ast })]\hfill \\ & +\frac{1}{N}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}[(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p+(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_{p+q}]\hfill \\ & \times {\eta }_p{\sigma }_q\frac{2{\eta }_{q+p}{(q+p)}^2-2{\sigma }_q(p·q)}{p^2+q^2+{(p+q)}^2}+{\mathcal{E}}_{{\mathcal{H}}_N}\hfill \end{array}\end{array}$$ 5.45

where, for every $\epsilon >0$ (s.t. $\epsilon >C(logN)/N$)

$$\begin{array}{c}\hfill \pm {\mathcal{E}}_{{\mathcal{H}}_N}⩽\epsilon \mathcal{K}+\frac{C}{N}\left[{(logN)}^{1/2},+,{\epsilon }^{-1}\right]({\mathcal{H}}_N+1){({\mathcal{N}}_{+}+1)}^4.\end{array}$$

Proof

From Lemma 4.1, we find, using the scattering equation (2.12) to combine ${[\mathcal{K},A]}_1$ and ${[{\mathcal{V}}_N,A]}_1$,

$$\begin{array}{c}\hfill [{\mathcal{H}}_N,A]=-{\mathcal{C}}_{{\mathcal{G}}_N}+\{{[\mathcal{K},A]}_2+{[{\mathcal{V}}_N,A]}_2+\text{h.c.}\}+{\mathcal{E}}_{[{\mathcal{H}}_N,A]}\end{array}$$

where

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_{[{\mathcal{H}}_N,A]}\xi ⟩|⩽\frac{C}{\sqrt{N}}\Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert \Vert ({\mathcal{N}}_{+}+1)\xi \Vert +\frac{C}{N}⟨\xi ,({\mathcal{H}}_N+1){({\mathcal{N}}_{+}+1)}^3\xi ⟩\end{array}$$

Here we used the estimate $\Vert {\widehat{\chi }}_{\ell }\ast {\widehat{f}}_{N,\ell }{\Vert }_2=\Vert {\chi }_{\ell }{f}_{N,\ell }{\Vert }_2⩽{\Vert {\chi }_{\ell }\Vert }_2⩽C$ (for fixed $\ell >0$, independent of N), to absorb into the error ${\mathcal{E}}_{[{\mathcal{H}}_N,A]}$ the contribution arising from the r.h.s. of the scattering equation (2.12), when combining ${[\mathcal{K},A]}_1$ and ${[{\mathcal{V}}_N,A]}_1$. Thus, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^{-A}{\mathcal{H}}_N{e}^A=& {\mathcal{H}}_N+{\int }_0^{1}ds{e}^{-sA}[{\mathcal{H}}_N,A]e^{\mathit{sA}}\hfill \\ \hfill =& {\mathcal{H}}_N-{\int }_0^1{e}^{-sA}{\mathcal{C}}_{{\mathcal{G}}_N}{e}^{\mathit{sA}}ds+{\int }_0^1{e}^{-sA}\left\{{[\mathcal{K},A]}_2,+,{[{\mathcal{V}}_N,A]}_2,+,\text{h.c.}\right\}{e}^{\mathit{sA}}ds\hfill \\ \hfill & +{\int }_0^1{e}^{-sA}{\mathcal{E}}_{[{\mathcal{H}}_N,A]}{e}^{\mathit{sA}}ds\hfill \end{array}\end{array}$$ 5.46

From Prop. 2.5, we find

$$\begin{array}{c}\hfill |⟨\xi ,e^{-sA}{\mathcal{E}}_{[{\mathcal{H}}_N,A]}{e}^{\mathit{sA}}\xi ⟩|⩽\frac{1}{\sqrt{N}}\Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert \Vert ({\mathcal{N}}_{+}+1)\xi \Vert +\frac{C}{N}⟨\xi ,({\mathcal{H}}_N+1){({\mathcal{N}}_{+}+1)}^4\xi ⟩.\end{array}$$

With Prop. 5.6 we obtain, after integration over s,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\int }_0^1{e}^{-sA}& {\mathcal{C}}_{{\mathcal{G}}_N}{e}^{\mathit{sA}}ds\hfill \\ \hfill =& {\mathcal{C}}_{{\mathcal{G}}_N}+\frac{1}{N}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }:p+q\ne 0}(\widehat{V}(p/N)+\widehat{V}((p+q)/N)){\eta }_p\hfill \\ & \times [{\sigma }_q^2+({\gamma }_q^2+{\sigma }_q^2)b_q^{\ast }{b}_q+{\gamma }_q{\sigma }_q(b_q{b}_{-q}+b_q^{\ast }{b}_{-q}^{\ast })]\hfill \\ & +\frac{1}{N}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}[\widehat{V}(p/N)+\widehat{V}((p+q)/N)]{\eta }_p{\sigma }_q\frac{2{\eta }_{q+p}{(q+p)}^2-2{\sigma }_q(p·q)}{p^2+q^2+{(p+q)}^2}\hfill \\ & +{\mathcal{E}}_{{\mathcal{C}}_{{\mathcal{G}}_N}},\hfill \end{array}\end{array}$$ 5.47

where, for every $\epsilon >0$,

$$\pm {\mathcal{E}}_{{\mathcal{C}}_{{\mathcal{G}}_N}}⩽\epsilon \mathcal{K}+\frac{C}{\epsilon N}({\mathcal{H}}_N+1){({\mathcal{N}}_{+}+1)}^4.$$

Finally, we compute

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^{-sA}& \left\{{[\mathcal{K},A]}_2,+,{[{\mathcal{V}}_N,A]}_2,+,\text{h.c.}\right\}{e}^{\mathit{sA}}\hfill \\ \hfill & ={[\mathcal{K},A]}_2+{[{\mathcal{V}}_N,A]}_2+\text{h.c.}+{\int }_0^s{e}^{-tA}\left\{[,{[\mathcal{K},A]}_2,,,A,],+,[,{[{\mathcal{V}}_N,A]}_2,,,A,],+,\text{h.c.}\right\}{e}^{\mathit{tA}}dt.\hfill \end{array}\end{array}$$

We estimate the terms in the integral over t using Lemma 5.7. We conclude that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\int }_0^1& e^{-sA}\left\{{[\mathcal{K},A]}_2,+,{[{\mathcal{V}}_N,A]}_2,+,\text{h.c.}\right\}{e}^{\mathit{sA}}ds\hfill \\ \hfill =& {[\mathcal{K},A]}_2+{[{\mathcal{V}}_N,A]}_2+\text{h.c.}\hfill \\ & +\frac{1}{N^2}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}[(\widehat{V}(·/N)\ast \eta {)}_p+(\widehat{V}(·/N)\ast \eta {)}_{p+q}]{\eta }_p{\sigma }_q\frac{2{\eta }_{q+p}{(q+p)}^2-2{\sigma }_q(p·q)}{p^2+q^2+{(p+q)}^2}\hfill \\ & +{\mathcal{E}}_{[[{\mathcal{H}}_N,A],A]}\hfill \end{array}\end{array}$$ 5.48

where

$$\begin{array}{c}\hfill \pm {\mathcal{E}}_{[[{\mathcal{H}}_N,A],A]}⩽\frac{C{(logN)}^{1/2}}{N}(\mathcal{K}+1){({\mathcal{N}}_{+}+1)}^2.\end{array}$$

Furthermore, from (4.15) we have

$$\begin{array}{cc}\hfill |⟨\xi ,& [{\mathcal{V}}_N,A{]}_2\xi ⟩|⩽\frac{C}{\sqrt{N}}\Vert {\mathcal{K}}^{1/2}\xi \Vert \Vert ({\mathcal{N}}_{+}+1)\xi \Vert .\hfill \end{array}$$

As for ${[\mathcal{K},A]}_2$, from (4.14) we also have

$$\begin{array}{c}\hfill |⟨\xi ,{[\mathcal{K},A]}_2\xi ⟩|⩽\frac{C}{\sqrt{N}}\Vert {\mathcal{K}}^{1/2}{\mathcal{N}}_{+}^{1/2}\xi \Vert \Vert {\mathcal{K}}^{1/2}\xi \Vert .\end{array}$$

Combining (5.48) with (5.47), and (5.46), and using the definition of ${\eta }_p=-N{\delta }_{p,0}+N{\widehat{f}}_{N,\ell }(p)$ we arrive at (5.45). $\square$

Proof of Theorem 2.6

We start from (2.30) in Theorem 2.4. We use Prop. 2.5 to estimate

$$\begin{array}{c}\hfill \pm e^{-A}{\mathcal{E}}_{{\mathcal{G}}_N}{e}^A⩽\epsilon {\mathcal{N}}_{+}+\frac{C}{\epsilon N}({\mathcal{H}}_N+1){({\mathcal{N}}_{+}+1)}^3\end{array}$$

Combining the expansion (2.30) for ${\mathcal{G}}_N$ with Prop. 2.5 and with the results of Prop. 5.2, Prop. 5.4, Prop. 5.6 and Prop. 5.8, we conclude that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{J}}_N& =C_{{\mathcal{G}}_N}+\frac{1}{N}\sum _{p,q}(\widehat{V}(p/N)+\widehat{V}((q+p)/N)){\eta }_p{\sigma }_q^2\hfill \\ & +\frac{1}{N}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}[(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p+(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_{p+q}]\hfill \\ & \times {\eta }_p{\sigma }_q\frac{2{\eta }_{q+p}{(q+p)}^2-2{\sigma }_q(p·q)}{p^2+q^2+{(p+q)}^2}\hfill \\ & +\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}\sqrt{{|p|}^4+2{(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell })}_p{p}^2}{a}_p^{\ast }{a}_p+{\mathcal{V}}_N+{\tilde{\mathcal{E}}}_{{\mathcal{J}}_N}\hfill \end{array}\end{array}$$ 5.49

where, for every $\epsilon >0$ (s.t. $\epsilon >C(logN)/N$), we have

$$\begin{array}{c}\hfill \pm {\tilde{\mathcal{E}}}_{{\mathcal{J}}_N}⩽\epsilon \mathcal{K}+\frac{C}{N}\left[{(logN)}^{1/2},+,{\epsilon }^{-1}\right]({\mathcal{H}}_N+1){({\mathcal{N}}_{+}+1)}^4.\end{array}$$

Notice here that the quadratic terms appearing on the second line of (2.30) cancel exactly the quadratic contribution arising from conjugation of the cubic term ${\mathcal{C}}_{{\mathcal{G}}_N}$ (as determined in Prop. 5.6) and of the Hamiltonian ${\mathcal{H}}_N$ (as determined in Prop. 5.45)1

To complete the proof of Theorem 2.6, we observe, first of all, that the terms on the second line of (5.49) produce, up to smaller errors, the terms on the second line of (2.38). To this end, we consider the difference

$$\begin{array}{cc}& \frac{1}{N}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }\\ p+q\ne 0\end{array}}[(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p+(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_{p+q}]\hfill \\ & [\frac{2{\eta }_{q+p}{\eta }_p({\sigma }_q-{\eta }_q){(q+p)}^2}{p^2+q^2+{(p+q)}^2}-\frac{2{\eta }_p({\sigma }_q^2-{\eta }_q^2)(p·q)}{p^2+q^2+{(p+q)}^2}].\hfill \end{array}$$

where we compare ${\sigma }_q$ and ${\sigma }_q^2$ with ${\eta }_q$ and, respectively, ${\eta }_q^2$. Using $|{\sigma }_q-{\eta }_q{|⩽C|q|}^{-4}$, the first contribution can be shown to be of order $N^{-1}$. Also the second contribution, containing the factor $p·q$, is of order $N^{-1}$; this can be proven proceeding similarly as we did after (5.23), switching $p\to -p$ and using the Lipschitz continuity of $\widehat{V}$ for $|p|⩽N$.

Finally, we claim that the constant terms on the first line on the r.h.s. of (5.49) match, up to negligible errors, the terms on the first line on the r.h.s. of (2.38). To this end, we set (recall (2.27) for the definition of $C_{{\mathcal{G}}_N}$)

$$\begin{array}{cc}\hfill C_{\mathcal{O}(1)}=& C_{{\mathcal{G}}_N}+\frac{1}{N}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}(\widehat{V}(p/N)+\widehat{V}((q+p)/N)){\eta }_p{\sigma }_q^2\hfill \\ \hfill =& \frac{(N-1)}{2}\widehat{V}(0)+\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}[p^2{\sigma }_p^2+\widehat{V}(p/N){\sigma }_p{\gamma }_p+(\widehat{V}(·/N)\ast {\widehat{f}}_N{)}_p{\sigma }_p^2]\hfill \\ \hfill & +\frac{1}{2N}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}((p-q)/N){\sigma }_q{\gamma }_q{\sigma }_p{\gamma }_p+\frac{1}{N}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}(p^2{\eta }_p^2+\frac{1}{2N}{\left(\widehat{V},\left(\frac{·}{N}\right),,\ast ,\eta \right)}_p{\eta }_p)\hfill \end{array}$$

with ${\gamma }_p=cosh({\mu }_p)$, ${\sigma }_p=sinh({\mu }_p)$, and ${\mu }_p={\eta }_p+{\tau }_p$. We now claim that

$$\begin{array}{c}\hfill C_{\mathcal{O}(1)}=4\pi \mathfrak{a}(N-1)+e_{\mathrm{\Lambda }}{\mathfrak{a}}^2-\frac{1}{2}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}[p^2+8\pi \mathfrak{a}-\sqrt{{|p|}^4+16\pi \mathfrak{a}{p}^2}-\frac{{(8\pi \mathfrak{a})}^2}{2p^2}]+\mathcal{O}(N^{-1})\end{array}$$ 5.50

To show (5.50), we consider the following quantities, defined in terms of the kernel $\eta$:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\tilde{C}}_{{\mathcal{J}}_N}=& \frac{N-1}{2}\widehat{V}(0)\hfill \\ \hfill & +\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}[p^{2}sinh{({\eta }_p)}^2+\widehat{V}(p/N)cosh({\eta }_p)sinh({\eta }_p)+(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_{p}sinh{({\eta }_p)}^2]\hfill \\ \hfill & +\frac{1}{2N}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}((p-q)/N)cosh({\eta }_q)sinh({\eta }_q)cosh({\eta }_p)sinh({\eta }_p)\hfill \\ \hfill & +\frac{1}{N}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}[p^2{\eta }_p^2+\frac{1}{2N}(\widehat{V}(·/N)\ast \eta {)}_p{\eta }_p]\hfill \end{array}\end{array}$$

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\tilde{F}}_p=& p^2(cosh{({\eta }_p)}^2+sinh{({\eta }_p)}^2)+(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p(cosh({\eta }_p)+sinh({\eta }_p){)}^2\hfill \\ \hfill {\tilde{G}}_p=& 2p^{2}cosh({\eta }_p)sinh({\eta }_p)+(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p(cosh({\eta }_p)+sinh({\eta }_p){)}^2.\hfill \end{array}\end{array}$$

In [5, Lemma 5.4 (i)] it was proven that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\tilde{C}}_{{\mathcal{J}}_N}& +\frac{1}{2}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}(-{\tilde{F}}_p+\sqrt{{\tilde{F}}_p^2-{\tilde{G}}_p^2})\hfill \\ \hfill =& 4\pi \mathfrak{a}(N-1)+e_{\mathrm{\Lambda }}{\mathfrak{a}}^2-\frac{1}{2}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}[p^2+8\pi \mathfrak{a}-\sqrt{{|p|}^4+16\pi \mathfrak{a}{p}^2}-\frac{{(8\pi \mathfrak{a})}^2}{2p^2}]+\mathcal{O}(N^{-1}).\hfill \end{array}\end{array}$$ 5.51

In fact, the statement in [5, Lemma 5.4 (i)] contains a larger remainder, of the order $\mathcal{O}(logN/N)$, emerging when controlling the difference

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}[& (\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p+p^2-\sqrt{p^4+2p^2(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p}+\frac{1}{2p^2}(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p^2\hfill \\ \hfill & -(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_0-p^2+\sqrt{p^4+2p^2(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_0}+\frac{1}{2p^2}(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_0^2].\hfill \end{array}\end{array}$$ 5.52

We can however show that this remainder is indeed smaller, of the order $\mathcal{O}(N^{-1})$, consistently with (5.51). Taylor expansion of the square root easily implies that the whole square bracket is bounded, in absolute value, by ${C|p|}^{-4}$. Hence, the sum over momenta $|p|>N$ can be estimated by $C{N}^{-1}$. To treat $|p|⩽N$, we notice that the function

$$\begin{array}{c}\hfill g_p(x)=x-p^2\sqrt{1+2x/p^2}-x^2/(2p^2)\end{array}$$

has derivative

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill g_p^{\prime }(x)=& 1-{(1+2x/p^2)}^{-1/2}-x/p^2\hfill \\ \hfill \end{array}\end{array}$$

satisfying $|g_p^{\prime }(x)|⩽C{x}^2/{|p|}^4$ (as it follows again by expansion of the square root). This implies that $|g_p(x)-g_p{(y)|⩽C|x-y|/|p|}^4$, for all x, y varying in a bounded interval. Estimating

$$\begin{array}{c}\hfill |{(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell })}_p-{(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell })}_0|⩽|N^3{\int }_{\mathrm{\Lambda }}dxV(Nx)f_{N,\ell }(x)(e^{-ip·x}-1)|⩽C{p}^2/N^2\end{array}$$ 5.53

since ${\int }_{\mathrm{\Lambda }}xV(x)f_{N,\ell }(x)dx=0$ by symmetry, we conclude that the bracket in (5.52) is bounded, in absolute value, by $C/p^2{N}^2$. Thus, the sum over all $|p|<N$ can also be estimated by C/N.

From (5.51), it follows that, in order to show (5.50), it is enough to prove that

$$\begin{array}{c}\hfill C_{\mathcal{O}(1)}={\tilde{C}}_{{\mathcal{J}}_N}+\frac{1}{2}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}[-{\tilde{F}}_p+\sqrt{{\tilde{F}}_p^2-{\tilde{G}}_p^2}]+\mathcal{O}(N^{-1})\end{array}$$ 5.54

To this end, we consider

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_{\mathcal{O}(1)}& -{\tilde{C}}_{{\mathcal{J}}_N}\hfill \\ \hfill =& \sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}[(p^2+(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p)({\sigma }_p^2-sinh{({\eta }_p)}^2)\hfill \\ \hfill & +\widehat{V}(p/N)({\gamma }_p{\sigma }_p-cosh({\eta }_p)sinh({\eta }_p))\hfill \\ \hfill & +\frac{1}{2N}\sum _{q\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}((p-q)/N)({\gamma }_q{\sigma }_q{\gamma }_p{\sigma }_p-cosh({\eta }_p)sinh({\eta }_p)cosh({\eta }_q)sinh({\eta }_q))].\hfill \end{array}\end{array}$$ 5.55

Let us separately manipulate the three main contributions to the sum in the right hand side. Through elementary identities we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\sigma }_p^2-sinh{({\eta }_p)}^2=& \frac{1}{2}sinh(2{\eta }_p)sinh(2{\tau }_p)+{sinh}^2({\tau }_p)cosh(2{\eta }_p)\hfill \\ \hfill =& \frac{1}{2}sinh(2{\eta }_p)sinh(2{\tau }_p)+\frac{1}{2}cosh(2{\eta }_p)(cosh(2{\tau }_p)-1),\hfill \end{array}\end{array}$$

and therefore the first contribution to the right hand side of (5.55) is

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}(& p^2+(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p)({\sigma }_p^2-sinh{({\eta }_p)}^2)\hfill \\ \hfill =& \frac{1}{2}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}(p^2+(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p)[sinh(2{\eta }_p)sinh(2{\tau }_p)+cosh(2{\eta }_p)(cosh(2{\tau }_p)-1)].\hfill \end{array}\end{array}$$ 5.56

In a similar way we rewrite the second contribution as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}& (p/N)({\gamma }_p{\sigma }_p-cosh({\eta }_p)sinh({\eta }_p))\hfill \\ \hfill =& \frac{1}{2}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}(p/N)[sinh(2{\tau }_p)cosh(2{\eta }_p)+sinh(2{\eta }_p)(cosh(2{\tau }_p)-1)].\hfill \end{array}\end{array}$$ 5.57

In order to rearrange the third contribution to the right hand side of (5.55), notice that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill 4{\gamma }_q& {\sigma }_q{\gamma }_p{\sigma }_p\hfill \\ \hfill =& sinh(2{\eta }_q)cosh(2{\tau }_q)sinh(2{\eta }_p)cosh(2{\tau }_p)+sinh(2{\tau }_q)cosh(2{\eta }_q)cosh(2{\eta }_p)sinh(2{\tau }_p)\hfill \\ \hfill & +sinh(2{\eta }_q)cosh(2{\tau }_q)cosh(2{\eta }_p)sinh(2{\tau }_p)+cosh(2{\eta }_q)sinh(2{\tau }_q)sinh(2{\eta }_p)cosh(2{\tau }_p)\hfill \end{array}\end{array}$$

Since $|{\tau }_p{|⩽C|p|}^{-4}$, the second term on the r.h.s. (containing $sinh(2{\tau }_q)sinh(2{\tau }_p)$) yields a contribution bounded by $C{N}^{-1}$ when inserted in the sum over p, q in (5.55). Using the estimate $|sinh(2{\eta }_p)cosh(2{\tau }_p)-2{\eta }_p{|⩽C|p|}^{-6}$ to handle the last two terms (they give the same contribution), decomposing $cosh(2{\tau }_p)=1+(cosh(2{\tau }_p)-1)$ and similarly for $cosh(2{\tau }_q)$ and subtracting the terms $cosh({\eta }_p)sinh({\eta }_p)cosh({\eta }_q)sinh({\eta }_q)=(1/4)sinh(2{\eta }_p)sinh(2{\eta }_q)$, we find (adding also a negligible term proportional to ${\eta }_0$)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{1}{2N}& \sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}\widehat{V}((p-q)/N)({\gamma }_q{\sigma }_q{\gamma }_p{\sigma }_p-cosh({\eta }_p)sinh({\eta }_p)cosh({\eta }_q)sinh({\eta }_q))\hfill \\ \hfill =& \frac{1}{2N}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}(\widehat{V}(·/N)\ast \eta {)}_p[sinh(2{\tau }_p)cosh(2{\eta }_p)+sinh(2{\eta }_p)(cosh(2{\tau }_p)-1)]+\mathcal{O}(N^{-1}).\hfill \end{array}\end{array}$$ 5.58

Plugging (5.56), (5.57), and (5.58) into the right hand side of (5.55), the terms with a minus sign recombine into

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill -\frac{1}{2}(p^2+& (\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p)cosh(2{\eta }_p)-\frac{1}{2}(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_{p}sinh(2{\eta }_p)=-\frac{{\tilde{F}}_p}{2}.\hfill \end{array}\end{array}$$

As already discussed in (3.31), the other terms produce

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{1}{2}(p^2+& (\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p)[sinh(2{\eta }_p)sinh(2{\tau }_p)+cosh(2{\eta }_p)cosh(2{\tau }_p)]\hfill \\ \hfill & +\frac{1}{2}(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p[sinh(2{\tau }_p)cosh(2{\eta }_p)+sinh(2{\eta }_p)cosh(2{\tau }_p)]\hfill \\ \hfill =& \frac{1}{2}\sqrt{p^4+2p^2(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell }{)}_p}=\frac{1}{2}\sqrt{{\tilde{F}}_p-{\tilde{G}}_p}.\hfill \end{array}\end{array}$$

This concludes the proof of (5.54).

To conclude the proof of Theorem 2.6, we observe now that, combining Lemma 2.1 and (5.53), we find

$$\begin{array}{c}\hfill |{(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell })}_p-8\pi \mathfrak{a}|⩽C|p|/N\end{array}$$

This allows us to replace the dispersion of the term on the third line of (5.49) with ${(|p|}^4+16\pi \mathfrak{a}{p}^2{)}^{1/2}$, since

$$\begin{array}{c}\hfill \pm \sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}[\sqrt{{|p|}^4+2{(\widehat{V}(·/N)\ast {\widehat{f}}_{N,\ell })}_p{p}^2}-\sqrt{{|p|}^4+16\pi \mathfrak{a}{p}^2}]a_p^{\ast }{a}_p⩽\frac{C}{N}\mathcal{K}.\end{array}$$

can be absorbed in the error.

Bound for d-Operators

In this section we prove Lemma 3.3, without any assumptions on the smallness of the ${\ell }^2$-norm of the kernel of the Bogoliubov transformation. We start by proving the following Lemma

Lemma A.1

Let $B_{\mu }$ be defined as in (2.24). Then there exists $C>0$ such that the following bounds hold true:

$$\begin{array}{cc}\hfill \Vert {({\mathcal{N}}_{+}+1)}^{n/2}{a}_p{e}^{s{B}_{\mu }}\xi \Vert & ⩽C[\Vert {({\mathcal{N}}_{+}+1)}^{n/2}{a}_p\xi \Vert +|{\mu }_p|\Vert {({\mathcal{N}}_{+}+1)}^{(n+1)/2}\xi \Vert ]\hfill \end{array}$$ A.1

$$\begin{array}{cc}\hfill \Vert {({\mathcal{N}}_{+}+1)}^{n/2}{\check{a}}_x{e}^{s{B}_{\mu }}\xi \Vert & ⩽C[\Vert {({\mathcal{N}}_{+}+1)}^{n/2}{\check{a}}_x\xi \Vert +\Vert {({\mathcal{N}}_{+}+1)}^{n+1/2}\xi \Vert ]\hfill \end{array}$$ A.2

$$\begin{array}{cc}\hfill \Vert {({\mathcal{N}}_{+}+1)}^{n/2}{a}_p{a}_q{e}^{s{B}_{\mu }}\xi \Vert & ⩽C[\Vert {({\mathcal{N}}_{+}+1)}^{n/2}{a}_p{a}_q\xi \Vert +|{\mu }_q|\Vert {({\mathcal{N}}_{+}+1)}^{(n+1)/2}{a}_p\xi \Vert \hfill \\ \hfill & +|{\mu }_p|\Vert {({\mathcal{N}}_{+}+1)}^{(n+1)/2}{a}_q\xi \Vert +{\delta }_{p,-q}|{\mu }_p|\Vert {({\mathcal{N}}_{+}+1)}^{(n+3)/2}\xi \Vert ]\hfill \end{array}$$ A.3

$$\begin{array}{cc}\hfill \Vert {({\mathcal{N}}_{+}+1)}^{n/2}{\check{a}}_x{\check{a}}_y{e}^{s{B}_{\mu }}\xi \Vert & ⩽C[\Vert {({\mathcal{N}}_{+}+1)}^{n/2}{\check{a}}_x{\check{a}}_y\xi \Vert +|\check{\mu }(x-y)|\Vert {({\mathcal{N}}_{+}+1)}^{n/2}\xi \Vert \hfill \\ \hfill & +\Vert {({\mathcal{N}}_{+}+1)}^{n+1/2}{\check{a}}_x\xi \Vert +\Vert {({\mathcal{N}}_{+}+1)}^{n+1/2}{\check{a}}_y\xi \Vert ]\hfill \end{array}$$ A.4

for any $p\in {\mathrm{\Lambda }}_{+}^{\ast }$, any $x,y\in \mathrm{\Lambda }$, and $\xi \in {\mathcal{F}}_{\perp }^{⩽N}$.

Proof

The bounds (A.1)–(A.4) can all be shown using Gronwall’s Lemma. We focus on (A.1); the other estimates can be proven similarly. We consider

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{d}{\mathit{ds}}{\Vert {({\mathcal{N}}_{+}+1)}^{n/2}{a}_p{e}^{s{B}_{\mu }}\xi \Vert }^2=& ⟨\xi ,e^{-s{B}_{\mu }}[{\mathcal{N}}_{+}^n,B_{\mu }]a_p^{\ast }{a}_p{e}^{s{B}_{\mu }}\xi ⟩\hfill \\ \hfill & +⟨\xi ,e^{-s{B}_{\mu }}{\mathcal{N}}_{+}^n[a_p^{\ast }{a}_p,B_{\mu }]e^{s{B}_{\mu }}\xi ⟩=K_1+K_2.\hfill \end{array}\end{array}$$ A.5

With the commutation relation (2.2), we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |K_2|=& |⟨\xi ,e^{-s{B}_{\mu }}{\mathcal{N}}_{+}^n({\mu }_p{b}_p^{\ast }{b}_{-p}^{\ast }+\text{h.c.})e^{s{B}_{\mu }}\xi ⟩|\hfill \\ \hfill ⩽& C|{\mu }_p|\Vert {({\mathcal{N}}_{+}+1)}^{n/2}{a}_p{e}^{s{B}_{\mu }}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{(n+1)/2}\xi \Vert \hfill \\ \hfill ⩽& C\Vert {({\mathcal{N}}_{+}+1)}^{n/2}{a}_p{e}^{s{B}_{\mu }}{\xi \Vert }^2+C|{\mu }_p{|}^2{\Vert {({\mathcal{N}}_{+}+1)}^{(n+1)/2}\xi \Vert }^2\hfill \end{array}\end{array}$$ A.6

Now we consider $K_1$. We find

$$\begin{array}{c}\hfill [{\mathcal{N}}_{+}^n,B_{\mu }]=\frac{1}{2}\sum _{q\in {\mathrm{\Lambda }}_{+}^{\ast }}{\mu }_q{b}_q^{\ast }{b}_{-q}^{\ast }[{({\mathcal{N}}_{+}+2)}^n-{\mathcal{N}}_{+}^n]+\text{h.c.}\end{array}$$ A.7

Therefore

$$\begin{array}{cc}\hfill |K_1|⩽& C\sum _{q\in {\mathrm{\Lambda }}_{+}^{\ast }}|{\mu }_q|\Vert b_{-q}{b}_q{({\mathcal{N}}_{+}+1)}^{n/2-1}{a}_p{e}^{s{B}_{\mu }}\xi \Vert \hfill \\ \hfill & \times \Vert {({\mathcal{N}}_{+}+1)}^{-n/2+1}[{({\mathcal{N}}_{+}+3)}^n-{({\mathcal{N}}_{+}+1)}^n]a_p{e}^{s{B}_{\mu }}\xi \Vert \hfill \\ \hfill ⩽& C\Vert {({\mathcal{N}}_{+}+1)}^{n/2}{a}_p{e}^{s{B}_{\mu }}{\xi \Vert }^2\hfill \end{array}$$

Inserting the last equation and (A.6) in (A.5) and applying Gronwall’s Lemma, we obtain the desired bound. $\square$

With Lemma A.1, we are now ready to show Lemma 3.3.

Proof of Lemma 3.3

We start with the first bound in Eq. (3.5). From (3.4), we find

$$\begin{array}{cc}\hfill \Vert {({\mathcal{N}}_{+}+1)}^{n/2}{d}_p\xi \Vert ⩽& \frac{C}{N}{\int }_0^{1}ds[|{\mu }_p|\Vert {({\mathcal{N}}_{+}+1)}^{(n+3)/2}{e}^{(1-s)B_{\mu }}\xi \Vert \hfill \\ \hfill & +\Vert \sum _{q\in {\mathrm{\Lambda }}_{+}^{\ast }}{\mu }_q{({\mathcal{N}}_{+}+1)}^{n/2}{b}_q^{\ast }{a}_{-q}^{\ast }{a}_p{e}^{(1-s)B_{\mu }}\xi \Vert \hfill \\ \hfill & +|{\sigma }_p^{(s)}|\Vert \sum _{q\in {\mathrm{\Lambda }}_{+}^{\ast }}{\mu }_q{({\mathcal{N}}_{+}+1)}^{n/2}{a}_{-p}^{\ast }{a}_{-q}{b}_q{e}^{(1-s)B_{\mu }}\xi \Vert ].\hfill \end{array}$$

Observing that $|{\sigma }_p^{(s)}|⩽C|{\mu }_p|$ and that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \Vert \sum _{q\in {\mathrm{\Lambda }}_{+}^{\ast }}{\mu }_q& {({\mathcal{N}}_{+}+1)}^{n/2}{b}_q^{\ast }{a}_{-q}^{\ast }{a}_p{e}^{(1-s)B_{\mu }}\xi \Vert ⩽\Vert {({\mathcal{N}}_{+}+1)}^{(n+2)/2}{a}_p{e}^{(1-s)B_{\mu }}\xi \Vert \hfill \\ \hfill \Vert \sum _{q\in {\mathrm{\Lambda }}_{+}^{\ast }}{\mu }_q& {({\mathcal{N}}_{+}+1)}^{n/2}{a}_{-p}^{\ast }{a}_{-q}{b}_q{e}^{(1-s)B_{\mu }}\xi \Vert ⩽\Vert {({\mathcal{N}}_{+}+1)}^{(n+3)/2}{e}^{(1-s)B_{\mu }}\xi \Vert \hfill \end{array}\end{array}$$ A.8

we conclude, with the help of (3.1) and Lemma A.1, that

$$\begin{array}{c}\hfill \Vert {({\mathcal{N}}_{+}+1)}^{n/2}{d}_p\xi \Vert ⩽\frac{C}{N}[|{\mu }_p|\Vert {({\mathcal{N}}_{+}+1)}^{(n+3)/2}\xi \Vert +\Vert {({\mathcal{N}}_{+}+1)}^{(n+2)/2}{a}_p\xi \Vert ]\end{array}$$

The second estimate in (3.5) follows similarly (it is simpler, because we do not need to extract factors decaying in p). Let us now consider the third bound in (3.5). We proceed as above, applying (3.4). We find

$$\begin{array}{cc}\hfill \Vert {({\mathcal{N}}_{+}+1)}^{n/2}{a}_p{d}_q\xi \Vert ⩽& \frac{C}{N}{\int }_0^{1}ds[|{\mu }_q|\Vert {({\mathcal{N}}_{+}+1)}^{(n+2)/2}{a}_p{b}_{-q}^{\ast }{e}^{(1-s)B_{\mu }}\xi \Vert \hfill \\ \hfill & +\Vert \sum _{r\in {\mathrm{\Lambda }}_{+}^{\ast }}{\mu }_r{({\mathcal{N}}_{+}+1)}^{n/2}{a}_p{b}_r^{\ast }{a}_{-r}^{\ast }{a}_q{e}^{(1-s)B_{\mu }}\xi \Vert \hfill \\ \hfill & +|{\sigma }_q^{(s)}||{\mu }_q|\Vert {({\mathcal{N}}_{+}+1)}^{(n+2)/2}{a}_p{b}_q{e}^{(1-s)B_{\mu }}\xi \Vert \hfill \\ \hfill & +|{\sigma }_q^{(s)}|\Vert \sum _{r\in {\mathrm{\Lambda }}_{+}^{\ast }}{\mu }_r{({\mathcal{N}}_{+}+1)}^{n/2}{a}_p{a}_{-q}^{\ast }{a}_{-r}{b}_r{e}^{(1-s)B_{\mu }}\xi \Vert ]\hfill \end{array}$$

Commuting $a_p$ to the right of all creation operators, proceeding as in (A.8) to bound the terms in the second and fourth line and applying Lemma A.1, we conclude that

$$\begin{array}{cc}\hfill \Vert {({\mathcal{N}}_{+}+1)}^{n/2}{a}_p{d}_q\xi \Vert ⩽& \frac{C}{N}[|{\mu }_q|\Vert {({\mathcal{N}}_{+}+1)}^{(n+3)/2}{a}_p\xi \Vert +|{\mu }_p|\Vert {({\mathcal{N}}_{+}+1)}^{(n+3)/2}{a}_q\xi \Vert \hfill \\ \hfill & +{\delta }_{p,-q}\Vert {({\mathcal{N}}_{+}+1)}^{(n+3)/2}\xi \Vert +\Vert {({\mathcal{N}}_{+}+1)}^{(n+2)/2}{a}_p{a}_q\xi \Vert \hfill \\ \hfill & +|{\mu }_p||{\mu }_q|\Vert {({\mathcal{N}}_{+}+1)}^{(n+4)/2}\xi \Vert ].\hfill \end{array}$$

Also the fourth estimate in (3.5) can be shown analogously. To prove (3.6), we rewrite (3.4) in position space. We obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\check{d}}_x=& -\frac{1}{N}{\int }_0^{1}ds{e}^{-(1-s)B_{\mu }}[\int dz\check{\gamma }(x-z)({\mathcal{N}}_{+}{b}^{\ast }({\check{\eta }}_z)+\int drds\check{\eta }(r-s){\check{b}}_s^{\ast }{\check{a}}_r^{\ast }{\check{a}}_z)\hfill \\ \hfill & +\int dz\check{\sigma }(x-z)(b({\check{\eta }}_z){\mathcal{N}}_{+}+\int drds\check{\eta }(r-s){\check{a}}_z^{\ast }{\check{a}}_r{\check{b}}_s)]e^{(1-s)B_{\mu }}.\hfill \end{array}\end{array}$$

With the last identity, we can show the first two bounds in (3.6) similarly as we did above for the estimates in (3.5). The third bound in (3.6) follows directly from the first two (applying the first bound to ${\xi }^{\prime }={\check{d}}_y\xi$, and then the first and the second bound to the vector $\xi$). $\square$

A-Priori Bounds: Proof of Proposition 2.3

Proof

It follows from [5, Proposition 4.1] that for every $k\in \mathbb{N}$ there exists $C>0$ such that

$$\begin{array}{cc}& ⟨{\xi }_N^{\prime },(\mathcal{K}+1){({\mathcal{N}}_{+}+1)}^k{\xi }_N^{\prime }⟩⩽C.\hfill \end{array}$$ B.1

for ${\xi }_N^{\prime }=e^{-B_{\eta }}{U}_N{\psi }_N$. We are going to show that the same bounds hold true, if we replace ${\xi }_N^{\prime }$ with ${\xi }_N=e^{-B_{\mu }}{U}_N{\psi }_N=e^{-B_{\mu }}{e}^{B_{\eta }}{\xi }_N^{\prime }$. Since ${\mathcal{V}}_N⩽C\mathcal{K}{\mathcal{N}}_{+}$ (see [5, Lemma 6.2]), this will conclude the proof of Prop. 2.3. From (3.1) and from the corresponding bound for $B_{\eta }$, it follows immediately that

$$\begin{array}{c}\hfill ⟨{\xi }_N,{({\mathcal{N}}_{+}+1)}^k{\xi }_N⟩⩽C\end{array}$$

Thus, we focus on the expectation of $\mathcal{K}{({\mathcal{N}}_{+}+1)}^k$. For $s\in (0;1)$, we compute

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{d}{\mathit{ds}}⟨{\xi }_N^{\prime },e^{-s{B}_{\eta }}{e}^{s{B}_{\mu }}& \mathcal{K}{({\mathcal{N}}_{+}+1)}^k{e}^{-s{B}_{\mu }}{e}^{s{B}_{\eta }}{\xi }_N^{\prime }⟩\hfill \\ \hfill =& -⟨{\xi }_s,[\mathcal{K}{({\mathcal{N}}_{+}+1)}^k,B_{\mu }]{\xi }_s⟩+⟨{\xi }_s,[\mathcal{K}{({\mathcal{N}}_{+}+1)}^k,e^{-s{B}_{\mu }}{B}_{\eta }{e}^{s{B}_{\mu }}]{\xi }_s⟩\hfill \end{array}\end{array}$$

where we defined ${\xi }_s=e^{-s{B}_{\mu }}{e}^{s{B}_{\eta }}{\xi }_N^{\prime }$. Using (3.3), we write

$$\begin{array}{c}\hfill e^{-s{B}_{\mu }}{B}_{\eta }{e}^{s{B}_{\mu }}=B_{\eta }+J\end{array}$$

where

$$\begin{array}{c}\hfill J=\sum _{q\in {\mathrm{\Lambda }}_{+}^{\ast }}{\eta }_{q}d_q^{\ast (s)}({\gamma }_{-q}^{(s)}{b}_{-q}^{\ast }+{\sigma }_q^{(s)}{b}_q+d_{-q}^{\ast (s)})+\sum _{q\in {\mathrm{\Lambda }}_{+}^{\ast }}{\eta }_q({\gamma }_q^{(s)}{b}_q^{\ast }+{\sigma }_q^{(s)}{b}_{-q})d_{-q}^{\ast (s)}-\mathrm{h}.\mathrm{c}..\end{array}$$

Therefore,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{d}{\mathit{ds}}⟨{\xi }_s,\mathcal{K}{({\mathcal{N}}_{+}+1)}^k{\xi }_s⟩=& ⟨{\xi }_s,[\mathcal{K},B_{\eta }-B_{\mu }]{({\mathcal{N}}_{+}+1)}^k{\xi }_s⟩\hfill \\ \hfill & +⟨{\xi }_s,\mathcal{K}[{({\mathcal{N}}_{+}+1)}^k,B_{\eta }-B_{\mu }]{\xi }_s⟩+⟨{\xi }_s,[\mathcal{K}{({\mathcal{N}}_{+}+1)}^k,J]{\xi }_s⟩.\hfill \end{array}\end{array}$$ B.2

With (2.21), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨{\xi }_s,[\mathcal{K},B_{\eta }-B_{\mu }]{({\mathcal{N}}_{+}+1)}^k{\xi }_s⟩|& =2|\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{\tau }_p⟨{\xi }_s,b_p^{\ast }{b}_{-p}^{\ast }{({\mathcal{N}}_{+}+1)}^k{\xi }_s⟩|\hfill \\ \hfill & ⩽C⟨{\xi }_s,{({\mathcal{N}}_{+}+1)}^{k+1}{\xi }_s⟩⩽C⟨{\xi }_N^{\prime },{({\mathcal{N}}_{+}+1)}^{k+1}{\xi }_N^{\prime }⟩⩽C\hfill \end{array}\end{array}$$

where we applied (3.1) and then (B.1). Using (A.7), we can bound the second term on the r.h.s. of (B.2) by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨{\xi }_s,\mathcal{K}& [{({\mathcal{N}}_{+}+1)}^k,B_{\eta }-B_{\mu }]{\xi }_s⟩|\hfill \\ \hfill & ⩽C\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2|{\tau }_q||⟨{\xi }_s,a_p^{\ast }{a}_p{b}_q{b}_{-q}[{({\mathcal{N}}_{+}+3)}^k-{({\mathcal{N}}_{+}+1)}^k]{\xi }_s⟩|\hfill \\ \hfill & ⩽C⟨{\xi }_s,\mathcal{K}{({\mathcal{N}}_{+}+1)}^k{\xi }_s⟩.\hfill \end{array}\end{array}$$

We now focus on the last term on the r.h.s. of (B.2). Using $J^{\ast }=-J$, we have

$$\begin{array}{c}\hfill \begin{array}{c}\hfill |⟨{\xi }_s,[\mathcal{K}{({\mathcal{N}}_{+}+1)}^k,J]{\xi }_s⟩|⩽2|⟨{\xi }_s,\mathcal{K}{({\mathcal{N}}_{+}+1)}^{k}J{\xi }_s⟩|⩽2(|J_1|+|J_2|+|J_3|)\end{array}\end{array}$$

where we defined

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill J_1& =\sum _{q\in {\mathrm{\Lambda }}_{+}^{\ast }}{\eta }_q\langle {\xi }_s,\mathcal{K}{({\mathcal{N}}_{+}+1)}^k({\gamma }_q^{(s)}(b_{-q}{d}_q^{(s)}+d_q^{(s)}{b}_{-q})+{\sigma }_q^{(s)}(b_q^{\ast }{d}_q^{(s)}+d_q^{(s)}{b}_q^{\ast })){\xi }_s\rangle \hfill \\ \hfill J_2& =\sum _{q\in {\mathrm{\Lambda }}_{+}^{\ast }}{\eta }_q\langle {\xi }_s,({\gamma }_q^{(s)}(b_{-q}{d}_q^{(s)}+d_q^{(s)}{b}_{-q})+{\sigma }_q^{(s)}(b_q^{\ast }{d}_q^{(s)}+d_q^{(s)}{b}_q^{\ast }))\mathcal{K}{({\mathcal{N}}_{+}+1)}^k{\xi }_s\rangle \hfill \\ \hfill J_3& =\sum _{q\in {\mathrm{\Lambda }}_{+}^{\ast }}{\eta }_q\langle {\xi }_s,\mathcal{K}{({\mathcal{N}}_{+}+1)}^k(d_q^{(s)}{d}_{-q}^{(s)}+d_q^{\ast (s)}d_{-q}^{\ast (s)}){\xi }_s\rangle \hfill \end{array}\end{array}$$

We first consider $J_1$. Making use of the third bound in (3.5), the first contribution to $J_1$ is estimated in absolute value by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}& p^2|{\eta }_q|\Vert a_p{\xi }_s\Vert \Vert a_p{({\mathcal{N}}_{+}+1)}^{k+1/2}{d}_q^{(s)}{\xi }_s\Vert \hfill \\ \hfill & ⩽\frac{C}{N}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2|{\eta }_q|\Vert a_p{\xi }_s\Vert (|{\mu }_p|\Vert {({\mathcal{N}}_{+}+1)}^{k+2}{a}_q{\xi }_s\Vert +|{\mu }_q|\Vert {({\mathcal{N}}_{+}+1)}^{k+2}{a}_p{\xi }_s\Vert \hfill \\ \hfill & +|{\mu }_p||{\mu }_q|\Vert {({\mathcal{N}}_{+}+1)}^{k+5/2}{\xi }_s\Vert +\Vert {({\mathcal{N}}_{+}+1)}^{k+3/2}{a}_p{a}_q{\xi }_s\Vert \hfill \\ \hfill & +|{\eta }_p|{\delta }_{p,q}\Vert {({\mathcal{N}}_{+}+1)}^{k+3/2}{\xi }_s\Vert )\hfill \\ \hfill & ⩽\frac{C}{N}⟨{\xi }_s,\mathcal{K}{({\mathcal{N}}_{+}+1)}^{2k+4}{\xi }_s⟩+C⟨{\xi }_s,{({\mathcal{N}}_{+}+1)}^{2k+5}{\xi }_s⟩⩽C\hfill \end{array}\end{array}$$

where we used (B.1) after applying (3.2), (3.1), to control the growth of moments of ${\mathcal{N}}_{+}$ and of $\mathcal{K}$ w.r.t. the action of $B_{\eta }$ and $B_{\mu }$. The other contributions to $J_1$ can be bounded similarly. Also $J_3$ can be bounded using the estimates in (3.5); we skip the details. As for the term $J_2$, the contributions proportional to ${\sigma }_q^{(s)}$ can be bounded using the fourth bound in (3.5). The second contribution proportional to ${\gamma }_q^{(s)}$, on the other hand, can be estimated after normal ordering by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}& p^2|{\eta }_q|\Vert a_{p}d_q^{\ast (s)}{\xi }_s\Vert \Vert a_p{b}_{-q}{({\mathcal{N}}_{+}+1)}^k{\xi }_s\Vert +\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2|{\eta }_p|\Vert d_p^{\ast (s)}{\xi }_s\Vert \Vert a_p{({\mathcal{N}}_{+}+1)}^k{\xi }_s\Vert \hfill \\ \hfill & ⩽\frac{C}{N}⟨{\xi }_s,\mathcal{K}{({\mathcal{N}}_{+}+1)}^{2k+1}{\xi }_s⟩+\frac{C}{N}⟨{\xi }_s,\mathcal{K}{({\mathcal{N}}_{+}+1)}^3{\xi }_s⟩+C⟨{\xi }_s,{({\mathcal{N}}_{+}+1)}^4{\xi }_s⟩⩽C\hfill \end{array}\end{array}$$

where we applied again the fourth bound in (3.5). Finally, let us consider the first contribution to $J_2$. Using the expansion (3.4), we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |\sum _{q\in {\mathrm{\Lambda }}_{+}^{\ast }}& {\eta }_q{\gamma }_q^{(s)}⟨{\xi }_s,b_{-q}{d}_q^{(s)}\mathcal{K}{({\mathcal{N}}_{+}+1)}^k{\xi }_s⟩|\hfill \\ \hfill & ⩽\frac{C}{N}{\int }_0^{1}d\tau \sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2|{\eta }_q||\langle {\xi }_s,b_{-q}{e}^{-(1-\tau )s{B}_{\mu }}[{\gamma }_q^{(\tau )}({\mu }_q{\mathcal{N}}_{+}{b}_{-q}^{\ast }+\sum _{r\in {\mathrm{\Lambda }}_{+}^{\ast }}{\mu }_r{b}_r^{\ast }{a}_{-r}^{\ast }{a}_q)\hfill \\ \hfill & +{\sigma }_q^{(\tau )}({\mu }_q{\mathcal{N}}_{+}{b}_q+\sum _{r\in {\mathrm{\Lambda }}_{+}^{\ast }}{\mu }_r{a}_{-q}^{\ast }{a}_{-r}{b}_r)]e^{(1-\tau )s{B}_{\mu }}{a}_p^{\ast }{a}_p{({\mathcal{N}}_{+}+1)}^k{\xi }_s\rangle |.\hfill \end{array}\end{array}$$ B.3

With (A.1), (3.1), (3.2) and, again, (B.1), the first term is bounded by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{C}{N}& {\int }_0^{1}d\tau \sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2|{\eta }_q||{\mu }_q|\Vert a_p{e}^{-(1-\tau )s{B}_{\mu }}{b}_{-q}{\mathcal{N}}_{+}{e}^{(1-\tau )s{B}_{\mu }}{b}_{-q}^{\ast }{\xi }_s\Vert \Vert a_p{({\mathcal{N}}_{+}+1)}^k{\xi }_s\Vert \hfill \\ \hfill & ⩽\frac{C}{N}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2|{\eta }_q||{\mu }_q|\Vert a_p{({\mathcal{N}}_{+}+1)}^k{\xi }_s\Vert (|{\mu }_p|\Vert {({\mathcal{N}}_{+}+1)}^{5/2}{\xi }_s\Vert +\Vert {({\mathcal{N}}_{+}+1)}^2{a}_p{\xi }_s\Vert \hfill \\ \hfill & +{\delta }_{p,-q}\Vert {({\mathcal{N}}_{+}+1)}^{3/2}{\xi }_s\Vert )\hfill \\ \hfill & ⩽\frac{C}{N}⟨{\xi }_s,\mathcal{K}{({\mathcal{N}}_{+}+1)}^{2k}{\xi }_s⟩+\frac{C}{N}⟨{\xi }_s,\mathcal{K}{({\mathcal{N}}_{+}+1)}^4{\xi }_s⟩+C⟨{\xi }_s,{({\mathcal{N}}_{+}+1)}^5{\xi }_s⟩⩽C.\hfill \end{array}\end{array}$$

The contributions on the r.h.s. of (B.3) that are proportional to ${\sigma }_q^{(\tau )}$ can be handled similarly, using also the estimate

$$\begin{array}{c}\hfill \Vert {({\mathcal{N}}_{+}+1)}^n\sum _{r\in {\mathrm{\Lambda }}_{+}^{\ast }}{\mu }_r{b}_r^{\ast }{a}_{-r}^{\ast }\xi \Vert ⩽C\Vert {({\mathcal{N}}_{+}+1)}^{n+1}\xi \Vert \end{array}$$

As for the second term proportional to ${\gamma }_q^{(\tau )}$ on the r.h.s. of (B.3), we write $a_{-r}^{\ast }{a}_q=b_{-r}^{\ast }{b}_q+N^{-1}{a}_{-r}^{\ast }({\mathcal{N}}_{+}-1)a_q$ and we observe that the contribution associated with the error $N^{-1}{a}_{-r}^{\ast }({\mathcal{N}}_{+}+1)a_q$ is negligible (it can be bounded through Cauchy-Schwarz). The contribution associated with $b_{-r}^{\ast }{b}_q$, on the other hand, can be estimated using (3.3) by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{C}{N}& {\int }_0^{1}d\tau \sum _{p,q,r\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2|{\eta }_q||{\mu }_r||⟨{\xi }_s,b_{-q}{e}^{-(1-\tau )s{B}_{\mu }}{b}_r^{\ast }{b}_{-r}^{\ast }{e}^{(1-\tau )s{B}_{\mu }}\hfill \\ \hfill & ({\gamma }_q^{((1-\tau )s)}{b}_q+{\sigma }_q^{((1-\tau )s)}{b}_{-q}^{\ast }+d_q^{((1-\tau )s)})a_p^{\ast }{a}_p{({\mathcal{N}}_{+}+1)}^k{\xi }_s⟩|.\hfill \end{array}\end{array}$$

The term proportional to ${\gamma }_q^{((1-\tau )s)}$ is bounded by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{C}{N}& {\int }_0^{1}d\tau \sum _{p,q,r\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2|{\eta }_q||{\mu }_r|\Vert a_p{e}^{-(1-\tau )s{B}_{\mu }}{b}_r{b}_{-r}{e}^{(1-\tau )s{B}_{\mu }}{b}_{-q}^{\ast }{\xi }_s\Vert \Vert b_q{a}_p{({\mathcal{N}}_{+}+1)}^k{\xi }_s\Vert \hfill \\ \hfill & +\sum _{p,r\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2|{\eta }_p||{\mu }_r|\Vert b_r{b}_{-r}{e}^{(1-\tau )s{B}_{\mu }}{b}_{-p}^{\ast }{\xi }_s\Vert \Vert a_p{({\mathcal{N}}_{+}+1)}^k{\xi }_s\Vert \hfill \\ \hfill & ⩽\frac{C}{N}⟨{\xi }_s,\mathcal{K}{({\mathcal{N}}_{+}+1)}^{2k+1}{\xi }_s⟩+\frac{C}{N}⟨{\xi }_s,\mathcal{K}{({\mathcal{N}}_{+}+1)}^3{\xi }_s⟩+⟨{\xi }_s,{({\mathcal{N}}_{+}+1)}^4{\xi }_s⟩⩽C\hfill \end{array}\end{array}$$

where we used (A.1), (3.1), (3.2) and (B.1). The term proportional to ${\sigma }_q^{((1-\tau )s)}$ can be handled analogously. As for the contribution proportional to $d_q^{((1-\tau )s)}$, we can apply the fourth bound in (3.5) to conclude that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{C}{N^2}& {\int }_0^{1}d\tau \sum _{p,q,r\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2|{\eta }_q||{\mu }_r|\Vert a_p{({\mathcal{N}}_{+}+1)}^k\Vert \hfill \\ \hfill & \times [\Vert {({\mathcal{N}}_{+}+1)}^{3/2}{a}_p{e}^{-(1-\tau )s{B}_{\mu }}{b}_r{b}_{-r}{e}^{(1-\tau )s{B}_{\mu }}{b}_{-q}^{\ast }{\xi }_s\Vert \hfill \\ \hfill & +|{\mu }_p|\Vert {({\mathcal{N}}_{+}+1)}^2{e}^{-(1-\tau )s{B}_{\mu }}{b}_r{b}_{-r}{e}^{(1-\tau )s{B}_{\mu }}{b}_{-q}^{\ast }{\xi }_s\Vert \hfill \\ \hfill & +{\delta }_{p,-q}\Vert ({\mathcal{N}}_{+}+1)e^{-(1-\tau )s{B}_{\mu }}{b}_r{b}_{-r}{e}^{(1-\tau )s{B}_{\mu }}{b}_{-q}^{\ast }{\xi }_s\Vert ]\hfill \\ \hfill & ⩽\frac{C}{N}⟨{\xi }_s,\mathcal{K}{({\mathcal{N}}_{+}+1)}^{2k}{\xi }_s⟩+\frac{C}{N}⟨{\xi }_s,\mathcal{K}{({\mathcal{N}}_{+}+1)}^6{\xi }_s⟩+⟨{\xi }_s,{({\mathcal{N}}_{+}+1)}^7{\xi }_s⟩⩽C\hfill \end{array}\end{array}$$

using again (A.1) (twice, in the first term in the parenthesis, to pass $a_p$ through the unitary operators $e^{\pm (1-\tau )s{B}_{\mu }}$), (3.1) (in the second and third terms, to pass powers of $({\mathcal{N}}_{+}+1)$ through the unitary operators), (3.2) and (B.1). Combining all bounds, we arrive at

$$\begin{array}{c}\hfill |\frac{d}{\mathit{ds}}⟨{\xi }_s,\mathcal{K}{({\mathcal{N}}_{+}+1)}^k{\xi }_s⟩|⩽C⟨{\xi }_s,\mathcal{K}{({\mathcal{N}}_{+}+1)}^k{\xi }_s⟩+C\end{array}$$

and the claim follows from Gronwall’s Lemma. $\square$

Funding

Open access funding provided by University of Geneva.

Data availability statement

This manuscript has no associated data.

Declarations

Conflict of interest

All authors declare that they have no Conflict of interest.