Bose–Einstein Condensation for Two Dimensional Bosons in the Gross–Pitaevskii Regime

Abstract

We consider systems of N bosons trapped on the two-dimensional unit torus, in the Gross-Pitaevskii regime, where the scattering length of the repulsive interaction is exponentially small in the number of particles. We show that low-energy states exhibit complete Bose–Einstein condensation, with almost optimal bounds on the number of orthogonal excitations.

Introduction

We consider $N\in \mathbb{N}$ bosons trapped in the two-dimensional box $\Lambda ={[-1/2;1/2]}^2$ with periodic boundary conditions. In the Gross-Pitaevskii regime, particles interact through a repulsive pair potential, with a scattering length exponentially small in N. The Hamilton operator is given by

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

and acts on a dense subspace of $L_s^2({\Lambda }^N)$, the Hilbert space consisting of functions in $L^2({\Lambda }^N)$ that are invariant with respect to permutations of the N particles. We assume here $V\in L^3({\mathbb{R}}^2)$ to be compactly supported and pointwise non-negative (i.e. $V(x)\ge 0$ for almost all $x\in {\mathbb{R}}^2$).

We denote by $\mathfrak{a}$ the scattering length of the unscaled potential V. We recall that in two dimensions and for a potential V with finite range $R_0$, the scattering length is defined by

$$\begin{array}{c}\hfill \frac{2\pi }{log(R/\mathfrak{a})}=\underset{\varphi }{inf}{\int }_{B_R}\left[{|\mathrm{\nabla }\varphi |}^2,+,\frac{1}{2},V,{|\varphi |}^2\right]dx\end{array}$$ 2

where $R>R_0$, $B_R$ is the disk of radius R centered at the origin and the infimum is taken over functions $\varphi \in H^1(B_R)$ with $\varphi (x)=1$ for all x with $|x|=R$. The unique minimizer of the variational problem on the r.h.s. of (2) is non-negative, radially symmetric and satisfies the scattering equation

$$\begin{array}{c}\hfill -\Delta {\varphi }^{(R)}+\frac{1}{2}V{\varphi }^{(R)}=0\end{array}$$

in the sense of distributions. For $R_0<|x|\le R$, we have

$$\begin{array}{c}\hfill {\varphi }^{(R)}(x)=\frac{log(|x|/\mathfrak{a})}{log(R/\mathfrak{a})}.\end{array}$$

By scaling, ${\varphi }_N(x):={\varphi }^{(e^{N}R)}(e^{N}x)$ is such that

$$\begin{array}{c}\hfill -\Delta {\varphi }_N+\frac{1}{2}{e}^{2N}V(e^{N}x){\varphi }_N=0\end{array}$$

We have

$$\begin{array}{c}\hfill {\varphi }_N(x)=\frac{log(|x|/{\mathfrak{a}}_N)}{log(R/{\mathfrak{a}}_N)}\forall x\in {\mathbb{R}}^2:e^{-N}{R}_0<|x|\le R,\end{array}$$

for all $x\in {\mathbb{R}}^2$ with $e^{-N}{R}_0<|x|\le R$. Here ${\mathfrak{a}}_N=e^{-N}\mathfrak{a}$.

The spectral properties of trapped two dimensional bosons in the Gross-Pitaevskii regime (in the more general case where the bosons are confined by external trapping potentials) have been first studied in [13, 14, 16]. These results can be translated to the Hamilton operator (1), defined on the torus, with no external potential. They imply that the ground state energy $E_N$ of (1) is such that

$$\begin{array}{c}\hfill E_N=2\pi N(1+O(N^{-1/5})).\end{array}$$ 3

Moreover, they imply Bose–Einstein condensation in the zero-momentum mode ${\phi }_0(x)=1$ for all $x\in \Lambda$, for any approximate ground state of (1). More precisely, it follows from [13] that, for any sequence ${\psi }_N\in L_s^2({\Lambda }^N)$ with $\Vert {\psi }_N\Vert =1$ and

$$\begin{array}{c}\hfill \underset{N\to \infty }{lim}\frac{1}{N}⟨{\psi }_N,H_N{\psi }_N⟩=2\pi ,\end{array}$$ 4

the one-particle reduced density matrix ${\gamma }_N={\mathrm{tr}}_{2,\dots ,N}|{\psi }_N⟩⟨{\psi }_N|$ is such that

$$\begin{array}{c}\hfill 1-⟨{\phi }_0,{\gamma }_N{\phi }_0⟩\le C{N}^{-\overline{\delta }}\end{array}$$ 5

for a sufficiently small $\overline{\delta }>0$. The estimate (5) states that, in many-body states satisfying (4) (approximate ground states), almost all particles are described by the one-particle orbital ${\phi }_0$, with at most $N^{1-\delta }\ll N$ orthogonal excitations.

Similar results have been obtained starting from a three dimensional Bose gas, trapped by a potential which is strongly confining in one direction, so that the system becomes effectively two-dimensional [22]. Finally, let us also mention [5, 10], where rigorous results on the time-evolution in the two-dimensional Gross-Pitaevskii regime have been established (in [5], the focus is on the dynamics of a three-dimensional gas, with strong confinement in one direction).

For $V\in L^3({\mathbb{R}}^2)$, our main theorem improves (3) and (5) by providing more precise bounds on the ground state energy and on the number of excitations.

Theorem 1

Let $V\in L^3({\mathbb{R}}^2)$ have compact support, be spherically symmetric and pointwise non-negative. Then there exists a constant $C>0$ such that the ground state energy $E_N$ of (1) satisfies

$$\begin{array}{c}\hfill 2\pi N-C\le E_N\le 2\pi N+ClogN.\end{array}$$ 6

Furthermore, consider a sequence ${\psi }_N\in L_s^2({\Lambda }^N)$ with $\Vert {\psi }_N\Vert =1$ and such that

$$\begin{array}{c}\hfill ⟨{\psi }_N,H_N{\psi }_N⟩\le 2\pi N+K\end{array}$$ 7

for a $K>0$. Then the reduced density matrix ${\gamma }_N={\mathrm{tr}}_{2,\dots ,N}|{\psi }_N⟩⟨{\psi }_N|$ associated with ${\psi }_N$ is such that

$$\begin{array}{c}\hfill 1-⟨{\phi }_0,{\gamma }_N{\phi }_0⟩\le \frac{C(1+K)}{N}\end{array}$$ 8

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

Remark

We expect that the bounds of Theorem 1 can be extended to two-dimensional systems of bosons trapped by an external potential (in three dimensions, similar estimates have been recently established in [7, 19]). In this case, the system exhibits condensation in the minimizer of the Gross-Pitaevskii energy functional, as shown in [13, 14, 16].

It is interesting to compare the Gross-Pitaevskii regime with the thermodynamic limit, where a Bose gas of N particles interacting through a fixed potential with scattering length $\mathfrak{a}$ is confined in a box with area $L^2$, so that $N,L\to \infty$ with the density $\rho =N/L^2$ kept fixed. Let $b=|log(\rho {\mathfrak{a}}^2){|}^{-1}$. Then, in the dilute limit $\rho {\mathfrak{a}}^2\ll 1$, the ground state energy per particle in the thermodynamic limit is expected to satisfy

$$\begin{array}{c}\hfill e_0(\rho )=4\pi {\rho }^{2}b(1+blogb+(1/2+2\gamma +log\pi )b+o(b)),\end{array}$$ 9

with $\gamma$ the Euler’s constant. The leading order term on the r.h.s. of (9) has been first derived in [21] and then rigorously established in [15], with an error rate $b^{-1/5}$. The corrections up to order b have been predicted in [1, 18, 20]. To date, there is no rigorous proof of (9). Some partial result, based on the restriction to quasi-free states, has been recently obtained in [9, Theorem 1].

Extrapolating from (9), in the Gross-Pitaevskii regime we expect $|E_N-2\pi N|\le C$. While our estimate (6) captures the correct lower bound, the upper bound is off by a logarithmic correction. Eq. (8), on the other hand, is expected to be optimal (but of course, by (6), we need to choose $K=ClogN$ to be sure that (7) can be satisfied). This bound can be used as starting point to investigate the validity of Bogoliubov theory for two dimensional bosons in the Gross-Pitaevskii regime, following the strategy developed in [3] for the three dimensional case; we plan to proceed in this direction in a separate paper.

The proof of Theorem 1 follows the strategy that has been recently introduced in [4] to prove condensation for three-dimensional bosons in the Gross-Pitaevskii limit. There are, however, additional obstacles in the two-dimensional case, requiring new ideas. To appreciate the difference between the Gross-Pitaevskii regime in two- and three-dimensions, we can compute the energy of the trivial wave function ${\psi }_N\equiv 1$. The expectation of (1) in this state is of order $N^2$. It is only through correlations that the energy can approach (6). Also in three dimensions, uncorrelated many-body wave functions have large energy, but in that case the difference with respect to the ground state energy is only of order N ($N\widehat{V}(0)/2$ rather than $4\pi \mathfrak{a}N$). This observation is a sign that correlations in two-dimensions are stronger and play a more important role than in three dimensions (this creates problems in handling error terms that, in the three dimensional setting, were simply estimated in terms of the integral of the potential).

The paper is organized as follows. In Sect. 2 we introduce our setting, based on a description of orthogonal excitations of the condensate on a truncated Fock space. Factoring out the condensate, we introduce an excitation Hamiltonian ${\mathcal{L}}_N$, unitarily equivalent to $H_N$. In Sects. 3 and 4 we define two additional unitary maps, modelling the correlation structure characterising low-energy states. The first map is a generalized Bogoliubov transformation, given by the exponential of an anti-symmetric operator B, quadratic in creation and annihilation operators, see Eq. (33). Its action on ${\mathcal{L}}_N$ leads to a second excitation Hamiltonian ${\mathcal{G}}_{N,\alpha }$, whose vacuum expectation matches (6), at leading order. Unfortunately, ${\mathcal{G}}_{N,\alpha }$ is not coercive enough to directly show Bose–Einstein condensation. To overcome this difficulty, we conjugate the main part of ${\mathcal{G}}_{N,\alpha }$ (later denoted by ${\mathcal{G}}_{N,\alpha }^{\text{eff}}$) with a second unitary map, given by the exponential of an operator A, cubic in creation and annihilation operators, see Eq. (44). This defines a renormalized excitation Hamiltonian ${\mathcal{R}}_{N,\alpha }$, where the singular interaction is regularized. In Sect. 5 we combine the bounds on ${\mathcal{G}}_{N,\alpha }$ and ${\mathcal{R}}_{N,\alpha }$ with a localization argument proposed in [11] for the number of excitations to conclude the proof of Theorem 1. Section 6 and App. 1 are devoted to the proof of the bounds on ${\mathcal{G}}_{N,\alpha }$ and on ${\mathcal{R}}_{N,\alpha }$ stated in Sects. 3 and 4, respectively. Finally, in App. 1, we establish some properties of the solution of the Neumann problem associated with the two-body potential V.

The Excitation Hamiltonian

Low-energy states of (1) exhibit condensation in the zero-momentum mode ${\phi }_0$ defined by ${\phi }_0(x)=1$ for all $x\in \Lambda ={[-1/2;1/2]}^2$. Similarly as in [2, 4, 11], we are going to describe excitations of the condensate on the truncated bosonic Fock space

$$\begin{array}{c}\hfill {\mathcal{F}}_{+}^{\le N}=\underset{k=0}{\overset{N}{⨁}}{L}_{\perp }^2{(\Lambda )}^{{\otimes }_{s}k}\end{array}$$

constructed on the orthogonal complement $L_{\perp }^2(\Lambda )$ of ${\phi }_0$ in $L^2(\Lambda )$. To reach this goal, we define a unitary map $U_N:L_s^2({\Lambda }^N)\to {\mathcal{F}}_{+}^{\le N}$ by requiring that $U_N{\psi }_N=\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_N\}$, with ${\alpha }_j\in L_{\perp }^2{(\Lambda )}^{{\otimes }_{s}j}$, if

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

With the usual creation and annihilation operators, we can write

$$\begin{array}{c}\hfill U_N{\psi }_N=\underset{n=0}{\overset{N}{⨁}}(1-|{\phi }_0⟩{⟨{\phi }_0|)}^{\otimes n}\frac{a{({\phi }_0)}^{N-n}}{\sqrt{(N-n)!}}{\psi }_N\end{array}$$

for all ${\psi }_N\in L_s^2({\Lambda }^N)$. It is then easy to check that $U_N^{\ast }:{\mathcal{F}}_{+}^{\le N}\to L_s^2({\Lambda }^N)$ is given by

$$\begin{array}{c}\hfill U_N^{\ast }\{{\alpha }^{(0)},\dots ,{\alpha }^{(N)}\}=\sum _{n=0}^N\frac{a^{\ast }{({\phi }_0)}^{N-n}}{\sqrt{(N-n)!}}{\alpha }^{(n)}\end{array}$$

and that $U_N^{\ast }{U}_N=1$, i.e. $U_N$ is unitary.

With $U_N$, we can define the excitation Hamiltonian ${\mathcal{L}}_N:=U_N{H}_N{U}_N^{\ast }$, acting on a dense subspace of ${\mathcal{F}}_{+}^{\le N}$. To compute the operator ${\mathcal{L}}_N$, we first write the Hamiltonian (1) in momentum space, in terms of creation and annihilation operators $a_p^{\ast },a_p$, for momenta $p\in {\Lambda }^{\ast }=2\pi {\mathbb{Z}}^2$. We find

$$\begin{array}{c}\hfill H_N=\sum _{p\in {\Lambda }^{\ast }}{p}^2{a}_p^{\ast }{a}_p+\frac{1}{2}\sum _{p,q,r\in {\Lambda }^{\ast }}\widehat{V}(r/e^N)a_{p+r}^{\ast }{a}_q^{\ast }{a}_p{a}_{q+r}\end{array}$$ 10

where

$$\begin{array}{c}\hfill \widehat{V}(k)={\int }_{{\mathbb{R}}^2}V(x)e^{-ik·x}dx\end{array}$$

is the Fourier transform of V, defined for all $k\in {\mathbb{R}}^2$ (in fact, (1) is the restriction of (10) to the N-particle sector of the Fock space). We can now determine ${\mathcal{L}}_N$ using the following rules, describing the action of the unitary operator $U_N$ on products of a creation and an annihilation operator (products of the form $a_p^{\ast }{a}_q$ can be thought of as operators mapping $L_s^2({\Lambda }^N)$ to itself). For any $p,q\in {\Lambda }_{+}^{\ast }=2\pi {\mathbb{Z}}^2\backslash \{0\}$, we find (see [11]):

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill U_N{a}_0^{\ast }{a}_0{U}_N^{\ast }& =N-{\mathcal{N}}_{+}\hfill \\ \hfill U_N{a}_p^{\ast }{a}_0{U}_N^{\ast }& =a_p^{\ast }\sqrt{N-{\mathcal{N}}_{+}}\hfill \\ \hfill U_N{a}_0^{\ast }{a}_p{U}_N^{\ast }& =\sqrt{N-{\mathcal{N}}_{+}}{a}_p\hfill \\ \hfill U_N{a}_p^{\ast }{a}_q{U}_N^{\ast }& =a_p^{\ast }{a}_q.\hfill \end{array}\end{array}$$ 11

where ${\mathcal{N}}_{+}={\sum }_{p\in {\Lambda }_{+}^{\ast }}{a}_p^{\ast }{a}_p$ is the number of particles operator on ${\mathcal{F}}_{+}^{\le N}$. We conclude that

$$\begin{array}{c}\hfill {\mathcal{L}}_N={\mathcal{L}}_N^{(0)}+{\mathcal{L}}_N^{(2)}+{\mathcal{L}}_N^{(3)}+{\mathcal{L}}_N^{(4)}\end{array}$$ 12

with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{L}}_N^{(0)}=& \frac{1}{2}\widehat{V}(0)(N-1)(N-{\mathcal{N}}_{+})+\frac{1}{2}\widehat{V}(0){\mathcal{N}}_{+}(N-{\mathcal{N}}_{+})\hfill \\ \hfill {\mathcal{L}}_N^{(2)}=& \sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{a}_p^{\ast }{a}_p+N\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N)\left[b_p^{\ast },b_p,-,\frac{1}{N},a_p^{\ast },a_p\right]\hfill \\ \hfill & +\frac{N}{2}\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N)\left[b_p^{\ast },b_{-p}^{\ast },+,b_p,b_{-p}\right]\hfill \\ \hfill {\mathcal{L}}_N^{(3)}=& \sqrt{N}\sum _{p,q\in {\Lambda }_{+}^{\ast }:p+q\ne 0}\widehat{V}(p/e^N)\left[b_{p+q}^{\ast },a_{-p}^{\ast },a_q,+,a_q^{\ast },a_{-p},b_{p+q}\right]\hfill \\ \hfill {\mathcal{L}}_N^{(4)}=& \frac{1}{2}\sum _{\begin{array}{c}p,q\in {\Lambda }_{+}^{\ast },r\in {\Lambda }^{\ast }:\\ r\ne -p,-q\end{array}}\widehat{V}(r/e^N)a_{p+r}^{\ast }{a}_q^{\ast }{a}_p{a}_{q+r},\hfill \end{array}\end{array}$$ 13

where we introduced generalized creation and annihilation operators

$$\begin{array}{c}\hfill b_p^{\ast }=U_N{a}_p^{\ast }{U}_N^{\ast }=a_p^{\ast }\sqrt{\frac{N-{\mathcal{N}}_{+}}{N}},\text{and}{b}_p=U_N{a}_p{U}_N^{\ast }=\sqrt{\frac{N-{\mathcal{N}}_{+}}{N}}{a}_p\end{array}$$

for all $p\in {\Lambda }_{+}^{\ast }$.

On states exhibiting complete Bose–Einstein condensation in the zero-momentum mode ${\phi }_0$, we have $a_0,a_0^{\ast }\simeq \sqrt{N}$ and we can therefore expect that $b_p^{\ast }\simeq a_p^{\ast }$ and that $b_p\simeq a_p$. From the canonical commutation relations for the standard creation and annihilation operators $a_p,a_p^{\ast }$, we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [b_p,b_q^{\ast }]& =\left(1,-,\frac{{\mathcal{N}}_{+}}{N}\right){\delta }_{p,q}-\frac{1}{N}{a}_q^{\ast }{a}_p\hfill \\ \hfill [b_p,b_q]& =[b_p^{\ast },b_q^{\ast }]=0.\hfill \end{array}\end{array}$$ 14

Furthermore,

$$\begin{array}{c}\hfill [b_p,a_q^{\ast }{a}_r]={\delta }_{\mathit{pq}}{b}_r,[b_p^{\ast },a_q^{\ast }{a}_r]=-{\delta }_{\mathit{pr}}{b}_q^{\ast }\end{array}$$

for all $p,q,r\in {\Lambda }_{+}^{\ast }$; this implies in particular that $[b_p,{\mathcal{N}}_{+}]=b_p$, $[b_p^{\ast },{\mathcal{N}}_{+}]=-b_p^{\ast }$. It is also useful to notice that the operators $b_p^{\ast },b_p$, like the standard creation and annihilation operators $a_p^{\ast },a_p$, can be bounded by the square root of the number of particles operators; we find

$$\begin{array}{c}\hfill \Vert b_p\xi \Vert \le \Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert ,\Vert b_p^{\ast }\xi \Vert \le \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \end{array}$$

for all $\xi \in {\mathcal{F}}_{+}^{\le N}$. Since ${\mathcal{N}}_{+}\le N$ on ${\mathcal{F}}_{+}^{\le N}$, the operators $b_p^{\ast },b_p$ are bounded, with $\Vert b_p\Vert ,\Vert b_p^{\ast }{\Vert \le (N+1)}^{1/2}$.

Quadratic Renormalization

From (13) we see that conjugation with $U_N$ extracts, from the original quartic interaction in (10), some large constant and quadratic contributions, collected in ${\mathcal{L}}_N^{(0)}$ and ${\mathcal{L}}_N^{(2)}$ respectively. In particular, the expectation of ${\mathcal{L}}_N$ on the vacuum state $\Omega$ is of order $N^2$, this being an indication of the fact that there are still large contributions to the energy hidden among cubic and quartic terms in ${\mathcal{L}}_N^{(3)}$ and ${\mathcal{L}}_N^{(4)}$. Since $U_N$ only removes products of the zero-energy mode ${\phi }_0$, correlations among particles remain in the excitation vector $U_N{\psi }_N$. Indeed, correlations play a crucial role in the two dimensional Gross-Pitaevskii regime and carry an energy of order $N^2$.

To take into account the short scale correlation structure on top of the condensate, we consider the solution $f_{\ell }$ of the equation

$$\begin{array}{c}\hfill (-\Delta +\frac{1}{2}V(x))f_{\ell }(x)={\lambda }_{\ell }{f}_{\ell }(x)\end{array}$$ 15

associated with the smallest possible eigenvalue ${\lambda }_{\ell }$, on the ball $|x|\le e^N\ell$, with Neumann boundary conditions and normalized so that $f_{\ell }(x)=1$ for $|x|=e^N\ell$. Here and in the following we omit the N-dependence in the notation for $f_{\ell }$ and for ${\lambda }_{\ell }$. By scaling, we observe that $f_{\ell }(e^N·)$ satisfies

$$\begin{array}{c}\hfill (-\Delta +\frac{e^{2N}}{2}V(e^{N}x))f_{\ell }(e^{N}x)=e^{2N}{\lambda }_{\ell }{f}_{\ell }(e^{N}x)\end{array}$$

on the ball $|x|\le \ell$. We choose $\ell <1/2$, so that the ball of radius $\ell$ is contained in the box $\Lambda ={[-1/2;1/2]}^2$. We extend then $f_{\ell }(e^N.)$ to $\Lambda$, by setting $f_{N,\ell }(x)=f_{\ell }(e^{N}x)$, if $|x|\le \ell$ and $f_{N,\ell }(x)=1$ for $x\in \Lambda$, with $|x|>\ell$. Then, assuming also that $R_0{e}^{-N}<\ell$ (later we will choose $\ell =N^{-\alpha }$, so this condition is satisfied, for all N large enough),

$$\begin{array}{c}\hfill (-\Delta +\frac{e^{2N}}{2}V(e^{N}x))f_{N,\ell }(x)=e^{2N}{\lambda }_{\ell }{f}_{N,\ell }(x){\chi }_{\ell }(x),\end{array}$$ 16

where ${\chi }_{\ell }$ is the characteristic function of the ball of radius $\ell$. The Fourier coefficients of the function $f_{N,\ell }$ are given by

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

for all $p\in {\Lambda }^{\ast }$. We introduce also the function $w_{\ell }(x)=1-f_{\ell }(x)$ for $|x|\le e^N\ell$ and extend it by setting $w_{\ell }(x)=0$ for $|x|>e^N\ell$. Its re-scaled version is defined by $w_{N,\ell }:\Lambda \to \mathbb{R}$ $w_{N,\ell }(x)=w_{\ell }(e^{N}x)$ if $|x|\le \ell$ and $w_{N,\ell }=0$ if $x\in \Lambda$ with $|x|>\ell$.

The Fourier coefficients of the re-scaled function $w_{N,\ell }$ are given by

$$\begin{array}{c}\hfill {\widehat{w}}_{N,\ell }(p)={\int }_{\Lambda }{w}_{\ell }(e^{N}x)e^{-ip·x}dx=e^{-2N}{\widehat{w}}_{\ell }\left(e^{-N},p\right).\end{array}$$ 17

We find ${\widehat{f}}_{N,\ell }(p)={\delta }_{p,0}-e^{-2N}{\widehat{w}}_{\ell }(e^{-N}p)$. From the Neumann problem (16) we obtain

$$\begin{array}{c}\hfill -p^2{e}^{-2N}{\widehat{w}}_{\ell }(e^{-N}p)+\frac{1}{2}\sum _{q\in {\Lambda }^{\ast }}\widehat{V}(e^{-N}(p-q)){\widehat{f}}_{N,\ell }(q)=e^{2N}{\lambda }_{\ell }\sum _{q\in {\Lambda }^{\ast }}{\widehat{\chi }}_{\ell }(p-q){\widehat{f}}_{N,\ell }(q).\\ \hfill \end{array}$$ 18

where we used the notation ${\widehat{\chi }}_{\ell }$ for the Fourier coefficients of the characteristic function on the ball of radius $\ell$. Note that ${\widehat{\chi }}_{\ell }(p)={\ell }^2\widehat{\chi }(\ell p)$ with $\widehat{\chi }(p)$ the Fourier coefficients of the characteristic function on the ball of radius one.

In the next lemma, we collect some important properties of the solution of (15).

Lemma 1

Let $V\in L^3({\mathbb{R}}^2)$ be non-negative, compactly supported (with range $R_0$) and spherically symmetric, and denote its scattering length by $\mathfrak{a}$. Fix $0<\ell <1/2$, N sufficiently large and let $f_{\ell }$ denote the solution of (16). Then

• (i)
  $$\begin{array}{c}\hfill 0\le f_{\ell }(x)\le 1\forall |x|\le e^N\ell .\end{array}$$
• (ii)
  We have

  $$\begin{array}{c}\hfill \left|{\lambda }_{\ell },-,\frac{2}{{(e^N\ell )}^{2}log(e^N\ell /\mathfrak{a})}\right|\le \frac{C}{{(e^N\ell )}^2{log}^2(e^N\ell /\mathfrak{a})}\end{array}$$ 19
• (iii)
  There exists a constant $C>0$ such that

  $$\begin{array}{c}\hfill \left|\int ,\text{d},x,,V,(x),f_{\ell },(x),-,\frac{4\pi }{log(e^N\ell /\mathfrak{a})}\right|\le \frac{C}{{log}^2(e^N\ell /\mathfrak{a})}\end{array}$$ 20
• (iv)
  There exists a constant $C>0$ such that

  $$\begin{array}{c}\hfill \begin{array}{cc}\hfill |w_{\ell }(x)|& \le \left\{\begin{array}{cc}C\hfill & \text{if}|x|\le R_0\hfill \\ C\frac{log(e^N\ell /|x|)}{log(e^N\ell /\mathfrak{a})}\hfill & \text{if}{R}_0\le |x|\le e^N\ell \hfill \end{array}\right)\hfill \\ \hfill |\mathrm{\nabla }{w}_{\ell }(x)|& \le \frac{C}{log(e^N\ell /\mathfrak{a})}\frac{1}{|x|+1}\text{for all}|x|\le e^N\ell \hfill \end{array}\end{array}$$ 21
• (v)
  Let $w_{N,\ell }=1-f_{N,\ell }$ with $f_{\ell ,N}=f_{\ell }(e^{N}x)$. Then the Fourier coefficients of the function $w_{N,\ell }$ defined in (17) are such that

  $$\begin{array}{c}\hfill |{\widehat{w}}_{N,\ell }(p)|\le \frac{C}{p^{2}log(e^N\ell /\mathfrak{a})}.\end{array}$$ 22

Proof

The proof of points (i)–(iv) is deferred in Appendix B. To prove point v) we use the scattering equation (18):

$$\begin{array}{c}\hfill {\widehat{w}}_{\ell }(e^{-N}p)=\frac{e^{2N}}{2p^2}\sum _{q\in {\Lambda }^{\ast }}\widehat{V}(e^{-N}(p-q)){\widehat{f}}_{N,\ell }(q)-\frac{e^{4N}}{p^2}{\lambda }_{\ell }\sum _{q\in {\Lambda }^{\ast }}{\widehat{\chi }}_{\ell }(p-q){\widehat{f}}_{N,\ell }(q).\end{array}$$

Using the fact that $e^{2N}{\lambda }_{\ell }\le C{\ell }^{-2}{|ln(e^N\ell /\mathfrak{a})|}^{-1}$ and that $0\le f_{\ell }\le 1$, we end up with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |{\widehat{w}}_{\ell }(e^{-N}p)|& \le \frac{e^{2N}}{2p^2}\left[|,(\widehat{V}(e^{-N}·)\ast {\widehat{f}}_{N,\ell }),(p),|,+,2,e^{2N},{\lambda }_{\ell },,|,({\widehat{\chi }}_{\ell }\ast {\widehat{f}}_{N,\ell }),(p),|\right]\hfill \\ \hfill & \le \frac{e^{2N}}{2p^2}\left[\int ,V,(x),f_{\ell },(x),d,x,+,C,{\ell }^{-2},{|log(e^N\ell /\mathfrak{a})|}^{-1},,\int ,{\chi }_{\ell },(x),f_{\ell },(e^{N}x),d,x\right]\hfill \\ \hfill & \le \frac{C{e}^{2N}}{p^{2}log(e^N\ell /\mathfrak{a})}.\hfill \end{array}\end{array}$$

$\square$

We now define $\check{\eta }:\Lambda \to \mathbb{R}$ through

$$\begin{array}{c}\hfill \check{\eta }(x)=-N{w}_{N,\ell }(x)=-N{w}_{\ell }(e^{N}x).\end{array}$$ 23

With (21) we find

$$\begin{array}{c}\hfill |\check{\eta }(x)|\le \left\{\begin{array}{cc}CN\hfill & \text{if}|x|\le e^{-N}{R}_0\hfill \\ Clog(\ell /|x|)\hfill & \text{if}{e}^{-N}{R}_0\le |x|\le \ell \hfill \end{array}\right)\end{array}$$ 24

and in particular, recalling that $e^{-N}{R}_0<\ell \le 1/2$,

$$\begin{array}{c}\hfill |\check{\eta }(x)|\le Cmax(N,log(\ell /|x|))\le CN\end{array}$$ 25

for all $x\in \Lambda$. Using (24) we find

$$\begin{array}{c}\hfill {\Vert \eta \Vert }^2=\Vert \check{\eta }{\Vert }^2\le C{\int }_{|x|\le \ell }|log(\ell /{|x|)|}^2{d}^{2}x\le C{\ell }^2{\int }_0^1{(logr)}^{2}rdr\le C{\ell }^2.\end{array}$$

In the following we choose $\ell =N^{-\alpha }$, for some $\alpha >0$ to be fixed later, so that

$$\begin{array}{c}\hfill \Vert \eta \Vert \le C{N}^{-\alpha }.\end{array}$$ 26

This choice of $\ell$ will be crucial for our analysis, as commented below. Notice, on the other hand, that the $H^1$-norms of $\eta$ diverge, as $N\to \infty$. From (23) and Lemma 1, part iv) we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \Vert \check{\eta }{\Vert }_{H_1}^2={\int }_{|x|\le \ell }{e}^{2N}{N}^2{|(\mathrm{\nabla }{w}_{\ell })(e^{N}x)|}^2{d}^{2}x& ={\int }_{|x|\le e^N\ell }{N}^2{|\mathrm{\nabla }{w}_{\ell }(x)|}^2{d}^{2}x\hfill \\ \hfill & \le C{\int }_{|x|\le e^N\ell }\frac{1}{{(|x|+1)}^2}{d}^{2}x\le CN\hfill \end{array}\end{array}$$

for $N\in \mathbb{N}$ large enough. We denote with $\eta :{\Lambda }^{\ast }\to \mathbb{R}$ the Fourier transform of $\check{\eta }$, or equivalently

$$\begin{array}{c}\hfill {\eta }_p=-N{\widehat{w}}_{N,\ell }(p)=-N{e}^{-2N}{\widehat{w}}_{\ell }(p/e^N).\end{array}$$ 27

With (22) we can bound (since $\ell =N^{-\alpha }$)

$$\begin{array}{c}\hfill |{\eta }_p|\le \frac{C}{{|p|}^2}\end{array}$$ 28

for all $p\in {\Lambda }_{+}^{\ast }=2\pi {\mathbb{Z}}^2\backslash \{0\}$, and for some constant $C>0$ independent of N, if N is large enough. From (26) we also have

$$\begin{array}{c}\hfill {\Vert \eta \Vert }_{\infty }\le C{N}^{-\alpha }.\end{array}$$ 29

Moreover, (18) implies the relation

$$\begin{array}{c}\hfill \begin{array}{c}\hfill p^2{\eta }_p+\frac{N}{2}(\widehat{V}(./e^N)\ast {\widehat{f}}_{N,\ell })(p)=N{e}^{2N}{\lambda }_{\ell }({\widehat{\chi }}_{\ell }\ast {\widehat{f}}_{N,\ell })(p)\end{array}\end{array}$$ 30

or equivalently, expressing also the other terms through the coefficients ${\eta }_p$,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill p^2{\eta }_p+\frac{N}{2}\widehat{V}(p/e^N)& +\frac{1}{2}\sum _{q\in {\Lambda }^{\ast }}\widehat{V}((p-q)/e^N){\eta }_q\hfill \\ \hfill & =N{e}^{2N}{\lambda }_{\ell }{\widehat{\chi }}_{\ell }(p)+e^{2N}{\lambda }_{\ell }\sum _{q\in {\Lambda }^{\ast }}{\widehat{\chi }}_{\ell }(p-q){\eta }_q.\hfill \end{array}\end{array}$$ 31

We will mostly use the coefficients ${\eta }_p$ with $p\ne 0$. Sometimes, however, it will be useful to have an estimate on ${\eta }_0$ (because Eq. (31) involves ${\eta }_0$). From (27) and Lemma 1, part iv) we find

$$\begin{array}{c}\hfill |{\eta }_0|\le N{\int }_{|x|\le \ell }{w}_{\ell }(e^{N}x)d^{2}x\le C{\int }_{|x|\le \ell }log(\ell /|x|)d^{2}x+CN{e}^{-N}\le C{\ell }^2.\end{array}$$ 32

With the coefficients (27) we define the antisymmetric operator

$$\begin{array}{c}\hfill B=\frac{1}{2}\sum _{p\in {\Lambda }_{+}^{\ast }}\left({\eta }_p,b_p^{\ast },b_{-p}^{\ast },-,{\overline{\eta }}_p,b_p,b_{-p}\right)\end{array}$$ 33

and we consider the unitary operator

$$\begin{array}{c}\hfill e^B=exp\left[\frac{1}{2},\sum _{p\in {\Lambda }_{+}^{\ast }},\left({\eta }_p,b_p^{\ast },b_{-p}^{\ast },-,{\overline{\eta }}_p,b_p,b_{-p}\right)\right].\end{array}$$ 34

We refer to operators of the form (34) as generalized Bogoliubov transformations. In contrast with the standard Bogoliubov transformations

$$\begin{array}{c}\hfill e^{\tilde{B}}=exp\left[\frac{1}{2},\sum _{p\in {\Lambda }_{+}^{\ast }},\left({\eta }_p,a_p^{\ast },a_{-p}^{\ast },-,{\overline{\eta }}_p,a_p,a_{-p}\right)\right]\end{array}$$ 35

defined in terms of the standard creation and annihilation operators, operators of the form (34) leave the truncated Fock space ${\mathcal{F}}_{+}^{\le N}$ invariant. On the other hand, while the action of standard Bogoliubov transformation on creation and annihilation operators is explicitly given by

$$\begin{array}{c}\hfill e^{-\tilde{B}}{a}_p{e}^{\tilde{B}}=cosh({\eta }_p)a_p+sinh({\eta }_p)a_{-p}^{\ast }\end{array}$$

there is no such formula describing the action of generalized Bogoliubov transformations.

Conjugation with (34) leaves the number of particles essentially invariant, as confirmed by the following lemma.

Lemma 2

Assume B is defined as in (33), with $\eta \in {\ell }^2({\Lambda }^{\ast })$ and ${\eta }_p={\eta }_{-p}$ for all $p\in {\Lambda }_{+}^{\ast }$. Then, for every $n\in \mathbb{N}$ there exists a constant $C>0$ such that, on ${\mathcal{F}}_{+}^{\le N}$,

$$\begin{array}{c}\hfill e^{-B}{({\mathcal{N}}_{+}+1)}^n{e}^B\le C{e}^{C\Vert \eta \Vert }{({\mathcal{N}}_{+}+1)}^n.\end{array}$$ 36

as an operator inequality on ${\mathcal{F}}_{+}^{\le N}$.

The proof of (36) can be found in [6, Lemma 3.1] (a similar result has been previously established in [23]).

With the generalized Bogoliubov transformation $e^B:{\mathcal{F}}_{+}^{\le N}\to {\mathcal{F}}_{+}^{\le N}$, we define a new, renormalized, excitation Hamiltonian ${\mathcal{G}}_{N,\alpha }:{\mathcal{F}}_{+}^{\le N}\to {\mathcal{F}}_{+}^{\le N}$ by setting

$$\begin{array}{c}\hfill {\mathcal{G}}_{N,\alpha }=e^{-B}{\mathcal{L}}_N{e}^B=e^{-B}{U}_N{H}_N{U}_N^{\ast }{e}^B.\end{array}$$ 37

In the next proposition, we collect important properties ${\mathcal{G}}_{N,\alpha }$. We will use the notation

$$\begin{array}{c}\hfill \mathcal{K}=\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{a}_p^{\ast }{a}_p\text{and}{\mathcal{V}}_N=\frac{1}{2}\sum _{\begin{array}{c}p,q\in {\Lambda }_{+}^{\ast },r\in {\Lambda }^{\ast }:\\ r\ne -p,-q\end{array}}\widehat{V}(r/e^N)a_{p+r}^{\ast }{a}_q^{\ast }{a}_{q+r}{a}_p\end{array}$$ 38

for the kinetic and potential energy operators, restricted on ${\mathcal{F}}_{+}^{\le N}$, and ${\mathcal{H}}_N=\mathcal{K}+{\mathcal{V}}_N$. We also introduce a renormalized interaction potential ${\omega }_N\in L^{\infty }(\Lambda )$, which is defined as the function with Fourier coefficients ${\widehat{\omega }}_N$

$$\begin{array}{c}\hfill {\widehat{\omega }}_N(p):=g_N\widehat{\chi }(p/N^{\alpha }),g_N=2N^{1-2\alpha }{e}^{2N}{\lambda }_{\ell }\end{array}$$ 39

for any $p\in {\Lambda }_{+}^{\ast }$, and

$$\begin{array}{c}\hfill {\widehat{\omega }}_N(0)=g_N\widehat{\chi }(0)=\pi g_N.\end{array}$$ 40

with $\widehat{\chi }(p)$ the Fourier coefficients of the characteristic function of the ball of radius one. From (19) and $\ell =N^{-\alpha }$ one has $|g_N|\le C$. Note in particular that the potential ${\widehat{\omega }}_N(p)$ decays on momenta of order $N^{\alpha }$, which are much smaller than $e^N$. From Lemma 1 parts (i) and (iii) we find

$$\begin{array}{c}\hfill |{\widehat{\omega }}_N(0)-N{\Vert V{f}_{\ell }\Vert }_1|\le \frac{C}{N},|{\widehat{\omega }}_N(0)-4\pi \left(1,+,\alpha ,,\frac{logN}{N}\right)|\le \frac{C}{N}.\end{array}$$ 41

Proposition 1

Let $V\in L^3({\mathbb{R}}^2)$ be compactly supported, pointwise non-negative and spherically symmetric. Let ${\mathcal{G}}_{N,\alpha }$ be defined as in (37) and define

$$\begin{array}{cc}& {\mathcal{G}}_{N,\alpha }^{\text{eff}}:=\frac{1}{2}{\widehat{\omega }}_N(0)(N-1)\left(1,-,\frac{{\mathcal{N}}_{+}}{N}\right)+\left[2,N,\widehat{V},(0),-,\frac{1}{2},{\widehat{\omega }}_N,(0)\right]{\mathcal{N}}_{+}\left(1,-,\frac{{\mathcal{N}}_{+}}{N}\right)\hfill \\ \hfill & +\frac{1}{2}\sum _{p\in {\Lambda }_{+}^{\ast }}{\widehat{\omega }}_N(p)(b_p{b}_{-p}+\text{h.c.})+\sqrt{N}\sum _{\begin{array}{c}p,q\in {\Lambda }_{+}^{\ast }:\\ p+q\ne 0\end{array}}\widehat{V}(p/e^N)\left[b_{p+q}^{\ast },a_{-p}^{\ast },a_q,+,,\text{h.c.},\right]\hfill \\ \hfill & +{\mathcal{H}}_N.\hfill \end{array}$$ 42

Then there exists a constant $C>0$ such that ${\mathcal{E}}_{\mathcal{G}}={\mathcal{G}}_{N,\alpha }-{\mathcal{G}}_{N,\alpha }^{\text{eff}}$ is bounded by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\mathcal{E}}_{\mathcal{G}}\xi ⟩|\le & C(N^{1/2-\alpha }+N^{-1}{(logN)}^{1/2})\Vert {\mathcal{H}}_N^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \hfill \\ \hfill & +C{N}^{1-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\xi \Vert }^2+C{\Vert \xi \Vert }^2\hfill \end{array}\end{array}$$ 43

for all $\alpha >1$, $\xi \in {\mathcal{F}}_{+}^{\le N}$ and $N\in \mathbb{N}$ large enough.

The proof of Proposition 1 is very similar to the proof of [3, Prop. 4.2]. For completeness, we discuss the changes in Appendix A.

Cubic Renormalization

Conjugation with the generalized Bogoliubov transformation (35) renormalizes constant and off-diagonal quadratic terms on the r.h.s. of (42). In order to estimate the number of excitations ${\mathcal{N}}_{+}$ through the energy and show Bose–Einstein condensation, we still need to renormalize the diagonal quadratic term (the part proportional to $N\widehat{V}(0){\mathcal{N}}_{+}$, on the first line of (42)) and the cubic term on the second line of (42). To this end, we conjugate ${\mathcal{G}}_{N,\alpha }^{\text{eff}}$ with an additional unitary operator, given by the exponential of the anti-symmetric operator

$$\begin{array}{c}\hfill A:=\frac{1}{\sqrt{N}}\sum _{r,v\in {\Lambda }_{+}^{\ast }}{\eta }_r[b_{r+v}^{\ast }{a}_{-r}^{\ast }{a}_v-\text{h.c.}]\end{array}$$ 44

with ${\eta }_p$ defined in (27).

An important observation is that while conjugation with $e^A$ allows to renormalize the large terms in ${\mathcal{G}}_{N,\alpha }$, it does not substantially change the number of excitations. The following proposition can be proved similarly to [4, Proposition 5.1].

Proposition 2

Suppose that A is defined as in (44). Then, for any $k\in \mathbb{N}$ there exists a constant $C>0$ such that the operator inequality

$$\begin{array}{c}\hfill e^{-A}{({\mathcal{N}}_{+}+1)}^k{e}^A\le C{({\mathcal{N}}_{+}+1)}^k\end{array}$$

holds true on ${\mathcal{F}}_{+}^{\le N}$, for any $\alpha >0$ (recall the choice $\ell =N^{-\alpha }$ in the definition (27) of the coefficients ${\eta }_r$), and N large enough.

We will also need to control the growth of the expectation of the energy ${\mathcal{H}}_N$ with respect to the cubic conjugation. This is the content of the following proposition, which is proved in Sect. 6.1.

Proposition 3

Let A be defined as in (44). Then there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill e^{-sA}{\mathcal{H}}_N{e}^{\mathit{sA}}\le C{\mathcal{H}}_N+CN({\mathcal{N}}_{+}+1)\end{array}$$ 45

for all $\alpha \ge 1$, $s\in [0;1]$ and $N\in \mathbb{N}$ large enough.

We use now the cubic phase $e^A$ to introduce a new excitation Hamiltonian, obtained by conjugating the main part ${\mathcal{G}}_{N,\alpha }^{\text{eff}}$ of ${\mathcal{G}}_{N,\alpha }$. We define

$$\begin{array}{c}\hfill {\mathcal{R}}_{N,\alpha }:=e^{-A}{\mathcal{G}}_{N,\alpha }^{\text{eff}}{e}^A\end{array}$$ 46

on a dense subset of ${\mathcal{F}}_{+}^{\le N}$. Conjugation with $e^A$ renormalizes both the contribution proportional to ${\mathcal{N}}_{+}$ (in the first line on the r.h.s. of (42)) and the cubic term on the r.h.s. of (42), effectively replacing the singular potential $\widehat{V}(p/e^N)$ by the renormalized potential ${\widehat{\omega }}_N(p)$ defined in (39). This follows from the following proposition.

Proposition 4

Let $V\in L^3({\mathbb{R}}^2)$ be compactly supported, pointwise non-negative and spherically symmetric. Let ${\mathcal{R}}_{N,\alpha }$ be defined in (46) and define

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{R}}_{N,\alpha }^{\text{eff}}=& \frac{1}{2}(N-1){\widehat{\omega }}_N(0)(1-{\mathcal{N}}_{+}/N)+\frac{1}{2}{\widehat{\omega }}_N(0){\mathcal{N}}_{+}\left(1,-,{\mathcal{N}}_{+},/,N\right)\hfill \\ \hfill & +{\widehat{\omega }}_N(0)\sum _{p\in {\Lambda }_{+}^{\ast }}{a}_p^{\ast }{a}_p(1-\frac{{\mathcal{N}}_{+}}{N})+\frac{1}{2}\sum _{p\in {\Lambda }_{+}^{\ast }}{\widehat{\omega }}_N(p)[b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p}]\hfill \\ \hfill & +\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}r,v\in {\Lambda }_{+}^{\ast }:\\ r\ne -v\end{array}}{\widehat{\omega }}_N(r)[b_{r+v}^{\ast }{a}_{-r}^{\ast }{a}_v+\text{h.c.}]+{\mathcal{H}}_N.\hfill \end{array}\end{array}$$ 47

Then for $\ell =N^{-\alpha }$ and $\alpha >2$ there exists a constant $C>0$ such that ${\mathcal{E}}_{\mathcal{R}}={\mathcal{R}}_{N,\alpha }-{\mathcal{R}}_{N,\alpha }^{\text{eff}}$ is bounded by

$$\begin{array}{c}\hfill \pm {\mathcal{E}}_{\mathcal{R}}\le C[N^{2-\alpha }+N^{-1/2}{(logN)}^{1/2}]({\mathcal{H}}_N+1),\end{array}$$ 48

for $N\in \mathbb{N}$ sufficiently large.

The proof of Proposition 4 will be given in Sect. 6. We will also need more detailed information on ${\mathcal{R}}_{N,\alpha }^{\text{eff}}$, as contained in the following proposition.

Proposition 5

Let ${\mathcal{R}}_{N,\alpha }^{\text{eff}}$ be defined in (47). Then, for every $c>0$ there is a constant $C>0$ (large enough) such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{R}}_{N,\alpha }^{\text{eff}}\ge & 2\pi N+\frac{{\widehat{\omega }}_N(0)}{2}{\mathcal{N}}_{+}+\frac{c}{logN}{\mathcal{H}}_N-C{(logN)}^2\frac{{\mathcal{N}}_{+}^2}{N}-C\hfill \end{array}\end{array}$$ 49

for all $\alpha >2$ and $N\in \mathbb{N}$ large enough.

Moreover, let $f,g:\mathbb{R}\to [0;1]$ be smooth, with $f^2(x)+g^2(x)=1$ for all $x\in \mathbb{R}$. For $M\in \mathbb{N}$, let $f_M:=f({\mathcal{N}}_{+}/M)$ and $g_M:=g({\mathcal{N}}_{+}/M)$. Then there exists $C>0$ such that

$$\begin{array}{c}\hfill {\mathcal{R}}_{N,\alpha }^{\text{eff}}=f_M{\mathcal{R}}_{N,\alpha }^{\text{eff}}{f}_M+g_M{\mathcal{R}}_{N,\alpha }^{\text{eff}}{g}_M+{\Theta }_M\end{array}$$ 50

with

$$\begin{array}{c}\hfill \pm {\Theta }_M\le \frac{ClogN}{M^2}(\Vert f^{\prime }{\Vert }_{\infty }^2+{\Vert g^{\prime }\Vert }_{\infty }^2)({\mathcal{H}}_N+1)\end{array}$$

for all $\alpha >2$, $M\in \mathbb{N}$ and $N\in \mathbb{N}$ large enough.

Proof

From (47), using that $|{\widehat{\omega }}_N(0)|\le C$ we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{R}}_{N,\alpha }^{\text{eff}}\ge & \frac{N}{2}{\widehat{\omega }}_N(0)+{\widehat{\omega }}_N(0){\mathcal{N}}_{+}+\frac{1}{2}\sum _{p\in {\Lambda }_{+}^{\ast }}{\widehat{\omega }}_N(p)[b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p}]\hfill \\ & +\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}r,v\in {\Lambda }_{+}^{\ast }:\\ r\ne -v\end{array}}{\widehat{\omega }}_N(r)[b_{r+v}^{\ast }{a}_{-r}^{\ast }{a}_v+\text{h.c.}]+{\mathcal{H}}_N-C\frac{{\mathcal{N}}_{+}^2}{N}-C.\hfill \end{array}\end{array}$$ 51

For the cubic term on the r.h.s. of (51), with

$$\begin{array}{c}\hfill \sum _{p\in {\Lambda }_{+}^{\ast }}\frac{|{\widehat{\omega }}_N{(p)|}^2}{p^2}\le ClogN\end{array}$$ 52

we can bound

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |\frac{1}{\sqrt{N}}& \sum _{\begin{array}{c}r,v\in {\Lambda }_{+}^{\ast }\\ r\ne -v\end{array}}{\widehat{\omega }}_N(r)⟨\xi ,b_{r+v}^{\ast }{a}_{-r}^{\ast }{a}_v\xi ⟩|\hfill \\ \hfill & \le \frac{1}{\sqrt{N}}\sum _{\begin{array}{c}r,v\in {\Lambda }_{+}^{\ast }\\ r\ne -v\end{array}}|{\widehat{\omega }}_N(r)|\Vert {({\mathcal{N}}_{+}+1)}^{-1/2}{b}_{r+v}{a}_{-r}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}{a}_v\xi \Vert \hfill \\ \hfill & \le \frac{1}{\sqrt{N}}[\sum _{\begin{array}{c}r,v\in {\Lambda }_{+}^{\ast }\\ r\ne -v\end{array}}{|r|}^2{\Vert {({\mathcal{N}}_{+}+1)}^{-1/2}{b}_{r+v}{a}_{-r}\xi \Vert }^2{]}^{1/2}\hfill \\ \hfill & \times [\sum _{\begin{array}{c}r,v\in {\Lambda }_{+}^{\ast }\\ r\ne -v\end{array}}\frac{|{\widehat{\omega }}_N{(r)|}^2}{{|r|}^2}{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}{a}_v\xi \Vert }^2{]}^{1/2}\hfill \\ \hfill & \le \frac{C{(logN)}^{1/2}}{\sqrt{N}}\Vert {\mathcal{K}}^{1/2}\xi \Vert \Vert ({\mathcal{N}}_{+}+1)\xi \Vert .\hfill \end{array}\end{array}$$ 53

As for the off-diagonal quadratic term on the r.h.s of (51), we combine it with part of the kinetic energy to estimate. For any $0<\mu <1$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{1}{2}\sum _{p\in {\Lambda }_{+}^{\ast }}{\widehat{\omega }}_N(p)& [b_p^{\ast }{b}_{-p}^{\ast }+b_{-p}{b}_p]+(1-\mu )\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{a}_p^{\ast }{a}_p\hfill \\ \hfill =& (1-\mu )\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2\left[b_p^{\ast },+,\frac{{\widehat{\omega }}_N(p)}{2(1-\mu )p^2},b_{-p}\right]\left[b_p,+,\frac{{\widehat{\omega }}_N(p)}{2(1-\mu )p^2},b_{-p}^{\ast }\right]\hfill \\ \hfill & -\frac{1}{4(1-\mu )}\sum _{p\in {\Lambda }_{+}^{\ast }}\frac{|{\widehat{\omega }}_N{(p)|}^2}{p^2}{b}_p{b}_p^{\ast }+(1-\mu )\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{a}_p^{\ast }\frac{{\mathcal{N}}_{+}}{N}{a}_p\hfill \end{array}\end{array}$$ 54

since $a_p^{\ast }{a}_p-b_p^{\ast }{b}_p=a_p^{\ast }({\mathcal{N}}_{+}/N)a_p$. With (14), we conclude that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{1}{2}\sum _{p\in {\Lambda }_{+}^{\ast }}{\widehat{\omega }}_N(p)& [b_p^{\ast }{b}_{-p}^{\ast }+b_{-p}{b}_p]+(1-\mu )\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{a}_p^{\ast }{a}_p\hfill \\ \hfill \ge & -\frac{1}{4(1-\mu )}\sum _{p\in {\Lambda }_{+}^{\ast }}\frac{|{\widehat{\omega }}_N{(p)|}^2}{p^2}{a}_p^{\ast }{a}_p-\frac{1}{4(1-\mu )}\sum _{p\in {\Lambda }_{+}^{\ast }}\frac{|{\widehat{\omega }}_N{(p)|}^2}{p^2}.\hfill \end{array}\end{array}$$

With the choice $\mu =C/logN$ and with (52), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{1}{2}\sum _{p\in {\Lambda }_{+}^{\ast }}{\widehat{\omega }}_N(p)& [b_p^{\ast }{b}_{-p}^{\ast }+b_{-p}{b}_p]+(1-\mu )\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{a}_p^{\ast }{a}_p\hfill \\ \hfill \ge & -\frac{1}{4(1-\mu )}\sum _{p\in {\Lambda }_{+}^{\ast }}\frac{|{\widehat{\omega }}_N{(p)|}^2}{p^2}{a}_p^{\ast }{a}_p-\frac{1}{4}\sum _{p\in {\Lambda }_{+}^{\ast }}\frac{|{\widehat{\omega }}_N{(p)|}^2}{p^2}-C.\hfill \end{array}\end{array}$$ 55

To bound the first terms on the r.h.s. of the last equation, we use the term ${\widehat{\omega }}_N(0){\mathcal{N}}_{+}$, in (51). To this end, we observe that, with (41),

$$\begin{array}{c}\hfill \frac{|{\widehat{\omega }}_N{(p)|}^2}{4(1-\mu )p^2}\le \frac{|{\widehat{\omega }}_N{(0)|}^2}{4(1-\mu )p^2}\le \frac{{\widehat{\omega }}_N(0)}{4(1-\mu )\pi }\left(1,+,C,\frac{logN}{N}\right)\le \frac{{\widehat{\omega }}_N(0)}{2}\end{array}$$

for every $p\in {\Lambda }_{+}^{\ast }$ (notice that $|p|\ge 2\pi$, for every $p\in {\Lambda }_{+}^{\ast }$) and for N large enough (recall the choice $\mu =C/logN$). Inserting (53) and (55) in (51) and using the kinetic energy $\mu \mathcal{K}=C{(logN)}^{-1}\mathcal{K}$ (remaining after subtracting the term $(1-\mu )\mathcal{K}$ needed on the l.h.s. of (55)) to bound the r.h.s. of (53), we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{R}}_{N,\alpha }^{\text{eff}}\ge & \frac{N}{2}{\widehat{\omega }}_N(0)-\frac{1}{4}\sum _{p\in {\Lambda }_{+}^{\ast }}\frac{|{\widehat{\omega }}_N{(p)|}^2}{p^2}+\frac{{\widehat{\omega }}_N(0)}{2}{\mathcal{N}}_{+}+\frac{c}{logN}{\mathcal{H}}_N\hfill \\ \hfill & -C\frac{{(logN)}^2}{N}{\mathcal{N}}_{+}^2-C.\hfill \end{array}\end{array}$$ 56

Let us now consider the second term on the r.h.s more carefully. Using that, from (39), ${\widehat{\omega }}_N(p)=g_N\widehat{\chi }(p/N^{\alpha })$, we can bound, for any fixed $K>0$,

$$\begin{array}{c}\hfill \frac{1}{4}\sum _{p\in {\Lambda }_{+}^{\ast }}\frac{|{\widehat{\omega }}_N{(p)|}^2}{p^2}\le C+\frac{1}{4}\sum _{\begin{array}{c}p\in {\Lambda }_{+}^{\ast }:\\ K<|p|\le N^{\alpha }\end{array}}\frac{|{\widehat{\omega }}_N{(p)|}^2}{p^2}.\end{array}$$

With $|{\widehat{\omega }}_N(p)-{\widehat{\omega }}_N(0)|\le C|p|/N^{\alpha }$, we obtain

$$\begin{array}{c}\hfill \frac{1}{4}\sum _{p\in {\Lambda }_{+}^{\ast }}\frac{|{\widehat{\omega }}_N{(p)|}^2}{p^2}\le C+\frac{|{\widehat{\omega }}_N{(0)|}^2}{4}\sum _{\begin{array}{c}p\in {\Lambda }_{+}^{\ast }:\\ K<|p|\le N^{\alpha }\end{array}}\frac{1}{p^2}\le C+4{\pi }^2\sum _{\begin{array}{c}p\in {\Lambda }_{+}^{\ast }:\\ K<|p|\le N^{\alpha }\end{array}}\frac{1}{p^2}.\end{array}$$ 57

For $q\in {\mathbb{R}}^2$, let us define $h(q)=1/p^2$, if q is contained in the square of side length $2\pi$ centered at $p\in {\Lambda }_{+}^{\ast }$ (with an arbitrary choice on the boundary of the squares). We can then estimate, for K large enough,

$$\begin{array}{c}\hfill 4{\pi }^2\sum _{\begin{array}{c}p\in {\Lambda }_{+}^{\ast }:\\ K<|p|\le N^{\alpha }\end{array}}\frac{1}{p^2}\le {\int }_{K/2<|q|\le N^{\alpha }+K}h(q)dq.\end{array}$$

For q in the square centered at $p\in {\Lambda }_{+}^{\ast }$, we bound

$$\begin{array}{c}\hfill \left|h,(q),-,\frac{1}{q^2}\right|=\frac{|p^2-q^2|}{p^2{q}^2}\le \frac{C}{{|q|}^3}.\end{array}$$

Hence

$$\begin{array}{c}\hfill 4{\pi }^2\sum _{\begin{array}{c}p\in {\Lambda }_{+}^{\ast }:\\ K<|p|\le N^{\alpha }\end{array}}\frac{1}{p^2}\le {\int }_{K/2<|q|<N^{\alpha }+K}\frac{1}{q^2}dq+C\le 2\pi \alpha logN+C.\end{array}$$

Inserting in (57), we conclude that

$$\begin{array}{c}\hfill \frac{1}{4}\sum _{p\in {\Lambda }_{+}^{\ast }}\frac{|{\widehat{\omega }}_N{(p)|}^2}{p^2}\le 2\pi \alpha logN+C.\end{array}$$

Combining the last bound with (41) (and noticing that the contribution proportional to $logN$ cancels exactly), from (56) we obtain

$$\begin{array}{c}\hfill {\mathcal{R}}_{N,\alpha }^{\text{eff}}\ge 2\pi N+\frac{{\widehat{\omega }}_N(0)}{2}{\mathcal{N}}_{+}+\frac{c}{logN}{\mathcal{H}}_N-C\frac{{(logN)}^2}{N}{\mathcal{N}}_{+}^2-C\end{array}$$

which proves (49).

Next we prove (50). From (47), with $|{\widehat{\omega }}_N(0)|\le C$, the bound (53) and since, by (52),

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left|\sum _{p\in {\Lambda }_{+}^{\ast }},{\widehat{\omega }}_N,(p),⟨\xi ,b_p^{\ast }{b}_{-p}^{\ast }\xi ⟩\right|& \le \sum _{p\in {\Lambda }_{+}^{\ast }}|{\widehat{\omega }}_N(p)|\Vert b_p\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \hfill \\ \hfill & \le {\left[\sum _{p\in {\Lambda }_{+}^{\ast }},\frac{|{\widehat{\omega }}_N{(p)|}^2}{p^2}\right]}^{1/2}\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{K}}^{1/2}\xi \Vert \hfill \\ \hfill & \le C{(logN)}^{1/2}\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{K}}^{1/2}\xi \Vert \hfill \end{array}\end{array}$$

it follows that

$$\begin{array}{c}\hfill {\mathcal{R}}_{N,\alpha }^{\text{eff}}=2\pi N+{\mathcal{H}}_N+{\theta }_{N,\alpha }\end{array}$$ 58

where for arbitrary $\delta >0$, there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill \pm {\theta }_{N,\alpha }\le \delta {\mathcal{H}}_N+C(logN)({\mathcal{N}}_{+}+1).\end{array}$$ 59

We now note that for $f:\mathbb{R}\to \mathbb{R}$ smooth and bounded and ${\theta }_{N,\alpha }$ defined above, there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill \pm [f({\mathcal{N}}_{+}/M),[f({\mathcal{N}}_{+}/M),{\theta }_{N,\alpha }]]\le C\frac{logN}{M^2}{\Vert f^{\prime }\Vert }_{\infty }^2({\mathcal{H}}_N+1)\end{array}$$ 60

for all $\alpha >2$ and $N\in \mathbb{N}$ large enough. The proof of (60) follows analogously to the one for (59), since the bounds leading to (59) remain true if we replace the operators $b_p^{\#}$, $\#=\{·,\ast \}$, and $a_p^{\ast }{a}_q$ with $[f({\mathcal{N}}_{+}/M),[f({\mathcal{N}}_{+}/M),b_p^{\#}]]$ or $[f({\mathcal{N}}_{+}/M),[f({\mathcal{N}}_{+}/M),a_p^{\ast }{a}_q]]$ respectively, provided we multiply the r.h.s. by an additional factor $M^{-2}{\Vert f^{\prime }\Vert }_{\infty }^2$, since, for example

$$\begin{array}{c}\hfill [f({\mathcal{N}}_{+}/M),[f({\mathcal{N}}_{+}/M),b_p]]=(f({\mathcal{N}}_{+}/M)-f(({\mathcal{N}}_{+}+1)/M){)}^2{b}_p\end{array}$$

and $\Vert f({\mathcal{N}}_{+}/M)-f(({\mathcal{N}}_{+}+1)/M)\Vert \le C{M}^{-1}{\Vert f^{\prime }\Vert }_{\infty }$. With an explicit computation we obtain

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathcal{R}}_{N,\alpha }^{\text{eff}}=f_M{\mathcal{R}}_{N,\alpha }^{\text{eff}}{f}_M+g_M{\mathcal{R}}_{N,\alpha }^{\text{eff}}{g}_M+\frac{1}{2}([f_M,[f_M,{\mathcal{R}}_{N,\alpha }^{\text{eff}}]]+[g_M,[g_M,{\mathcal{R}}_{N,\alpha }^{\text{eff}}]])\end{array}\end{array}$$

Writing ${\mathcal{R}}_{N,\alpha }^{\text{eff}}$ as in (58) and using (60) we get

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \pm ([f_M,[f_M,{\mathcal{R}}_{N,\alpha }^{\text{eff}}]]+[g_M,[g_M,{\mathcal{R}}_{N,\alpha }^{\text{eff}}]])\le \frac{ClogN}{M^2}(\Vert f^{\prime }{\Vert }_{\infty }^2+{\Vert g^{\prime }\Vert }_{\infty }^2)({\mathcal{H}}_N+1).\end{array}\end{array}$$

$\square$

Proof of Theorem 1

The next proposition combines the results of Propositions 1, 4 and 5. Its proof makes use of localization in the number of particle and is an adaptation of the proof of [4, Proposition 6.1]. The main difference w.r.t. [4] is that here we need to localize on sectors of ${\mathcal{F}}^{\le N}$ where the number of particles is o(N), in the limit $N\to \infty$.

Proposition 6

Let $V\in L^3({\mathbb{R}}^2)$ be compactly supported, pointwise non-negative and spherically symmetric. Let ${\mathcal{G}}_{N,\alpha }$ be the renormalized excitation Hamiltonian defined as in (37). Then, for every $\alpha \ge 5/2$, there exist constants $C,c>0$ such that

$$\begin{array}{c}\hfill {\mathcal{G}}_{N,\alpha }-2\pi N\ge c{\mathcal{N}}_{+}-C\end{array}$$ 61

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

Proof

Let $f,g:\mathbb{R}\to [0;1]$ be smooth, with $f^2(x)+g^2(x)=1$ for all $x\in \mathbb{R}$. Moreover, assume that $f(x)=0$ for $x>1$ and $f(x)=1$ for $x<1/2$. For a small $\epsilon >0$, we fix $M=N^{1-\epsilon }$ and we set $f_M=f({\mathcal{N}}_{+}/M),g_M=g({\mathcal{N}}_{+}/M)$. It follows from Proposition 5 that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{R}}_{N,\alpha }^{\text{eff}}-2\pi N\ge & f_M({\mathcal{R}}_{N,\alpha }^{\text{eff}}-2\pi N)f_M+g_M({\mathcal{R}}_{N,\alpha }^{\text{eff}}-2\pi N)g_M\hfill \\ \hfill & -C{N}^{2\epsilon -2}(logN)({\mathcal{H}}_N+1)\hfill \end{array}\end{array}$$ 62

Let us consider the first term on the r.h.s. of (62). From Proposition 5, for all $\alpha >2$ there exist $c,C>0$ such that

$$\begin{array}{c}\hfill {\mathcal{R}}_{N,\alpha }^{\text{eff}}-2\pi N\ge c{\mathcal{N}}_{+}-\frac{C}{N}{(logN)}^2{\mathcal{N}}_{+}^2-C.\end{array}$$ 63

On the other hand, with (58) and (59) we also find

$$\begin{array}{c}\hfill {\mathcal{R}}_{N,\alpha }^{\text{eff}}-2\pi N\ge c{\mathcal{H}}_N-C(logN)({\mathcal{N}}_{+}+1)\end{array}$$ 64

for all $\alpha >2$ and N large enough. Moreover, due to the choice $M=N^{1-\epsilon }$, we have

$$\begin{array}{c}\hfill \frac{{(logN)}^2}{N}{f}_M{\mathcal{N}}_{+}^2{f}_M\le \frac{{(logN)}^2}{N^{\epsilon }}{f}_M^2{\mathcal{N}}_{+}.\end{array}$$

With the last bound, Eq. (63) implies that

$$\begin{array}{c}\hfill f_M({\mathcal{R}}_{N,\alpha }^{\text{eff}}-2\pi N)f_M\ge c{f}_M^2{\mathcal{N}}_{+}-C\end{array}$$ 65

for N large enough.

Let us next consider the second term on the r.h.s. of (62). We claim that there exists a constant $c>0$ such that

$$\begin{array}{c}\hfill g_M({\mathcal{R}}_{N,\alpha }^{\text{eff}}-2\pi N)g_M\ge cN{g}_M^2\end{array}$$ 66

for all N sufficiently large. To prove (66) we observe that, since $g(x)=0$ for all $x\le 1/2$,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill g_M({\mathcal{R}}_{N,\alpha }^{\text{eff}}-2\pi N& )g_M\ge \left[\underset{\xi \in {\mathcal{F}}_{\ge M/2}^{\le N}:\Vert \xi \Vert =1}{inf},\frac{1}{N},⟨\xi ,{\mathcal{R}}_{N,\alpha }^{\text{eff}}\xi ⟩,-,2,\pi \right]N{g}_M^2\hfill \end{array}\end{array}$$

where ${\mathcal{F}}_{\ge M/2}^{\le N}=\{\xi \in {\mathcal{F}}_{+}^{\le N}:\xi =\chi ({\mathcal{N}}_{+}\ge M/2)\xi \}$ is the subspace of ${\mathcal{F}}_{+}^{\le N}$ where states with at least M/2 excitations are described (recall that $M=N^{1-\epsilon }$). To prove (66) it is enough to show that there exists $C>0$ with

$$\begin{array}{c}\hfill \underset{\xi \in {\mathcal{F}}_{\ge M/2}^{\le N}:\Vert \xi \Vert =1}{inf}\frac{1}{N}⟨\xi ,{\mathcal{R}}_{N,\alpha }^{\text{eff}}\xi ⟩-2\pi \ge C\end{array}$$ 67

for all N large enough. On the other hand, using the definitions of ${\mathcal{G}}_{N,\alpha }$ in (42), ${\mathcal{R}}_{N,\alpha }$ and ${\mathcal{R}}_{N,\alpha }^{\text{eff}}$ in (47), we obtain that the ground state energy $E_N$ of the system is given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill E_N& =\underset{\xi \in {\mathcal{F}}_{+}^{\le N}:\Vert \xi \Vert =1}{inf}\langle \xi ,e^{-A}{\mathcal{G}}_{N,\alpha }{e}^A\xi \rangle =\underset{\xi \in {\mathcal{F}}_{+}^{\le N}:\Vert \xi \Vert =1}{inf}\langle \xi ,({\mathcal{R}}_{N,\alpha }^{\text{eff}}+{\mathcal{E}}_L)\xi \rangle \hfill \end{array}\end{array}$$

with ${\mathcal{E}}_L={\mathcal{E}}_{\mathcal{R}}+e^{-A}{\mathcal{E}}_{\mathcal{G}}{e}^A$. The bounds (43) and (48), together with Propositions 2 and 3, imply that for any $\alpha \ge 5/2$ there exists $C>0$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \pm {\mathcal{E}}_L& \le C{N}^{-1/2}{(logN)}^{1/2}[({\mathcal{H}}_N+1)+e^{-A}(N^{-1}({\mathcal{H}}_N+1)+({\mathcal{N}}_{+}+1))e^A]+C\hfill \\ \hfill & \le C{N}^{-1/2}{(logN)}^{1/2}({\mathcal{H}}_N+1)+C\hfill \end{array}\end{array}$$

With (64) we obtain

$$\begin{array}{c}\hfill \pm {\mathcal{E}}_L\le C{N}^{-1/2}{(logN)}^{1/2}({\mathcal{R}}_{N,\alpha }^{\text{eff}}-2\pi N)+C{N}^{-1/2}{(logN)}^{3/2}{\mathcal{N}}_{+}+C,\end{array}$$ 68

and therefore, with ${\mathcal{N}}_{+}\le N$

$$\begin{array}{c}\hfill E_N-2\pi N\le C\underset{\xi \in {\mathcal{F}}_{+}^{\le N}:\Vert \xi \Vert =1}{inf}\langle \xi ,({\mathcal{R}}_{N,\alpha }^{\text{eff}}-2\pi N)\xi \rangle +C{N}^{1/2}{(logN)}^{3/2}+C.\end{array}$$

From the result (3) of [13, 14, 16]

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \underset{\xi \in {\mathcal{F}}_{\ge M/2}^{\le N}:\Vert \xi \Vert =1}{inf}\frac{1}{N}⟨\xi ,{\mathcal{R}}_{N,\alpha }^{\text{eff}}\xi ⟩-2\pi & \ge \underset{\xi \in {\mathcal{F}}_{+}^{\le N}:\Vert \xi \Vert =1}{inf}\frac{1}{N}⟨\xi ,({\mathcal{R}}_{N,\alpha }^{\text{eff}}-2\pi N)\xi ⟩\hfill \\ \hfill & \ge c\left(\frac{E_N}{N},-,2,\pi \right)-\frac{C}{\sqrt{N}}{(logN)}^{3/2}-C{N}^{-1}\to 0\hfill \end{array}\end{array}$$

as $N\to \infty$. If we assume by contradiction that (67) does not hold true, then we can find a subsequence $N_j\to \infty$ with

$$\begin{array}{c}\hfill \underset{\xi \in {\mathcal{F}}_{\ge M_j/2}^{\le N_j}:\Vert \xi \Vert =1}{inf}\frac{1}{N_j}⟨\xi ,{\mathcal{R}}_{N_j,\alpha }^{\text{eff}}\xi ⟩-2\pi \to 0\end{array}$$

as $j\to \infty$ (here we used the notation $M_j=N_j^{1-\epsilon }$). This implies that there exists a sequence ${\tilde{\xi }}_{N_j}\in {\mathcal{F}}_{\ge M_j/2}^{\le N_j}$ with $\Vert {\tilde{\xi }}_{N_j}\Vert =1$ for all $j\in \mathbb{N}$ such that

$$\begin{array}{c}\hfill \underset{j\to \infty }{lim}\frac{1}{N_j}⟨{\tilde{\xi }}_{N_j},{\mathcal{R}}_{N_j,\alpha }^{\text{eff}}{\tilde{\xi }}_{N_j}⟩=2\pi .\end{array}$$

On the other hand, using the relation ${\mathcal{R}}_{N_j,\alpha }^{\text{eff}}=e^{-A}{\mathcal{G}}_{N_j,\alpha }{e}^A-{\mathcal{E}}_{L,j}$ with ${\mathcal{E}}_{L,j}$ satisfying the bound (68) (with ${\mathcal{N}}_{+}\le N_j$), we obtain that there exist constants $c_1,c_2,C>0$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}& c_1⟨{\tilde{\xi }}_{N_j},({\mathcal{R}}_{N,\alpha }^{\text{eff}}-2\pi N_j){\tilde{\xi }}_{N_j}⟩-C{N}_j^{1/2}{(log{N}_j)}^{3/2}\hfill \\ \hfill & \le ⟨e^A{\tilde{\xi }}_{N_j},({\mathcal{G}}_{N_j,\alpha }-2\pi N_j)e^A{\tilde{\xi }}_{N_j}⟩\hfill \\ \hfill & \le c_2⟨{\tilde{\xi }}_{N_j},({\mathcal{R}}_{N,\alpha }^{\text{eff}}-2\pi N_j){\tilde{\xi }}_{N_j}⟩+C{N}_j^{1/2}{(log{N}_j)}^{3/2}\hfill \end{array}\end{array}$$

Hence for ${\xi }_{N_j}=e^A{\tilde{\xi }}_{N_j}$ we have

$$\begin{array}{c}\hfill \underset{N_j\to \infty }{lim}\frac{1}{N_j}⟨{\xi }_{N_j},{\mathcal{G}}_{N_j,\alpha }{\xi }_{N_j}⟩=2\pi .\end{array}$$

Let now $S:=\{N_j:j\in \mathbb{N}\}\subset \mathbb{N}$ and denote by ${\xi }_N$ a normalized minimizer of ${\mathcal{G}}_{N,\alpha }$ for all $N\in \mathbb{N}\backslash S$. Setting ${\psi }_N=U_N^{\ast }{e}^B{\xi }_N$, for all $N\in \mathbb{N}$, we obtain that $\Vert {\psi }_N\Vert =1$ and that

$$\begin{array}{c}\hfill \underset{N\to \infty }{lim}\frac{1}{N}⟨{\psi }_N,H_N{\psi }_N⟩=\underset{N\to \infty }{lim}\frac{1}{N}⟨{\xi }_N,{\mathcal{G}}_{N,\alpha }{\xi }_N⟩=2\pi \end{array}$$ 69

Eq. (69) shows that the sequence ${\psi }_N$ is an approximate ground state of $H_N$. From (5), we conclude that ${\psi }_N$ exhibits complete Bose–Einstein condensation in the zero-momentum mode ${\phi }_0$, and in particular that there exists $\overline{\delta }>0$ such that

$$\begin{array}{c}\hfill \left|1,-,⟨,{\phi }_0,,,{\gamma }_N,{\phi }_0,⟩\right|\le C{N}^{-\overline{\delta }}.\end{array}$$

Using Lemma 2, Proposition 2 and the rules (11), we observe that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{1}{N}⟨{\xi }_N,{\mathcal{N}}_{+}{\xi }_N⟩& =\frac{1}{N}⟨e^{-B}{U}_N{\psi }_N,{\mathcal{N}}_{+}{e}^{-B}{U}_N{\psi }_N⟩\hfill \\ \hfill & \le \frac{C}{N}⟨{\psi }_N,U_N^{\ast }({\mathcal{N}}_{+}+1)U_N{\psi }_N⟩\hfill \\ \hfill & =\frac{C}{N}+C\left[1,-,\frac{1}{N},⟨{\psi }_N,a^{\ast }({\phi }_0)a({\phi }_0){\psi }_N⟩\right]\hfill \\ \hfill & =\frac{C}{N}+C\left[1,-,⟨,{\phi }_0,,,{\gamma }_N,{\phi }_0,⟩\right]\le C{N}^{-\overline{\delta }}\hfill \end{array}\end{array}$$ 70

as $N\to \infty$.

On the other hand, for $N\in S=\{N_j:j\in \mathbb{N}\}$, we have ${\xi }_N=\chi ({\mathcal{N}}_{+}\ge M/2){\xi }_N$ and therefore

$$\begin{array}{c}\hfill \frac{1}{N}⟨{\xi }_N,{\mathcal{N}}_{+}{\xi }_N⟩\ge \frac{M}{2N}=\frac{N^{-\epsilon }}{2}.\end{array}$$

Choosing $\epsilon <\overline{\delta }$ and N large enough we get a contradiction with (70). This proves (67), (66) and therefore also

$$\begin{array}{c}\hfill g_M({\mathcal{R}}_{N,\alpha }^{\text{eff}}-2\pi N)g_M\ge c{\mathcal{N}}_{+}{g}_M^2.\end{array}$$ 71

Inserting (65) and (71) on the r.h.s. of (62), we obtain that

$$\begin{array}{c}\hfill {\mathcal{R}}_{N,\alpha }^{\text{eff}}-2\pi N\ge c{\mathcal{N}}_{+}-C(logN)N^{2\epsilon -2}({\mathcal{H}}_N+1)-C\end{array}$$ 72

for N large enough. With (64), (72) implies

$$\begin{array}{c}\hfill {\mathcal{R}}_{N,\alpha }^{\text{eff}}-2\pi N\ge c{\mathcal{N}}_{+}-C.\end{array}$$

To conclude, we use the relation $e^{-A}{\mathcal{G}}_{N,\alpha }{e}^A={\mathcal{R}}_{N,\alpha }^{\text{eff}}+{\mathcal{E}}_L$ and the bound (68). We have that for $\alpha \ge 5/2$ there exist $c,C>0$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{G}}_{N,\alpha }-2\pi N& \ge c{e}^A({\mathcal{R}}_{N,\alpha }^{\text{eff}}-2\pi N)e^{-A}-C{N}^{-1/2}{(logN)}^{3/2}{e}^A{\mathcal{N}}_{+}{e}^A-C\hfill \\ \hfill & \ge c{e}^A{\mathcal{N}}_{+}{e}^{-A}-C\ge c{\mathcal{N}}_{+}-C\hfill \end{array}\end{array}$$

where we used (72) and Proposition 2. $\square$

We are now ready to show our main theorem.

Proof of Theorem 1

Let $E_N$ be the ground state energy of $H_N$. Evaluating (42) and (43) on the vacuum $\Omega \in {\mathcal{F}}_{+}^{\le N}$ and using (40), we obtain the upper bound

$$\begin{array}{c}\hfill E_N\le 2\pi N+ClogN.\end{array}$$

Notice that we cannot reach the expected optimal upper bound $E_N\le 2\pi N+C$ because of the logarithmic correction in ${\widehat{\omega }}_N(0)$ (see (40)). In the lower bound, this logarithmic factor is compensated by the contribution arising from the off-diagonal quadratic term, extracted starting from (54). To obtain the same term for the upper bound, we would have to modify our trial state (diagonalizing the quadratic terms in ${\mathcal{R}}_{N,\alpha }$); this, however, would produce even larger contributions arising from the potential energy.

With Eq. (61) we also find the lower bound $E_N\ge 2\pi N-C$. This proves (6).

Let now ${\psi }_N\in L_s^2({\Lambda }^N)$ with $\Vert {\psi }_N\Vert =1$ and

$$\begin{array}{c}\hfill ⟨{\psi }_N,H_N{\psi }_N⟩\le 2\pi N+K.\end{array}$$ 73

We define the excitation vector ${\xi }_N=e^{-B}{U}_N{\psi }_N$. Then $\Vert {\xi }_N\Vert =1$ and, recalling that ${\mathcal{G}}_{N,\alpha }=e^{-B}{U}_N{H}_N{U}_N^{\ast }{e}^B$ we have, with (61),

74

From Eqs. (73) and (74) we conclude that

$$\begin{array}{c}\hfill ⟨{\xi }_N,{\mathcal{N}}_{+}{\xi }_N⟩\le C(1+K).\end{array}$$

If ${\gamma }_N$ denotes the one-particle reduced density matrix associated with ${\psi }_N$, using Lemma 2 we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill 1-⟨{\phi }_0,{\gamma }_N{\phi }_0⟩& =1-\frac{1}{N}⟨{\psi }_N,a^{\ast }({\phi }_0)a({\phi }_0){\psi }_N⟩\hfill \\ \hfill & =1-\frac{1}{N}⟨U_N^{\ast }{e}^B{\xi }_N,a^{\ast }({\phi }_0)a({\phi }_0)U_N^{\ast }{e}^B{\xi }_N⟩\hfill \\ \hfill & =\frac{1}{N}⟨e^B{\xi }_N,{\mathcal{N}}_{+}{e}^B{\xi }_N⟩\le \frac{C}{N}⟨{\xi }_N,{\mathcal{N}}_{+}{\xi }_N⟩\le \frac{C(1+K)}{N}\hfill \end{array}\end{array}$$

which concludes the proof of (8). $\square$

Analysis of the Excitation Hamiltonian ${\mathcal{R}}_N$

In this section, we show Proposition 4, where we establish a lower bound for the operator ${\mathcal{R}}_{N,\alpha }=e^{-A}{\mathcal{G}}_{N,\alpha }^{\text{eff}}{e}^A$, with ${\mathcal{G}}_{N,\alpha }^{\text{eff}}$ as defined in (42) and with

$$\begin{array}{c}\hfill A=\frac{1}{\sqrt{N}}\sum _{r,v\in {\Lambda }_{+}^{\ast }}{\eta }_r[b_{r+v}^{\ast }{a}_{-r}^{\ast }{a}_v-\text{h.c.}].\end{array}$$ 75

We decompose

$$\begin{array}{c}\hfill {\mathcal{G}}_{N,\alpha }^{\text{eff}}={\mathcal{O}}_N+\mathcal{K}+{\mathcal{Z}}_N+{\mathcal{C}}_N+{\mathcal{V}}_N\end{array}$$ 76

with $\mathcal{K}$ and ${\mathcal{V}}_N$ as in (38), and with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{O}}_N=& \frac{1}{2}{\widehat{\omega }}_N(0)(N-1)(1-\frac{{\mathcal{N}}_{+}}{N})+[2N\widehat{V}(0)-\frac{1}{2}{\widehat{\omega }}_N(0)]{\mathcal{N}}_{+}(1-\frac{{\mathcal{N}}_{+}}{N}),\hfill \\ \hfill {\mathcal{Z}}_N=& \frac{1}{2}\sum _{p\in {\Lambda }_{+}^{\ast }}{\widehat{\omega }}_N(p)(b_p{b}_{-p}+\text{h.c.})\hfill \\ \hfill {\mathcal{C}}_N=& \sqrt{N}\sum _{p,q\in {\Lambda }_{+}^{\ast }:p+q\ne 0}\widehat{V}(p/e^N)\left[b_{p+q}^{\ast },a_{-p}^{\ast },a_q,+,,\text{h.c.},\right].\hfill \end{array}\end{array}$$ 77

We will analyze the conjugation of all terms on the r.h.s. of (76) in Sects. 6.2–6.6. The estimates emerging from these subsections will then be combined in Sect. 6.6 to conclude the proof of Proposition 4. Throughout the section, we will need Proposition 3 to control the growth of the expectation of the energy ${\mathcal{H}}_N=\mathcal{K}+{\mathcal{V}}_N$ under the action of (75); the proof of Proposition 3 is contained in Sect. 6.1.

In this section, we will always assume that $V\in L^3({\mathbb{R}}^2)$ is compactly supported, pointwise non-negative and spherically symmetric.

A Priori Bounds on the Energy

In this section, we show Proposition 3. To this end, we will need the following proposition.

Proposition 7

Let ${\mathcal{V}}_N$ and A be defined in (38) and (44) respectively. Then, there exists a constant $C>0$ such that

$$\begin{array}{ccc}& [\hfill & \hfill {\mathcal{V}}_N,A]=\frac{1}{N^{1/2}}\sum _{\begin{array}{c}u,r,v\in {\Lambda }_{+}^{\ast }\\ u\ne -v\end{array}}\widehat{V}((u-r)/e^N){\eta }_r[b_{u+v}^{\ast }{a}_{-u}^{\ast }{a}_v+\text{h.c.}]+{\delta }_{{\mathcal{V}}_N}\end{array}$$

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\delta }_{{\mathcal{V}}_N}\xi ⟩|\le & C{(logN)}^{1/2}{N}^{1/2-\alpha }{\Vert {\mathcal{H}}_N^{1/2}\xi \Vert }^2\hfill \end{array}\end{array}$$ 78

for any $\alpha >0$, for all $\xi \in {\mathcal{F}}_{+}^{\le N}$, and $N\in \mathbb{N}$ large enough.

Proof

We proceed as in [4, Prop. 8.1], computing $[a_{p+u}^{\ast }{a}_q^{\ast }{a}_p{a}_{q+u},b_{r+v}^{\ast }{a}_{-r}^{\ast }{a}_v]$. We obtain

$$\begin{array}{c}\hfill [{\mathcal{V}}_N,A]=\frac{1}{N^{1/2}}\sum _{u\in {\Lambda }^{\ast },r,v\in {\Lambda }_{+}^{\ast }}^{\ast }\widehat{V}((u-r)/e^N){\eta }_r{b}_{u+v}^{\ast }{a}_{-u}^{\ast }{a}_v+{\Theta }_1+{\Theta }_2+{\Theta }_3+\text{h.c.}\end{array}$$

with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\Theta }_1& :=\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}u\in {\Lambda }^{\ast }\\ r,p,v\in {\Lambda }_{+}^{\ast }\end{array}}^{\ast }\widehat{V}(u/e^N){\eta }_r{b}_{p+u}^{\ast }{a}_{r+v-u}^{\ast }{a}_{-r}^{\ast }{a}_p{a}_v,\hfill \\ \hfill {\Theta }_2& :=\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}u\in {\Lambda }^{\ast }\\ p,r,v\in {\Lambda }_{+}^{\ast }\end{array}}^{\ast }\widehat{V}(u/e^N){\eta }_r{b}_{r+v}^{\ast }{a}_{p+u}^{\ast }{a}_{-r-u}^{\ast }{a}_p{a}_v,\hfill \\ \hfill {\Theta }_3& :=-\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}u\in {\Lambda }^{\ast },p,r,v\in {\Lambda }_{+}^{\ast }\end{array}}^{\ast }\widehat{V}(u/e^N){\eta }_r{b}_{r+v}^{\ast }{a}_{-r}^{\ast }{a}_{p+u}^{\ast }{a}_p{a}_{v+u}.\hfill \end{array}\end{array}$$ 79

and with ${\sum }^{\ast }$ running over all momenta, except choices for which the argument of a creation or annihilation operator vanishes. We conclude that ${\delta }_{{\mathcal{V}}_N}={\Theta }_1+{\Theta }_2+{\Theta }_3+\text{h.c.}$. Next, we show that each error term ${\Theta }_j$, with $j=1,2,3$, satisfies (78). To bound ${\Theta }_1$ we switch to position space and apply Cauchy–Schwarz. We find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\Theta }_1\xi ⟩|\le & \frac{1}{\sqrt{N}}{\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y))\Vert \check{a}({\check{\eta }}_y){\check{a}}_y{\check{a}}_x\xi \Vert \Vert {\check{a}}_y{\check{a}}_x\xi \Vert \hfill \\ \hfill \le & C\Vert \eta \Vert {\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y)){\Vert {\check{a}}_y{\check{a}}_x\xi \Vert }^2\hfill \\ \hfill \le & C{N}^{-\alpha }{\Vert {\mathcal{V}}_N^{1/2}\xi \Vert }^2,\hfill \end{array}\end{array}$$

for any $\xi \in {\mathcal{F}}_{+}^{\le N}$ The term ${\Theta }_3$ can be controlled similarly. We find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\Theta }_3\xi ⟩|=& |\frac{1}{\sqrt{N}}{\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y))⟨\xi ,{\check{b}}_x^{\ast }{\check{a}}^{\ast }({\check{\eta }}_x){\check{a}}_y^{\ast }{\check{a}}_x{\check{a}}_y\xi ⟩|\hfill \\ \hfill & \le C{N}^{-\alpha }{\Vert {\mathcal{V}}_N^{1/2}\xi \Vert }^2.\hfill \end{array}\end{array}$$

It remains to bound the term ${\Theta }_2$ on the r.h.s. of (79). Passing to position space we obtain, by Cauchy–Schwarz,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\Theta }_2\xi ⟩|=& |\frac{1}{\sqrt{N}}{\int }_{{\Lambda }^3}dxdydz{e}^{2N}V(e^N(y-z))\check{\eta }(x-z)⟨\xi ,{\check{b}}_x^{\ast }{\check{a}}_y^{\ast }{\check{a}}_z^{\ast }{\check{a}}_x{\check{a}}_y\xi ⟩|\hfill \\ \hfill \le & C{N}^{-1/2}{\int }_{{\Lambda }^3}dxdydz{e}^{2N}V(e^N(y-z))|\check{\eta }(x-z)|\Vert {\check{a}}_x{\check{a}}_y{\check{a}}_z\xi \Vert \Vert {\check{a}}_x{\check{a}}_y\xi \Vert \hfill \\ \hfill \le & C{N}^{-1/2}\Vert {\mathcal{V}}_N^{1/2}{\mathcal{N}}_{+}^{1/2}\xi \Vert {\left[{\int }_{{\Lambda }^3},d,x,d,y,d,z,,e^{2N},V,(e^N(y-z)),|,\check{\eta },{(x-z)|}^2,{\Vert {\check{a}}_x{\check{a}}_y\xi \Vert }^2\right]}^{1/2},\hfill \end{array}\end{array}$$

To bound the term in the square bracket, we write it in first quantized form and, for any $2<q<\infty$, we apply Hölder inequality and the Sobolev inequality ${\Vert u\Vert }_q\le C\sqrt{q}{\Vert u\Vert }_{H^1}$ to estimate (denoting by $1<q^{\prime }<2$ the dual index to q),

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \sum _{n=2}^N& \sum _{i<j}^n\int \left[e^{2N},V,(e^N·),,\ast ,{|\check{\eta }|}^2\right](x_i-x_j){|{\xi }^{(n)}(x_1,\dots ,x_n)|}^{2}d{x}_1\dots d{x}_n\hfill \\ \hfill & \le Cq\Vert e^{2N}V(e^N·)\ast |\check{\eta }{{|}^2\Vert }_{q^{\prime }}\hfill \\ \hfill & \times \sum _{n=2}^{N}n\sum _{i=1}^n\int \left[|,{\mathrm{\nabla }}_{x_i},{\xi }^{(n)},(x_1,\dots ,x_n),{|}^2,+,{|{\xi }^{(n)}(x_1,\dots ,x_n)|}^2\right]d{x}_1\dots d{x}_n\hfill \\ \hfill & \le Cq\Vert \check{\eta }{\Vert }_{2q^{\prime }}^2{\Vert {(\mathcal{K}+{\mathcal{N}}_{+})}^{1/2}{\mathcal{N}}_{+}^{1/2}\xi \Vert }^2.\hfill \end{array}\end{array}$$ 80

With the bounds (25), (26),

$$\begin{array}{c}\hfill \Vert \check{\eta }{\Vert }_{2q^{\prime }}^2\le \Vert \check{\eta }{\Vert }_2^{2/q^{\prime }}{\Vert \check{\eta }\Vert }_{\infty }^{2(q^{\prime }-1)/q^{\prime }}\le N^{-2\alpha /q^{\prime }}{N}^{2(q^{\prime }-1)/q^{\prime }}\end{array}$$

we conclude that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\Theta }_2\xi ⟩|& \le C{q}^{1/2}{N}^{-1/2}{N}^{-\alpha /q^{\prime }}{N}^{1/q}\Vert {\mathcal{V}}_N^{1/2}{\mathcal{N}}_{+}^{1/2}\xi \Vert \Vert {(\mathcal{K}+{\mathcal{N}}_{+})}^{1/2}{\mathcal{N}}_{+}^{1/2}\xi \Vert \hfill \\ \hfill & \le C{q}^{1/2}{N}^{1/2}{N}^{-\alpha /q^{\prime }}{N}^{1/q}\Vert {\mathcal{V}}_N^{1/2}\xi \Vert \Vert {\mathcal{K}}^{1/2}\xi \Vert \hfill \end{array}\end{array}$$

for any $2<q<\infty$, if $1/q+1/q^{\prime }=1$. Choosing $q=logN$, we obtain that

$$\begin{array}{c}\hfill |⟨\xi ,{\Theta }_2\xi ⟩|\le C{(logN)}^{1/2}{N}^{1/2-\alpha }{\Vert {\mathcal{H}}_N^{1/2}\xi \Vert }^2.\end{array}$$

$\square$

Using Proposition 7, we can now show Proposition 3.

Proof of Proposition 3

The proof follows a strategy similar to [4, Lemma 8.2]. For fixed $\xi \in {\mathcal{F}}_{+}^{\le N}$ and $s\in [0;1]$, we define

$$\begin{array}{c}\hfill f_{\xi }(s):=⟨\xi ,e^{-sA}{\mathcal{H}}_N{e}^{\mathit{sA}}\xi ⟩.\end{array}$$

We compute

$$\begin{array}{c}\hfill f_{\xi }^{\prime }(s)=⟨\xi ,e^{-sA}[\mathcal{K},A]e^{\mathit{sA}}\xi ⟩+⟨\xi ,e^{-sA}[{\mathcal{V}}_N,A]e^{\mathit{sA}}\xi ⟩.\end{array}$$ 81

With Proposition 7, we have

$$\begin{array}{c}\hfill [{\mathcal{V}}_N,A]=\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}u,v\in {\Lambda }_{+}^{\ast },u\ne -v\end{array}}(\widehat{V}(·/e^N)\ast \eta )(u)\left[b_{u+v}^{\ast },a_{-u}^{\ast },a_v,+,,\text{h.c.},\right]+{\delta }_{{\mathcal{V}}_N}\end{array}$$

with ${\delta }_{{\mathcal{V}}_N}$ satisfying (78). Switching to position space and using Proposition 2 we find , using (25) to bound $\Vert \check{\eta }{\Vert }_{\infty }\le CN$,

$$\begin{array}{c}\hfill \begin{array}{cc}& |\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}u,v\in {\Lambda }_{+}^{\ast }\end{array}}(\widehat{V}(·/e^N)\ast \eta )(u)⟨\xi ,e^{-sA}{b}_{u+v}^{\ast }{a}_{-u}^{\ast }{a}_v{e}^{\mathit{sA}}\xi ⟩|\hfill \\ \hfill & =|\frac{1}{\sqrt{N}}{\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y))\check{\eta }(x-y)⟨\xi ,e^{-sA}{\check{a}}_x^{\ast }{\check{a}}_y^{\ast }{\check{a}}_y{e}^{\mathit{sA}}\xi ⟩|\hfill \\ \hfill & \le N^{1/2}[{\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y)){\Vert {\check{a}}_x{\check{a}}_y{e}^{\mathit{sA}}\xi \Vert }^2{]}^{1/2}\hfill \\ \hfill & \times [{\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y)){\Vert {\check{a}}_y{e}^{\mathit{sA}}\xi \Vert }^2{]}^{1/2}\hfill \\ \hfill & \le C{N}^{1/2}\Vert {\mathcal{V}}_N^{1/2}{e}^{\mathit{sA}}\xi \Vert \Vert {\mathcal{N}}_{+}^{1/2}{e}^{\mathit{sA}}\xi \Vert \hfill \end{array}\end{array}$$ 82

Together with (78) we conclude that for any $\alpha >1/2$

$$\begin{array}{c}\hfill |⟨\xi ,e^{-sA}[{\mathcal{V}}_N,A]e^{\mathit{sA}}\xi ⟩|\le C⟨\xi ,e^{-sA}{\mathcal{H}}_N{e}^{\mathit{sA}}\xi ⟩+CN⟨\xi ,e^{-sA}({\mathcal{N}}_{+}+1)e^{\mathit{sA}}\xi ⟩\end{array}$$ 83

if N is large enough. Next, we analyze the first term on the r.h.s. of (81). We compute

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [\mathcal{K},A]& =\frac{1}{\sqrt{N}}\sum _{r,v\in {\Lambda }_{+}^{\ast }}2r^2{\eta }_r[b_{r+v}^{\ast }{a}_{-r}^{\ast }{a}_v+\text{h.c.}]\hfill \\ \hfill & +\frac{2}{\sqrt{N}}\sum _{r,v\in {\Lambda }_{+}^{\ast }}r·v{\eta }_r[b_{r+v}^{\ast }{a}_{-r}^{\ast }{a}_v+\text{h.c.}]\hfill \\ \hfill & =:{\text{T}}_1+{\text{T}}_2.\hfill \end{array}\end{array}$$ 84

With (31), we write

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\text{T}}_1=& -\sqrt{N}\sum _{\begin{array}{c}r,v\in {\Lambda }_{+}^{\ast }\\ r\ne -v\end{array}}(\widehat{V}(·/e^N)\ast {\widehat{f}}_{N,\ell })(r)[b_{r+v}^{\ast }{a}_{-r}^{\ast }{a}_v+\text{h.c.}]\hfill \\ \hfill & +2\sqrt{N}\sum _{r,v\in {\Lambda }_{+}^{\ast }}{e}^{2N}{\lambda }_{\ell }({\widehat{\chi }}_{\ell }\ast {\widehat{f}}_{N,\ell })(r)[b_{r+v}^{\ast }{a}_{-r}^{\ast }{a}_v+\text{h.c.}]\hfill \\ \hfill =:& {\text{T}}_{11}+{\text{T}}_{12}.\hfill \end{array}\end{array}$$ 85

The contribution of ${\text{T}}_{11}$ can be estimated similarly as in (82); switching to position space and using (20), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨{\xi }_1,{\text{T}}_{11}{\xi }_2⟩|& \le C\sqrt{N}\int dxdy{e}^{2N}V(e^N(x-y))f_{\ell }(e^N(x-y))\Vert {\check{a}}_x{\check{a}}_y\xi \Vert \Vert a_y\xi \Vert \hfill \\ \hfill & \le C\sqrt{N}[\int dxdy{e}^{2N}V(e^N(x-y)){\Vert {\check{a}}_x{\check{a}}_y\xi \Vert }^2{]}^{1/2}\hfill \\ \hfill & \times [\int dxdy{e}^{2N}V(e^N(x-y))f_{\ell }(e^N(x-y)){\Vert a_y\xi \Vert }^2{]}^{1/2}\hfill \\ \hfill & \le C\Vert {\mathcal{V}}_N^{1/2}\xi \Vert \Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$ 86

for any $\xi \in {\mathcal{F}}_{+}^{\le N}$. The second term in (85) can be controlled using that for any $\xi \in {\mathcal{F}}_{+}^{\le N}$ and $2\le q<\infty$ we have

$$\begin{array}{c}\hfill \begin{array}{cc}& N^{2\alpha }{\int }_{{\Lambda }^2}dxdy\chi (|x-y|\le N^{-\alpha })\Vert {\check{a}}_x{\check{a}}_y\xi \Vert \Vert {\check{a}}_x\xi \Vert \hfill \\ \hfill & \le N^{2\alpha }{\int }_{{\Lambda }^2}dx\Vert {\check{a}}_x\xi \Vert {\left(\int ,d,y,,\chi ,(,|,x,-,y,|,\le ,N^{-\alpha },)\right)}^{1-1/q}{\left(\int dy\Vert ,{\check{a}}_x,{\check{a}}_y,{\xi \Vert }^q\right)}^{1/q}\hfill \\ \hfill & \le C{N}^{2\alpha /q}{q}^{1/2}{\left[\int dx\Vert ,{\check{a}}_x,{\xi \Vert }^2\right]}^{1/2}{\left[\int dxdy\Vert ,{\check{a}}_x,{\mathrm{\nabla }}_y,{\check{a}}_y,{\xi \Vert }^2,+,\int ,d,x,d,y,{\Vert {\check{a}}_x{\check{a}}_y\xi \Vert }^2\right]}^{1/2}\hfill \\ \hfill & \le C{N}^{2\alpha /q}{q}^{1/2}\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \left[\Vert ,{\mathcal{K}}^{1/2},{({\mathcal{N}}_{+}+1)}^{1/2},\xi \Vert +\Vert ,({\mathcal{N}}_{+}+1),\xi \Vert \right].\hfill \end{array}\end{array}$$ 87

Hence, choosing $q=logN$,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,& {\text{T}}_{12}\xi ⟩|\hfill \\ \hfill & =|\sqrt{N}{e}^{2N}{\lambda }_{\ell }{\int }_{{\Lambda }^2}dxdy\chi (|x-y|\le N^{-\alpha })f_{N,\ell }(x-y)\langle \xi ,{\check{b}}_x^{\ast }{\check{a}}_y^{\ast }{\check{a}}_x\xi \rangle |\hfill \\ \hfill & \le C{N}^{2\alpha -1/2}{\int }_{{\Lambda }^2}dxdy\chi (|x-y|\le N^{-\alpha })\Vert {\check{a}}_x{\check{a}}_y\xi \Vert \Vert {\check{a}}_x\xi \Vert \hfill \\ \hfill & \le C{(logN)}^{1/2}\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \left[\Vert ,{\mathcal{K}}^{1/2},\xi \Vert +\Vert ,{({\mathcal{N}}_{+}+1)}^{1/2},\xi \Vert \right],\hfill \end{array}\end{array}$$ 88

With (86) and (88) we conclude that

$$\begin{array}{c}\hfill |⟨\xi ,e^{-A}{\text{T}}_1{e}^A\xi ⟩|\le C{(logN)}^{1/2}\Vert {({\mathcal{H}}_N+1)}^{1/2}{e}^{\mathit{sA}}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}{e}^{\mathit{sA}}\xi \Vert .\end{array}$$ 89

for all $\xi \in {\mathcal{F}}_{+}^{\le N}$. As for the second term on the r.h.s. of (84) we have

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,{\text{T}}_2\xi ⟩|\hfill \\ \hfill & \le \frac{C}{\sqrt{N}}[\sum _{r\in {\Lambda }_{+}^{\ast }}{|r|}^2\Vert {\mathcal{N}}_{+}^{1/2}{a}_{-r}{\xi \Vert }^2{]}^{1/2}[\sum _{r,v\in {\Lambda }_{+}^{\ast }}{|v|}^2{\eta }_r^2{\Vert a_v\xi \Vert }^2{]}^{1/2}\hfill \\ \hfill & \le C{N}^{-\alpha }{\Vert {\mathcal{K}}^{1/2}\xi \Vert }^2.\hfill \end{array}\end{array}$$ 90

for any $\xi \in {\mathcal{F}}_{+}^{\le N}$. With (89) and Proposition 2, we conclude that

$$\begin{array}{c}\hfill |⟨\xi ,e^{-sA}[\mathcal{K},A]e^{\mathit{sA}}\xi ⟩|\le C⟨\xi ,e^{-sA}{\mathcal{H}}_N{e}^{\mathit{sA}}\xi ⟩+ClogN⟨\xi ,e^{-sA}{\mathcal{N}}_{+}{e}^{\mathit{sA}}\xi ⟩.\end{array}$$

Combining with Eq. (83) we obtain

$$\begin{array}{c}\hfill |⟨\xi ,e^{-sA}[{\mathcal{H}}_N,A]e^{\mathit{sA}}\xi ⟩|\le C⟨\xi ,e^{-sA}{\mathcal{H}}_N{e}^{\mathit{sA}}\xi ⟩+CN⟨\xi ,e^{-sA}{\mathcal{N}}_{+}{e}^{\mathit{sA}}\xi ⟩.\end{array}$$

With Proposition 2 we obtain the differential inequality

$$\begin{array}{c}\hfill |f_{\xi }^{\prime }(s)|\le C{f}_{\xi }(s)+CN⟨\xi ,({\mathcal{N}}_{+}+1)\xi ⟩.\end{array}$$

By Gronwall’s Lemma, we find (45). $\square$

Analysis of $e^{-A}{\mathcal{O}}_N{e}^A$

In this section we study the contribution to ${\mathcal{R}}_{N,\alpha }$ arising from the operator ${\mathcal{O}}_N$, defined in (77). To this end, it is convenient to use the following lemma.

Lemma 3

Let A be defined in (44). Then, there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill \sum _{p\in {\Lambda }_{+}^{\ast }}{F}_p{e}^{-A}{a}_p^{\ast }{a}_p{e}^A=\sum _{p\in {\Lambda }_{+}^{\ast }}{F}_p{a}_p^{\ast }{a}_p+{\mathcal{E}}_F\end{array}$$

where

$$\begin{array}{c}\hfill |\langle {\xi }_1,{\mathcal{E}}_F{\xi }_2\rangle |\le C{N}^{-\alpha }{\Vert F\Vert }_{\infty }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\xi }_1\Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\xi }_2\Vert \end{array}$$

for all $\alpha >0$, ${\xi }_1,{\xi }_2\in {\mathcal{F}}_{+}^{\le N}$, $F\in {\ell }^{\infty }({\Lambda }_{+}^{\ast })$, and $N\in \mathbb{N}$ large enough.

Proof

The lemma is analogous to [4, Lemma 8.6]. We estimate

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |\sum _{p\in {\Lambda }_{+}^{\ast }}{F}_p& (⟨{\xi }_1,e^{-A}{a}_p^{\ast }{a}_p{e}^A{\xi }_2⟩-⟨{\xi }_1,a_p^{\ast }{a}_p{\xi }_2⟩)|\hfill \\ \hfill & =|{\int }_0^{1}ds\sum _{p\in {\Lambda }_{+}^{\ast }}{F}_p⟨{\xi }_1,e^{-sA}[a_p^{\ast }{a}_p,A]e^{\mathit{sA}}{\xi }_2⟩|\hfill \\ \hfill & \le \frac{1}{\sqrt{N}}{\int }_0^{1}ds\sum _{r,v\in {\Lambda }_{+}^{\ast }}|F_{r+v}+F_{-r}-F_v||{\eta }_r||⟨e^{\mathit{sA}}{\xi }_1,b_{r+v}^{\ast }{a}_{-r}^{\ast }{a}_v{e}^{\mathit{sA}}{\xi }_2⟩|\hfill \\ \hfill & \le {C\Vert \eta \Vert \Vert F\Vert }_{\infty }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\xi }_1\Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\xi }_2\Vert .\hfill \end{array}\end{array}$$

where we used Proposition 2. $\square$

We consider now the action of $e^A$ on the operator ${\mathcal{O}}_N$, as defined in (77).

Proposition 8

Let A be defined in (44). Then there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^{-A}{\mathcal{O}}_N{e}^A=& \frac{1}{2}{\widehat{\omega }}_N(0)(N-1)\left(1,-,\frac{{\mathcal{N}}_{+}}{N}\right)+[2N\widehat{V}(0)-\frac{1}{2}{\widehat{\omega }}_N(0)]{\mathcal{N}}_{+}(1-{\mathcal{N}}_{+}/N)\hfill \\ \hfill & +{\delta }_{{\mathcal{O}}_N}\hfill \end{array}\end{array}$$

where

$$\begin{array}{c}\hfill \pm {\delta }_{{\mathcal{O}}_N}\le C{N}^{1-\alpha }({\mathcal{N}}_{+}+1)\end{array}$$

for all $\alpha >0$, and $N\in \mathbb{N}$ large enough.

Proof

The proof is very similar to [4, Prop. 8.7]. First of all, with Lemma 3 we can bound

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \pm \{e^{-A}& \left[\frac{1}{2},{\widehat{\omega }}_N,(0),(N-1),\left(1,-,\frac{{\mathcal{N}}_{+}}{N}\right),+,[,2,N,\widehat{V},(0),-,\frac{1}{2},{\widehat{\omega }}_N,(0),],{\mathcal{N}}_{+}\right]e^A\hfill \\ \hfill & -\left[\frac{1}{2},{\widehat{\omega }}_N,(0),(N-1),\left(1,-,\frac{{\mathcal{N}}_{+}}{N}\right),+,[,2,N,\widehat{V},(0),-,\frac{1}{2},{\widehat{\omega }}_N,(0),],{\mathcal{N}}_{+}\right]\}\hfill \\ \hfill & \le C{N}^{1-\alpha }({\mathcal{N}}_{+}+1).\hfill \end{array}\end{array}$$

Moreover, for the contribution quadratic in ${\mathcal{N}}_{+}$, we can decompose

$$\begin{array}{c}\hfill \begin{array}{cc}& 〈\xi ,,,\left[e^{-A},{\mathcal{N}}_{+}^2,e^A,-,{\mathcal{N}}_{+}^2\right],\xi 〉\hfill \\ \hfill & =〈{\xi }_1,,,\left[e^{-A},{\mathcal{N}}_{+},e^A,-,{\mathcal{N}}_{+}\right],\xi 〉+〈\xi ,,,\left[e^{-A},{\mathcal{N}}_{+},e^A,-,{\mathcal{N}}_{+}\right],{\xi }_2〉\hfill \end{array}\end{array}$$

with ${\xi }_1=e^{-A}{\mathcal{N}}_{+}{e}^A\xi$ and ${\xi }_2={\mathcal{N}}_{+}\xi$, and estimate, again with Lemma 3,

$$\begin{array}{c}\hfill \begin{array}{cc}& \left|〈\xi ,,,\left[e^{-A},{\mathcal{N}}_{+}^2,e^A,-,{\mathcal{N}}_{+}^2\right],\xi 〉\right|\hfill \\ \hfill & \le C{N}^{-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \left[\Vert ,{({\mathcal{N}}_{+}+1)}^{1/2},{\xi }_1,\Vert +\Vert ,{({\mathcal{N}}_{+}+1)}^{1/2},{\xi }_2,\Vert \right].\hfill \end{array}\end{array}$$

With Proposition 2, we have $\Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\xi }_1\Vert \le C\Vert {({\mathcal{N}}_{+}+1)}^{3/2}\xi \Vert$. $\square$

Contributions from $e^{-A}\mathcal{K}{e}^A$

In Sect. 6.6 we will analyse the contributions to ${\mathcal{R}}_{N,\alpha }$ arising from conjugation of the kinetic energy operator $\mathcal{K}={\sum }_{p\in {\Lambda }_{+}^{\ast }}{p}^2{a}_p^{\ast }{a}_p$. To this aim we will exploit properties of the commutator $[\mathcal{K},A]$, collected in the following proposition.

Proposition 9

Let A be defined as in (44) and ${\widehat{\omega }}_N(r)$ be defined in (39). Then there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [\mathcal{K},A]=& -\sqrt{N}\sum _{p,q\in {\Lambda }_{+}^{\ast },p\ne -q}(\widehat{V}(·/e^N)\ast {\widehat{f}}_{N,\ell })(p)(b_{p+q}^{\ast }{a}_{-p}^{\ast }{a}_q+\text{h.c.})\hfill \\ \hfill & +\frac{1}{\sqrt{N}}\sum _{p,q\in {\Lambda }_{+}^{\ast },p\ne -q}{\widehat{\omega }}_N(p)[b_{p+q}^{\ast }{a}_{-p}^{\ast }{a}_q+\text{h.c.}]+{\delta }_{\mathcal{K}}\hfill \end{array}\end{array}$$

where

$$\begin{array}{c}\hfill |⟨\xi ,{\delta }_{\mathcal{K}}\xi ⟩|\le C{N}^{-1}{(logN)}^{1/2}\Vert {\mathcal{K}}^{1/2}\xi \Vert \Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert +C{N}^{-\alpha }{\Vert {\mathcal{K}}^{1/2}\xi \Vert }^2\end{array}$$ 91

for all $\alpha >1$, $\xi \in {\mathcal{F}}_{+}^{\le N}$, and $N\in \mathbb{N}$ large enough. Moreover, the operator

$$\begin{array}{c}\hfill {\Delta }_{\mathcal{K}}=\frac{1}{\sqrt{N}}\sum _{p,q\in {\Lambda }_{+}^{\ast },p\ne -q}{\widehat{\omega }}_N(p)[b_{p+q}^{\ast }{a}_{-p}^{\ast }{a}_q,A]\end{array}$$

satisfies

$$\begin{array}{c}\hfill |⟨\xi ,{\Delta }_{\mathcal{K}}\xi ⟩|\le C{N}^{-\alpha }{(logN)}^{1/2}\Vert {\mathcal{K}}^{1/2}{\xi \Vert }^2+C{N}^{-1}{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2\end{array}$$ 92

for all $\alpha >1$, $\xi \in {\mathcal{F}}_{+}^{\le N}$, and $N\in \mathbb{N}$ large enough.

Proof

To show (91) we recall from Eqs. (84), (85) that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [\mathcal{K},A]=& -\sqrt{N}\sum _{\begin{array}{c}r,v\in {\Lambda }_{+}^{\ast }\\ r\ne -v\end{array}}(\widehat{V}(·/e^N)\ast {\widehat{f}}_{N,\ell })(r)[b_{r+v}^{\ast }{a}_{-r}^{\ast }{a}_v+\text{h.c.}]\hfill \\ & +2\sqrt{N}\sum _{r,v\in {\Lambda }_{+}^{\ast }}{e}^{2N}{\lambda }_{\ell }({\widehat{\chi }}_{\ell }\ast {\widehat{f}}_{N,\ell })(r)[b_{r+v}^{\ast }{a}_{-r}^{\ast }{a}_v+\text{h.c.}]\hfill \\ & +\frac{2}{\sqrt{N}}\sum _{r,v\in {\Lambda }_{+}^{\ast }}r·v{\eta }_r[b_{r+v}^{\ast }{a}_{-r}^{\ast }{a}_v+\text{h.c.}]\hfill \\ \hfill =& {\text{T}}_{11}+{\text{T}}_{12}+{\text{T}}_2.\hfill \end{array}\end{array}$$

with ${\text{T}}_2$ satisfying (90). Using the definition ${\widehat{\omega }}_N(p)=2N{e}^{2N}{\lambda }_{\ell }{\widehat{\chi }}_{\ell }(p)$ we write

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\text{T}}_{12}=& \frac{1}{\sqrt{N}}\sum _{p,q\in {\Lambda }_{+}^{\ast },p\ne -q}{\widehat{\omega }}_N(p)[b_{p+q}^{\ast }{a}_{-p}^{\ast }{a}_q+\text{h.c.}]\hfill \\ \hfill & +\frac{2}{\sqrt{N}}{e}^{2N}{\lambda }_{\ell }\sum _{p,q\in {\Lambda }_{+}^{\ast },p\ne -q}(\widehat{{\chi }_{\ell }}\ast \eta )(p)[b_{p+q}^{\ast }{a}_{-p}^{\ast }{a}_q+\text{h.c.}]\hfill \\ \hfill =& {\text{T}}_{121}+{\text{T}}_{122}.\hfill \end{array}\end{array}$$

Hence, ${\delta }_K=T_2+T_{122}$. To bound $T_{122}$ we switch to position space:

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,{\text{T}}_{122}\xi ⟩|\hfill \\ \hfill & \le C{N}^{2\alpha -3/2}{\int }_{{\Lambda }^2}{\chi }_{\ell }(x-y)\check{\eta }(x-y)\Vert {\check{a}}_x{\check{a}}_y\xi \Vert \Vert {\check{a}}_x\xi \Vert \hfill \\ \hfill & \le C{N}^{2\alpha -3/2}{\left[{\int }_{{\Lambda }^2},{\chi }_{\ell },(x-y),{\Vert {\check{a}}_x{\check{a}}_y\xi \Vert }^2,d,x,d,y\right]}^{1/2}{\left[{\int }_{{\Lambda }^2},|,\check{\eta },{(x-y)|}^2,{\Vert {\check{a}}_x\xi \Vert }^2,d,x,d,y\right]}^{1/2}\hfill \\ \hfill & \le C{N}^{\alpha -3/2}\Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert {\left[{\int }_{{\Lambda }^2},{\chi }_{\ell },(x-y),{\Vert {\check{a}}_x{\check{a}}_y\xi \Vert }^2,d,x,d,y\right]}^{1/2}.\hfill \end{array}\end{array}$$

To bound the term in the parenthesis, we proceed similarly as in (80). We find

$$\begin{array}{c}\hfill {\int }_{{\Lambda }^2}{\chi }_{\ell }(x-y)\Vert {\check{a}}_x{\check{a}}_y{\xi \Vert }^{2}dxdy\le Cq\Vert {\chi }_{\ell }{\Vert }_{q^{\prime }}\Vert {\mathcal{K}}^{1/2}{\mathcal{N}}_{+}^{1/2}{\xi \Vert }^2\le Cq{N}^{1-2\alpha /q^{\prime }}{\Vert {\mathcal{K}}^{1/2}\xi \Vert }^2\end{array}$$

for any $q>2$ and $1<q^{\prime }<2$ with $1/q+1/q^{\prime }=1$. Choosing $q=logN$, we obtain

$$\begin{array}{c}\hfill |⟨\xi ,{\text{T}}_{122}\xi ⟩|\le C{N}^{-1}{(logN)}^{1/2}\Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert \Vert {\mathcal{K}}^{1/2}\xi \Vert \end{array}$$

With (90), this implies (91).

Let us now focus on (92). We have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{1}{\sqrt{N}}& \sum _{p,q\in {\Lambda }_{+}^{\ast },p\ne -q}{\widehat{\omega }}_N(p)[b_{p+q}^{\ast }{a}_{-p}^{\ast }{a}_q,A]\hfill \\ \hfill & =\frac{1}{N}\sum _{\begin{array}{c}r,p,q,v\in {\Lambda }_{+}^{\ast },\\ p\ne -q,r\ne -v\end{array}}{\widehat{\omega }}_N(p){\eta }_r[b_{p+q}^{\ast }{a}_{-p}^{\ast }{a}_q,b_{r+v}^{\ast }{a}_{-r}^{\ast }{a}_v-a_v^{\ast }{a}_{-r}{b}_{r+v}].\hfill \end{array}\end{array}$$

With the commutators from the proof of Proposition 8.8 in [4], we arrive at

$$\begin{array}{c}\hfill \frac{1}{\sqrt{N}}\sum _{p,q\in {\Lambda }_{+}^{\ast },p\ne -q}{\widehat{\omega }}_N(p)[b_{p+q}^{\ast }{a}_{-p}^{\ast }{a}_q,A]+\text{h.c.}=\sum _{j=1}^{12}{{\rm Y}}_j+\text{h.c.}\end{array}$$

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {{\rm Y}}_1:=& -\frac{1}{N}\sum _{\begin{array}{c}q,r,v\in {\Lambda }_{+}^{\ast },\\ q\ne v,r\ne -v\end{array}}({\widehat{\omega }}_N(v-q)+{\widehat{\omega }}_N(v)){\eta }_r{b}_{r+v}^{\ast }{b}_{-r}^{\ast }{a}_{q-v}^{\ast }{a}_q,\hfill \\ \hfill {{\rm Y}}_2:=& \frac{1}{N}\sum _{\begin{array}{c}q,r,v\in {\Lambda }_{+}^{\ast },\\ r\ne -v,r\ne -q\end{array}}{\widehat{\omega }}_N(r+q){\eta }_r(1-{\mathcal{N}}_{+}/N)a_v^{\ast }{a}_{r+q}^{\ast }{a}_q{a}_{r+v},\hfill \\ \hfill {{\rm Y}}_3:=& \frac{1}{N}\sum _{\begin{array}{c}r,v\in {\Lambda }_{+}^{\ast },\\ r\ne -v\end{array}}({\widehat{\omega }}_N(r+v)+{\widehat{\omega }}_N(r)){\eta }_r(1-{\mathcal{N}}_{+}/N)a_v^{\ast }{a}_v,\hfill \\ \hfill {{\rm Y}}_4:=& \frac{1}{N}\sum _{\begin{array}{c}q,r,v\in {\Lambda }_{+}^{\ast },\\ q\ne v,r\ne -v\end{array}}{\widehat{\omega }}_N(r+v-q){\eta }_r(1-{\mathcal{N}}_{+}/N)a_v^{\ast }{a}_{q-r-v}^{\ast }{a}_{-r}{a}_q,\hfill \\ \hfill {{\rm Y}}_5:=& -\frac{1}{N^2}\sum _{\begin{array}{c}p,q,r,v\in {\Lambda }_{+}^{\ast },\\ p\ne -q,r\ne -v\end{array}}{\widehat{\omega }}_N(p){\eta }_r{a}_v^{\ast }{a}_{p+q}^{\ast }{a}_{-p}^{\ast }{a}_{-r}{a}_{r+v}{a}_q,\hfill \\ \hfill {{\rm Y}}_6:=& -\frac{1}{N^2}\sum _{\begin{array}{c}q,r,v\in {\Lambda }_{+}^{\ast },\\ q\ne r+v\end{array}}{\widehat{\omega }}_N(r+v){\eta }_r{a}_v^{\ast }{a}_{q-r-v}^{\ast }{a}_{-r}{a}_q,\hfill \\ \hfill {{\rm Y}}_7:=& -\frac{1}{N^2}\sum _{\begin{array}{c}q,r,v\in {\Lambda }_{+}^{\ast },\\ q\ne -r,r\ne -v\end{array}}{\widehat{\omega }}_N(r){\eta }_r{a}_v^{\ast }{a}_{q+r}^{\ast }{a}_{r+v}{a}_q,\hfill \\ \hfill {{\rm Y}}_8:=& \frac{1}{N}\sum _{\begin{array}{c}r,v,p\in {\Lambda }_{+}^{\ast },\\ p\ne -r-v\end{array}}{\widehat{\omega }}_N(p){\eta }_r{b}_{p+r+v}^{\ast }{b}_{-p}^{\ast }{a}_{-r}^{\ast }{a}_v,\hfill \\ \hfill {{\rm Y}}_9:=& \frac{1}{N}\sum _{\begin{array}{c}p,r,v\in {\Lambda }_{+}^{\ast },\\ p\ne r,r\ne -v\end{array}}{\widehat{\omega }}_N(p){\eta }_r{b}_{p-r}^{\ast }{b}_{r+v}^{\ast }{a}_{-p}^{\ast }{a}_v,\hfill \\ \hfill {{\rm Y}}_{10}:=& \frac{1}{N}\sum _{\begin{array}{c}q,r,v\in {\Lambda }_{+}^{\ast },\\ q\ne -r,r\ne -v\end{array}}{\widehat{\omega }}_N(r){\eta }_r{b}_{q+r}^{\ast }{a}_v^{\ast }{a}_q{b}_{r+v},\hfill \\ \hfill {{\rm Y}}_{11}:=& -\frac{1}{N}\sum _{\begin{array}{c}p,r,v\in {\Lambda }_{+}^{\ast },\\ p\ne -v,r\ne -v\end{array}}{\widehat{\omega }}_N(p){\eta }_r{b}_{p+v}^{\ast }{a}_{-p}^{\ast }{a}_{-r}{b}_{r+v},\hfill \\ \hfill {{\rm Y}}_{12}:=& \frac{1}{N}\sum _{\begin{array}{c}q,r,v\in {\Lambda }_{+}^{\ast }\\ r\ne q-v,-v\end{array}}{\widehat{\omega }}_N(r+v){\eta }_r{b}_{q-r-v}^{\ast }{a}_v^{\ast }{a}_{-r}{b}_q.\hfill \\ \hfill \end{array}\end{array}$$ 93

To conclude the proof of Proposition 9, we show that all operators in (93) satisfy (92). To study all these terms it is convenient to switch to position space. We recall that ${\widehat{\omega }}_N(p)=g_N\widehat{\chi }(\ell p)$ with $|g_N|\le C$ and $\ell =N^{-\alpha }$. Using (87) we find:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{{\rm Y}}_1\xi ⟩|& \le C{N}^{2\alpha -1}{\int }_{{\Lambda }^2}dxdy{\chi }_{\ell }(x-y)\Vert \check{b}({\check{\eta }}_x){\check{b}}_x{\check{a}}_y\xi \Vert \left[\Vert ,{\check{a}}_x,\xi \Vert +\Vert ,{\check{a}}_y,\xi \Vert \right]\hfill \\ \hfill & \le C{N}^{2\alpha -1}\Vert \eta \Vert {\int }_{{\Lambda }^2}dxdy{\chi }_{\ell }(x-y)\Vert {\check{b}}_x{\check{a}}_y{({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\check{a}}_x\xi \Vert \hfill \\ \hfill & \le C{N}^{-\alpha }{(logN)}^{1/2}\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{K}}^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$

The expectation of ${{\rm Y}}_2$ is bounded following the same strategy used to show (87). For any $2\le q<\infty$ we have

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,{{\rm Y}}_2\xi ⟩|\hfill \\ \hfill & \le C{N}^{2\alpha -1}{\int }_{{\Lambda }^3}dxdydz{\chi }_{\ell }(z-y)|\check{\eta }(z-x)|\Vert {\check{a}}_x{\check{a}}_y\xi \Vert \Vert {\check{a}}_z{\check{a}}_x\xi \Vert \hfill \\ \hfill & \le C{N}^{2\alpha -1}{\int }_{{\Lambda }^2}dxdz|\check{\eta }(z-x)|\Vert {\check{a}}_z{\check{a}}_x\xi \Vert \hfill \\ \hfill & \times {\left({\int }_{\Lambda },dy\chi (|z-y|,\le ,N^{-\alpha },)\right)}^{1-1/q}{\left({\int }_{\Lambda },d,y,{\Vert {\check{a}}_x{\check{a}}_y\xi \Vert }^q\right)}^{1/q}\hfill \\ \hfill & \le C{q}^{1/2}{N}^{2\alpha /q-1}\Vert \eta \Vert \Vert ({\mathcal{N}}_{+}+1)\xi \Vert {\left[{\int }_{{\Lambda }^2},dxdy\Vert ,{\check{a}}_x,{\mathrm{\nabla }}_y,{\check{a}}_y,{\xi \Vert }^2,+,{\int }_{{\Lambda }^2},d,x,d,y,{\Vert {\check{a}}_x{\check{a}}_y\xi \Vert }^2\right]}^{1/2}\hfill \\ \hfill & \le C{N}^{-\alpha }{(logN)}^{1/2}\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{K}}^{1/2}\xi \Vert ,\hfill \end{array}\end{array}$$

where in the last line we chose $q=logN$. The term ${{\rm Y}}_3$ is of lower order; using that $|{\sum }_r{\widehat{\omega }}_N(r){\eta }_r|\le \Vert \widehat{\chi }(./N^{\alpha }){\Vert }_2{\Vert \eta \Vert }_2\le C$ and Cauchy–Schwarz, we easily obtain

$$\begin{array}{c}\hfill |⟨\xi ,{{\rm Y}}_3\xi ⟩|\le C{N}^{-1}{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2.\end{array}$$

The term ${{\rm Y}}_4$ can be estimated as ${{\rm Y}}_1$ using (87):

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{{\rm Y}}_4\xi ⟩|& \le C{N}^{2\alpha -1}{\int }_{{\Lambda }^2}dxdy{\chi }_{\ell }(x-y)\Vert {\check{a}}_x{\check{a}}_y\xi \Vert \Vert \check{a}({\check{\eta }}_y){\check{a}}_y\xi \Vert \hfill \\ \hfill & \le C{N}^{2\alpha -1}\Vert \eta \Vert {\int }_{{\Lambda }^2}dxdy{\chi }_{\ell }(x-y)\Vert {\check{a}}_x{\check{a}}_y\xi \Vert \Vert {\check{a}}_y{({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \hfill \\ \hfill & \le C{N}^{-\alpha }{(logN)}^{1/2}\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{K}}^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$

The term ${{\rm Y}}_5$ is bounded similarly to ${{\rm Y}}_2$; with $q=logN$ we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{{\rm Y}}_5\xi ⟩|& \le C{N}^{2\alpha -2}\Vert \eta \Vert {\int }_{{\Lambda }^3}dxdydz{\chi }_{\ell }(y-z)\Vert {\check{a}}_x{\check{a}}_y{\check{a}}_z\xi \Vert \Vert {\mathcal{N}}_{+}^{1/2}{\check{a}}_x{\check{a}}_y\xi \Vert \hfill \\ \hfill & \le C{N}^{2\alpha -3/2}\Vert \eta \Vert {\int }_{{\Lambda }^2}dxdy\Vert {\check{a}}_x{\check{a}}_y\xi \Vert \hfill \\ \hfill & \times {\left({\int }_{\Lambda },dz\chi (|y-z|,\le ,N^{-\alpha },)\right)}^{1-1/q}{\left({\int }_{\Lambda },d,z,,{\Vert {\check{a}}_x{\check{a}}_y{\check{a}}_z\xi \Vert }^q\right)}^{1/q}\hfill \\ \hfill & \le C{N}^{-\alpha }{(logN)}^{1/2}\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{K}}^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$

The terms ${{\rm Y}}_6$ and ${{\rm Y}}_7$ are of smaller order and can be bounded with Cauchy–Schwarz; we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{{\rm Y}}_6\xi ⟩|& \le C{N}^{2\alpha -2}{\int }_{{\Lambda }^2}dxdydz{\chi }_{\ell }(x-y)\Vert {\check{a}}_x{\check{a}}_y\xi \Vert \Vert \check{a}({\check{\eta }}_x){\check{a}}_y\xi \Vert \hfill \\ \hfill & \le C{N}^{\alpha -3/2}{\left({\int }_{{\Lambda }^2},d,x,d,y,,{\Vert {\check{a}}_x{\check{a}}_y\xi \Vert }^2\right)}^{1/2}{\left({\int }_{{\Lambda }^2},dxdy\chi (|x-y|,\le ,N^{-\alpha },)\Vert ,{\check{a}}_y,{\xi \Vert }^2\right)}^{1/2}\hfill \\ \hfill & \le C{N}^{-1}{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2,\hfill \end{array}\end{array}$$

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{{\rm Y}}_7\xi ⟩|& \le C{N}^{2\alpha -2}{\int }_{{\Lambda }^3}dxdydz{\chi }_{\ell }(y-z)|\check{\eta }(z-x)|\Vert {\check{a}}_x{\check{a}}_y{\xi \Vert }^2\hfill \\ \hfill & \le C{N}^{2\alpha -2}{\left({\int }_{{\Lambda }^3},d,x,d,y,d,z,,{\chi }_{\ell },(y-z),{\Vert {\check{a}}_x{\check{a}}_y\xi \Vert }^2\right)}^{1/2}\hfill \\ \hfill & \times {\left({\int }_{{\Lambda }^3},dxdydz|,\check{\eta },{(z-x)|}^2,{\Vert {\check{a}}_x{\check{a}}_y\xi \Vert }^2\right)}^{1/2}\hfill \\ \hfill & \le C{N}^{-1}{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2.\hfill \end{array}\end{array}$$

The terms ${{\rm Y}}_8,{{\rm Y}}_{11},{{\rm Y}}_{12}$ are again bounded, as ${{\rm Y}}_1$, using (87). We find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,({{\rm Y}}_8+{{\rm Y}}_{11}+{{\rm Y}}_{12})\xi ⟩|& \le C{N}^{2\alpha -1}\Vert \eta \Vert {\int }_{{\Lambda }^2}dxdy{\chi }_{\ell }(x-y)\Vert {\mathcal{N}}_{+}^{1/2}{\check{a}}_x{\check{a}}_y\xi \Vert \Vert {\check{a}}_x\xi \Vert \hfill \\ \hfill & \le C{N}^{-\alpha }{(logN)}^{1/2}\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{K}}^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$

It remains to bound ${{\rm Y}}_9$ and ${{\rm Y}}_{10}$. The term ${{\rm Y}}_9$ is bounded analogously to ${{\rm Y}}_2$:

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,{{\rm Y}}_9\xi ⟩|\hfill \\ \hfill & \le C{N}^{2\alpha -1}{\int }_{{\Lambda }^3}dxdydz{\chi }_{\ell }(x-z)|\check{\eta }(x-y)|\Vert {\check{a}}_x{\check{a}}_y{\check{a}}_z\xi \Vert \Vert {\check{a}}_y\xi \Vert \hfill \\ \hfill & \le C{N}^{2\alpha -1}{\int }_{{\Lambda }^2}dxdy|\check{\eta }(x-y)|\Vert {\check{a}}_y\xi \Vert {\left({\int }_{\Lambda },dz\chi (|y-z|,\le ,N^{-\alpha },)\right)}^{1-1/q}\hfill \\ \hfill & \times {\left({\int }_{\Lambda },d,z,,{\Vert {\check{a}}_x{\check{a}}_y{\check{a}}_z\xi \Vert }^q\right)}^{1/q}\hfill \\ \hfill & \le C{q}^{1/2}{N}^{2\alpha /q-1}{\left[{\int }_{{\Lambda }^2},dxdy|,\check{\eta },{(x-y)|}^2,{\Vert {\check{a}}_y\xi \Vert }^2\right]}^{1/2}{\left[{\int }_{{\Lambda }^3},d,x,d,y,,\Vert ,\Vert {\check{a}}_x{\check{a}}_y{\check{a}}_z\xi \Vert ,{\Vert }_{L_z^q}^2\right]}^{1/2}\hfill \\ \hfill & \le C{N}^{-\alpha }{(logN)}^{1/2}\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{K}}^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$

As for ${{\rm Y}}_{10}$, we find

$$\begin{array}{c}\hfill |⟨\xi ,{{\rm Y}}_{10}\xi ⟩|\le C{N}^{2\alpha -1}{\int }_{{\Lambda }^3}dxdydz{\chi }_{\ell }(y-z)|\check{\eta }(x-z)|\Vert {\check{a}}_x{\check{a}}_y{\xi \Vert }^2\end{array}$$

Proceeding as in (80), we obtain

$$\begin{array}{c}\hfill |⟨\xi ,{{\rm Y}}_{10}\xi ⟩|\le Cq{N}^{2\alpha }\Vert {\chi }_{\ell }\ast |\check{\eta }{|\Vert }_{q^{\prime }}\Vert {\mathcal{K}}^{1/2}{\xi \Vert }^2\le Cq\Vert \check{\eta }{\Vert }_{q^{\prime }}{\Vert {\mathcal{K}}^{1/2}\xi \Vert }^2\end{array}$$

for any $q>2$, and $q^{\prime }<2$ with $1/q+1/q^{\prime }=1$. Since, for an arbitrary $q^{\prime }<2$, $\Vert \check{\eta }{\Vert }_{q^{\prime }}\le \Vert \check{\eta }{\Vert }_2={\Vert \eta \Vert }_2\le N^{-\alpha }$, we obtain

$$\begin{array}{c}\hfill |⟨\xi ,{{\rm Y}}_{10}\xi ⟩|\le C{N}^{-\alpha }{\Vert {\mathcal{K}}^{1/2}\xi \Vert }^2\end{array}$$

We conclude that for any $\alpha >1$

$$\begin{array}{c}\hfill \left|⟨,\xi ,,,\sum _{j=1}^{12},{{\rm Y}}_i,\xi ,⟩\right|\le C{N}^{-\alpha }{(logN)}^{1/2}{\Vert (\mathcal{K}+1)}^{1/2}{\xi \Vert }^2+C{N}^{-1}{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2.\end{array}$$

$\square$

Analysis of $e^{-A}{\mathcal{Z}}_N{e}^A$

In this subsection, we consider contributions to ${\mathcal{R}}_{N,\alpha }$ arising from conjugation of ${\mathcal{Z}}_N$, as defined in (77).

Proposition 10

Let A be defined in (44). Then, there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill e^A{\mathcal{Z}}_N{e}^{-A}=\frac{1}{2}\sum _{\begin{array}{c}p\in {\Lambda }_{+}^{\ast }\end{array}}{\widehat{\omega }}_N(p)(b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p})+{\delta }_{{\mathcal{Z}}_N}\end{array}$$

where

$$\begin{array}{c}\hfill \pm {\delta }_{{\mathcal{Z}}_N}\le C{N}^{1-\alpha }({\mathcal{H}}_N+1)\end{array}$$

for all $\alpha >0$, and $N\in \mathbb{N}$ large enough.

Proof

We have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{1}{2}\sum _{\begin{array}{c}p\in {\Lambda }_{+}^{\ast }\end{array}}{\widehat{\omega }}_N(p)& [e^{-A}(b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p})e^A-(b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p})]\hfill \\ \hfill & =\frac{1}{2}{\int }_0^{1}ds\sum _{\begin{array}{c}p\in {\Lambda }_{+}^{\ast }\end{array}}{\widehat{\omega }}_N(p)e^{-sA}[b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p},A]e^{\mathit{sA}}.\hfill \end{array}\end{array}$$ 94

We compute

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{1}{2}\sum _{\begin{array}{c}p\in {\Lambda }_{+}^{\ast }\end{array}}& {\widehat{\omega }}_N(p)[b_p^{\ast }{b}_{-p}^{\ast },b_{r+v}^{\ast }{a}_{-r}^{\ast }{a}_v-a_v^{\ast }{a}_{-r}{b}_{r+v}]\hfill \\ \hfill =& -{\widehat{\omega }}_N(v)b_{r+v}^{\ast }{b}_{-v}^{\ast }{b}_{-r}^{\ast }+{\widehat{\omega }}_N(r)b_v^{\ast }(b_r^{\ast }{b}_{r+v}-\frac{2}{N}{a}_r^{\ast }{a}_{r+v})\hfill \\ \hfill & +{\widehat{\omega }}_N(r+v)(1-\frac{{\mathcal{N}}_{+}}{N})b_{-r-v}^{\ast }{a}_v^{\ast }{a}_{-r}-\frac{1}{N}\sum _{\begin{array}{c}p\in {\Lambda }^{\ast }\end{array}}{\widehat{\omega }}_N(p)b_p^{\ast }{a}_{-p}^{\ast }{a}_v^{\ast }{a}_{-r}{a}_{r+v}.\hfill \end{array}\end{array}$$ 95

With (95) we write

$$\begin{array}{c}\hfill \frac{1}{2}\sum _{\begin{array}{c}p\in {\Lambda }^{\ast }\end{array}}{\widehat{\omega }}_N(p)[b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p},A]=\sum _{j=1}^4{\Pi }_j+\text{h.c.}\end{array}$$

with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\Pi }_1=& -\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}r,v\in {\Lambda }_{+}^{\ast }\\ r\ne -v\end{array}}{\widehat{\omega }}_N(v){\eta }_r{b}_{r+v}^{\ast }{b}_{-v}^{\ast }{b}_{-r}^{\ast },\hfill \\ \hfill {\Pi }_2=& \frac{1}{\sqrt{N}}\sum _{\begin{array}{c}r,v\in {\Lambda }_{+}^{\ast }:\\ r\ne -v\end{array}}{\widehat{\omega }}_N(r){\eta }_r{b}_v^{\ast }(b_r^{\ast }{b}_{r+v}-\frac{2}{N}{a}_r^{\ast }{a}_{r+v}),\hfill \\ \hfill {\Pi }_3=& \frac{1}{\sqrt{N}}\sum _{\begin{array}{c}r,v\in {\Lambda }_{+}^{\ast }\\ r\ne -v\end{array}}{\widehat{\omega }}_N(r+v){\eta }_r(1-\frac{{\mathcal{N}}_{+}}{N})b_{-r-v}^{\ast }{a}_v^{\ast }{a}_{-r},\hfill \\ \hfill {\Pi }_4=& -\frac{1}{N^{3/2}}\sum _{\begin{array}{c}r,v,p\in {\Lambda }_{+}^{\ast }:\\ r\ne -v\end{array}}{\widehat{\omega }}_N(p){\eta }_r{b}_p^{\ast }{a}_{-p}^{\ast }{a}_v^{\ast }{a}_{-r}{a}_{r+v}.\hfill \\ \hfill \end{array}\end{array}$$

To bound the first term, we observe, with (52),

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\Pi }_1\xi ⟩|& \le \frac{\Vert \eta \Vert }{\sqrt{N}}\Vert {\mathcal{K}}^{1/2}{\mathcal{N}}_{+}^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert {\left[\sum _{v\in {\Lambda }_{+}^{\ast }},\frac{|{\widehat{\omega }}_N{(v)|}^2}{v^2}\right]}^{1/2}\hfill \\ \hfill & \le C{N}^{-\alpha }{(logN)}^{1/2}\Vert {\mathcal{K}}^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$

The term ${\Pi }_3$ can be bounded similarly to ${\Pi }_1$, with (52). We find

$$\begin{array}{c}\hfill |⟨\xi ,{\Pi }_3\xi ⟩|\le C{N}^{-\alpha }{(logN)}^{1/2}\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{K}}^{1/2}\xi \Vert .\end{array}$$

With $|{\widehat{\omega }}_N(r)|\le C$, we similarly obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\Pi }_2\xi ⟩|& \le N^{-1/2}\Vert \eta \Vert \Vert {\mathcal{K}}^{1/2}{\mathcal{N}}_{+}^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \hfill \\ \hfill & \le C{N}^{-\alpha }\Vert {\mathcal{K}}^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$

Finally, we estimate, using again (52),

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\Pi }_4\xi ⟩|& \le N^{-3/2}(\sum _{r,v,p\in {\Lambda }_{+}^{\ast }}{p}^2|{\eta }_r{|}^2{\Vert a_{-p}{a}_v{({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill & \times (\sum _{r,v,p\in {\Lambda }_{+}^{\ast }}\frac{|{\widehat{\omega }}_N{(p)|}^2}{p^2}{\Vert a_{-r}{a}_{r+v}\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill & \le C{N}^{-3/2}{\Vert \eta \Vert (logN)}^{1/2}\Vert {\mathcal{K}}^{1/2}({\mathcal{N}}_{+}+1)\xi \Vert \Vert ({\mathcal{N}}_{+}+1)\xi \Vert \hfill \\ \hfill & \le C{N}^{-\alpha }{(logN)}^{1/2}\Vert {\mathcal{K}}^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$

With (94), we conclude that

$$\begin{array}{c}\hfill \begin{array}{cc}& |\frac{1}{2}\sum _{\begin{array}{c}p\in {\Lambda }^{\ast }\end{array}}{\widehat{\omega }}_N(p)[⟨\xi ,e^{-A}(b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p})e^A\xi ⟩-⟨\xi ,(b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p})\xi ⟩]|\hfill \\ \hfill & \le C{N}^{-\alpha }{(logN)}^{1/2}{\int }_0^{1}ds\Vert {\mathcal{K}}^{1/2}{e}^{\mathit{sA}}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}{e}^{\mathit{sA}}\xi \Vert .\hfill \end{array}\end{array}$$

With Proposition 2, Lemma 3, we conclude that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |\frac{1}{2}\sum _{\begin{array}{c}p\in {\Lambda }^{\ast }\end{array}}{\widehat{\omega }}_N(p)& [⟨\xi ,e^{-A}(b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p})e^A\xi ⟩-⟨\xi ,(b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p})\xi ⟩]|\hfill \\ \hfill & \le C{N}^{-\alpha }{(logN)}^{1/2}\left[\Vert ,{\mathcal{H}}_N^{1/2},\xi \Vert +,N^{1/2},\Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert \right]\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \hfill \\ \hfill & \le C{N}^{1-\alpha }{\Vert {({\mathcal{H}}_N+1)}^{1/2}\xi \Vert }^2.\hfill \end{array}\end{array}$$

$\square$

Contributions from $e^{-A}{\mathcal{C}}_N{e}^A$

In Sect. 6.6 we will analyse the contributions to ${\mathcal{R}}_{N,\alpha }$ arising from conjugation of the cubic operator ${\mathcal{C}}_N$ defined in (77). To this aim we will need some properties of the commutator $[{\mathcal{C}}_N,A]$, as established in the following proposition.

Proposition 11

Let A be defined in (44). Then, there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [{\mathcal{C}}_N,A]=& 2\sum _{r,v\in {\Lambda }_{+}^{\ast }}[\widehat{V}(r/e^N){\eta }_r+\widehat{V}((r+v)/e^N){\eta }_r]a_v^{\ast }{a}_v(1-\frac{{\mathcal{N}}_{+}}{N})+{\delta }_{{\mathcal{C}}_N}\hfill \\ \hfill \end{array}\end{array}$$

where

$$\begin{array}{c}\hfill |⟨\xi ,{\delta }_{{\mathcal{C}}_N}\xi ⟩|\le C{N}^{3/2-\alpha }\Vert {\mathcal{H}}_N^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \end{array}$$ 96

for all $\alpha >0$, $\xi \in {\mathcal{F}}_{+}^{\le N}$, and $N\in \mathbb{N}$ large enough.

Proof

We consider the commutator

$$\begin{array}{c}\hfill [{\mathcal{C}}_N,A]=\sum _{\begin{array}{c}p,q\in {\Lambda }_{+}^{\ast }:p+q\ne 0\\ r,v\in {\Lambda }_{+}^{\ast }\end{array}}\widehat{V}(p/e^N){\eta }_r[b_{p+q}^{\ast }{a}_{-p}^{\ast }{a}_q,b_{r+v}^{\ast }{a}_{-r}^{\ast }{a}_v-a_v^{\ast }{a}_{-r}{b}_{r+v}]+\text{h.c.}.\end{array}$$

As in the proof of Proposition 9, we use the commutators from the proof of Proposition 8.8 in [4] to conclude that

$$\begin{array}{c}\hfill [{\mathcal{C}}_N,A]=2\sum _{r,v\in {\Lambda }_{+}^{\ast }}[\widehat{V}(r/e^N){\eta }_r+\widehat{V}((r+v)/e^N){\eta }_r]a_v^{\ast }{a}_v\frac{N-{\mathcal{N}}_{+}}{N}+\sum _{j=1}^{12}({\Xi }_j+\text{h.c.})\end{array}$$

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\Xi }_1:=& -\sum _{\begin{array}{c}r,v,p\in {\Lambda }_{+}^{\ast },\\ p\ne v\end{array}}\widehat{V}(p/e^N){\eta }_r{b}_{r+v}^{\ast }{b}_{-r}^{\ast }{a}_{-p}^{\ast }{a}_{v-p},\hfill \\ \hfill {\Xi }_2:=& \sum _{\begin{array}{c}r,v,p\in {\Lambda }_{+}^{\ast }\\ r\ne -p\end{array}}\widehat{V}(p/e^N){\eta }_r(1-{\mathcal{N}}_{+}/N)a_v^{\ast }{a}_{-p}^{\ast }{a}_{-r-p}{a}_{r+v},\hfill \\ \hfill {\Xi }_3:=& \sum _{\begin{array}{c}r,v,p\in {\Lambda }_{+}^{\ast }:\\ r+v\ne p\end{array}}\widehat{V}(p/e^N){\eta }_r(1-{\mathcal{N}}_{+}/N)a_v^{\ast }{a}_{-p}^{\ast }{a}_{-r}{a}_{r+v-p},\hfill \\ \hfill {\Xi }_4:=& -\frac{1}{N}\sum _{\begin{array}{c}r,v,p,q\in {\Lambda }_{+}^{\ast }:p+q\ne 0\end{array}}\widehat{V}(p/e^N){\eta }_r{a}_v^{\ast }{a}_{p+q}^{\ast }{a}_{-p}^{\ast }{a}_{-r}{a}_{r+v}{a}_q,\hfill \\ \hfill {\Xi }_5:=& -\frac{1}{N}\sum _{\begin{array}{c}r,v,q\in {\Lambda }_{+}^{\ast }:\\ r+v\ne q\end{array}}\widehat{V}((r+v)/e^N){\eta }_r{a}_v^{\ast }{a}_{q-r-v}^{\ast }{a}_{-r}{a}_q,\hfill \\ \hfill {\Xi }_6:=& -\frac{1}{N}\sum _{\begin{array}{c}r,v,q\in {\Lambda }_{+}^{\ast }:\\ r\ne -q\end{array}}\widehat{V}(r/e^N){\eta }_r{a}_v^{\ast }{a}_{q+r}^{\ast }{a}_{r+v}{a}_q\hfill \\ \hfill {\Xi }_7:=& \sum _{\begin{array}{c}r,v,p\in {\Lambda }_{+}^{\ast }:\\ r+v\ne -p\end{array}}\widehat{V}(p/e^N){\eta }_r{b}_{p+r+v}^{\ast }{b}_{-p}^{\ast }{a}_{-r}^{\ast }{a}_v,\hfill \\ \hfill {\Xi }_8:=& \sum _{\begin{array}{c}r,v,p\in {\Lambda }_{+}^{\ast }:\\ r\ne -p\end{array}}\widehat{V}(p/e^N){\eta }_r{b}_{p-r}^{\ast }{b}_{r+v}^{\ast }{a}_{-p}^{\ast }{a}_v,\hfill \\ \hfill {\Xi }_9:=& -\sum _{\begin{array}{c}r,v,q\in {\Lambda }_{+}^{\ast }:\\ q\ne v\end{array}}\widehat{V}(v/e^N){\eta }_r{b}_{q-v}^{\ast }{b}_{r+v}^{\ast }{a}_{-r}^{\ast }{a}_q,\hfill \\ \hfill {\Xi }_{10}:=& \sum _{\begin{array}{c}r,v,q\in {\Lambda }_{+}^{\ast }:\\ r\ne -q\end{array}}\widehat{V}(r/e^N){\eta }_r{b}_{q+r}^{\ast }{a}_v^{\ast }{a}_q{b}_{r+v},\hfill \\ \hfill {\Xi }_{11}:=& -\sum _{\begin{array}{c}r,v,p\in {\Lambda }_{+}^{\ast }:\\ p\ne -v\end{array}}\widehat{V}(p/e^N){\eta }_r{b}_{p+v}^{\ast }{a}_{-p}^{\ast }{a}_{-r}{b}_{r+v},\hfill \\ \hfill {\Xi }_{12}:=& \sum _{\begin{array}{c}r,v,q\in {\Lambda }_{+}^{\ast }:\\ q\ne r+v\end{array}}\widehat{V}((r+v)/e^N){\eta }_r{b}_{q-r-v}^{\ast }{a}_v^{\ast }{a}_{-r}{b}_q.\hfill \end{array}\end{array}$$

To prove the proposition, we have to show that all terms ${\Xi }_j$, $j=1,\dots ,12$, satisfy the bound (96). We bound ${\Xi }_1$ in position space, with Cauchy–Schwarz, by

$$\begin{array}{cc}\hfill |⟨\xi ,{\Xi }_1\xi ⟩|\le & C{\int }_{{\Lambda }^3}dxdydz{e}^{2N}V(e^N(x-y))|\check{\eta }(x-z)|\Vert {\check{a}}_x\xi \Vert \Vert {\check{a}}_x{\check{a}}_y{\check{a}}_z\xi \Vert \hfill \\ \hfill & \le C{\left[{\int }_{{\Lambda }^3},d,x,d,y,d,z,,e^{2N},V,(e^N(x-y)),{\Vert {\check{a}}_x{\check{a}}_y{\check{a}}_z\xi \Vert }^2\right]}^{1/2}\hfill \\ \hfill & \times {\left[{\int }_{{\Lambda }^3},d,x,d,y,d,z,,e^{2N},V,(e^N(x-y)),|,\check{\eta },{(x-z)|}^2,{\Vert {\check{a}}_x\xi \Vert }^2\right]}^{1/2}\hfill \\ \hfill & \le C\Vert \eta \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{V}}_N^{1/2}{\mathcal{N}}_{+}^{1/2}\xi \Vert \hfill \\ \hfill & \le C{N}^{1/2-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{V}}_N^{1/2}\xi \Vert .\hfill \end{array}$$

We can proceed similarly to control ${\Xi }_9$. We obtain

$$\begin{array}{c}\hfill |⟨\xi ,{\Xi }_9\xi ⟩|\le C{N}^{1/2-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{V}}_N^{1/2}\xi \Vert .\end{array}$$

The expectations of the terms ${\Xi }_3$ and ${\Xi }_{12}$ can be bounded analogously:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |& ⟨\xi ,{\Xi }_3\xi ⟩|+|⟨\xi ,{\Xi }_{12}\xi ⟩|\hfill \\ \hfill & \le C{\int }_{{\Lambda }^3}dxdydz{e}^{2N}V(e^N(x-y))(|\eta (x-z)|+|\eta (y-z)|)\Vert {\check{a}}_x{\check{a}}_y\xi \Vert \Vert {\check{a}}_x{\check{a}}_z\xi \Vert \hfill \\ \hfill & \le C{\left[{\int }_{{\Lambda }^3},d,x,d,y,d,z,,e^{2N},V,(e^N(x-y)),\Vert ,{\check{a}}_x,{\check{a}}_y,{\xi \Vert }^2,{(|\eta (x-z)|}^2,+,{|\eta (y-z)|}^2,)\right]}^{1/2}\hfill \\ \hfill & \times {\left[{\int }_{{\Lambda }^3},d,x,d,y,d,z,,e^{2N},V,(e^N(x-y)),{\Vert {\check{a}}_x{\check{a}}_z\xi \Vert }^2\right]}^{1/2}\hfill \\ \hfill & \le C\Vert \eta \Vert \Vert ({\mathcal{N}}_{+}+1)\xi \Vert \Vert {\mathcal{V}}_N^{1/2}\xi \Vert \hfill \\ \hfill & \le C{N}^{1/2-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{V}}_N^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$

As for ${\Xi }_4$, we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\Xi }_4\xi ⟩|=& |\frac{1}{N}{\int }_{{\Lambda }^2}dxdydz{e}^{2N}V(e^N(y-z))⟨\xi ,{\check{a}}_x^{\ast }{\check{a}}_y^{\ast }{\check{a}}_z^{\ast }\check{a}({\check{\eta }}_x){\check{a}}_x{\check{a}}_y\xi ⟩|\hfill \\ \hfill & \le C{N}^{-1}\Vert \eta \Vert {\int }_{{\Lambda }^2}dxdydz{e}^{2N}V(e^N(y-z))\Vert {\check{a}}_x{\check{a}}_y{\check{a}}_z\xi \Vert \Vert {\mathcal{N}}_{+}^{1/2}{\check{a}}_x{\check{a}}_y\xi \Vert \hfill \\ \hfill & \le C{N}^{-1}\Vert \eta \Vert {\left[{\int }_{{\Lambda }^2},d,x,d,y,d,z,,e^{2N},V,(e^N(y-z)),{\Vert {\check{a}}_x{\check{a}}_y{\check{a}}_z\xi \Vert }^2\right]}^{1/2}\hfill \\ \hfill & \times {\left[{\int }_{{\Lambda }^2},d,x,d,y,d,z,,e^{2N},V,(e^N(y-z)),{\Vert {\mathcal{N}}_{+}^{1/2}{\check{a}}_x{\check{a}}_y\xi \Vert }^2\right]}^{1/2}\hfill \\ \hfill & \le C{N}^{1/2-\alpha }\Vert {\mathcal{V}}_N^{1/2}\xi \Vert \Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$

The terms ${\Xi }_5$ and ${\Xi }_6$ can be bounded in momentum space, using (154). Hence,

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,{\Xi }_5\xi ⟩|+|⟨\xi ,{\Xi }_6\xi ⟩|\hfill \\ \hfill & \le C{N}^{-1}\sum _{r,v,q\in {\Lambda }_{+}^{\ast }}(\frac{\widehat{V}((v+r)/e^N)}{|v|}|{\eta }_r||v|\Vert a_v{a}_{q-r-v}\xi \Vert \Vert a_{-r}{a}_q\xi \Vert \hfill \\ \hfill & +\frac{\widehat{V}(r/e^N)}{|r+v|}|{\eta }_r||r+v|\Vert a_{r+q}{a}_v\xi \Vert \Vert a_q{a}_{r+v}\xi \Vert )\hfill \\ \hfill & \le C{N}^{1/2-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{K}}^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$

Similarly we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\Xi }_2\xi ⟩|+|⟨\xi ,{\Xi }_{10}\xi ⟩|& \le \sum _{r,v,p\in {\Lambda }_{+}^{\ast }}(\frac{\widehat{V}(p/e^N)}{|p|}|{\eta }_r||p|\Vert a_v{a}_{-p}\xi \Vert \Vert a_{r+v}{a}_{-r-p}\xi \Vert \hfill \\ \hfill & +\frac{\widehat{V}(r/e^N)}{|r+v|}|{\eta }_r||r+v|\Vert a_q{a}_{r+v}\xi \Vert \Vert a_{r+q}{a}_v\xi \Vert )\hfill \\ \hfill & \le C{N}^{3/2-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{K}}^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$

Next, we rewrite ${\Xi }_7$, ${\Xi }_8$ and ${\Xi }_{11}$ as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\Xi }_7=& {\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y)){\check{b}}_x^{\ast }{\check{b}}_y^{\ast }{a}^{\ast }({\check{\eta }}_x){\check{a}}_x,\hfill \\ \hfill {\Xi }_8=& {\int }_{{\Lambda }^2}dxdydz{e}^{2N}V(e^N(x-y))\check{\eta }(z-x){\check{b}}_x^{\ast }{\check{b}}_z^{\ast }{\check{a}}_y^{\ast }{\check{a}}_z,\hfill \\ \hfill {\Xi }_{11}=& -{\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y)){\check{b}}_x^{\ast }{\check{a}}_y^{\ast }\check{a}({\check{\eta }}_x){\check{b}}_x.\hfill \end{array}\end{array}$$

Thus, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\Xi }_7\xi ⟩|& \le C\Vert \eta \Vert {\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y))\Vert {\mathcal{N}}_{+}^{1/2}{\check{a}}_x{\check{a}}_y\xi \Vert \Vert {\check{a}}_x\xi \Vert \hfill \\ \hfill & \le C\Vert \eta \Vert \Vert {\mathcal{N}}_{+}^{1/2}{\mathcal{V}}_N^{1/2}\xi \Vert \Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert \hfill \\ \hfill & \le C{N}^{1/2-\alpha }\Vert {\mathcal{V}}_N^{1/2}\xi \Vert \Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert ,\hfill \end{array}\end{array}$$

as well as

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,{\Xi }_8\xi ⟩|\hfill \\ \hfill & \le C{\int }_{{\Lambda }^2}dxdydz{e}^{2N}V(e^N(x-y))|\check{\eta }(x-z)|\Vert {\check{a}}_x{\check{a}}_y{\check{a}}_z\xi \Vert \Vert {\check{a}}_z\xi \Vert \hfill \\ \hfill & \le C{\left[{\int }_{{\Lambda }^2},d,x,d,y,d,z,,e^{2N},V,(e^N(x-y)),{\Vert {\check{a}}_x{\check{a}}_y{\check{a}}_z\xi \Vert }^2\right]}^{1/2}\hfill \\ \hfill & \times {\left[{\int }_{{\Lambda }^2},d,x,d,y,d,z,,e^{2N},V,(e^N(x-y)),{|\eta (x-z)|}^2,{\Vert {\check{a}}_z\xi \Vert }^2\right]}^{1/2}\hfill \\ \hfill & \le C{N}^{1/2-\alpha }\Vert {\mathcal{V}}_N^{1/2}\xi \Vert \Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert ,\hfill \end{array}\end{array}$$

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\Xi }_{11}\xi ⟩|& \le C\Vert \eta \Vert {\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y))\Vert {\check{a}}_x{\check{a}}_y\xi \Vert \Vert {\mathcal{N}}_{+}^{1/2}{\check{a}}_x\xi \Vert \hfill \\ \hfill & \le C\Vert \eta \Vert \Vert {\mathcal{V}}_N^{1/2}\xi \Vert \Vert {\mathcal{N}}_{+}\xi \Vert \le C{N}^{1/2-\alpha }\Vert {\mathcal{V}}_N^{1/2}\xi \Vert \Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$

Collecting all the bounds above, we arrive at (96). $\square$

Proof of Proposition 4

With the results of Sects. 6.1–6.5, we can now show Proposition 4. We assume $\alpha >2$. From Eq. (76), Propositions 8 and 10 we obtain that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{R}}_{N,\alpha }=& e^{-A}{\mathcal{G}}_{N,\alpha }^{\text{eff}}{e}^A\hfill \\ \hfill =& \frac{1}{2}{\widehat{\omega }}_N(0)(N-1)(1-{\mathcal{N}}_{+}/N)+\left[2,N,\widehat{V},(0),-,\frac{1}{2},{\widehat{\omega }}_N,(0)\right]{\mathcal{N}}_{+}(1-{\mathcal{N}}_{+}/N)\hfill \\ \hfill & +\frac{1}{2}\sum _{p\in {\Lambda }_{+}^{\ast }}{\widehat{\omega }}_N(p)[b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p}]+\mathcal{K}+{\mathcal{C}}_N+{\mathcal{V}}_N\hfill \\ \hfill & +{\int }_0^{1}ds{e}^{-sA}[\mathcal{K}+{\mathcal{C}}_N+{\mathcal{V}}_N,A]e^{\mathit{sA}}+{\mathcal{E}}_{\mathcal{R}}^{(1)}\hfill \end{array}\end{array}$$

with

$$\begin{array}{c}\hfill \pm {\mathcal{E}}_{\mathcal{R}}^{(1)}\le C{N}^{1-\alpha }({\mathcal{H}}_N+1).\end{array}$$

From Propositions 7, 9 and 11, we can write, for N large enough,

$$\begin{array}{c}\hfill \begin{array}{cc}& [\mathcal{K}+{\mathcal{C}}_N+{\mathcal{V}}_N,A]\hfill \\ \hfill & =\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}r,v\in {\Lambda }_{+}^{\ast }\end{array}}{\widehat{\omega }}_N(r)[b_{r+v}^{\ast }{a}_{-r}^{\ast }{a}_v+\text{h.c.}]-\sqrt{N}\sum _{\begin{array}{c}r,v,\in {\Lambda }_{+}^{\ast },\\ p\ne -q\end{array}}\widehat{V}(r/e^N)[b_{r+v}^{\ast }{a}_{-r}^{\ast }{a}_v+\text{h.c.}]\hfill \\ \hfill & +2\sum _{r,v\in {\Lambda }_{+}^{\ast }}[\widehat{V}(r/e^N){\eta }_r+\widehat{V}((r+v)/e^N){\eta }_r]a_v^{\ast }{a}_v(1-{\mathcal{N}}_{+}/N)+{\mathcal{E}}_{\mathcal{R}}^{(2)}\hfill \end{array}\end{array}$$

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\mathcal{E}}_{\mathcal{R}}^{(2)}\xi ⟩|\le & C{N}^{1/2-\alpha }{(logN)}^{1/2}\Vert {\mathcal{H}}_N^{1/2}{\xi \Vert }^2+C{N}^{3/2-\alpha }\Vert {\mathcal{H}}_N^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \hfill \\ \hfill & +C{N}^{-1}{(logN)}^{1/2}\Vert {\mathcal{H}}_N^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$

for all $\xi \in {\mathcal{F}}_{+}^{\le N}$. From Proposions 2, 3 and recalling the definition (77) of the operator ${\mathcal{C}}_N$, we deduce that

$$\begin{array}{c}\hfill \begin{array}{cc}& {\int }_0^{1}ds{e}^{-sA}[\mathcal{K}+{\mathcal{C}}_N+{\mathcal{V}}_N,A]e^{\mathit{sA}}\hfill \\ \hfill & ={\int }_0^{1}ds{e}^{-sA}[-{\mathcal{C}}_N+\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}r,v\in {\Lambda }_{+}^{\ast }\end{array}}{\widehat{\omega }}_N(r)[b_{r+v}^{\ast }{a}_{-r}^{\ast }{a}_v+\text{h.c.}]\hfill \\ \hfill & +2\sum _{r,v\in {\Lambda }_{+}^{\ast }}[\widehat{V}(r/e^N){\eta }_r+\widehat{V}((r+v)/e^N){\eta }_r]a_v^{\ast }{a}_v(1-\frac{{\mathcal{N}}_{+}}{N})]e^{\mathit{sA}}+{\mathcal{E}}_{\mathcal{R}}^{(3)}\hfill \end{array}\end{array}$$ 97

with

$$\begin{array}{c}\hfill \pm {\mathcal{E}}_{\mathcal{R}}^{(3)}\le C[N^{2-\alpha }+N^{-1/2}{(logN)}^{1/2}]({\mathcal{H}}_N+1)\end{array}$$

for $N\in \mathbb{N}$ sufficiently large.

We now rewrite

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill 2\sum _{r,v\in {\Lambda }_{+}^{\ast }}& [\widehat{V}(r/e^N){\eta }_r+\widehat{V}((r+v)/e^N){\eta }_r]a_v^{\ast }{a}_v(1-\frac{{\mathcal{N}}_{+}}{N})\hfill \\ \hfill =& 4\sum _{r,v\in {\Lambda }_{+}^{\ast }}\widehat{V}(r/e^N){\eta }_r{a}_v^{\ast }{a}_v(1-\frac{{\mathcal{N}}_{+}}{N})\hfill \\ \hfill & +2\sum _{r,v\in {\Lambda }_{+}^{\ast }}[\widehat{V}((r+v)/e^N)-\widehat{V}(r/e^N)]{\eta }_r{a}_v^{\ast }{a}_v(1-\frac{{\mathcal{N}}_{+}}{N}):={\text{Q}}_1+{\text{Q}}_2.\hfill \end{array}\end{array}$$ 98

With Lemma 1, part (iii) we get

$$\begin{array}{c}\hfill \begin{array}{cc}& |2\sum _{r\in {\Lambda }^{\ast }}\widehat{V}(r/e^N){\eta }_r-[2{\widehat{\omega }}_N(0)-2N\widehat{V}(0)]|\le \frac{C}{N},\hfill \end{array}\end{array}$$ 99

and therefore, using Lemma 3 and (99)

$$\begin{array}{c}\hfill \begin{array}{cc}& \pm [e^{-sA}{\text{Q}}_1{e}^{\mathit{sA}}-2[2{\widehat{\omega }}_N(0)-2N\widehat{V}(0)]\sum _{v\in {\Lambda }_{+}^{\ast }}{a}_v^{\ast }{a}_v(1-\frac{{\mathcal{N}}_{+}}{N})]\hfill \\ \hfill & \le C{N}^{1-\alpha }({\mathcal{N}}_{+}+1)+\frac{C}{N}{\mathcal{N}}_{+}.\hfill \end{array}\end{array}$$ 100

On the other hand it is easy to check that $e^{-sA}{\text{Q}}_2{e}^{\mathit{sA}}$ is an error term; to this aim we notice that

$$\begin{array}{c}\hfill \begin{array}{cc}& |\sum _{r\in {\Lambda }^{\ast }}[\widehat{V}(r/e^N){\eta }_r-\widehat{V}((r+v)/e^N){\eta }_r]|\le CN|v|e^{-N}.\hfill \end{array}\end{array}$$

Hence with Props. 2 and 3 we find

$$\begin{array}{c}\hfill \pm [e^{-sA}{\text{Q}}_2{e}^{\mathit{sA}}]\le CN{e}^{-N}{e}^{-sA}{\mathcal{N}}_{+}^{1/2}{\mathcal{K}}^{1/2}{e}^{\mathit{sA}}\le C{N}^2{e}^{-N}({\mathcal{H}}_N+1).\end{array}$$ 101

To handle the second term on the second line of (97), we apply Proposition 9 and then Propositions 2 and 3

$$\begin{array}{c}\hfill \begin{array}{cc}& \pm (\frac{1}{\sqrt{N}}{\int }_0^{1}ds\sum _{\begin{array}{c}r,v\in {\Lambda }_{+}^{\ast }\end{array}}{\widehat{\omega }}_N(r)[e^{-sA}{b}_{r+v}^{\ast }{a}_{-r}^{\ast }{a}_v{e}^{\mathit{sA}}-b_{r+v}^{\ast }{a}_{-r}^{\ast }{a}_v]+\text{h.c.})\hfill \\ \hfill & =\pm (\frac{1}{\sqrt{N}}{\int }_0^{1}ds{\int }_0^{s}dt\sum _{\begin{array}{c}r,v\in {\Lambda }_{+}^{\ast }\end{array}}{\widehat{\omega }}_N(r)e^{-tA}[b_{r+v}^{\ast }{a}_{-r}^{\ast }{a}_v,A]e^{\mathit{tA}})\hfill \\ \hfill & \le C{\int }_0^{1}ds{\int }_0^{s}dt{e}^{-tA}(N^{-\alpha }(logN)\mathcal{K}+N^{-1}({\mathcal{N}}_{+}+1))e^{\mathit{tA}}\hfill \\ \hfill & \le C{N}^{1-\alpha }logN({\mathcal{H}}_N+1).\hfill \end{array}\end{array}$$ 102

As for the first term on the second line of (97), we use again Proposition 11. Using (98), (100) and (101) we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\int }_0^{1}ds{e}^{-sA}{\mathcal{C}}_N{e}^{\mathit{sA}}-{\mathcal{C}}_N& ={\int }_0^{1}ds{\int }_0^{s}dt{e}^{-tA}[{\mathcal{C}}_N,A]e^{\mathit{tA}}\hfill \\ \hfill & =[2{\widehat{\omega }}_N(0)-2N\widehat{V}(0)]\sum _{p\in {\Lambda }_{+}^{\ast }}{a}_p^{\ast }{a}_p(1-\frac{{\mathcal{N}}_{+}}{N})+{\mathcal{E}}_{\mathcal{R}}^{(4)}\hfill \end{array}\end{array}$$ 103

with $\pm {\mathcal{E}}_{\mathcal{R}}^{(4)}\le C{N}^{2-\alpha }({\mathcal{H}}_N+1)+C{N}^{-1}({\mathcal{N}}_{+}+1)$.

Inserting the bounds (100), (101), (102) and (103) into (97) we arrive at

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{R}}_{N,\alpha }=& \frac{1}{2}(N-1){\widehat{\omega }}_N(0)(1-{\mathcal{N}}_{+}/N)+\frac{1}{2}{\widehat{\omega }}_N(0){\mathcal{N}}_{+}\left(1,-,{\mathcal{N}}_{+},/,N\right)\hfill \\ & +{\widehat{\omega }}_N(0)\sum _{p\in {\Lambda }_{+}^{\ast }}{a}_p^{\ast }{a}_p(1-\frac{{\mathcal{N}}_{+}}{N})+\frac{1}{2}\sum _{p\in {\Lambda }_{+}^{\ast }}{\widehat{\omega }}_N(p)[b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p}]\hfill \\ & +\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}r,v\in {\Lambda }_{+}^{\ast }:\\ r\ne -v\end{array}}{\widehat{\omega }}_N(r)[b_{r+v}^{\ast }{a}_{-r}^{\ast }{a}_v+\text{h.c.}]+{\mathcal{H}}_N+{\mathcal{E}}_{\mathcal{R}}\hfill \end{array}\end{array}$$

with

$$\begin{array}{c}\hfill \pm {\mathcal{E}}_{\mathcal{R}}\le C[N^{2-\alpha }+N^{-1/2}{(logN)}^{1/2}]({\mathcal{H}}_N+1)\end{array}$$

for $N\in \mathbb{N}$ sufficiently large.

Appendix A: Analysis of ${\mathcal{G}}_{N,\alpha }$

The aim of this section is to show Proposition 1. From (12) and (37), we can decompose

$$\begin{array}{c}\hfill {\mathcal{G}}_{N,\alpha }=e^{-B}{\mathcal{L}}_N{e}^B={\mathcal{G}}_{N,\alpha }^{(0)}+{\mathcal{G}}_{N,\alpha }^{(2)}+{\mathcal{G}}_{N,\alpha }^{(3)}+{\mathcal{G}}_{N,\alpha }^{(4)}\end{array}$$

with

$$\begin{array}{c}\hfill {\mathcal{G}}_{N,\alpha }^{(j)}=e^{-B}{\mathcal{L}}_N^{(j)}{e}^B.\end{array}$$

To analyse ${\mathcal{G}}_{N,\alpha }$ we will need precise informations on the action of the generalized Bogoliubov transformation $e^B$ with B the antisymmetric operator defined in (33), which are summarized in Sect. 1. Then, in the Sects. 1–1 we prove separate bounds for the operators ${\mathcal{G}}_{N,\alpha }^{(j)}$, $j=0,2,3,4$, which we combine in Sect. 1 to prove Proposition 1.

The analysis in this section follows closely that of [4, Sect. 7] with some slight modifications due to the different scaling of the interaction potential and the fact that the kernel ${\eta }_p$ of $e^B$ is different from zero for all $p\in {\Lambda }_{+}^{\ast }$ (in [4] ${\eta }_p$ is different from zero only for momenta larger than a sufficiently large cutoff of order one). Moreover, while in three dimensions it was sufficient to choose the function ${\eta }_p$ appearing in the generalized Bogoliubov transformation with $\Vert \eta \Vert$ sufficiently small but of order one, we need here $\Vert \eta \Vert$ to be of order $N^{-\alpha }$ for some $\alpha >0$ large enough. As discussed in the introduction this is achieved by considering the Neumann problem for the scattering equation in (16) on a ball of radius $\ell =N^{-\alpha }$; as a consequence some terms depending on $\ell$ will be large, compared to the analogous terms in [4].

Appendix A.1: Generalized Bogoliubov Transformations

In this subsection we collect important properties about the action of unitary operators of the form $e^B$, as defined in (34). As shown in [2, Lemma 2.5 and 2.6], we have, if $\Vert \eta \Vert$ is sufficiently small,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^{-B}{b}_p{e}^B& =\sum _{n=0}^{\infty }\frac{{(-1)}^n}{n!}{\text{ad}}_B^{(n)}(b_p)\hfill \\ \hfill e^{-B}{b}_p^{\ast }{e}^B& =\sum _{n=0}^{\infty }\frac{{(-1)}^n}{n!}{\text{ad}}_B^{(n)}(b_p^{\ast })\hfill \end{array}\end{array}$$ 104

where the series converge absolutely. To confirm the expectation that generalized Bogoliubov transformation act similarly to standard Bogoliubov transformations, on states with few excitations, we define (for $\Vert \eta \Vert$ small enough) the remainder operators

$$\begin{array}{c}\hfill d_q=\sum _{m\ge 0}\frac{1}{m!}[{\text{ad}}_{-B}^{(m)}(b_q)-{\eta }_q^m{b}_{{\alpha }_{m}q}^{{♯}_m}],d_q^{\ast }=\sum _{m\ge 0}\frac{1}{m!}[{\text{ad}}_{-B}^{(m)}(b_q^{\ast })-{\eta }_q^m{b}_{{\alpha }_{m}q}^{{♯}_{m+1}}]\end{array}$$ 105

where $q\in {\Lambda }_{+}^{\ast }$, $({♯}_m,{\alpha }_m)=(·,+1)$ if m is even and $({♯}_m,{\alpha }_m)=(\ast ,-1)$ if m is odd. It follows then from (104) that

$$\begin{array}{c}\hfill e^{-B}{b}_q{e}^B={\gamma }_q{b}_q+{\sigma }_q{b}_{-q}^{\ast }+d_q,e^{-B}{b}_q^{\ast }{e}^B={\gamma }_q{b}_q^{\ast }+{\sigma }_q{b}_{-q}+d_q^{\ast }\end{array}$$ 106

where we introduced the notation ${\gamma }_q=cosh({\eta }_q)$ and ${\sigma }_q=sinh({\eta }_q)$. It will also be useful to introduce remainder operators in position space. For $x\in \Lambda$, we define the operator valued distributions ${\check{d}}_x,{\check{d}}_x^{\ast }$ through

$$\begin{array}{c}\hfill e^{-B}{\check{b}}_x{e}^B=b({\check{\gamma }}_x)+b^{\ast }({\check{\sigma }}_x)+{\check{d}}_x,e^{-B}{\check{b}}_x^{\ast }{e}^B=b^{\ast }({\check{\gamma }}_x)+b({\check{\sigma }}_x)+{\check{d}}_x^{\ast }\end{array}$$ 107

where ${\check{\gamma }}_x(y)={\sum }_{q\in {\Lambda }^{\ast }}cosh({\eta }_q)e^{iq·(x-y)}$ and ${\check{\sigma }}_x(y)={\sum }_{q\in {\Lambda }^{\ast }}sinh({\eta }_q)e^{iq·(x-y)}$. The next lemma is taken from [4, Lemma 3.4].

Lemma 4

Let $\eta \in {\ell }^2({\Lambda }_{+}^{\ast })$, $n\in \mathbb{Z}$. For $p\in {\Lambda }_{+}^{\ast }$, let $d_p$ be defined as in (106). If $\Vert \eta \Vert$ is small enough, there exists $C>0$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \Vert {({\mathcal{N}}_{+}+1)}^{n/2}{d}_p\xi \Vert & \le \frac{C}{N}\left[|,{\eta }_p,|\Vert ,{({\mathcal{N}}_{+}+1)}^{(n+3)/2},\xi \Vert +\Vert \eta \Vert \Vert ,b_p,{({\mathcal{N}}_{+}+1)}^{(n+2)/2},\xi \Vert \right],\hfill \\ \hfill \Vert {({\mathcal{N}}_{+}+1)}^{n/2}{d}_p^{\ast }\xi \Vert & \le \frac{C}{N}\Vert \eta \Vert \Vert {({\mathcal{N}}_{+}+1)}^{(n+3)/2}\xi \Vert \hfill \end{array}\end{array}$$ 108

for all $p\in {\Lambda }_{+}^{\ast },\xi \in {\mathcal{F}}_{+}^{\le N}$. In position space, with ${\check{d}}_x$ defined as in (107), we find

$$\begin{array}{c}\hfill \Vert {({\mathcal{N}}_{+}+1)}^{n/2}{\check{d}}_x\xi \Vert \le \frac{C}{N}\Vert \eta \Vert [\Vert {({\mathcal{N}}_{+}+1)}^{(n+3)/2}\xi \Vert +\Vert b_x{({\mathcal{N}}_{+}+1)}^{(n+2)/2}\xi \Vert .]\end{array}$$ 109

Furthermore, letting ${\check{\overline{d}}}_x={\check{d}}_x+({\mathcal{N}}_{+}/N)b^{\ast }({\check{\eta }}_x)$, we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \Vert ({\mathcal{N}}_{+}& {+1)}^{n/2}{\check{a}}_y{\check{\overline{d}}}_x\xi \Vert \hfill \\ \hfill & \le \frac{C}{N}{[\Vert \eta \Vert }^2\Vert {({\mathcal{N}}_{+}+1)}^{(n+2)/2}\xi \Vert +\Vert \eta \Vert |\check{\eta }(x-y)|\Vert {(\mathcal{N}+1)}^{(n+2)/2}\xi \Vert \hfill \\ \hfill & +\Vert \eta \Vert \Vert {\check{a}}_x{({\mathcal{N}}_{+}+1)}^{(n+1)/2}{\xi \Vert +\Vert \eta \Vert }^2\Vert {\check{a}}_y{({\mathcal{N}}_{+}+1)}^{(n+3)/2}\xi \Vert \hfill \\ \hfill & +\Vert \eta \Vert \Vert {\check{a}}_x{\check{a}}_y{(\mathcal{N}+1)}^{(n+2)/2}\xi \Vert ]\hfill \end{array}\end{array}$$ 110

and, finally,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \Vert ({\mathcal{N}}_{+}& {+1)}^{n/2}{\check{d}}_x{\check{d}}_y\xi \Vert \hfill \\ \hfill & \le \frac{C}{N^2}{[\Vert \eta \Vert }^2\Vert {({\mathcal{N}}_{+}+1)}^{(n+6)/2}\xi \Vert +\Vert \eta \Vert |\check{\eta }(x-y)|\Vert {({\mathcal{N}}_{+}+1)}^{(n+4)/2}\xi \Vert \hfill \\ \hfill & +{\Vert \eta \Vert }^2\Vert a_x{({\mathcal{N}}_{+}+1)}^{(n+5)/2}{\xi \Vert +\Vert \eta \Vert }^2\Vert a_y{({\mathcal{N}}_{+}+1)}^{(n+5)/2}\xi \Vert \hfill \\ \hfill & +{\Vert \eta \Vert }^2\Vert a_x{a}_y{({\mathcal{N}}_{+}+1)}^{(n+4)/2}\xi \Vert ]\hfill \end{array}\end{array}$$ 111

for all $\xi \in {\mathcal{F}}_{+}^{\le n}$.

A first simple application of Lemma 4 is the following bound on the growth of the expectation of ${\mathcal{N}}_{+}$.

Lemma 5

Assume B is defined as in (33), with $\eta \in {\ell }^2({\Lambda }^{\ast })$ and ${\eta }_p={\eta }_{-p}$ for all $p\in {\Lambda }_{+}^{\ast }$. Then, there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill |⟨\xi ,[e^{-B}{\mathcal{N}}_{+}{e}^B-{\mathcal{N}}_{+}]\xi ⟩|\le \Vert \eta \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\xi \Vert }^2\end{array}$$

for all $\xi \in {\mathcal{F}}_{+}^{\le N}$.

Proof

With (106) we write

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^{-B}& {\mathcal{N}}_{+}{e}^B-{\mathcal{N}}_{+}\hfill \\ \hfill =& {\int }_0^1{e}^{-sB}[{\mathcal{N}}_{+},B]e^{\mathit{sB}}ds\hfill \\ \hfill =& {\int }_0^1\sum _{p\in {\Lambda }_{+}^{\ast }}{\eta }_p{e}^{-sB}(b_p{b}_{-p}+b_p^{\ast }{b}_{-p}^{\ast })e^{\mathit{sB}}ds\hfill \\ \hfill =& {\int }_0^1\sum _{p\in {\Lambda }_{+}^{\ast }}{\eta }_p\left[({\gamma }_p^{(s)}{b}_p+{\sigma }_p^{(s)}{b}_{-p}^{\ast }+d_p^{(s)}),({\gamma }_p^{(s)}{b}_{-p}+{\sigma }_p^{(s)}{b}_{-p}^{\ast }+d_{-p}^{(s)}),+,,\text{h.c.},\right]ds\hfill \end{array}\end{array}$$

with ${\gamma }_p^{(s)}=cosh(s{\eta }_p)$, ${\sigma }_p^{(s)}=sinh(s{\eta }_p)$. Using $|{\gamma }_p^{(s)}|\le C$ and $|{\sigma }_p^{(s)}|\le C|{\eta }_p|$, (108) in Lemma 4 we arrive at

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |& ⟨\xi ,[e^{-B}{\mathcal{N}}_{+}{e}^B-{\mathcal{N}}_{+}]\xi ⟩|\hfill \\ \hfill & \le C\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \sum _{p\in {\Lambda }_{+}^{\ast }}|{\eta }_p|\left[|,{\eta }_p,|\Vert ,{({\mathcal{N}}_{+}+1)}^{1/2},\xi \Vert +\Vert ,b_p,\xi \Vert \right]\le C\Vert \eta \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\xi \Vert }^2\hfill \end{array}\end{array}$$

$\square$

Appendix A.2: Analysis of ${\mathcal{G}}_{N,\alpha }^{(0)}=e^{-B}{\mathcal{L}}_N^{(0)}{e}^B$

We define ${\mathcal{E}}_N^{(0)}$ so that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{G}}_{N,\alpha }^{(0)}& =e^{-B}{\mathcal{L}}_N^{(0)}{e}^B=\frac{1}{2}\widehat{V}(0)(N+{\mathcal{N}}_{+}-1)(N-{\mathcal{N}}_{+})+{\mathcal{E}}_{N,\alpha }^{(0)}.\hfill \end{array}\end{array}$$

where we recall from (13) that

$$\begin{array}{c}\hfill {\mathcal{L}}_N^{(0)}=\frac{1}{2}\widehat{V}(0)(N-1+{\mathcal{N}}_{+})(N-{\mathcal{N}}_{+}).\end{array}$$

Proposition 12

Under the assumptions of Proposition 1, there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill \pm {\mathcal{E}}_{N,\alpha }^{(0)}\le C{N}^{1-\alpha }({\mathcal{N}}_{+}+1)\end{array}$$

for all $\alpha >0$ and $N\in \mathbb{N}$ large enough.

Proof

The proof follows [4, Prop. 7.1].

We write

$$\begin{array}{c}\hfill {\mathcal{L}}_N^{(0)}=\frac{N(N-1)}{2}\widehat{V}(0)+\frac{N}{2}\widehat{V}(0)[\sum _{q\in {\Lambda }_{+}^{\ast }}{b}_q^{\ast }{b}_q-{\mathcal{N}}_{+}].\end{array}$$

Hence,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{E}}_N^{(0)}& =\frac{N}{2}\widehat{V}(0)\sum _{q\in {\Lambda }_{+}^{\ast }}\left[e^{-B},b_q^{\ast },b_q,e^B,-,b_q^{\ast },b_q\right]-\frac{N}{2}\widehat{V}(0)\left[e^{-B},{\mathcal{N}}_{+},e^B,-,{\mathcal{N}}_{+}\right].\hfill \end{array}\end{array}$$

To bound the first term we use (106), $|{\gamma }_q^2-1|\le C{\eta }_q^2$, $|{\sigma }_q|\le C|{\eta }_q|$, the first bound in (108), Cauchy–Schwarz and the estimate $\Vert \eta \Vert \le C{N}^{-\alpha }$. To bound the second term, we use Lemma 5. We conclude that

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_N^{(0)}\xi ⟩|\le C{N}^{1-\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2.\end{array}$$

$\square$

Appendix A.3: Analysis of ${\mathcal{G}}_{N,\alpha }^{(2)}=e^{-B}{\mathcal{L}}_N^{(2)}{e}^B$

We consider first conjugation of the kinetic energy operator.

Proposition 13

Under the assumptions of Proposition 1, there exists $C>0$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^{-B}\mathcal{K}{e}^B=& \mathcal{K}+\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{\eta }_p(b_p{b}_{-p}+b_p^{\ast }{b}_{-p}^{\ast })\hfill \\ \hfill & +\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{\eta }_p^2(\frac{N-{\mathcal{N}}_{+}}{N})(\frac{N-{\mathcal{N}}_{+}-1}{N})+{\mathcal{E}}_{N,\alpha }^{(K)}\hfill \end{array}\end{array}$$ 112

where

$$\begin{array}{c}\hfill \begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_{N,\alpha }^{(K)}\xi ⟩|\le C{N}^{1/2-\alpha }\Vert {\mathcal{H}}_N^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert +C{N}^{1-\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2\end{array}\end{array}$$ 113

for any $\alpha >1$, $\xi \in {\mathcal{F}}_{+}^{\le N}$ and $N\in \mathbb{N}$ large enough.

Proof

We proceed as in the proof of [4, Prop. 7.2]. We write

$$\begin{array}{c}\hfill \begin{array}{cc}& e^{-B}\mathcal{K}{e}^B-\mathcal{K}\hfill \\ \hfill & ={\int }_0^{1}ds\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{\eta }_p[({\gamma }_p^{(s)}{b}_p+{\sigma }_p^{(s)}{b}_{-p}^{\ast })({\gamma }_p^{(s)}{b}_{-p}+{\sigma }_p^{(s)}{b}_p^{\ast })+\text{h.c.}]\hfill \\ \hfill & +{\int }_0^{1}ds\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{\eta }_p[({\gamma }_p^{(s)}{b}_p+{\sigma }_p^{(s)}{b}_{-p}^{\ast })d_{-p}^{(s)}+d_p^{(s)}({\gamma }_p^{(s)}{b}_{-p}+{\sigma }_p^{(s)}{b}_p^{\ast })+\text{h.c.}]\hfill \\ \hfill & +{\int }_0^{1}ds\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{\eta }_p[d_p^{(s)}{d}_{-p}^{(s)}+\text{h.c.}]\hfill \\ \hfill & =:{\text{G}}_1+{\text{G}}_2+{\text{G}}_3\hfill \end{array}\end{array}$$ 114

with ${\gamma }_p^{(s)}=cosh(s{\eta }_p)$, ${\sigma }_p^{(s)}=sinh(s{\eta }_p)$ and where $d_p^{(s)}$ is defined as in (105), with ${\eta }_p$ replaced by $s{\eta }_p$. We find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\text{G}}_1& =\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{\eta }_p(b_p{b}_{-p}+b_{-p}^{\ast }{b}_p^{\ast })+\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{\eta }_p^2\left(1,-,\frac{{\mathcal{N}}_{+}}{N}\right)+{\mathcal{E}}_1^K\hfill \end{array}\end{array}$$

with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{E}}_1^K=& 2{\int }_0^{1}ds\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{\eta }_p{({\sigma }_p^{(s)})}^2(b_p{b}_{-p}+b_{-p}^{\ast }{b}_p^{\ast })\hfill \\ \hfill & +{\int }_0^{1}ds\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{\eta }_p{\gamma }_p^{(s)}{\sigma }_p^{(s)}(4b_p^{\ast }{b}_p-2N^{-1}{a}_p^{\ast }{a}_p)\hfill \\ \hfill & +2{\int }_0^{1}ds\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{\eta }_p\left[({\gamma }_p^{(s)}-1),{\sigma }_p^{(s)},+,({\sigma }_p^{(s)}-s{\eta }_p)\right](1-\frac{{\mathcal{N}}_{+}}{N}).\hfill \end{array}\end{array}$$

Since $|({({\gamma }_p^{(s)})}^2-1)|\le C{\eta }_p^2$, ${({\sigma }_p^{(s)})}^2\le C{\eta }_p^2$, $p^2|{\eta }_p|\le C$, ${\Vert \eta \Vert }_{\infty }\le N^{-\alpha }$, we can estimate

$$\begin{array}{cc}\hfill |⟨\xi ,& {\mathcal{E}}_1^K\xi ⟩|\hfill \\ \hfill \le & C\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2|{\eta }_p{|}^3\Vert b_p\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert +C\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{\eta }_p^2\Vert a_p{\xi \Vert }^2+C\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{\eta }_p^4{\Vert \xi \Vert }^2\hfill \\ \hfill \\ \hfill \le & C\Vert \eta \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\xi \Vert }^2\le C{N}^{-\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2,\hfill \end{array}$$ 115

for any $\xi \in {\mathcal{F}}_{+}^{\le N}$. To bound the term ${\text{G}}_3$ in (114), we switch to position space:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\text{G}}_3\xi ⟩|\le & CN{\int }_0^{1}ds{\int }_{{\Lambda }^2}dxdy\left[e^{2N},V,(e^N(x-y)),+,N^{2\alpha -1},\chi (|x-y|,\le ,N^{-\alpha },)\right]\hfill \\ \hfill & \times \Vert {({\mathcal{N}}_{+}+1)}^{-1/2}{\check{d}}_x^{(s)}{\check{d}}_y^{(s)}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \hfill \end{array}\end{array}$$

With (111), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,{\text{G}}_3\xi ⟩|\hfill \\ \hfill & \le C{N}^{1-\alpha }{\int }_{{\Lambda }^2}dxdy\left[e^{2N},V,(e^N(x-y)),+,N^{2\alpha -1},\chi (|x-y|,\le ,N^{-\alpha },)\right]{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2\hfill \\ \hfill & +C{N}^{-2\alpha }{\int }_{{\Lambda }^2}dxdy\left[e^{2N},V,(e^N(x-y)),+,N^{2\alpha -1},\chi (|x-y|,\le ,N^{-\alpha },)\right]\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \hfill \\ \hfill & \times [\Vert {\check{a}}_x({\mathcal{N}}_{+}+1)\xi \Vert +\Vert {\check{a}}_y({\mathcal{N}}_{+}+1)\xi \Vert +\Vert {\check{a}}_x{\check{a}}_y{({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert ]\hfill \\ \hfill & \le C{N}^{1-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\xi \Vert }^2+C{N}^{1/2-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{V}}_N^{1/2}\xi \Vert .\hfill \end{array}\\ \hfill \end{array}$$ 116

Finally, we consider ${\text{G}}_2$ in (114). We split it as ${\text{G}}_2={\text{G}}_{21}+{\text{G}}_{22}+{\text{G}}_{23}+{\text{G}}_{24}$, with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\text{G}}_{21}& ={\int }_0^{1}ds\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{\eta }_p\left({\gamma }_p^{(s)},b_p,d_{-p}^{(s)},+,,\text{h.c.},\right),\hfill \\ \hfill {\text{G}}_{22}& ={\int }_0^{1}ds\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{\eta }_p\left({\sigma }_p^{(s)},b_{-p}^{\ast },d_{-p}^{(s)},+,,\text{h.c.},\right)\hfill \\ \hfill {\text{G}}_{23}& ={\int }_0^{1}ds\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{\eta }_p\left({\gamma }_p^{(s)},d_p^{(s)},b_{-p},+,,\text{h.c.},\right),\hfill \\ \hfill {\text{G}}_{24}& ={\int }_0^{1}ds\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{\eta }_p\left({\sigma }_p^{(s)},d_p^{(s)},b_p^{\ast },+,,\text{h.c.},\right).\hfill \end{array}\end{array}$$ 117

We consider ${\text{G}}_{21}$ first. We write

$$\begin{array}{c}\hfill {\text{G}}_{21}=-\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{\eta }_p^2\frac{{\mathcal{N}}_{+}+1}{N}\frac{N-{\mathcal{N}}_{+}}{N}+\left[{\mathcal{E}}_2^K,+,,\text{h.c.},\right]\end{array}$$

where ${\mathcal{E}}_2^K={\sum }_{j=1}^3{\mathcal{E}}_{2j}^K$, with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{E}}_{21}^K=& \frac{1}{2N}\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{\eta }_p^2({\mathcal{N}}_{+}+1)(b_p^{\ast }{b}_p-\frac{1}{N}{a}_p^{\ast }{a}_p),\hfill \\ \hfill {\mathcal{E}}_{22}^K=& {\int }_0^{1}ds\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{\eta }_p({\gamma }_p^{(s)}-1)b_p{d}_{-p}^{(s)},\hfill \\ \hfill {\mathcal{E}}_{23}^K=& {\int }_0^{1}ds\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{\eta }_p{b}_p{\overline{d}}_{-p}^{(s)}.\hfill \end{array}\end{array}$$ 118

and where we introduced the notation ${\overline{d}}_{-p}^{(s)}=d_{-p}^{(s)}+s{\eta }_p({\mathcal{N}}_{+}/N)b_p^{\ast }$. With (29), we find

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_{21}^K\xi ⟩|\le C\sum _{p\in {\Lambda }_{+}^{\ast }}{\eta }_p\Vert a_p{\xi \Vert }^2\le C{N}^{-\alpha }{\Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert }^2\end{array}$$ 119

Using $|{\gamma }_p^{(s)}-1|\le C{\eta }_p^2$ and (108), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\mathcal{E}}_{22}^K\xi ⟩|& \le \sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2|{\eta }_p{|}^3\Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert \Vert d_{-p}^{(s)}\xi \Vert \le C{N}^{-3\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2.\hfill \end{array}\end{array}$$ 120

To control the third term in (118), we use (30) and we switch to position space. We find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{E}}_{23}^K=& -N{\int }_0^{1}ds{\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y))f_{N,\ell }(x-y){\check{b}}_x{\check{\overline{d}}}_y^{(s)}\hfill \\ \hfill & +N{\int }_0^{1}ds{e}^{2N}{\lambda }_{\ell }{\int }_{{\Lambda }^2}dxdy{\chi }_{\ell }(x-y)f_{N,\ell }(x-y){\check{b}}_x{\check{\overline{d}}}_y^{(s)}\hfill \\ \hfill =;& {\mathcal{E}}_{231}^K+{\mathcal{E}}_{232}^K.\hfill \end{array}\end{array}$$ 121

With (110) and $|\check{\eta }(x-y)|\le CN$, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\mathcal{E}}_{231}^K\xi ⟩|\le & N{\int }_0^{1}ds{\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y))\hfill \\ \hfill & \times \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{-1/2}{\check{a}}_x{\check{\overline{d}}}_y^{(s)}\xi \Vert \hfill \\ \hfill \le & C{N}^{1-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\xi \Vert }^2+C{N}^{1/2-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{V}}_N^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$ 122

As for ${\mathcal{E}}_{232}^K$, with (110) and Lemma 1 (recalling $\ell =N^{-\alpha })$, we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\mathcal{E}}_{232}^K\xi ⟩|\le & C{N}^{-\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2\hfill \\ \hfill & +{\int }_{{\Lambda }^2}dxdy\chi (|x-y|\le N^{-\alpha })\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\check{a}}_x{\check{a}}_y{\mathcal{N}}_{+}^{1/2}\xi \Vert \hfill \end{array}\end{array}$$ 123

To bound the last term on the r.h.s. of (123) we use Hölder’s and Sobolev inequality ${\Vert u\Vert }_q\le C{q}^{1/2}{\Vert u\Vert }_{H^1}$, valid for any $2\le q<\infty$. We find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\int }_{{\Lambda }^2}& dxdy\chi (|x-y|\le N^{-\alpha })\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\check{a}}_x{\check{a}}_y{\mathcal{N}}_{+}^{1/2}\xi \Vert \hfill \\ \hfill & \le C\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert {\int }_{\Lambda }dx{\left({\int }_{\Lambda },dy\chi (|x-y|,\le ,N^{-\alpha },)\right)}^{1-1/q}{\left({\int }_{\Lambda },d,y,,{\Vert {\check{a}}_x{\check{a}}_y{\mathcal{N}}_{+}^{1/2}\xi \Vert }^q\right)}^{1/q}\hfill \\ \hfill & \le C{N}^{2\alpha /q-2\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert {\int }_{\Lambda }dx{\left({\int }_{\Lambda },d,y,,{\Vert {\check{a}}_x{\check{a}}_y{\mathcal{N}}_{+}^{1/2}\xi \Vert }^q\right)}^{1/q}\hfill \\ \hfill & \le C{q}^{1/2}{N}^{2\alpha /q-2\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \hfill \\ \hfill & \times {\left[{\int }_{{\Lambda }^2},dxdy\Vert ,{\check{a}}_x,{\mathrm{\nabla }}_y,{\check{a}}_y,{\mathcal{N}}_{+}^{1/2},{\xi \Vert }^2,+,{\int }_{{\Lambda }^2},d,x,d,y,,{\Vert {\check{a}}_x{\check{a}}_y{\mathcal{N}}_{+}^{1/2}\xi \Vert }^2\right]}^{1/2}\hfill \\ \hfill & \le C{q}^{1/2}{N}^{2\alpha /q-2\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \left[\Vert ,{\mathcal{K}}^{1/2},{\mathcal{N}}_{+},\xi \Vert +\Vert ,{\mathcal{N}}_{+}^{3/2},\xi \Vert \right].\hfill \end{array}\end{array}$$

Choosing $q=logN$, we get

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\int }_{{\Lambda }^2}& dxdy\chi (|x-y|\le N^{-\alpha })\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\check{a}}_x{\check{a}}_y{({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \hfill \\ \hfill \le & C{N}^{1-2\alpha }{(logN)}^{1/2}\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{K}}^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$ 124

Therefore, for any $\xi \in {\mathcal{F}}_{+}^{\le N}$,

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_{232}^K\xi ⟩|\le N^{1-2\alpha }{(logN)}^{1/2}\Vert {\mathcal{K}}^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert +N^{-\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2.\end{array}$$

Combining the last bound with (119), (120) and (122), we conclude that

$$\begin{array}{c}\hfill \begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_2^K\xi ⟩|\le C{N}^{1-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\xi \Vert }^2+C{N}^{1/2-\alpha }\Vert {\mathcal{H}}_N^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert .\end{array}\end{array}$$ 125

for any $\alpha >1$, $N\in \mathbb{N}$ large enough, $\xi \in {\mathcal{F}}_{+}^{\le N}$.

The term ${\text{G}}_{22}$ in (117) can be bounded using (108). We find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\text{G}}_{22}\xi ⟩|& \le C{N}^{-2\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2.\hfill \end{array}\end{array}$$ 126

We split ${\text{G}}_{23}={\mathcal{E}}_{31}^K+{\mathcal{E}}_{32}^K+\text{h.c.}$, with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{E}}_{31}^K=& {\int }_0^{1}ds\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{\eta }_p({\gamma }_p^{(s)}-1)d_p^{(s)}{b}_{-p},{\mathcal{E}}_{32}^K={\int }_0^{1}ds\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{\eta }_p{d}_p^{(s)}{b}_{-p}\hfill \end{array}\end{array}$$

With (108), we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\mathcal{E}}_{31}^K\xi ⟩|& \le C{\int }_0^{1}ds\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2|{\eta }_p{|}^3\Vert {(d_p^{(s)})}^{\ast }\xi \Vert \Vert b_{-p}\xi \Vert ds\le C{N}^{-3\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2\hfill \end{array}\end{array}$$

To estimate ${\mathcal{E}}_{32}^K$, we use (30) and we switch to position space. Proceeding as we did in (121), (122), (123), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\mathcal{E}}_{32}^K\xi ⟩|& \le CN{\int }_0^{1}ds{\int }_{{\Lambda }^2}dxdy\left[e^{2N},V,(e^N(x-y)),+,N^{2\alpha -1},\chi (|x-y|,\le ,N^{-\alpha },)\right]\hfill \\ \hfill & \times \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{-1/2}{\check{d}}_x^{(s)}{\check{b}}_y\xi \Vert .\hfill \end{array}\end{array}$$

With (109) and (124) we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\mathcal{E}}_{32}^K\xi ⟩|& \le C{N}^{-\alpha }{\int }_{{\Lambda }^2}dxdy\left[e^{2N},V,(e^N(x-y)),+,N^{2\alpha -1},\chi (|x-y|,\le ,N^{-\alpha },)\right]\hfill \\ \hfill & \times \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \left[\Vert ,{\check{a}}_y,({\mathcal{N}}_{+}+1),\xi \Vert +\Vert ,{\check{a}}_x,{\check{a}}_y,{({\mathcal{N}}_{+}+1)}^{1/2},\xi \Vert \right]\hfill \\ \hfill & \le C{N}^{1-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\xi \Vert }^2+C{N}^{1/2-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{V}}_N^{1/2}\xi \Vert \hfill \\ \hfill & +C{N}^{1-2\alpha }{(logN)}^{1/2}\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{K}}^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$

Combining the bounds for ${\mathcal{E}}_{31}^K$ and ${\mathcal{E}}_{32}^K$ , we conclude that, if $\alpha >1$,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill |⟨\xi ,{\text{G}}_{23}\xi ⟩|\le C{N}^{1/2-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{H}}_N^{1/2}\xi \Vert +C{N}^{1-\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2\end{array}\end{array}$$ 127

To bound ${\text{G}}_{24}$ in (117), we use (108), the bounds (28) and ${\Vert \eta \Vert }_{H_1}^2\le CN$, and the commutator (14):

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨& \xi ,{\text{G}}_{24}\xi ⟩|\hfill \\ \hfill & \le C{\int }_0^{1}ds\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{\eta }_p^2\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{-1/2}{d}_p^{(s)}{b}_p^{\ast }\xi \Vert \hfill \\ \hfill & \le C\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{\eta }_p^2\left[|,{\eta }_p,|\Vert ,{({\mathcal{N}}_{+}+1)}^{1/2},\xi \Vert +,N^{-1},\Vert \eta \Vert \Vert ,b_p,b_p^{\ast },{({\mathcal{N}}_{+}+1)}^{1/2},\xi \Vert \right]\hfill \\ \hfill & \le C{N}^{-\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2.\hfill \end{array}\end{array}$$

Together with (117), (125), (126) and (127), this implies that

$$\begin{array}{c}\hfill {\text{G}}_2=-\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2{\eta }_p^2\frac{{\mathcal{N}}_{+}+1}{N}\frac{N-{\mathcal{N}}_{+}}{N}+{\mathcal{E}}_4^K\end{array}$$

with

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_4^K\xi ⟩|\le C{N}^{1/2-\alpha }\Vert {\mathcal{H}}_N^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert +C{N}^{1-\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2.\end{array}$$ 128

Combining (115), (116) and (128), we obtain (112) and (113). $\square$

In the next proposition, we consider the conjugation of the operator

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{L}}_N^{(2,V)}=& N\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N)\left[b_p^{\ast },b_p,-,\frac{1}{N},a_p^{\ast },a_p\right]+\frac{N}{2}\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N)\left[b_p^{\ast },b_{-p}^{\ast },+,b_p,b_{-p}\right]\hfill \end{array}\end{array}$$

Proposition 14

Under the assumptions of Proposition 1, there is a constant $C>0$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^{-B}{\mathcal{L}}_N^{(2,V)}{e}^B=& N\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N){\eta }_p(\frac{N-{\mathcal{N}}_{+}}{N})(\frac{N-{\mathcal{N}}_{+}-1}{N})\hfill \\ \hfill & +N\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N)a_p^{\ast }{a}_p\left(1,-,\frac{{\mathcal{N}}_{+}}{N}\right)\hfill \\ \hfill & +\frac{N}{2}\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N)(b_p{b}_{-p}+b_{-p}^{\ast }{b}_p^{\ast })+{\mathcal{E}}_N^{(V)}\hfill \end{array}\end{array}$$ 129

where

$$\begin{array}{c}\hfill \begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_N^{(V)}\xi ⟩|\le C{N}^{1/2-\alpha }\Vert {\mathcal{H}}_N^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert +C{N}^{1-\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2.\end{array}\end{array}$$ 130

for any $\alpha >1$, $\xi \in {\mathcal{F}}_{+}^{\le N}$ and $N\in \mathbb{N}$ large enough.

Proof

We write

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^{-B}{\mathcal{L}}_N^{(2,V)}{e}^B=& N\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N)e^{-B}{b}_p^{\ast }{b}_p{e}^B-\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N)e^{-B}{a}_p^{\ast }{a}_p{e}^B\hfill \\ \hfill & +\frac{N}{2}\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N)e^{-B}[b_p{b}_{-p}+b_p^{\ast }{b}_{-p}^{\ast }]e^B\hfill \\ \hfill =:& {\text{F}}_1+{\text{F}}_2+{\text{F}}_3.\hfill \end{array}\end{array}$$ 131

With (106), we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\text{F}}_1=& N\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N)[{\gamma }_p{b}_p^{\ast }+{\sigma }_p{b}_{-p}][{\gamma }_p{b}_p+{\sigma }_p{b}_{-p}^{\ast }]\hfill \\ \hfill & +N\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N)[({\gamma }_p{b}_p^{\ast }+{\sigma }_p{b}_{-p})d_p+d_p^{\ast }({\gamma }_p{b}_p+{\sigma }_p{b}_{-p}^{\ast })+d_p^{\ast }{d}_p]\hfill \end{array}\end{array}$$

where ${\gamma }_p=cosh{\eta }_p$, ${\sigma }_p=sinh{\eta }_p$ and the operators $d_p$ are defined in (105). Using $|1-{\gamma }_p|\le {\eta }_p^2$, $|{\sigma }_p|\le C|{\eta }_p|$ and using Lemma 4 for the terms on the second line, we find

$$\begin{array}{c}\hfill {\text{F}}_1=N\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N)b_p^{\ast }{b}_p+{\mathcal{E}}_1^V\end{array}$$ 132

with $\pm {\mathcal{E}}_1^V\le C{N}^{1-\alpha }({\mathcal{N}}_{+}+1)$.

Let us now consider the second contribution on the r.h.s. of (131). We find

$$\begin{array}{c}\hfill -{\text{F}}_2=\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N)a_p^{\ast }{a}_p+{\mathcal{E}}_2^V\end{array}$$ 133

with

$$\begin{array}{c}\hfill {\mathcal{E}}_2^V=\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N){\int }_0^1{e}^{-sB}({\eta }_p{b}_{-p}{b}_p+\text{h.c.})e^{\mathit{sB}}ds.\end{array}$$

With Lemma 2, we easily find $\pm {\mathcal{E}}_2^V\le C{N}^{-\alpha }({\mathcal{N}}_{+}+1)$.

Finally, we consider the last term on the r.h.s. of (131). With (106), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\text{F}}_3=& \frac{N}{2}\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N)\left[{\gamma }_p,b_p,+,{\sigma }_p,b_{-p}^{\ast }\right]\left[{\gamma }_p,b_{-p},+,{\sigma }_p,b_p^{\ast }\right]+\text{h.c.}\hfill \\ \hfill & +\frac{N}{2}\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N)\left[({\gamma }_p{b}_p+{\sigma }_p{b}_{-p}^{\ast }),,d_{-p},+,d_p,,({\gamma }_p{b}_{-p}+{\sigma }_p{b}_p^{\ast })\right]+\text{h.c.}\hfill \\ \hfill & +\frac{N}{2}\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N)d_p{d}_{-p}+\text{h.c.}\hfill \\ \hfill =:& {\text{F}}_{31}+{\text{F}}_{32}+{\text{F}}_{33}.\hfill \end{array}\end{array}$$ 134

Using $|1-{\gamma }_p|\le C{\eta }_p^2$, $|{\sigma }_p|\le C|{\eta }_p|$, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\text{F}}_{31}=& \frac{N}{2}\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N)(b_p{b}_{-p}+b_{-p}^{\ast }{b}_p^{\ast })+N\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N){\eta }_p\frac{N-{\mathcal{N}}_{+}}{N}+{\mathcal{E}}_3^V\hfill \end{array}\end{array}$$ 135

with $\pm {\mathcal{E}}_3^V\le C{N}^{1-\alpha }({\mathcal{N}}_{+}+1)$. As for ${\text{F}}_{32}$ in (134), we divide it into four parts

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\text{F}}_{32}=& \frac{N}{2}\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N)\left[({\gamma }_p{b}_p+{\sigma }_p{b}_{-p}^{\ast }),,d_{-p},+,d_p,,({\gamma }_p{b}_{-p}+{\sigma }_p{b}_p^{\ast })\right]+\text{h.c.}\hfill \\ \hfill =:& {\text{F}}_{321}+{\text{F}}_{322}+{\text{F}}_{323}+{\text{F}}_{324}.\hfill \end{array}\end{array}$$ 136

We start with ${\text{F}}_{321}$, which we write as

$$\begin{array}{c}\hfill {\text{F}}_{321}=-N\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N){\eta }_p\left(\frac{N-{\mathcal{N}}_{+}}{N}\right)\left(\frac{{\mathcal{N}}_{+}+1}{N}\right)+{\mathcal{E}}_4^V\end{array}$$

where ${\mathcal{E}}_4^V={\mathcal{E}}_{41}^V+{\mathcal{E}}_{42}^V+{\mathcal{E}}_{43}^V+\text{h.c.}$, with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{E}}_{41}^V=& \frac{N}{2}\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N)({\gamma }_p-1)b_p{d}_{-p},{\mathcal{E}}_{42}^V=\frac{N}{2}\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N)b_p{\overline{d}}_{-p}\hfill \\ \hfill {\mathcal{E}}_{43}^V=& -\frac{N}{2}\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N){\eta }_p\frac{{\mathcal{N}}_{+}+1}{N}(b_p^{\ast }{b}_p-N^{-1}{a}_p^{\ast }{a}_p)\hfill \end{array}\end{array}$$

and with the notation ${\overline{d}}_{-p}=d_{-p}+N^{-1}{\eta }_p{\mathcal{N}}_{+}{b}_p^{\ast }$. Since $|{\gamma }_p-1|\le C{\eta }_p^2$, ${\Vert \eta \Vert }_{\infty }\le C{N}^{-\alpha }$, we find easily with (108) that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\mathcal{E}}_{41}^V\xi ⟩|& \le C{N}^{1-3\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2.\hfill \end{array}\end{array}$$

Moreover

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_{43}^V\xi ⟩|\le CN\sum _{p\in {\Lambda }_{+}^{\ast }}{\eta }_p\Vert a_p{\xi \Vert }^2\le C{N}^{1-\alpha }{\Vert {\mathcal{N}}_{+}^{1/2}\xi \Vert }^2.\end{array}$$

As for ${\mathcal{E}}_{42}^V$, we switch to position space and we use (110). We obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\mathcal{E}}_{42}^V\xi ⟩|& \le CN{\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y))\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{-1/2}{\check{a}}_x{\check{\overline{d}}}_y\xi \Vert \hfill \\ \hfill & \le C{N}^{1-\alpha }{\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y))\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \hfill \\ \hfill & \times [\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert +\Vert {\check{a}}_x\xi \Vert +\Vert {\check{a}}_y\xi \Vert +N^{-1/2}\Vert {\check{a}}_x{\check{a}}_y\xi \Vert ]\hfill \\ \hfill & \le C{N}^{1-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\xi \Vert }^2+C{N}^{1/2-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{V}}_N^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$

We conclude that

$$\begin{array}{c}\hfill \begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_4^V\xi ⟩|\le C{N}^{1/2-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{V}}_N^{1/2}\xi \Vert +C{N}^{1-\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2.\end{array}\end{array}$$

To bound the term ${\text{F}}_{322}$ in (136), we use (108) and $|{\sigma }_p|\le C|{\eta }_p|$; we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\text{F}}_{322}\xi ⟩|\le & CN\sum _{p\in {\Lambda }_{+}^{\ast }}|{\eta }_p|\Vert b_{-p}\xi \Vert \left[|,{\eta }_p,|\Vert ,{({\mathcal{N}}_{+}+1)}^{1/2},\xi \Vert +\Vert \eta \Vert \Vert ,b_{-p},\xi \Vert \right]\hfill \\ \hfill \le & C{N}^{1-2\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2.\hfill \end{array}\end{array}$$

Let us now consider the term ${\text{F}}_{323}$ on the r.h.s. of (136). We write ${\text{F}}_{323}={\mathcal{E}}_{51}^V+{\mathcal{E}}_{52}^V+\text{h.c.}$, with

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathcal{E}}_{51}^V=\frac{N}{2}\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N)({\gamma }_p-1)d_p{b}_{-p},{\mathcal{E}}_{52}^V=\frac{N}{2}\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N)d_p{b}_{-p}.\end{array}\end{array}$$

With $|{\gamma }_p-1|\le C{\eta }_p^2$ and (108) we obtain

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_{51}^V\xi ⟩|\le CN\sum _{p\in {\Lambda }_{+}^{\ast }}{\eta }_p^2\Vert d_p^{\ast }\xi \Vert \Vert a_p\xi \Vert \le C{N}^{1-3\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2.\end{array}$$

We find, switching to position space and using (109),

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,{\mathcal{E}}_{52}^V\xi ⟩|\hfill \\ \hfill & \le CN{\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y))\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{-1/2}{\check{d}}_x{\check{a}}_y\xi \Vert \hfill \\ \hfill & \le C{N}^{1-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert {\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y))\left[\Vert ,{\check{a}}_y,\xi \Vert +,N^{-1/2},\Vert {\check{a}}_x{\check{a}}_y\xi \Vert \right]\hfill \\ \hfill & \le C{N}^{1-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\xi \Vert }^2+C{N}^{1/2-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{V}}_N^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$

Hence,

$$\begin{array}{c}\hfill |⟨\xi ,{\text{F}}_{323}\xi ⟩|\le C{N}^{1-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\xi \Vert }^2+C{N}^{1/2-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{V}}_N^{1/2}\xi \Vert \end{array}$$

To estimate the term ${\text{F}}_{324}$ in (136) we use (108) and the bound

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \sum _{p\in {\Lambda }_{+}^{\ast }}|\widehat{V}(p/e^N)||{\eta }_p|& \le C\sum _{\begin{array}{c}p\in {\Lambda }_{+}^{\ast },|p|\le e^N\end{array}}\frac{1}{p^2}+C\sum _{\begin{array}{c}p\in {\Lambda }_{+}^{\ast },|p|>e^N\end{array}}\frac{|\widehat{V}(p/e^N)|}{p^2}\hfill \\ \hfill & \le CN+C(\sum _{\begin{array}{c}p\in {\Lambda }_{+}^{\ast }\end{array}}{|\widehat{V}(p/e^N)|}^2{)}^{1/2}(\sum _{\begin{array}{c}p\in {\Lambda }_{+}^{\ast },|p|>e^N\end{array}}\frac{1}{p^4}{)}^{1/2}\hfill \\ \hfill & \le CN\hfill \end{array}\end{array}$$

We find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\text{F}}_{324}\xi ⟩|& \le CN\sum _{p\in {\Lambda }_{+}^{\ast }}|\widehat{V}(p/e^N)||{\eta }_p|\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{-1/2}{d}_p{b}_p^{\ast }\xi \Vert \hfill \\ \hfill & \le CN\sum _{p\in {\Lambda }_{+}^{\ast }}|\widehat{V}(p/e^N)||{\eta }_p|\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \hfill \\ \hfill & \times \left[|,{\eta }_p,|\Vert ,{({\mathcal{N}}_{+}+1)}^{1/2},\xi \Vert +,N^{-1},\Vert \eta \Vert \Vert ,b_p,b_p^{\ast },{({\mathcal{N}}_{+}+1)}^{1/2},\xi \Vert \right]\hfill \\ \hfill & \le CN\sum _{p\in {\Lambda }_{+}^{\ast }}|\widehat{V}(p/e^N)||{\eta }_p|\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \hfill \\ \hfill & \times \left[|,{\eta }_p,|\Vert ,{({\mathcal{N}}_{+}+1)}^{1/2},\xi \Vert +,N^{-1},\Vert \eta \Vert \Vert ,{({\mathcal{N}}_{+}+1)}^{1/2},\xi \Vert +\Vert \eta \Vert \Vert ,a_p,\xi \Vert \right]\hfill \\ \hfill & \le C{N}^{1-\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2.\hfill \end{array}\end{array}$$

Combining the last bounds, we arrive at

$$\begin{array}{c}\hfill {\text{F}}_{32}=N\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N){\eta }_p\left(\frac{N-{\mathcal{N}}_{+}}{N}\right)\left(\frac{-{\mathcal{N}}_{+}-1}{N}\right)+{\mathcal{E}}_6^V\end{array}$$

with

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_6^V\xi ⟩|\le C{N}^{1-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\xi \Vert }^2+C{N}^{1/2-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{V}}_N^{1/2}\xi \Vert .\end{array}$$ 137

To control the last contribution ${\text{F}}_{33}$ in (134), we switch to position space. With (111) and (25) we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\text{F}}_{33}\xi ⟩|& \le CN\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert {\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y))\Vert {({\mathcal{N}}_{+}+1)}^{-1/2}{\check{d}}_x{\check{d}}_y\xi \Vert \hfill \\ \hfill & \le C{N}^{1-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\xi \Vert }^2+C{N}^{1/2-2\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{V}}_N^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$

The last equation, combined with (134), (135) and (137), implies that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\text{F}}_3=& \frac{N}{2}\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N)(b_p{b}_{-p}+b_{-p}^{\ast }{b}_p^{\ast })\hfill \\ \hfill & +N\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N){\eta }_p\left(\frac{N-{\mathcal{N}}_{+}}{N}\right)\left(\frac{N-{\mathcal{N}}_{+}-1}{N}\right)+{\mathcal{E}}_7^V\hfill \end{array}\end{array}$$

with

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_7^V\xi ⟩|\le C{N}^{1-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\xi \Vert }^2+C{N}^{1/2-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{V}}_N^{1/2}\xi \Vert .\end{array}$$

Together with (132) and with (133), and recalling that $b_p^{\ast }{b}_p-N^{-1}{a}_p^{\ast }{a}_p=a_p^{\ast }{a}_p(1-{\mathcal{N}}_{+}/N)$, we obtain (129) with (130). $\square$

Appendix A.4: Analysis of ${\mathcal{G}}_{N,\alpha }^{(3)}=e^{-B}{\mathcal{L}}_N^{(3)}{e}^B$

We consider here the conjugation of the cubic term ${\mathcal{L}}_N^{(3)}$, defined in (13).

Proposition 15

Under the assumptions of Proposition 1, there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill {\mathcal{G}}_{N,\alpha }^{(3)}=e^{-B}{\mathcal{L}}_N^{(3)}{e}^B=\sqrt{N}\sum _{p,q\in {\Lambda }_{+}^{\ast }:p+q\ne 0}\widehat{V}(p/e^N)\left[b_{p+q}^{\ast },a_{-p}^{\ast },a_q,+,,\text{h.c.},\right]+{\mathcal{E}}_N^{(3)}\end{array}$$

where

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_N^{(3)}\xi ⟩|\le C{N}^{1/2-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{V}}_N^{1/2}\xi \Vert +C{N}^{1-\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2\end{array}$$ 138

for any $\alpha >1$ and $N\in \mathbb{N}$ large enough.

Proof

This proof is similar to the proof of [4, Prop. 7.5]. Expanding $e^{-B}{a}_{-p}^{\ast }{a}_q{e}^B$, we arrive at

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{E}}_N^{(3)}& =\sqrt{N}\sum _{p,q\in {\Lambda }_{+}^{\ast }:p+q\ne 0}\widehat{V}(p/e^N)(({\gamma }_{p+q}-1)b_{p+q}^{\ast }+{\sigma }_{p+q}{b}_{-p-q}+d_{p+q}^{\ast })a_{-p}^{\ast }{a}_q\hfill \\ \hfill & +\sqrt{N}\sum _{p,q\in {\Lambda }_{+}^{\ast },p+q\ne 0}\widehat{V}(p/e^N){\eta }_p{e}^{-B}{b}_{p+q}^{\ast }{e}^B{\int }_0^{1}ds{e}^{-sB}{b}_p{b}_q{e}^{\mathit{sB}}\hfill \\ \hfill & +\sqrt{N}\sum _{p,q\in {\Lambda }_{+}^{\ast },p+q\ne 0}\widehat{V}(p/e^N){\eta }_q{e}^{-B}{b}_{p+q}^{\ast }{e}^B{\int }_0^{1}ds{e}^{-sB}{b}_{-p}^{\ast }{b}_{-q}^{\ast }{e}^{\mathit{sB}}\hfill \\ \hfill & +\text{h.c.}\hfill \\ \hfill & =:{\mathcal{E}}_1^{(3)}+{\mathcal{E}}_2^{(3)}+{\mathcal{E}}_3^{(3)}+\text{h.c.}\hfill \end{array}\end{array}$$ 139

where, as usual, ${\gamma }_p=cosh\eta (p)$, ${\sigma }_p=sinh\eta (p)$ and $d_p$ is as in (105). We consider ${\mathcal{E}}_1^{(3)}$. To this end, we write

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{E}}_1^{(3)}=& \sqrt{N}\sum _{p,q\in {\Lambda }_{+}^{\ast }:p+q\ne 0}\widehat{V}(p/e^N)(({\gamma }_{p+q}-1)b_{p+q}^{\ast }+{\sigma }_{p+q}{b}_{-p-q}+d_{p+q}^{\ast })a_{-p}^{\ast }{a}_q\hfill \\ \hfill =:& {\mathcal{E}}_{11}^{(3)}+{\mathcal{E}}_{12}^{(3)}+{\mathcal{E}}_{13}^{(3)}.\hfill \end{array}\end{array}$$

Since $|{\gamma }_{p+q}-1|\le |{\eta }_{p+q}{|}^2$ and $\Vert \eta \Vert \le C{N}^{-\alpha }$, we find

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_{11}^{(3)}\xi ⟩{|\le CN\Vert \eta \Vert }^2\Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\xi \Vert }^2\le C{N}^{1-2\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2.\end{array}$$ 140

As for ${\mathcal{E}}_{12}^{(3)}$, we commute $a_{-p}^{\ast }$ through $b_{-p-q}$ (recall $q\ne 0$). With $|{\sigma }_{p+q}|\le C|{\eta }_{p+q}|$, we obtain

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_{12}^{(3)}\xi ⟩|\le C{N}^{1-\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2.\end{array}$$ 141

We decompose now ${\mathcal{E}}_{13}^{(3)}={\mathcal{E}}_{131}^{(3)}+{\mathcal{E}}_{132}^{(3)}$, with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{E}}_{131}^{(3)}=& \sqrt{N}\sum _{p,q\in {\Lambda }_{+}^{\ast }:p+q\ne 0}\widehat{V}(p/e^N){\overline{d}}_{p+q}^{\ast }{a}_{-p}^{\ast }{a}_q\hfill \\ \hfill {\mathcal{E}}_{132}^{(3)}=& -\frac{({\mathcal{N}}_{+}+1)}{N}\sqrt{N}\sum _{p,q\in {\Lambda }_{+}^{\ast }:p+q\ne 0}\widehat{V}(p/e^N){\eta }_{p+q}{b}_{-p-q}{a}_{-p}^{\ast }{a}_q.\hfill \end{array}\end{array}$$

where we defined $d_{p+q}^{\ast }={\overline{d}}_{p+q}^{\ast }-\frac{({\mathcal{N}}_{+}+1)}{N}{\eta }_{p+q}{b}_{-p-q}$. The term ${\mathcal{E}}_{132}^{(3)}$ is estimated similarly to ${\mathcal{E}}_{12}^{(3)}$, moving $a_{-p}^{\ast }$ to the left of $b_{-p-q}$; we find $\pm {\mathcal{E}}_{132}^{(3)}\le C{N}^{1-\alpha }({\mathcal{N}}_{+}+1)$. We bound ${\mathcal{E}}_{131}^{(3)}$ in position space. We find

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,{\mathcal{E}}_{131}^{(3)}\xi ⟩|\hfill \\ \hfill & \le N^{1/2}{\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y))\Vert {\check{a}}_x\xi \Vert \Vert {\check{a}}_y{\check{\overline{d}}}_x\xi \Vert \hfill \\ \hfill & \le C{N}^{1/2-\alpha }{\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y))\Vert {\check{a}}_x\xi \Vert \hfill \\ \hfill & \times [\Vert ({\mathcal{N}}_{+}+1)\xi \Vert +N^{-1}\Vert {\check{a}}_x{({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert +\Vert \eta \Vert \Vert {\check{a}}_y{({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert +\Vert {\check{a}}_x{\check{a}}_y\xi \Vert ]\hfill \\ \hfill & \le C{N}^{1-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\xi \Vert }^2+C{N}^{1/2-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{V}}_N^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$

With (140) and (141) we obtain

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_1^{(3)}\xi ⟩|\le C{N}^{1/2-\alpha }\Vert {\mathcal{V}}_N^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert +C{N}^{1-\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2.\end{array}$$ 142

Next, we focus on ${\mathcal{E}}_2^{(3)}$, defined in (139). With Eq. (106), we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{E}}_2^{(3)}=& \sqrt{N}\sum _{p,q\in {\Lambda }_{+}^{\ast },p+q\ne 0}\widehat{V}(p/e^N){\eta }_p{e}^{-B}{b}_{p+q}^{\ast }{e}^B\hfill \\ \hfill & \times {\int }_0^{1}ds({\gamma }_p^{(s)}{\gamma }_q^{(s)}{b}_p{b}_q+{\sigma }_p^{(s)}{\sigma }_q^{(s)}{b}_{-p}^{\ast }{b}_{-q}^{\ast }+{\gamma }_p^{(s)}{\sigma }_q^{(s)}{b}_{-q}^{\ast }{b}_p+{\sigma }_p^{(s)}{\gamma }_q^{(s)}{b}_{-p}^{\ast }{b}_q)\hfill \\ \hfill & +\sqrt{N}\sum _{p,q\in {\Lambda }_{+}^{\ast },p+q\ne 0}\widehat{V}(p/e^N){\eta }_p{e}^{-B}{b}_{p+q}^{\ast }{e}^B{\int }_0^{1}ds{\gamma }_p^{(s)}{\sigma }_q^{(s)}[b_p,b_{-q}^{\ast }]\hfill \\ \hfill & +\sqrt{N}\sum _{p,q\in {\Lambda }_{+}^{\ast },p+q\ne 0}\widehat{V}(p/e^N){\eta }_p{e}^{-B}{b}_{p+q}^{\ast }{e}^B\hfill \\ \hfill & \times {\int }_0^{1}ds[d_p^{(s)}({\gamma }_q^{(s)}{b}_q+{\sigma }_q^{(s)}{b}_{-q}^{\ast })+({\gamma }_p^{(s)}{b}_p+{\sigma }_p^{(s)}{b}_{-p}^{\ast })d_q^{(s)}+d_p^{(s)}{d}_q^{(s)}]\hfill \\ \hfill =:& {\mathcal{E}}_{21}^{(3)}+{\mathcal{E}}_{22}^{(3)}+{\mathcal{E}}_{23}^{(3)}\hfill \end{array}\end{array}$$ 143

with ${\gamma }_p^{(s)}=cosh(s{\eta }_p)$, ${\sigma }_p^{(s)}=sinh(s{\eta }_p)$ and $d_p^{(s)}$ defined as in (105), with $\eta$ replaced by $s\eta$. With Lemma 2, we get

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_{21}^{(3)}\xi ⟩|\le C{N}^{1-\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2.\end{array}$$ 144

Since $[b_p,b_{-q}^{\ast }]=-a_{-q}^{\ast }{a}_p/N$ for $p\ne -q$, we find

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_{22}^{(3)}\xi ⟩|\le C{N}^{-2\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2.\end{array}$$ 145

As for the third term on the r.h.s. of (143), we switch to position space. We find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{E}}_{23}^{(3)}=& \sqrt{N}{\int }_{{\Lambda }^3}dxdydz{e}^{2N}V(e^N(x-z))\check{\eta }(y-z)e^{-B}{\check{b}}_x^{\ast }{e}^B\hfill \\ \hfill & \times {\int }_0^{1}ds[{\check{d}}_y^{(s)}(b({\check{\gamma }}_x^{(s)})+b^{\ast }({\check{\sigma }}_x^{(s)}))+(b({\check{\gamma }}_y^{(s)})+b^{\ast }({\check{\sigma }}_y^{(s)})){\check{d}}_x^{(s)}+{\check{d}}_y^{(s)}{\check{d}}_x^{(s)}].\hfill \end{array}\end{array}$$

Using the bounds (109), (110), (111) and Lemma 2 we arrive at

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,{\mathcal{E}}_{23}^{(3)}\xi ⟩|\hfill \\ \hfill & \le C\sqrt{N}{\int }_{{\Lambda }^3}dxdydz{e}^{2N}V(e^N(x-z))|\check{\eta }(y-z)|\Vert {\check{b}}_x{e}^B\xi \Vert {\int }_0^{1}ds\hfill \\ \hfill & \times [\Vert {\check{d}}_y^{(s)}({\check{b}}_x+b({\check{r}}_x^{(s)})+b^{\ast }({\check{\sigma }}_x^{(s)}))\xi \Vert +\Vert ({\check{b}}_y+b({\check{r}}_y^{(s)})+b^{\ast }({\check{\sigma }}_y^{(s)})){\check{d}}_x^{(s)}\xi \Vert +\Vert {\check{d}}_x^{(s)}{\check{d}}_y^{(s)}\xi \Vert ]\hfill \\ \hfill & \le C\sqrt{N}{\int }_{{\Lambda }^3}dxdydz{e}^{2N}V(e^N(x-z))|\check{\eta }(y-z)|\Vert {\check{b}}_x{e}^B\xi \Vert [N^{-1}|\check{\eta }(x-y)|\Vert ({\mathcal{N}}_{+}+1)\xi \Vert \hfill \\ \hfill & +\Vert \eta \Vert \Vert {\check{b}}_x{\check{b}}_y\xi \Vert +\Vert \eta \Vert \Vert ({\mathcal{N}}_{+}+1)\xi \Vert +\Vert \eta \Vert \Vert {\check{b}}_x{({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert +\Vert \eta \Vert \Vert {\check{b}}_y{({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert ]\hfill \\ \hfill & \le C{N}^{1-\alpha }\Vert {\mathcal{N}}_{+}^{1/2}{e}^B\xi \Vert \Vert ({\mathcal{N}}_{+}+1)\xi \Vert \hfill \\ \hfill & \le C{N}^{1-\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2\hfill \end{array}\end{array}$$

where $\check{r}$ indicates the function in $L^2(\Lambda )$ with Fourier coefficients $r_p=1-{\gamma }_p$, and the fact that $\Vert \check{\eta }\Vert ,\Vert \check{r}\Vert ,\Vert \check{\sigma }\Vert \le C{N}^{-\alpha }$. Combined with (144) and (145), the last bound implies that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \pm {\mathcal{E}}_2^{(3)}\le & C{N}^{1-\alpha }({\mathcal{N}}_{+}+1).\hfill \end{array}\end{array}$$ 146

To bound the last contribution on the r.h.s. of (139), it is convenient to bound (in absolute value) the expectation of its adjoint

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{E}}_3^{(3)\ast }=& \sqrt{N}\sum _{p,q\in {\Lambda }_{+}^{\ast },p+q\ne 0}\widehat{V}(p/e^N){\eta }_q{\int }_0^{1}ds{e}^{-sB}{b}_{-q}{e}^{\mathit{sB}}\hfill \\ \hfill & \times ({\gamma }_p^{(s)}{b}_{-p}+{\sigma }_p^{(s)}{b}_p^{\ast }+d_{-p}^{(s)})({\gamma }_{p+q}{b}_{p+q}+{\sigma }_{p+q}{b}_{-p-q}^{\ast }+d_{p+q})\hfill \\ \hfill =& \sqrt{N}\sum _{p,q\in {\Lambda }_{+}^{\ast },p+q\ne 0}\widehat{V}(p/e^N){\eta }_q{\int }_0^{1}ds{e}^{-sB}{b}_{-q}{e}^{\mathit{sB}}\hfill \\ \hfill & \times [{\gamma }_p^{(s)}{\gamma }_{p+q}{b}_{-p}{b}_{p+q}+{\sigma }_p^{(s)}{\sigma }_{p+q}{b}_p^{\ast }{b}_{-p-q}^{\ast }+{\gamma }_p^{(s)}{\sigma }_{p+q}{b}_{-p-q}^{\ast }{b}_{-p}+{\gamma }_{p+q}{\sigma }_p^{(s)}{b}_p^{\ast }{b}_{p+q}\hfill \\ \hfill & +d_{-p}^{(s)}({\gamma }_{p+q}{b}_{p+q}+{\sigma }_{p+q}{b}_{-p-q}^{\ast })+({\gamma }_p^{(s)}{b}_{-p}+{\sigma }_p^{(s)}{b}_p^{\ast })d_{p+q}+d_{-p}^{(s)}{d}_{p+q}]\hfill \\ \hfill & +\sqrt{N}\sum _{p,q\in {\Lambda }_{+}^{\ast },p+q\ne 0}\widehat{V}(p/e^N){\eta }_q{\int }_0^{1}ds{e}^{-sB}{b}_{-q}{e}^{\mathit{sB}}{\gamma }_p^{(s)}{\sigma }_{p+q}[b_{-p},b_{-p-q}^{\ast }]\hfill \\ \hfill =:& {\mathcal{E}}_{31}^{(3)}+{\mathcal{E}}_{32}^{(3)}.\hfill \end{array}\end{array}$$

Since $q\ne 0$, $[b_{-p},b_{-p-q}^{\ast }]=-a_{-p-q}^{\ast }{a}_{-p}/N$. Thus, we can estimate

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,& {\mathcal{E}}_{32}^{(3)}\xi ⟩|\hfill \\ \hfill & \le C{N}^{-1/2}{\int }_0^{1}ds\sum _{p,q\in {\Lambda }_{+}^{\ast },p+q\ne 0}|{\eta }_q||{\eta }_{p+q}|\Vert a_{-p-q}{e}^{-sB}{b}_{-q}^{\ast }{e}^{\mathit{sB}}\xi \Vert \Vert a_{-p}\xi \Vert \hfill \\ \hfill & \le {C\Vert \eta \Vert }^2\Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\xi \Vert }^2\le C{N}^{-2\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2.\hfill \end{array}\end{array}$$ 147

To bound the expectation of ${\mathcal{E}}_{31}^{(3)}$, we switch to position space. We find

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,{\mathcal{E}}_{31}^{(3)}\xi ⟩|\hfill \\ \hfill & \le N^{1/2}{\int }_0^{1}ds{\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y))\Vert b^{\ast }({\check{\eta }}_x)e^{\mathit{sB}}\xi \Vert [\Vert {\check{b}}_x{\check{b}}_y\xi \Vert \hfill \\ \hfill & +\Vert \eta \Vert \Vert {\check{b}}_x{({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert +\Vert \eta \Vert \Vert {\check{b}}_y{({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert +N^{-1}|\check{\eta }(x-y)|\Vert ({\mathcal{N}}_{+}+1)\xi \Vert ].\hfill \end{array}\end{array}$$

With Lemma 2, we conclude that

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_{31}^{(3)}\xi ⟩|\le C{N}^{1/2-\alpha }\Vert {\mathcal{V}}_N^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert +C{N}^{1-\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2.\end{array}$$ 148

From (147) and (148) we obtain

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_3^{(3)}\xi ⟩|\le C{N}^{1/2-\alpha }\Vert {\mathcal{V}}_N^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert +C{N}^{1-\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2.\end{array}$$

Together with (139), (142) and (146), we arrive at (138). $\square$

Appendix A.5: Analysis of ${\mathcal{G}}_{N,\alpha }^{(4)}=e^{-B}{\mathcal{L}}_N^{(4)}{e}^B$

Finally, we consider the conjugation of the quartic term ${\mathcal{L}}_N^{(4)}$. We define the error operator ${\mathcal{E}}_N^{(4)}$ through

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{G}}_{N,\alpha }^{(4)}=e^{-B}{\mathcal{L}}_N^{(4)}{e}^B=& {\mathcal{V}}_N+\frac{1}{2}\sum _{\begin{array}{c}q\in {\Lambda }_{+}^{\ast },r\in {\Lambda }^{\ast }\\ r\ne -q\end{array}}\widehat{V}(r/e^N){\eta }_{q+r}{\eta }_q\left(1,-,\frac{{\mathcal{N}}_{+}}{N}\right)\left(1,-,\frac{{\mathcal{N}}_{+}+1}{N}\right)\hfill \\ \hfill & +\frac{1}{2}\sum _{\begin{array}{c}q\in {\Lambda }_{+}^{\ast },r\in {\Lambda }^{\ast }:\\ r\ne -q\end{array}}\widehat{V}(r/e^N){\eta }_{q+r}\left(b_q,b_{-q},+,b_q^{\ast },b_{-q}^{\ast }\right)+{\mathcal{E}}_N^{(4)}\hfill \end{array}\end{array}$$

Proposition 16

Under the assumptions of Proposition 1 there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_N^{(4)}\xi ⟩|\le C{N}^{1/2-\alpha }\Vert {\mathcal{V}}_N^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert +C{N}^{1-\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2\end{array}$$ 149

for any $\alpha >1$, $\xi \in {\mathcal{F}}_{+}^{\le N}$ and $N\in \mathbb{N}$ large enough.

To show Proposition 16, we use the following lemma, whose proof can be obtained as in [4, Lemma 7.7].

Lemma 6

Let $\eta \in {\ell }^2({\Lambda }^{\ast })$ as defined in (27). Then there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \Vert {({\mathcal{N}}_{+}+1)}^{n/2}{e}^{-B}& {\check{b}}_x{\check{b}}_y{e}^B\xi \Vert \hfill \\ \hfill & \le C[N\Vert {({\mathcal{N}}_{+}+1)}^{n/2}\xi \Vert +\Vert {\check{a}}_y{({\mathcal{N}}_{+}+1)}^{(n+1)/2}\xi \Vert \hfill \\ \hfill & +\Vert {\check{a}}_x{({\mathcal{N}}_{+}+1)}^{(n+1)/2}\xi \Vert +\Vert {\check{a}}_x{\check{a}}_y{({\mathcal{N}}_{+}+1)}^{n/2}\xi \Vert ]\hfill \end{array}\end{array}$$

for all $\xi \in {\mathcal{F}}_{+}^{\le N}$, $n\in \mathbb{Z}$.

Proof of Proposition 16

We follow the proof of [4, Prop. 7.6]. We write

$$\begin{array}{c}\hfill {\mathcal{G}}_{N,\alpha }^{(4)}={\mathcal{V}}_N+W_1+W_2+W_3+W_4\end{array}$$

with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\text{W}}_1=& \frac{1}{2}\sum _{q\in {\Lambda }_{+}^{\ast },r\in {\Lambda }^{\ast }:r\ne -q}\widehat{V}(r/e^N){\eta }_{q+r}{\int }_0^{1}ds(e^{-sB}{b}_q{b}_{-q}{e}^{\mathit{sB}}+\text{h.c.})\hfill \\ \hfill {\text{W}}_2=& \sum _{p,q\in {\Lambda }_{+}^{\ast },r\in {\Lambda }^{\ast }:r\ne p,-q}\widehat{V}(r/e^N){\eta }_{q+r}{\int }_0^{1}ds(e^{-sB}{b}_{p+r}^{\ast }{b}_q^{\ast }{e}^{\mathit{sB}}{a}_{-q-r}^{\ast }{a}_p+\text{h.c.})\hfill \\ \hfill {\text{W}}_3=& \sum _{p,q\in {\Lambda }_{+}^{\ast },r\in {\Lambda }^{\ast }:r\ne -p-q}\widehat{V}(r/e^N){\eta }_{q+r}{\eta }_p\hfill \\ \hfill & \times {\int }_0^{1}ds{\int }_0^{s}d\tau (e^{-sB}{b}_{p+r}^{\ast }{b}_q^{\ast }{e}^{\mathit{sB}}{e}^{-\tau B}{b}_{-p}^{\ast }{b}_{-q-r}^{\ast }{e}^{\tau B}+\text{h.c.})\hfill \\ \hfill {\text{W}}_4=& \sum _{p,q\in {\Lambda }_{+}^{\ast },r\in {\Lambda }^{\ast }:r\ne -p-q}\widehat{V}(r/e^N){\eta }_{q+r}^2\hfill \\ \hfill & \times {\int }_0^{1}ds{\int }_0^{s}d\tau (e^{-sB}{b}_{p+r}^{\ast }{b}_q^{\ast }{e}^{\mathit{sB}}{e}^{-\tau B}{b}_p{b}_{q+r}{e}^{\tau B}+\text{h.c.}).\hfill \end{array}\end{array}$$ 150

Let us first consider the term ${\text{W}}_1$. With (106), we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\text{W}}_1=& \frac{1}{2}\sum _{q\in {\Lambda }_{+}^{\ast },r\in {\Lambda }^{\ast }:r\ne -q}\widehat{V}(r/e^N){\eta }_{q+r}{\int }_0^{1}ds{({\gamma }_q^{(s)})}^2(b_q{b}_{-q}+\text{h.c.})\hfill \\ \hfill & +\frac{1}{2}\sum _{q\in {\Lambda }_{+}^{\ast },r\in {\Lambda }^{\ast }:r\ne -q}\widehat{V}(r/e^N){\eta }_{q+r}{\int }_0^{1}ds{\gamma }_q^{(s)}{\sigma }_q^{(s)}([b_q,b_q^{\ast }]+\text{h.c.})\hfill \\ \hfill & +\frac{1}{2}\sum _{q\in {\Lambda }_{+}^{\ast },r\in {\Lambda }^{\ast }:r\ne -q}\widehat{V}(r/e^N){\eta }_{q+r}{\int }_0^{1}ds{\gamma }_q^{(s)}(b_q{d}_{-q}^{(s)}+\text{h.c.})+{\mathcal{E}}_{10}^{(4)}\hfill \\ \hfill =:& {\text{W}}_{11}+{\text{W}}_{12}+{\text{W}}_{13}+{\mathcal{E}}_{10}^{(4)}\hfill \end{array}\end{array}$$ 151

where

$$\begin{array}{c}\hfill {\mathcal{E}}_{10}^{(4)}={\mathcal{E}}_{101}^{(4)}+{\mathcal{E}}_{102}^{(4)}+{\mathcal{E}}_{103}^{(4)}+{\mathcal{E}}_{104}^{(4)}+{\mathcal{E}}_{105}^{(4)}\end{array}$$ 152

with

$$\begin{array}{cc}\hfill {\mathcal{E}}_{101}^{(4)}=& \frac{1}{2}\sum _{q\in {\Lambda }_{+}^{\ast },r\in {\Lambda }^{\ast }:r\ne -q}\widehat{V}(r/e^N){\eta }_{q+r}{\int }_0^{1}ds[2{\gamma }_q^{(s)}{\sigma }_q^{(s)}{b}_q^{\ast }{b}_q+{({\sigma }_q^{(s)})}^2{b}_{-q}^{\ast }{b}_q^{\ast }+\text{h.c.}]\hfill \\ \hfill {\mathcal{E}}_{102}^{(4)}=& \frac{1}{2}\sum _{q\in {\Lambda }_{+}^{\ast },r\in {\Lambda }^{\ast }:r\ne -q}\widehat{V}(r/e^N){\eta }_{q+r}{\int }_0^{1}ds{\sigma }_q^{(s)}(b_{-q}^{\ast }{d}_{-q}^{(s)}+\text{h.c.})\hfill \\ \hfill {\mathcal{E}}_{103}^{(4)}=& \frac{1}{2}\sum _{q\in {\Lambda }_{+}^{\ast },r\in {\Lambda }^{\ast }:r\ne -q}\widehat{V}(r/e^N){\eta }_{q+r}{\int }_0^{1}ds{\sigma }_q^{(s)}(d_q^{(s)}{b}_q^{\ast }+\text{h.c.})\hfill \\ \hfill {\mathcal{E}}_{104}^{(4)}=& \frac{1}{2}\sum _{q\in {\Lambda }_{+}^{\ast },r\in {\Lambda }^{\ast }:r\ne -q}\widehat{V}(r/e^N){\eta }_{q+r}{\int }_0^{1}ds{\gamma }_q^{(s)}(d_q^{(s)}{b}_{-q}+\text{h.c.})\hfill \\ \hfill {\mathcal{E}}_{105}^{(4)}=& \frac{1}{2}\sum _{q\in {\Lambda }_{+}^{\ast },r\in {\Lambda }^{\ast }:r\ne -q}\widehat{V}(r/e^N){\eta }_{q+r}{\int }_0^{1}ds(d_q^{(s)}{d}_{-q}^{(s)}+\text{h.c.}).\hfill \end{array}$$ 153

With

$$\begin{array}{c}\hfill \frac{1}{N}\underset{q\in {\Lambda }_{+}^{\ast }}{sup}\sum _{r\in {\Lambda }_{+}^{\ast }}|\widehat{V}(r/e^N)||{\eta }_{q+r}|\le C<\infty \end{array}$$ 154

uniformly in $N\in \mathbb{N}$, we can estimate the first term in (153) by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\mathcal{E}}_{101}^{(4)}\xi ⟩|& \le C{N}^{1-\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2.\hfill \end{array}\end{array}$$

Using (154) and (108) we also find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\mathcal{E}}_{102}^{(4)}\xi ⟩|& \le C{N}^{1-2\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2.\hfill \end{array}\end{array}$$

For the third term in (153) we switch to position space and use (109):

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\mathcal{E}}_{103}^{(4)}\xi ⟩|& \le \frac{1}{2}\int dxdy{e}^{2N}V(e^N(x-y))|\check{\eta }(x-y)|\hfill \\ \hfill & \times {\int }_0^1{ds\Vert (\mathcal{N}+1)}^{-1/2}{\check{d}}_y{b}^{\ast }({\check{\sigma }}_x^{(s)})\xi \Vert \Vert {(\mathcal{N}+1)}^{1/2}\xi \Vert \hfill \\ \hfill & \le C\Vert \check{\eta }{\Vert }_{\infty }\Vert \eta \Vert \int dxdy{e}^{2N}V(e^N(x-y))\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert {\int }_0^{1}ds\hfill \\ \hfill & \times [\Vert b^{\ast }({\check{\sigma }}_x^{(s)})\xi \Vert +\frac{1}{N}|{\check{\eta }}^{(s)}(x-y)|\Vert {(\mathcal{N}+1)}^{1/2}\xi \Vert +\frac{1}{\sqrt{N}}\Vert b^{\ast }({\check{\sigma }}_x^{(s)}){\check{b}}_y\xi \Vert ]\hfill \\ \hfill & \le C{N}^{1-\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2.\hfill \end{array}\end{array}$$

Consider now the fourth term in (153). We write ${\mathcal{E}}_{104}^{(4)}={\mathcal{E}}_{1041}^{(4)}+{\mathcal{E}}_{1042}^{(4)}$, with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{E}}_{1041}^{(4)}& =\frac{1}{2}\sum _{q\in {\Lambda }_{+}^{\ast },r\in {\Lambda }^{\ast }:r\ne -q}\widehat{V}(r/e^N){\eta }_{q+r}{\int }_0^{1}ds({\gamma }_q^{(s)}-1)d_q^{(s)}{b}_{-q}\hfill \\ \hfill {\mathcal{E}}_{1042}^{(4)}& =\frac{1}{2}\sum _{q\in {\Lambda }_{+}^{\ast },r\in {\Lambda }^{\ast }:r\ne -q}\widehat{V}(r/e^N){\eta }_{q+r}{\int }_0^{1}ds{d}_q^{(s)}{b}_{-q}\hfill \end{array}\end{array}$$

With $|{\gamma }_q^{(s)}-1|\le C|{\eta }_q{|}^2$, (154) and $\Vert d_q^{\ast }\xi \Vert \le C\Vert \eta \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert$, we find

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_{1041}^{(4)}\xi ⟩|\le C{N}^{1-3\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2\end{array}$$

As for ${\mathcal{E}}_{1042}^{(4)}$, we switch to position space. Using (25) and (109), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,{\mathcal{E}}_{1042}^{(4)}\xi ⟩|\hfill \\ \hfill & =|\frac{1}{2}{\int }_0^{1}ds{\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y))\check{\eta }(x-y)⟨\xi ,{\check{d}}_x^{(s)}{\check{b}}_y\xi ⟩|\hfill \\ \hfill & \le CN{\int }_0^{1}ds{\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y))\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{-1/2}{\check{d}}_x^{(s)}{\check{b}}_y\xi \Vert \hfill \\ \hfill & \le CN\Vert \eta \Vert {\int }_0^{1}ds{\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y))\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \hfill \\ \hfill & \times N^{-1}\left[\Vert ,{\check{a}}_y,{\mathcal{N}}_{+},\xi \Vert +\Vert ,{\check{a}}_x,{\check{a}}_y,{\mathcal{N}}_{+}^{1/2},\xi \Vert \right]\hfill \\ \hfill & \le C{N}^{1-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\xi \Vert }^2+C{N}^{1/2-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{V}}_N^{1/2}\xi \Vert \hfill \end{array}\end{array}$$

Let us consider the last term in (153). Switching to position space and using (111) in Lemma 4 and again (25), we arrive at

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,{\mathcal{E}}_{105}^{(4)}\xi ⟩|\hfill \\ \hfill & \le CN{\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y))\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert {\int }_0^{1}ds\Vert {({\mathcal{N}}_{+}+1)}^{-1/2}{\check{d}}_x^{(s)}{\check{d}}_y^{(s)}\xi \Vert \hfill \\ \hfill & \le CN\Vert \eta \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert {\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y))\hfill \\ \hfill & \times \left[\Vert ,{({\mathcal{N}}_{+}+1)}^{1/2},\xi \Vert +\Vert \eta \Vert \Vert ,{\check{a}}_x,\xi \Vert +\Vert \eta \Vert \Vert ,{\check{a}}_y,\xi \Vert +,N^{-1/2},\Vert \eta \Vert \Vert ,{\check{a}}_x,{\check{a}}_y,\xi \Vert \right]\hfill \\ \hfill & \le C{N}^{1-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\xi \Vert }^2+C{N}^{1/2-2\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{V}}_N^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$

Summarizing, we have shown that (152) can be bounded by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\mathcal{E}}_{10}^{(4)}\xi ⟩|\le & C{N}^{1/2-\alpha }\Vert {\mathcal{V}}_N^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert +C{N}^{1-\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2\hfill \end{array}\end{array}$$ 155

for any $\alpha >1$, $\xi \in {\mathcal{F}}_{+}^{\le N}$. Next, we come back to the terms ${\text{W}}_{11},{\text{W}}_{12},{\text{W}}_{13}$ introduced in (151). Using (154) and $|{\gamma }_q^{(s)}-1|\le C{\eta }_q^2$, we can write

$$\begin{array}{c}\hfill {\text{W}}_{11}=\frac{1}{2}\sum _{q\in {\Lambda }_{+}^{\ast },r\in {\Lambda }^{\ast }:r\ne -q}\widehat{V}(r/e^N){\eta }_{q+r}(b_q{b}_{-q}+\text{h.c.})+{\mathcal{E}}_{11}^{(4)},\end{array}$$ 156

where ${\mathcal{E}}_{11}^{(4)}$ is such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\mathcal{E}}_{11}^{(4)}\xi ⟩|& \le C{N}^{1-2\alpha }{\Vert ({\mathcal{N}}_{+}+1)\xi \Vert }^2.\hfill \end{array}\end{array}$$

Next, we can decompose the second term in (151) as

$$\begin{array}{c}\hfill {\text{W}}_{12}=\frac{1}{2}\sum _{q\in {\Lambda }_{+}^{\ast },r\in {\Lambda }^{\ast }:r\ne -q}\widehat{V}(r/e^N){\eta }_{q+r}{\eta }_q\left(1,-,\frac{{\mathcal{N}}_{+}}{N}\right)+{\mathcal{E}}_{12}^{(4)}\end{array}$$ 157

where $\pm {\mathcal{E}}_{12}^{(4)}\le C{N}^{-\alpha }{\mathcal{N}}_{+}+N^{1-3\alpha }$.

The third term on the r.h.s. of (151) can be written as

$$\begin{array}{c}\hfill {\text{W}}_{13}=-\frac{1}{2}\sum _{q\in {\Lambda }_{+}^{\ast },r\in {\Lambda }^{\ast }:r\ne -q}\widehat{V}(r/e^N){\eta }_{q+r}{\eta }_q\left(1,-,\frac{{\mathcal{N}}_{+}}{N}\right)\frac{{\mathcal{N}}_{+}+1}{N}+{\mathcal{E}}_{13}^{(4)}\end{array}$$ 158

where ${\mathcal{E}}_{13}^{(4)}={\mathcal{E}}_{131}^{(4)}+{\mathcal{E}}_{132}^{(4)}+{\mathcal{E}}_{133}^{(4)}+{\mathcal{E}}_{134}^{(4)}$, with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{E}}_{131}^{(4)}=& \frac{1}{2}\sum _{q\in {\Lambda }_{+}^{\ast },r\in {\Lambda }^{\ast }:r\ne -q}\widehat{V}(r/e^N){\eta }_{q+r}{\int }_0^{1}ds({\gamma }_q^{(s)}-1)b_q{d}_{-q}^{(s)}+\text{h.c.}\hfill \\ \hfill {\mathcal{E}}_{132}^{(4)}=& \frac{1}{2}\sum _{q\in {\Lambda }_{+}^{\ast },r\in {\Lambda }^{\ast }:r\ne -q}\widehat{V}(r/e^N){\eta }_{q+r}{\int }_0^{1}ds{b}_q\left[d_{-q}^{(s)},+,s,{\eta }_q,\frac{{\mathcal{N}}_{+}}{N},b_q^{\ast }\right]+\text{h.c.}\hfill \\ \hfill {\mathcal{E}}_{133}^{(4)}=& -\frac{1}{2}\sum _{q\in {\Lambda }_{+}^{\ast },r\in {\Lambda }^{\ast }:r\ne -q}\widehat{V}(r/e^N){\eta }_{q+r}{\eta }_q{b}_q^{\ast }{b}_q\frac{{\mathcal{N}}_{+}+1}{N}\hfill \\ \hfill {\mathcal{E}}_{134}^{(4)}=& \frac{1}{2N}\sum _{q\in {\Lambda }_{+}^{\ast },r\in {\Lambda }^{\ast }:r\ne -q}\widehat{V}(r/e^N){\eta }_{q+r}{\eta }_q{a}_q^{\ast }{a}_q\frac{{\mathcal{N}}_{+}+1}{N}.\hfill \end{array}\end{array}$$

With (154), we immediately find

$$\begin{array}{c}\hfill \pm {\mathcal{E}}_{133}^{(4)}\le C{N}^{1-\alpha }({\mathcal{N}}_{+}+1),\pm {\mathcal{E}}_{134}^{(4)}\le C{N}^{-\alpha }({\mathcal{N}}_{+}+1).\end{array}$$

With $|{\gamma }_q^{(s)}-1|\le C{\eta }_q^2$, Lemma 4 and, again, (154), we also obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\mathcal{E}}_{131}^{(4)}\xi ⟩|& \le C{N}^{1-3\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2.\hfill \end{array}\end{array}$$

Let us now consider ${\mathcal{E}}_{132}^{(4)}$. In position space, with ${\check{\overline{d}}}_y^{(s)}=d_y^{(s)}+({\mathcal{N}}_{+}/N)b^{\ast }({\check{\eta }}_y)$ and using (110), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\mathcal{E}}_{132}^{(4)}\xi ⟩|& =|\frac{1}{2}{\int }_0^{1}ds{\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y))\check{\eta }(x-y)⟨\xi ,{\check{b}}_x{\check{\overline{d}}}_y^{(s)}\xi ⟩|\hfill \\ \hfill & \le C{N}^{1-\alpha }{\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y))\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \hfill \\ \hfill & \times \left[\Vert ,{({\mathcal{N}}_{+}+1)}^{1/2},\xi \Vert +\Vert ,{\check{a}}_y,\xi \Vert +\Vert ,{\check{a}}_x,\xi \Vert +,N^{-1},\Vert {\check{a}}_x{\check{a}}_y{\mathcal{N}}_{+}^{1/2}\xi \Vert \right]\hfill \\ \hfill & \le C{N}^{1-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\xi \Vert }^2+C{N}^{1/2-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{V}}_N^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$

It follows that

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_{13}^{(4)}⟩|\le C{N}^{1/2-\alpha }\Vert {\mathcal{V}}_N^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert +C{N}^{1-\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2.\end{array}$$

With (155), (156), (157), (158), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\text{W}}_1=& \frac{1}{2}\sum _{q\in {\Lambda }_{+}^{\ast },r\in {\Lambda }^{\ast }:r\ne -q}\widehat{V}(r/e^N){\eta }_{q+r}(b_q{b}_{-q}+\text{h.c.})\hfill \\ \hfill & +\frac{1}{2}\sum _{q\in {\Lambda }_{+}^{\ast },r\in {\Lambda }^{\ast }:r\ne -q}\widehat{V}(r/e^N){\eta }_{q+r}{\eta }_q\left(1,-,\frac{{\mathcal{N}}_{+}}{N}\right)\left(1,-,\frac{{\mathcal{N}}_{+}+1}{N}\right)+{\mathcal{E}}_1^{(4)}\hfill \end{array}\end{array}$$ 159

where

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_1^{(4)}\xi ⟩|\le C{N}^{1/2-\alpha }\Vert {\mathcal{V}}_N^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert +C{N}^{1-\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2,\end{array}$$

Next, we control the term ${\text{W}}_2$, from (150). In position space, we find

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\text{W}}_2={\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y)){\int }_0^{1}ds(e^{-sB}{\check{b}}_x^{\ast }{\check{b}}_y^{\ast }{e}^{\mathit{sB}}{a}^{\ast }({\check{\eta }}_x){\check{a}}_y+\text{h.c.})\end{array}\end{array}$$

with ${\check{\eta }}_x(z)=\check{\eta }(x-z)$. By Cauchy–Schwarz, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\text{W}}_2\xi ⟩|& \le {\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y)){\int }_0^{1}ds\hfill \\ \hfill & \times \Vert {({\mathcal{N}}_{+}+1)}^{1/2}{e}^{-sB}{\check{b}}_x{\check{b}}_y{e}^{\mathit{sB}}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{-1/2}{a}^{\ast }({\check{\eta }}_x){\check{a}}_y\xi \Vert .\hfill \end{array}\end{array}$$

With

$$\begin{array}{c}\hfill \Vert {({\mathcal{N}}_{+}+1)}^{-1/2}{a}^{\ast }({\check{\eta }}_x){\check{a}}_y\xi \Vert \le C\Vert \eta \Vert \Vert {\check{a}}_y\xi \Vert \le C{N}^{-\alpha }\Vert {\check{a}}_y\xi \Vert \end{array}$$

and using Lemma 6, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\text{W}}_2\xi ⟩|& \le C{N}^{-\alpha }{\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y))\Vert {\check{a}}_y\xi \Vert \hfill \\ \hfill & \times \{N\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert +N\Vert {\check{a}}_x\xi \Vert +N\Vert {\check{a}}_y\xi \Vert +N^{1/2}\Vert {\check{a}}_x{\check{a}}_y\xi \Vert \}\hfill \\ \hfill & \le C{N}^{1-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\xi \Vert }^2+C{N}^{1/2-\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{V}}_N^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$ 160

Also for the term ${\text{W}}_3$ in (150), we switch to position space. We find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\text{W}}_3=& {\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y))\hfill \\ \hfill & \times {\int }_0^{1}ds{\int }_0^{s}d\tau (e^{-sB}{\check{b}}_x^{\ast }{\check{b}}_y^{\ast }{e}^{\mathit{sB}}{e}^{-\tau B}{b}^{\ast }({\check{\eta }}_x)b^{\ast }({\check{\eta }}_y)e^{\tau B}+\text{h.c.}).\hfill \end{array}\end{array}$$

and thus

$$\begin{array}{c}\hfill \begin{array}{cc}& \left|⟨,\xi ,,,{\text{W}}_3,\xi ,⟩\right|\le {\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y)){\int }_0^{1}ds{\int }_0^{s}d\tau \Vert {({\mathcal{N}}_{+}+1)}^{1/2}{e}^{-sB}{\check{b}}_x{\check{b}}_y{e}^{\mathit{sB}}\xi \Vert \hfill \\ \hfill & \times \Vert {({\mathcal{N}}_{+}+1)}^{-1/2}{e}^{-\tau B}{b}^{\ast }({\check{\eta }}_x)b^{\ast }({\check{\eta }}_y)e^{\tau B}\xi \Vert .\hfill \end{array}\end{array}$$

With Lemma 2, we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \Vert {({\mathcal{N}}_{+}+1)}^{-1/2}{e}^{-\tau B}{b}^{\ast }({\check{\eta }}_x)b^{\ast }({\check{\eta }}_y)e^{\tau B}\xi \Vert & \le {C\Vert \eta \Vert }^2\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$

Using Lemma 6, we conclude that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\text{W}}_3\xi ⟩|& \le {C\Vert \eta \Vert }^2{\int }_{{\Lambda }^2}dxdy{e}^{2N}V(e^N(x-y))\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \hfill \\ \hfill & \times \{N\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert +N\Vert {\check{a}}_x\xi \Vert +N\Vert {\check{a}}_y\xi \Vert +N^{1/2}\Vert {\check{a}}_x{\check{a}}_y\xi \Vert \}\hfill \\ \hfill & \le C{N}^{1-2\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\xi \Vert }^2+C{N}^{1/2-2\alpha }\Vert {\mathcal{V}}_N^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$ 161

The term ${\text{W}}_4$ in (150) can be bounded similarly. In position space, we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\text{W}}_4=& \int dxdy{e}^{2N}V(e^N(x-y))\hfill \\ \hfill & \times {\int }_0^{1}ds{\int }_0^{s}d\tau (e^{-sB}{\check{b}}_x^{\ast }{\check{b}}_y^{\ast }{e}^{\mathit{sB}}{e}^{-\tau B}b(\check{{\eta }_x^2}){\check{b}}_y{e}^{\tau B}+\text{h.c.})\hfill \end{array}\end{array}$$

with $\check{{\eta }^2}$ the function with Fourier coefficients ${\eta }_q^2$, for $q\in {\Lambda }^{\ast }$, and where $\check{{\eta }_x^2}(y):=\check{{\eta }^2}(x-y)$. Clearly $\Vert \check{{\eta }_x^2}\Vert \le C\Vert \check{\eta }{\Vert }^2\le C{N}^{-2\alpha }$. With Cauchy–Schwarz and Lemma 2, we obtain

$$\begin{array}{c}\hfill |⟨\xi ,{\text{W}}_4\xi ⟩|\le C{N}^{-2\alpha }{\int }_0^{1}ds{\int }_0^{s}d\tau \int dxdy{e}^{2N}V(e^N(x-y))\Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\check{b}}_y{\check{b}}_x{e}^{\mathit{sB}}\xi \Vert \Vert {\check{b}}_y{e}^{\tau B}\xi \Vert .\end{array}$$

Applying Lemma 6 and then Lemma 2, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\text{W}}_4\xi ⟩|\le & C{N}^{-2\alpha }{\int }_0^{1}ds{\int }_0^{s}d\tau \int dxdy{e}^{2N}V(e^N(x-y))\Vert {\check{b}}_y{e}^{\tau B}\xi \Vert \hfill \\ \hfill & \times \left\{N\Vert ,{({\mathcal{N}}_{+}+1)}^{1/2},\xi \Vert +N\Vert ,{\check{a}}_x,\xi \Vert +N\Vert ,{\check{a}}_y,\xi \Vert +,N^{1/2},\Vert {\check{a}}_x{\check{a}}_y\xi \Vert \right\}\hfill \\ \hfill \le & C{N}^{1-2\alpha }\Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\xi \Vert }^2+C{N}^{1/2-2\alpha }\Vert {\mathcal{V}}_N^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$

From (159), (160), (161) and the last bound, we conclude that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{G}}_{N,\alpha }^{(4)}=& {\mathcal{V}}_N+\frac{1}{2}\sum _{q\in {\Lambda }_{+}^{\ast },r\in {\Lambda }^{\ast }:r\ne -q}\widehat{V}(r/e^N){\eta }_{q+r}(b_q{b}_{-q}+\text{h.c.})\hfill \\ \hfill & +\frac{1}{2}\sum _{q\in {\Lambda }_{+}^{\ast },r\in {\Lambda }^{\ast }:r\ne -q}\widehat{V}(r/e^N){\eta }_{q+r}{\eta }_q\left(1,-,\frac{{\mathcal{N}}_{+}}{N}\right)\left(1,-,\frac{{\mathcal{N}}_{+}+1}{N}\right)+{\mathcal{E}}_{N,\alpha }^{(4)}\hfill \end{array}\end{array}$$

where ${\mathcal{E}}_{N,\alpha }^{(4)}$ satisfies (149). $\square$

Appendix A.6: Proof of Proposition 1

With the results established in Sects. 1–1, we cam now show Proposition 1. Propositions 12, 13, 14, 15, 16, imply that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{G}}_{N,\alpha }& =\frac{\widehat{V}(0)}{2}(N+{\mathcal{N}}_{+}-1)(N-{\mathcal{N}}_{+})\hfill \\ \hfill & +\sum _{p\in {\Lambda }_{+}^{\ast }}{\eta }_p[p^2{\eta }_p+N\widehat{V}(p/e^N)+\frac{1}{2}\sum _{\begin{array}{c}r\in {\Lambda }^{\ast }\\ p+r\ne 0\end{array}}\widehat{V}(r/e^N){\eta }_{p+r}](\frac{N-{\mathcal{N}}_{+}}{N})(\frac{N-{\mathcal{N}}_{+}-1}{N})\hfill \\ \hfill & +\mathcal{K}+N\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N)a_p^{\ast }{a}_p(1-\frac{{\mathcal{N}}_{+}}{N})\hfill \\ \hfill & +\sum _{p\in {\Lambda }_{+}^{\ast }}[p^2{\eta }_p+\frac{N}{2}\widehat{V}(p/e^N)+\frac{1}{2}\sum _{r\in {\Lambda }^{\ast }:p+r\ne 0}\widehat{V}(r/e^N){\eta }_{p+r}](b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p})\hfill \\ \hfill & +\sqrt{N}\sum _{p,q\in {\Lambda }_{+}^{\ast }:p+q\ne 0}\widehat{V}(p/e^N)\left[b_{p+q}^{\ast },a_{-p}^{\ast },a_q,+,,\text{h.c.},\right]+{\mathcal{V}}_N+{\mathcal{E}}_1\hfill \end{array}\end{array}$$ 162

where

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_1\xi ⟩|\le C{N}^{1/2-\alpha }\Vert {\mathcal{H}}_N^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert +C{N}^{1-\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2\end{array}$$

for any $\alpha >1$ and $\xi \in {\mathcal{F}}_{+}^{\le N}$. With (31), we find

$$\begin{array}{c}\hfill \begin{array}{cc}& \sum _{p\in {\Lambda }_{+}^{\ast }}{\eta }_p[p^2{\eta }_p+N\widehat{V}(p/e^N)+\frac{1}{2}\sum _{r\in {\Lambda }^{\ast }:p+r\ne 0}\widehat{V}(r/e^N){\eta }_{p+r}]\hfill \\ \hfill & =\sum _{p\in {\Lambda }_{+}^{\ast }}{\eta }_p[\frac{N}{2}\widehat{V}(p/e^N)+N{e}^{2N}{\lambda }_{\ell }{\widehat{\chi }}_{\ell }(p)+e^{2N}{\lambda }_{\ell }\sum _{q\in {\Lambda }^{\ast }}{\widehat{\chi }}_{\ell }(p-q){\eta }_q-\frac{1}{2}\widehat{V}(p/e^N){\eta }_0]\hfill \end{array}\end{array}$$

From Lemma 1 and estimating $\Vert {\widehat{\chi }}_{\ell }\Vert =\Vert {\chi }_{\ell }\Vert \le C{N}^{-\alpha }$, $\Vert \eta \Vert \le C{N}^{-\alpha }$ and $\Vert {\widehat{\chi }}_{\ell }\ast \eta \Vert =\Vert {\chi }_{\ell }\check{\eta }\Vert \le \Vert \check{\eta }\Vert \le C{N}^{-\alpha }$, we have

$$\begin{array}{c}\hfill |N{e}^{2N}{\lambda }_{\ell }\sum _{p\in {\Lambda }_{+}^{\ast }}{\eta }_p{\widehat{\chi }}_{\ell }(p)|\le C{N}^{2\alpha }\Vert {\widehat{\chi }}_{\ell }\Vert \Vert \eta \Vert \le C,\end{array}$$

and

$$\begin{array}{c}\hfill |e^{2N}{\lambda }_{\ell }\sum _{\begin{array}{c}p\in {\Lambda }_{+}^{\ast },q\in {\Lambda }^{\ast }\end{array}}{\widehat{\chi }}_{\ell }(p-q){\eta }_q{\eta }_p|\le C{N}^{2\alpha -1}\Vert {\widehat{\chi }}_{\ell }\ast \eta \Vert \Vert \eta \Vert \le C{N}^{-1}.\end{array}$$

Moreover, using (154) and the bound (32) we find

$$\begin{array}{c}\hfill |\frac{1}{2}\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N){\eta }_p{\eta }_0|\le C{N}^{1-2\alpha }.\end{array}$$

We obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& \sum _{p\in {\Lambda }_{+}^{\ast }}{\eta }_p[p^2{\eta }_p+N\widehat{V}(p/e^N)+\frac{1}{2}\sum _{\begin{array}{c}r\in {\Lambda }^{\ast }\\ p+r\in {\Lambda }_{+}^{\ast }\end{array}}\widehat{V}(r/e^N){\eta }_{p+r}](\frac{N-{\mathcal{N}}_{+}}{N})(\frac{N-{\mathcal{N}}_{+}-1}{N})\hfill \\ \hfill & =\frac{N}{2}\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N){\eta }_p\left(\frac{N-{\mathcal{N}}_{+}}{N}\right)\left(\frac{N-{\mathcal{N}}_{+}-1}{N}\right)+{\mathcal{E}}_2\hfill \end{array}\end{array}$$

with $\pm {\mathcal{E}}_2\le C$ for all $\alpha \ge 1/2$. On the other hand, using (32) we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{N}{2}\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N){\eta }_p=& \frac{N}{2}(\widehat{V}(·/e^N)\ast \eta )(0)-\frac{N}{2}\widehat{V}(0){\eta }_0\hfill \\ \hfill =& \frac{N^2}{2}(\int dxV(x)f_{\ell }(x)-\widehat{V}(0))+{\tilde{\mathcal{E}}}_2\hfill \end{array}\end{array}$$

with $\pm {\tilde{\mathcal{E}}}_2\le C{N}^{1-2\alpha }$. With the first bound in (41) we conclude that

$$\begin{array}{c}\hfill \begin{array}{cc}& \sum _{p\in {\Lambda }_{+}^{\ast }}{\eta }_p[p^2{\eta }_p+N\widehat{V}(p/e^N)+\frac{1}{2}\sum _{\begin{array}{c}r\in {\Lambda }^{\ast }\\ p+r\in {\Lambda }_{+}^{\ast }\end{array}}\widehat{V}(r/e^N){\eta }_{p+r}](\frac{N-{\mathcal{N}}_{+}}{N})(\frac{N-{\mathcal{N}}_{+}-1}{N})\hfill \\ & =\frac{1}{2N}\left[{\widehat{\omega }}_N,(0),-,N,\widehat{V},(0)\right](N-{\mathcal{N}}_{+}-1)\left(N,-,{\mathcal{N}}_{+}\right)+{\mathcal{E}}_3\hfill \end{array}\end{array}$$ 163

where $\pm {\mathcal{E}}_3\le C$, if $\alpha \ge 1/2$. Using (31), we can also handle the fourth line of (162); we find

$$\begin{array}{c}\hfill \begin{array}{cc}& \sum _{p\in {\Lambda }_{+}^{\ast }}[p^2{\eta }_p+\frac{N}{2}\widehat{V}(p/e^N)+\frac{1}{2}\sum _{r\in {\Lambda }^{\ast }:p+r\in {\Lambda }_{+}^{\ast }}\widehat{V}(r/e^N){\eta }_{p+r}](b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p})\hfill \\ \hfill & =\sum _{p\in {\Lambda }_{+}^{\ast }}[N{e}^{2N}{\lambda }_{\ell }{\widehat{\chi }}_{\ell }(p)+e^{2N}{\lambda }_{\ell }\sum _{q\in {\Lambda }^{\ast }}{\widehat{\chi }}_{\ell }(p-q){\eta }_q-\frac{1}{2}\widehat{V}(p/e^N){\eta }_0](b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p}).\hfill \end{array}\end{array}$$ 164

The last two terms on the right hand side of (164) are error terms. With (32) and (154) we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |\sum _{p\in {\Lambda }_{+}^{\ast }}& \widehat{V}(p/e^N){\eta }_0(b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p})|\hfill \\ \hfill & \le C{N}^{-2\alpha }[\sum _{p\in {\Lambda }_{+}^{\ast }}\frac{|\widehat{V}(p/e^N){|}^2}{p^2}{]}^{1/2}[\sum _{p\in {\Lambda }_{+}^{\ast }}{p}^2\Vert a_p{\xi \Vert }^2{]}^{1/2}\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \hfill \\ \hfill & \le C{N}^{1/2-2\alpha }\Vert {\mathcal{K}}^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$

The second term on the right hand side of (164) can be bounded in position space:

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,e^{2N}{\lambda }_{\ell }\sum _{p\in {\Lambda }_{+}^{\ast }}({\widehat{\chi }}_{\ell }\ast \eta )(p)(b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p})\xi ⟩|\hfill \\ \hfill & \le C{N}^{2\alpha -1}\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert {\int }_{{\Lambda }^2}dxdy{\chi }_{\ell }(x-y)|\check{\eta }(x-y)|\Vert {({\mathcal{N}}_{+}+1)}^{-1/2}{\check{b}}_x{\check{b}}_y\xi \Vert \hfill \\ \hfill & \le C{N}^{\alpha -1}\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert {\left[{\int }_{{\Lambda }^2},d,x,d,y,,{\chi }_{\ell },(x-y),{\Vert {({\mathcal{N}}_{+}+1)}^{-1/2}{\check{a}}_x{\check{a}}_y\xi \Vert }^2\right]}^{1/2}.\hfill \end{array}\end{array}$$

The term in parenthesis can be bounded similarly as in (80). Namely,

$$\begin{array}{c}\hfill {\int }_{{\Lambda }^2}dxdy{\chi }_{\ell }(x-y)\Vert {({\mathcal{N}}_{+}+1)}^{-1/2}{\check{a}}_x{\check{a}}_y{\xi \Vert }^2\le Cq{N}^{-2\alpha /q^{\prime }}{\Vert {\mathcal{K}}^{1/2}\xi \Vert }^2\end{array}$$

for any $q>2$ and $1<q^{\prime }<2$ with $1/q+1/q^{\prime }=1$. Choosing $q=logN$, we get

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,e^{2N}{\lambda }_{\ell }& \sum _{p\in {\Lambda }_{+}^{\ast }}({\widehat{\chi }}_{\ell }\ast \eta )(p)(b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p})\xi ⟩|\hfill \\ \hfill & \le C{N}^{-1}{(logN)}^{1/2}\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{K}}^{1/2}\xi \Vert ,\hfill \end{array}\end{array}$$

and, from (164), we conclude that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \sum _{p\in {\Lambda }_{+}^{\ast }}& [p^2{\eta }_p+\frac{N}{2}\widehat{V}(p/e^N)+\frac{1}{2}\sum _{\begin{array}{c}r\in {\Lambda }^{\ast }:\\ p+r\in {\Lambda }_{+}^{\ast }\end{array}}\widehat{V}(r/e^N){\eta }_{p+r}](b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p})\hfill \\ \hfill & =\sum _{p\in {\Lambda }_{+}^{\ast }}N{e}^{2N}{\lambda }_{\ell }{\widehat{\chi }}_{\ell }(p)(b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p})+{\mathcal{E}}_4,\hfill \end{array}\end{array}$$ 165

with

$$\begin{array}{c}\hfill |⟨\xi ,{\mathcal{E}}_4\xi ⟩|\le C{N}^{-1}{(logN)}^{1/2}\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert \Vert {\mathcal{K}}^{1/2}\xi \Vert .\end{array}$$

if $\alpha >1$. Combining (162) with (163) and (165), and using the definition (39) we conclude that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{G}}_{N,\alpha }=& \frac{1}{2}{\widehat{\omega }}_N(0)(N-1)(1-\frac{{\mathcal{N}}_{+}}{N})+[N\widehat{V}(0)-\frac{1}{2}{\widehat{\omega }}_N(0)]{\mathcal{N}}_{+}(1-\frac{{\mathcal{N}}_{+}}{N})\hfill \\ \hfill & +N\sum _{p\in {\Lambda }_{+}^{\ast }}\widehat{V}(p/e^N)a_p^{\ast }{a}_p\left(1,-,\frac{{\mathcal{N}}_{+}}{N}\right)+\frac{1}{2}\sum _{p\in {\Lambda }_{+}^{\ast }}{\widehat{\omega }}_N(p)(b_p{b}_{-p}+\text{h.c.})\hfill \\ \hfill & +\sqrt{N}\sum _{p,q\in {\Lambda }_{+}^{\ast }:p+q\ne 0}\widehat{V}(p/e^N)\left[b_{p+q}^{\ast },a_{-p}^{\ast },a_q,+,,\text{h.c.},\right]\hfill \\ \hfill & +\mathcal{K}+{\mathcal{V}}_N+{\mathcal{E}}_5,\hfill \end{array}\end{array}$$ 166

with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\mathcal{E}}_5\xi ⟩|\le & C{N}^{1/2-\alpha }\Vert {\mathcal{H}}_N^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert +C{N}^{1-\alpha }{\Vert {({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2\hfill \\ \hfill & +C{N}^{-1}{(logN)}^{1/2}\Vert {\mathcal{K}}^{1/2}\xi \Vert \Vert {({\mathcal{N}}_{+}+1)}^{1/2}{\xi \Vert +C\Vert \xi \Vert }^2,\hfill \end{array}\end{array}$$

for any $\alpha >1$. Observing that $|\widehat{V}(p/e^N)-\widehat{V}(0)|\le C|p|e^{-N}$ in the second line on the r.h.s. of (166), we arrive at ${\mathcal{G}}_{N,\alpha }={\mathcal{G}}_{N,\alpha }^{\text{eff}}+{\mathcal{E}}_{\mathcal{G}}$, with ${\mathcal{G}}_{N,\alpha }^{\text{eff}}$ defined as in (42) and with ${\mathcal{E}}_{\mathcal{G}}$ that satisfies (43).

Appendix B: Properties of the Scattering Function

Let V be a potential with finite range $R_0>0$ and scattering length $\mathfrak{a}$. For a fixed $R>R_0$, we study properties of the ground state $f_R$ of the Neumann problem

$$\begin{array}{c}\hfill (-\Delta +\frac{1}{2}V(x))f_R(x)={\lambda }_R{f}_R(x)\end{array}$$ 167

on the ball $|x|\le R$, normalized so that $f_R(x)=1$ for $|x|=R$. Lemma 1, parts (i)–(iv), follows by setting $R=e^N\ell$ in the following lemma.

Lemma 7

Let $V\in L^3({\mathbb{R}}^2)$ be non-negative, compactly supported and spherically symmetric, and denote its scattering length by $\mathfrak{a}$. Fix $R>0$ sufficiently large and denote by $f_R$ the Neumann ground state of (167). Set $w_R=1-f_R$. Then we have

$$\begin{array}{c}\hfill 0\le f_R(x)\le 1\end{array}$$

Moreover, for R large enough there is a constant $C>0$ independent of R such that

$$\begin{array}{c}\hfill \left|,{\lambda }_R,-,\frac{2}{R^{2}log(R/\mathfrak{a})},\left(1,+,\frac{3}{4},\frac{1}{log(R/\mathfrak{a})}\right),\right|\le \frac{C}{R^2}\frac{1}{{log}^3(R/\mathfrak{a})}.\end{array}$$ 168

and

$$\begin{array}{c}\hfill \left|\int ,\text{d},x,,V,(x),f_R,(x),-,\frac{4\pi }{log(R/\mathfrak{a})}\right|\le \frac{C}{{log}^2(R/\mathfrak{a})}.\end{array}$$ 169

Finally, there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |w_R(x)|& \le \chi (|x|\le R_0)+C\frac{log(|x|/R)}{log(\mathfrak{a}/R)}\chi (R_0\le |x|\le R)\hfill \\ \hfill |\mathrm{\nabla }{w}_R(x)|& \le \frac{C}{log(R/\mathfrak{a})}\frac{\chi (|x|\le R)}{|x|+1}\hfill \end{array}\end{array}$$ 170

for R large enough.

To show Lemma 7 we adapt to the two dimensional case the strategy used in [8, Lemma A.1] for the three dimensional problem. We will use some well known properties of the zero energy scattering equation in two dimensions, summarized in the following lemma.

Lemma 8

Let $V\in L^3({\mathbb{R}}^2)$ non-negative, with $\text{supp}V\subset B_{R_0}(0)$ for an $R_0>0$. Let $\mathfrak{a}\le R_0$ denote the scattering length of V. For $R>R_0$, let ${\varphi }_R:{\mathbb{R}}^2\to \mathbb{R}$ be the radial solution of the zero energy scattering equation

$$\begin{array}{c}\hfill \left[-,\Delta ,+,\frac{1}{2},V\right]{\varphi }_R=0\end{array}$$ 171

normalized such that ${\varphi }_R(x)=1$ for $|x|=R$. Then

$$\begin{array}{c}\hfill {\varphi }_R(x)=\frac{log(|x|/\mathfrak{a})}{log(R/\mathfrak{a})}\end{array}$$ 172

for all $|x|>R_0$. Moreover, $|x|\to {\varphi }_R(x)$ is monotonically increasing and there exists a constant $C>0$ (depending only on V) such that

$$\begin{array}{c}\hfill {\varphi }_R(x)\ge {\varphi }_R(0)\ge \frac{C}{log(R/\mathfrak{a})}\end{array}$$ 173

for all $x\in {\mathbb{R}}^2$. Furthermore, there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill |\mathrm{\nabla }{\varphi }_R(x)|\le \frac{C}{|log(R/\mathfrak{a})|}\frac{1}{|x|+1}\end{array}$$ 174

for all $x\in {\mathbb{R}}^2$.

Proof

The existence of the solution of (171), the expression (172), the fact that ${\varphi }_R(x)\ge 0$ and the monotonicity are standard (see, for example, Theorem C.1 and Lemma C.2 in [17]). The bound (173) for ${\varphi }_R(0)$ follows from (172), comparing ${\varphi }_R(0)$ with ${\varphi }_R(x)$ at $|x|=R_0$, with Harnack’s inequality (see [24, Theorem C.1.3]). Finally, (174) follows by rewriting (171) in integral form

$$\begin{array}{c}\hfill {\varphi }_R(x)=1-\frac{1}{4\pi }{\int }_{{\mathbb{R}}^2}log(R/|x-y|)V(y){\varphi }_R(y)\text{d}y.\end{array}$$

For $|x|\le R_0$, this leads (using that ${\varphi }_R(y)\le log(R_0/\mathfrak{a})/log(R/\mathfrak{a})$ for all $|y|\le R_0$ and the local integrability of ${|.|}^{-3/2}$) to

$$\begin{array}{c}\hfill |\mathrm{\nabla }{\varphi }_R(x)|\le C\int \frac{V(y){\varphi }_R(y)}{|x-y|}dy\le \frac{{C\Vert V\Vert }_3}{log(R/\mathfrak{a})}\end{array}$$

Combining with the bound for $|x|>R_0$ obtained differentiating (172), we obtain the desired estimate. $\square$

Proof of Lemma 7

By standard arguments (see for example [17, proof of theorem C1]), $f_R(x)$ is spherically symmetric and non-negative. We now start by proving an upper bound for ${\lambda }_R$, consistent with (168). To this end, we calculate the energy of a suitable trial function. For $k\in \mathbb{R}$ we define

$$\begin{array}{c}\hfill {\psi }_k(x)=J_0(k|x|)-\frac{J_0(k\mathfrak{a})}{Y_0(k\mathfrak{a})}{Y}_0(k|x|).\end{array}$$

with $J_0$ and $Y_0$ the zero Bessel functions of first and second type, respectively. Note that

$$\begin{array}{c}\hfill -\Delta {\psi }_k(x)=k^2{\psi }_k(x).\end{array}$$

and ${\psi }_k(x)=0$ if $|x|=a$. We define $k=k(R)$ to be the smallest positive real number satisfying ${\partial }_r{\psi }_R(x)=0$ for $|x|=R$. One can check that

$$\begin{array}{c}\hfill \left|,k^2,-,\frac{2}{R^{2}log(R/\mathfrak{a})},\left(1,+,\frac{3}{4},\frac{1}{log(R/\mathfrak{a})}\right),\right|\le \frac{C}{R^2}\frac{1}{{log}^3(R/\mathfrak{a})}\end{array}$$ 175

in the limit $R\to \infty$. To prove (175), we observe that

$$\begin{array}{c}\hfill {\partial }_r{\psi }_k(x){|}_{|x|=R}=-k{J}_1(kR)+k\frac{J_0(k\mathfrak{a})}{Y_0(k\mathfrak{a})}{Y}_1(kR),\end{array}$$ 176

and we expand for $kR,k\mathfrak{a}\ll 1$ using (with $\gamma$ the Euler constant)

$$\begin{array}{c}\hfill \begin{array}{cc}& \left|J_0,(r),-,1,+,\frac{r^2}{4}\right|\le C{r}^4,\left|J_1,(r),-,\frac{r}{2},(,1,-,\frac{r^2}{8},)\right|\le C{r}^5,\hfill \\ \hfill & \left|Y_0,(r),-,\frac{2}{\pi },log,(r{e}^{\gamma }/2)\right|\le C{r}^{2}log(r),\hfill \\ \hfill & \left|Y_1,(r),+,\frac{2}{\pi },\frac{1}{r},\left(1,-,\frac{r^2}{2},(,1,-,\frac{r^2}{8},),log,(r{e}^{\gamma }/2),+,\frac{r^2}{4}\right)\right|\le C{r}^3.\hfill \end{array}\end{array}$$ 177

With (177) one finds that (176)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\partial }_r{\psi }_R& (x){|}_{|x|=R}\hfill \\ \hfill =& -\frac{1}{2kRlog(k\mathfrak{a}{e}^{\gamma }/2)}\hfill \\ \hfill & ·\left\{\frac{{(kR)}^4}{8},log,(R/\mathfrak{a}),-,{(kR)}^2,\left[log,(R/\mathfrak{a}),-,\frac{1}{2}\right],+,2,+,\mathcal{O},({(kR)}^4+{(k\mathfrak{a})}^2)\right\}\hfill \end{array}\end{array}$$ 178

The smallest solution of

$$\begin{array}{c}\hfill \frac{{(kR)}^4}{8}log(R/\mathfrak{a})-{(kR)}^2\left[log,(R/\mathfrak{a}),-,\frac{1}{2}\right]+2=0\end{array}$$

is such that

$$\begin{array}{c}\hfill {(kR)}^2=\frac{2}{log(R/\mathfrak{a})}\left[1,+,\frac{3}{4log(R/\mathfrak{a})}\right]+\mathcal{O}({log}^{-3}(R/\mathfrak{a}))\end{array}$$ 179

in the limit of large R. Inserting in (178), we find that the r.h.s. changes sign around the value of k defined in (179). By the intermediate value theorem, we conclude that there is a $k=k(R)>0$ satisfying (175), such that ${\partial }_r{\psi }_{k(R)}(x)=0$ if $|x|=R$.

Now, let ${\varphi }_R(x)$ be the solution of the zero energy scattering equation (171), with ${\varphi }_R(x)=1$ for $|x|=R$. We set

$$\begin{array}{c}\hfill {\Psi }_R(x):={\psi }_k(m_R(x))=J_0(k{m}_R(x))-\frac{J_0(k\mathfrak{a})}{Y_0(k\mathfrak{a})}{Y}_0(k{m}_R(x)),\end{array}$$ 180

with $k=k(R)$ satisfying (175) and

$$\begin{array}{c}\hfill m_R(x):=\mathfrak{a}exp(log(R/\mathfrak{a}){\varphi }_R(x)).\end{array}$$

With this choice we have $m_R(x)=|x|$ outside the range of the potential; hence ${\Psi }_R(x)={\psi }_k(x)$ for $R_0\le |x|\le R$. In particular, ${\Psi }_R$ satisfies Neumann boundary conditions at $|x|=R$.

From (172), (173) and the monotonicity of ${\varphi }_R$, we get

$$\begin{array}{c}\hfill C\mathfrak{a}\le m_R(x)\le R_0\text{for all}0\le |x|\le R_0\end{array}$$ 181

and for a constant $C>1$, independent of R. From (174) we also get

$$\begin{array}{c}\hfill |\mathrm{\nabla }{m}_R(x)|\le C\text{for all}0\le |x|\le R.\end{array}$$ 182

With the notation $\mathfrak{h}=-\Delta +\frac{1}{2}V$, we now evaluate $\langle {\Psi }_R,\mathfrak{h}{\Psi }_R\rangle$. To this end we note that

$$\begin{array}{c}\hfill \langle {\Psi }_R,\mathfrak{h}{\Psi }_R\rangle ={\int }_{|x|<R_0}\overline{{\Psi }_R(x)}(\mathfrak{h}{\Psi }_R(x))dx+k^2{\int }_{|x|\ge R_0}{|{\Psi }_R(x)|}^{2}dx.\end{array}$$ 183

Let us consider the region $|x|<R_0$. From (180) and (177) we find, first of all,

$$\begin{array}{c}\hfill \left|{\Psi }_R,(x),+,\frac{log(m_R(x)/\mathfrak{a})}{log(k\mathfrak{a}{e}^{\gamma }/2)}\right|\le C{(k{m}_R(x))}^2,\end{array}$$ 184

Next, we compute $-\Delta {\Psi }_R(x)$. With

$$\begin{array}{ccc}\hfill J_0^{\prime }(r)& =-J_1(r)\hfill & \hfill J_1^{\prime }(r)=\frac{1}{2}(J_0(r)-J_2(r))\\ \hfill Y_0^{\prime }(r)& =-Y_1(r)\hfill & \hfill Y_1^{\prime }(r)=\frac{1}{2}(Y_0(r)-Y_2(r)).\end{array}$$

we obtain (here, we use the notation $m_R^{\prime }$ and $m_R^{\prime \prime }$ for the radial derivatives of the radial function $m_R$)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill -\Delta {\Psi }_R(x)=& -{\partial }_r^2{\Psi }_R(x)-\frac{1}{|x|}{\partial }_r{\Psi }_R(x)\hfill \\ \hfill =& -k{m}_R^{\prime \prime }(x)[-J_1(k{m}_R(x))+\frac{J_0(k\mathfrak{a})}{Y_0(k\mathfrak{a})}{Y}_1(k{m}_R(x))]\hfill \\ \hfill & -\frac{1}{2}{k}^2(m_R^{\prime }(x){)}^2[J_2(k{m}_R(x))-\frac{J_0(k\mathfrak{a})}{Y_0(k\mathfrak{a})}{Y}_2(k{m}_R(x))]\hfill \\ \hfill & -\frac{1}{2}{k}^2(m_R^{\prime }(x){)}^2[-J_0(k{m}_R(x))+\frac{J_0(k\mathfrak{a})}{Y_0(k\mathfrak{a})}{Y}_0(k{m}_R(x))]\hfill \\ \hfill & -\frac{k{m}_R^{\prime }(x)}{|x|}[-J_1(k{m}_R(x))+\frac{J_0(k\mathfrak{a})}{Y_0(k\mathfrak{a})}{Y}_1(k{m}_R(x))].\hfill \end{array}\end{array}$$

We note that, using the scattering equation (171),

$$\begin{array}{c}\hfill m_R^{\prime \prime }-\frac{{(m_R^{\prime })}^2}{m_R}+\frac{1}{|x|}{m}_R^{\prime }=\frac{1}{2}V{m}_R{\varphi }_{R}log(R/\mathfrak{a})=\frac{1}{2}V{m}_{R}log(m_R/\mathfrak{a}).\end{array}$$ 185

Now we write

$$\begin{array}{c}\hfill \begin{array}{cc}& -\Delta {\Psi }_R(x)\hfill \\ \hfill & =[-k(m_R^{\prime \prime }(x)+\frac{m_R^{\prime }(x)}{|x|})Y_1(k{m}_R(x))+\frac{k^2}{2}{(m_R^{\prime }(x))}^2{Y}_2(k{m}_R(x))]\frac{J_0(k\mathfrak{a})}{Y_0(k\mathfrak{a})}+g_R(x)\hfill \end{array}\end{array}$$ 186

where $g_R(x)={\sum }_{i=1}^3{g}_R^{(i)}(x)$ with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill g_R^{(1)}(x)& =k(m_R^{\prime \prime }(x)+\frac{m_R^{\prime }(x)}{|x|})J_1(k{m}_R(x))\hfill \\ \hfill g_R^{(2)}(x)& =-\frac{1}{2}{k}^2{(m_R^{\prime }(x))}^2{J}_2(k{m}_R(x))\hfill \\ \hfill g_R^{(3)}(x)& =-\frac{1}{2}{k}^2{(m_R^{\prime }(x))}^2(-J_0(k{m}_R(x)+\frac{J_0(k\mathfrak{a})}{Y_0(k\mathfrak{a})}{Y}_0(k{m}_R(x)))=\frac{k^2}{2}{(m_R^{\prime }(x))}^2{\Psi }_R(x).\hfill \end{array}\end{array}$$

With (185), (177) and (181), (182), we find

$$\begin{array}{c}\hfill |g_R^{(1)}(x)|\le C{k}^2({(m_R^{\prime }(x))}^2+\frac{1}{2}V(x)m_R^2(x)log(m_R(x)/\mathfrak{a}))\le C{k}^2(1+V(x)).\end{array}$$

Next, with $|J_2(r)-r^2/8|\le C{r}^4$ we get

$$\begin{array}{c}\hfill |g_R^{(2)}(x)|\le C{k}^4{(m_R^{\prime }(x))}^2{(m_R(x))}^2\le C{k}^4.\end{array}$$

With (184), we can also bound

$$\begin{array}{c}\hfill |g_R^{(3)}(x)|\le C{k}^2{(m_R^{\prime }(x))}^2\frac{log(m_R(x)/\mathfrak{a})}{log(k\mathfrak{a})}\le C{k}^2{log}^{-1}(k\mathfrak{a}).\end{array}$$

We conclude that $|g_R(r)|\le C(1+V(x))k^2$ for all $r\le R_0$ and R large enough. Finally, using Eq. (185), the expansion for $Y_1(r)$ in Eq. (177), and the bound

$$\begin{array}{c}\hfill |Y_2(r)+\frac{4}{\pi }\frac{1}{r^2}|\le C,\end{array}$$

we can rewrite the first term on the r.h.s. of (186) as

$$\begin{array}{c}\hfill \begin{array}{cc}& [-k(m_R^{\prime \prime }(x)+\frac{m_R^{\prime }(x)}{|x|})Y_1(k{m}_R(x))+\frac{k^2}{2}{(m_R^{\prime }(x))}^2{Y}_2(k{m}_R(x))]\frac{J_0(k\mathfrak{a})}{Y_0(k\mathfrak{a})}\hfill \\ \hfill & =\frac{1}{\pi }V(x)log(m_R(x)/\mathfrak{a})\frac{J_0(k\mathfrak{a})}{Y_0(k\mathfrak{a})}+h_R(x)\hfill \end{array}\end{array}$$ 187

with $|h_R(x)|\le C(1+V(x))k^2$ for all $r\le R_0$, R large enough. With the identities (186) and (187) we obtain

$$\begin{array}{c}\hfill |-\Delta {\Psi }_R(x)-\frac{1}{\pi }\frac{J_0(k\mathfrak{a})}{Y_0(k\mathfrak{a})}V(x)log(m_R(x)/\mathfrak{a})|\le C(1+V(x))k^2,\end{array}$$

for all $|x|\le R_0$ and for R sufficiently large. With (184), we conclude that, for $0\le |x|\le R_0$,

$$\begin{array}{c}\hfill |(-\Delta +\frac{1}{2}V){\Psi }_R(x)|\le C(1+V(x))k^2.\end{array}$$ 188

With (183), (188) and the upper bound

$$\begin{array}{c}\hfill |{\Psi }_R(r)|\le \frac{C}{|log(k\mathfrak{a})|}\end{array}$$ 189

for all $|x|\le R_0$ (which follows from (184) and (181)), we get

$$\begin{array}{c}\hfill \langle {\Psi }_R,\mathfrak{h}{\Psi }_R\rangle \le k^2\langle {\Psi }_R,{\Psi }_R\rangle +\frac{C{k}^2}{|log(k\mathfrak{a})|}{\int }_{|x|\le R_0}(1+V(x))\text{d}x.\end{array}$$

On the other hand, Eq.(184), together with $m_R(x)=|x|$ for $|x|\ge R_0$, implies the lower bound

$$\begin{array}{c}\hfill \langle {\Psi }_R,{\Psi }_R\rangle \ge {\int }_{R_0\le |x|\le R}|{\Psi }_R{(x)|}^2\text{d}x\ge \frac{C}{{|log(k\mathfrak{a})|}^2}{\int }_{R_0\le |x|\le R}{log}^2(|x|/\mathfrak{a})\text{d}x\ge C{R}^2.\end{array}$$

Hence, with (175), we conclude that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\lambda }_R\le \frac{\langle {\Psi }_R,\mathfrak{h}{\Psi }_R\rangle }{\langle {\Psi }_R,{\Psi }_R\rangle }& \le k^2\left(1,+,\frac{C|log(k\mathfrak{a})|}{R^2}\right)\hfill \\ \hfill & \le \frac{2}{R^{2}log(R/\mathfrak{a})}\left(1,+,\frac{3}{4},\frac{1}{log(R/\mathfrak{a})},+,\frac{C}{{log}^2(R/\mathfrak{a})}\right)\hfill \end{array}\end{array}$$ 190

in agreement with (168).

To prove the lower bound for ${\lambda }_R$ it is convenient to show some upper and lower bounds for $f_R$. We start by considering $f_R$ outside the range of the potential. We denote ${\epsilon }_R=\sqrt{{\lambda }_R}R$. Keeping into account the boundary conditions at $|x|=R$, we find, for $R_0\le |x|\le R$,

$$\begin{array}{c}\hfill f_R(x)=A_R{J}_0({\epsilon }_R|x|/R)+B_R{Y}_0({\epsilon }_R|x|/R),\end{array}$$

with

$$\begin{array}{c}\hfill A_R={\left(J_0,({\epsilon }_R),-,J_1,({\epsilon }_R),,\frac{Y_0({\epsilon }_R)}{Y_1({\epsilon }_R)}\right)}^{-1},\end{array}$$

and

$$\begin{array}{c}\hfill B_R={\left(Y_0,({\epsilon }_R),-,\frac{J_0({\epsilon }_R)}{J_1({\epsilon }_R)},Y_1,({\epsilon }_R)\right)}^{-1}.\end{array}$$

From (190), we have $|{\epsilon }_R{|\le C|log(R/\mathfrak{a})|}^{-1/2}$. Thus, we can expand $f_R$ for large R, using (177) and, for $Y_0$, the improved bound

$$\begin{array}{c}\hfill \left|Y_0,(r),-,\frac{2}{\pi },log,(r{e}^{\gamma }/2),\left(1,-,\frac{1}{4},r^2\right)\right|\le C{r}^2,\end{array}$$

we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |A_R-1+\frac{{\epsilon }_R^2}{4}(2log({\epsilon }_R{e}^{\gamma }/2)-1)|& \le C{\epsilon }_R^4{(log{\epsilon }_R)}^2,\hfill \\ \hfill |B_R-\frac{\pi }{4}{\epsilon }_R^2\left(1,-,\frac{{\epsilon }_R^2}{8}\right)|& \le C{\epsilon }_R^6.\hfill \end{array}\end{array}$$ 191

which leads to

$$\begin{array}{c}\hfill \begin{array}{cc}& \left|f_R,(x),-,1,+,\frac{{\epsilon }_R^2}{4},\left(2,log,(R/|x|),-,1,+,\frac{x^2}{R^2}\right),-,\frac{{\epsilon }_R^4}{16},log,(R/|x|),\left(1,+,\frac{2x^2}{R^2}\right)\right|\hfill \\ \hfill & \le C{\epsilon }_R^4{(log{\epsilon }_R)}^2.\hfill \end{array}\end{array}$$ 192

We can also compute the radial derivative

$$\begin{array}{c}\hfill {\partial }_r{f}_R(x)=-\frac{{\epsilon }_R}{R}(A_R{J}_1({\epsilon }_{R}r/R)+B_R{Y}_1({\epsilon }_{R}r/R)).\end{array}$$

With the expansions (177) and (191) we conclude that for all $R_0\le |x|<R$ we have

$$\begin{array}{c}\hfill \left|{\partial }_r,f_R,(x),-,\frac{{\epsilon }_R^2}{2|x|},\left(1,-,\frac{x^2}{R^2},+,\frac{{\epsilon }_R^2{x}^2}{2R^2},log,(R/|x|)\right)\right|\le C{\epsilon }_R^{4}log{\epsilon }_R.\end{array}$$ 193

The bound (193) shows that ${\partial }_r{f}_R(x)$ is positive, for, say, $R_0<|x|<R/2$. Since ${\partial }_r{f}_R(x)$ must have its first zero at $|x|=R$, we conclude that $f_R$ is increasing in |x|, on $R_0\le |x|\le R$. From the normalization $f_R(x)=1$, for $|x|=R$, we conclude therefore that $f_R(x)\le 1$, for all $R_0\le |x|\le R$.

From (192) and (190) we obtain, on the other hand, the lower bound

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill f_R(x)& \ge 1-\frac{{\epsilon }_R^2}{2}log(R/|x|)-C{\epsilon }_R^4{(log{\epsilon }_R)}^2\hfill \\ \hfill & \ge 1-\frac{log(R/|x|)}{log(R/\mathfrak{a})}\left(1,+,\frac{3}{4},\frac{1}{log(R/\mathfrak{a})},+,\frac{C}{{log}^2(R/\mathfrak{a})}\right)-C\frac{{(loglog(R/\mathfrak{a}))}^2}{{log}^2(R/\mathfrak{a})}\hfill \\ \hfill & \ge \frac{log(|x|/\mathfrak{a})}{log(R/\mathfrak{a})}-\frac{3}{4}\frac{log(R/|x|)}{{log}^2(R/\mathfrak{a})}-C\frac{log(R/|x|)}{{log}^3(R/a)}-C\frac{{(loglog(R/\mathfrak{a}))}^2}{{log}^2(R/\mathfrak{a})},\hfill \end{array}\end{array}$$ 194

for R sufficiently large. Let $R_{\ast }=max\{R_0,e\mathfrak{a}\}$. Then Eq. (194) implies in particular that, for R large enough,

$$\begin{array}{c}\hfill f_R(x)\ge \frac{C}{log(R/\mathfrak{a})}.\end{array}$$ 195

for all $R_{\ast }<|x|\le R$.

Finally, we show that $f_R(x)\le 1$ also for $|x|\le R_0$. First of all, we observe that, by elliptic regularity, as stated for example in [12, Theorem 11.7, part iv)], there exists $0<\alpha <1$ and $C>0$ such that

$$\begin{array}{c}\hfill \left|f_R,(x),-,f_R,(y)\right|\le C\Vert (V-2{\lambda }_R)f_R{\Vert }_2{|x-y|}^{\alpha }\end{array}$$

With $\Vert V{f}_R{\Vert }_2\le {\Vert V\Vert }_3\Vert f_R{\Vert }_6\le C\Vert f_R{\Vert }_{H^1}\le C(1+{\lambda }_R){\Vert f_R\Vert }_2$, we conclude that $0\le f_R(x)\le 1+C{\Vert f\Vert }_2$ for all $|x|\le R_0$ (because we know that $f_R(x)\le 1$ for $R_0\le |x|\le R$). To improve this bound, we go back to the differential equation (167), to estimate

$$\begin{array}{c}\hfill \Delta f_R=\frac{1}{2}V{f}_R-{\lambda }_R{f}_R\ge -{\lambda }_R{(1+C\Vert f\Vert }_2)\end{array}$$ 196

This implies that $f_R(x)+{\lambda }_R{(1+C\Vert f\Vert }_2)x^2/2$ is subharmonic. Using (192), we find $f_R(x)\le 1-C{\epsilon }_R^2$ for $|x|=R_0$. From the maximum principle, we obtain therefore that

$$\begin{array}{c}\hfill f_R(x)\le 1-C{\epsilon }_R^2+C{\lambda }_R(1+C\Vert f_R{\Vert }_2)\end{array}$$ 197

for all $|x|\le R_0$. In particular, this implies that $\Vert f_R{\mathbf{1}}_{|x|\le R_0}{\Vert }_2\le C+C{\lambda }_R{\Vert f_R\Vert }_2$, and therefore that

$$\begin{array}{c}\hfill \Vert f_R{\mathbf{1}}_{R_0\le |x|\le R}{\Vert }_2\ge {\Vert f_R\Vert }_2(1-C{\lambda }_R)-C\end{array}$$

With $f_R(x)\le 1$ for $R_0\le |x|\le R$, we find, on the other hand, that $\Vert f_R{\mathbf{1}}_{R_0\le |x|\le R}{\Vert }_2\le CR$. We conclude therefore that $\Vert f_R{\Vert }_2\le CR$ and, from (197), that $f_R(x)\le 1-C{\epsilon }_R^2+C/R\le 1$, for all $|x|\le R_0$, if R is large enough.

We are now ready to prove the lower bound for ${\lambda }_R$. We use now that any function $\Phi$ satisfying Neumann boundary conditions at $|x|=R$ can be written as $\Phi (x)=q(x){\Psi }_R(x)$, with ${\Psi }_R(x)$ the trial function used for the upper bound and $q>0$ a function that satisfies Neumann boundary condition at $|x|=R$ as well. This is in particular true for the solution $f_R(x)$ of (167). In the following we write

$$\begin{array}{c}\hfill f_R(x)=q_R(x){\Psi }_R(x)\end{array}$$

where $q_R$ satisfies Neumann boundary conditions at $|x|=R$. From (184), we find $|{\Psi }_R(x)|\ge C/log(ka)$. The bound $f_R(x)\le 1$ implies therefore that there exists $c>0$ such that

$$\begin{array}{c}\hfill q_R(x)\le Clog(k\mathfrak{a})\forall |x|\le R_0.\end{array}$$ 198

From the identity

$$\begin{array}{c}\hfill \mathfrak{h}{f}_R=(\mathfrak{h}{\Psi }_R)q_R-(\Delta q_R){\Psi }_R-2\mathrm{\nabla }{q}_R\mathrm{\nabla }{\Psi }_R\end{array}$$

we have

$$\begin{array}{c}\hfill {\int }_{|x|\le R}\text{d}x{f}_R\mathfrak{h}{f}_R={\int }_{|x|\le R}\text{d}x|\mathrm{\nabla }{q}_R{|}^2{\Psi }_R^2+{\int }_{|x|\le R}\text{d}x{|q_R|}^2{\Psi }_R\mathfrak{h}{\Psi }_R.\end{array}$$

From (188) and (189), we have

$$\begin{array}{c}\hfill \left|{\Psi }_R,(x),(\mathfrak{h}{\Psi }_R),(x),-,k^2,{\Psi }_R^2,(x)\right|\le C\frac{k^2}{|logka|}(1+V(x))\chi (|x|\le R_0).\end{array}$$

Hence

$$\begin{array}{c}\hfill {\int }_{|x|\le R}\text{d}x{f}_R\mathfrak{h}{f}_R\ge k^2\Vert f_R{\Vert }^2-\frac{C{k}^2}{|logk|}{\int }_{|x|\le R_0}\text{d}x(1+V(x)){|q_R(x)|}^2.\end{array}$$ 199

With (198), we obtain

$$\begin{array}{c}\hfill {\int }_{|x|\le R}\text{d}x{f}_R\mathfrak{h}{f}_R\ge k^2{\Vert f_R\Vert }^2-C{k}^{2}log(k\mathfrak{a}).\end{array}$$

With (195) (recalling that $R_{\ast }=max\{R_0,e\mathfrak{a}\}$), we bound

$$\begin{array}{c}\hfill \Vert f_R{\Vert }^2\ge {\int }_{R_{\ast }\le |x|\le R}{|f_R(x)|}^2\text{d}x\ge \frac{C{R}^2}{{log}^2(R/\mathfrak{a})}\end{array}$$

and, inserting in (199), we conclude that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\lambda }_R=\frac{\langle f_R,\mathfrak{h}{f}_R\rangle }{\langle f_R,f_R\rangle }& \ge k^2\left(1,-,\frac{C{log}^3(R/\mathfrak{a})}{R^2}\right)\hfill \\ \hfill & \ge \frac{2}{R^{2}log(R/\mathfrak{a})}\left(1,+,\frac{3}{4},\frac{1}{log(R/\mathfrak{a})},-,,\frac{C}{{log}^2(R/\mathfrak{a})}\right),\hfill \end{array}\end{array}$$

where in the last inequality we used (175).

To prove (169) we use the scattering equation (167) to write

$$\begin{array}{c}\hfill \int \text{d}xV(x)f_R(x)=2{\int }_{|x|\le R}\text{d}x\Delta f_R(x)+2{\int }_{|x|\le R}\text{d}x{\lambda }_R{f}_R(x).\end{array}$$

Passing to polar coordinates, and using that $\Delta f_R(x)={|x|}^{-1}{\partial }_r|x|{\partial }_r{f}_R(x)$, we find that the first term vanishes. Hence

$$\begin{array}{c}\hfill \int \text{d}xV(x)f_R(x)=2{\lambda }_R\int \text{d}x{f}_R(x).\end{array}$$

With the upper bound $f_R(r)\le 1$ and with (168), we find

$$\begin{array}{c}\hfill \int \text{d}xV(x)f_R(x)\le 2\pi R^2{\lambda }_R\le \frac{4\pi }{log(R/\mathfrak{a})}\left(1,+,\frac{C}{log(R/\mathfrak{a})}\right).\end{array}$$

To obtain a lower bound for the same integral we use that $f_R(r)\ge 0$ inside the range of the potential. Outside the range of V, we use (192). We find

$$\begin{array}{c}\hfill \int \text{d}xV(x)f_R(x)\ge 4\pi {\lambda }_R{\int }_{R_0}^R\text{d}rr(1-C{\epsilon }_R^{2}log(R/r))\ge \frac{4\pi }{log(R/\mathfrak{a})}\left(1,-,\frac{C}{log(R/\mathfrak{a})}\right)\end{array}$$

We conclude that

$$\begin{array}{c}\hfill \left|\int ,\text{d},x,,V,(x),f_R,(x),-,\frac{4\pi }{log(R/\mathfrak{a})}\right|\le \frac{C}{{log}^2(R/\mathfrak{a})}.\end{array}$$

Finally, we show the bounds in (170). For $r\in [R_0,R]$, from (192) we have

$$\begin{array}{c}\hfill \left|,w_R,(x),-,\frac{log(R/|x|)}{log(R/\mathfrak{a})},\right|\le \frac{C}{log(R/\mathfrak{a})}.\end{array}$$ 200

As for the derivative of $w_R$ we use (193) to compute

$$\begin{array}{c}\hfill \left|{\partial }_r,f_R,(x)\right|\le \frac{C}{|x|}\frac{1}{log(R/\mathfrak{a})}.\end{array}$$

Moreover ${\partial }_r{f}_R(x)=0$ if $|x|=R$, by construction.

On the other hand, if $|x|\le R_0$, we have $w_R(x)=1-f_R(x)\le 1$. As for the derivative, we define ${\tilde{f}}_R$ on ${\mathbb{R}}_{+}$ through ${\tilde{f}}_R(r)=f_R(x)$, if $|x|=r$, and we use the representation

$$\begin{array}{c}\hfill {\tilde{f}}_R^{\prime }(r)=\frac{1}{r}{\int }_0^r\text{d}s({\tilde{f}}_R^{\prime \prime }(s)s+{\tilde{f}}_R^{\prime }(s)).\end{array}$$

With (167), we have (with $\tilde{V}$ defined on ${\mathbb{R}}_{+}$ through $V(x)=\tilde{V}(r)$, if $|x|=r$)

$$\begin{array}{c}\hfill {\tilde{f}}_R^{\prime \prime }(r)+\frac{1}{r}{\tilde{f}}_R^{\prime }(r)={\lambda }_R{\tilde{f}}_R(r)-\frac{1}{2}\tilde{V}(r){\tilde{f}}_R(r),\end{array}$$

By (200), we can estimate ${\tilde{f}}_R(R_0)\le C/log(R/\mathfrak{a})$. From (196), we also recall that

$$\begin{array}{c}\hfill {\tilde{f}}_R(r)\le {\tilde{f}}_R(R_0)+CR{\lambda }_R\le C/log(R/\mathfrak{a})\end{array}$$

for any $r<R_0$. We conclude therefore that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |{\tilde{f}}_R^{\prime }(r)|& =|\frac{1}{r}{\int }_0^r\text{d}ss({\lambda }_R{\tilde{f}}_R(s)-\frac{1}{2}\tilde{V}(s){\tilde{f}}_R(s))|\hfill \\ \hfill & \le \frac{{\lambda }_R}{r}{\int }_0^{r}rdr+\frac{C}{rlog(R/\mathfrak{a})}{\int }_0^{r}drr\tilde{V}(r)\hfill \\ \hfill & \le \frac{C}{log(R/\mathfrak{a})}+C{\Vert V\Vert }_2\frac{log(R_0/\mathfrak{a})}{log(R/\mathfrak{a})}\le \frac{C}{log(R/\mathfrak{a})}.\hfill \end{array}\end{array}$$

$\square$

Funding

Open Access funding provided by University Zurich.