Bose–Einstein Condensation Beyond the Gross–Pitaevskii Regime

Abstract

We consider N bosons in a box with volume one, interacting through a two-body potential with scattering length of the order $N^{-1+\kappa }$, for $\kappa >0$. Assuming that $\kappa \in (0;1/43)$, we show that low-energy states exhibit Bose–Einstein condensation and we provide bounds on the expectation and on higher moments of the number of excitations.

Introduction

We consider systems of $N\in \mathbb{N}$ bosons trapped in the box $\mathrm{\Lambda }={[0;1]}^3$ with periodic boundary conditions (the three-dimensional torus with volume one) and interacting through a repulsive potential with scattering length of the order $N^{-1+\kappa }$, for $\kappa \in (0;1/43)$. We are interested in the limit of large N. The Hamilton operator has the form

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

and acts on a dense subspace of $L_s^2({\mathrm{\Lambda }}^N)$, the Hilbert space consisting of functions in $L^2({\mathrm{\Lambda }}^N)$ that are invariant with respect to permutations of the $N\in \mathbb{N}$ particles. Here, we assume the interaction potential $V\in L^3({\mathbb{R}}^3)$ to have compact support and to be nonnegative, ie. $V(x)\ge 0$ for almost all $x\in {\mathbb{R}}^3$.

For $\kappa =0$, the Hamilton operator (1.1) describes bosons in the so-called Gross–Pitaevskii limit. This regime is frequently used to model trapped Bose gases observed in recent experiments. Another important regime is the thermodynamic limit, where N bosons interacting through a fixed potential V (independent of N) are trapped in the box ${\mathrm{\Lambda }}_L={[0;L]}^3$ and where the limits $N,L\to \infty$ are taken, keeping the density $\rho =N/L^3$ fixed. After rescaling lengths (introducing new coordinates $x^{\prime }=x/L$), the Hamilton operator of the Bose gas in the thermodynamic limit is given (up to a multiplicative constant) by (1.1), with $\kappa =2/3$. Choosing $0<\kappa <2/3$, we are interpolating therefore between the Gross–Pitaevskii and the thermodynamic limits.

The goal of this paper is to show that low-energy states of (1.1) exhibit Bose–Einstein condensation in the zero-momentum mode ${\phi }_0\in L^2(\mathrm{\Lambda })$ defined by ${\phi }_0(x)=1$ for all $x\in \mathrm{\Lambda }$ and to give bounds on the number of excitations of the condensate. To achieve this goal, it is convenient to switch to an equivalent representation of the bosonic system, removing the condensate and focusing instead on its orthogonal excitations. To this end, we notice that every ${\psi }_N\in L_s^2({\mathrm{\Lambda }}^N)$ can be uniquely decomposed as

$$\begin{array}{c}\hfill {\psi }_N={\alpha }_0{\phi }_0^{\otimes N}+{\alpha }_1{\otimes }_s{\phi }_0^{\otimes (N-1)}+{\alpha }_2{\otimes }_s{\phi }_0^{\otimes (N-2)}+\cdots +{\alpha }_N\end{array}$$

where ${\otimes }_s$ denotes the symmetric tensor product and ${\alpha }_j\in L_{\perp }^2{(\mathrm{\Lambda })}^{{\otimes }_{s}j}$ for all $j=0,\dots ,N$, with $L_{\perp }^2(\mathrm{\Lambda })$ the orthogonal complement in $L^2(\mathrm{\Lambda })$ of ${\phi }_0$. This observation allows us to define a unitary map $U_N:L_s^2({\mathrm{\Lambda }}^N)\to {\mathcal{F}}_{+}^{\le N}={⨁}_{j=0}^N{L}_{\perp }^2{(\mathrm{\Lambda })}^{{\otimes }_{s}j}$ by setting

$$\begin{array}{c}\hfill U_N{\psi }_N=\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_N\}.\end{array}$$ 1.2

The truncated Fock space ${\mathcal{F}}_{+}^{\le N}={⨁}_{j=0}^N{L}_{\perp }^2{(\mathrm{\Lambda })}^{{\otimes }_{s}j}$ is used to describe orthogonal excitations of the condensate (some properties of the map $U_N$ will be discussed in Sect. 2 below). On ${\mathcal{F}}_{+}^{\le N}$, we introduce the number of particles operator, defining ${({\mathcal{N}}_{+}\xi )}^{(n)}=n{\xi }^{(n)}$ for every $\xi =\{{\xi }^{(0)},\dots {\xi }^{(N)}\}\in {\mathcal{F}}_{+}^{\le N}$.

We are now ready to state our main theorem, which provides estimates of the expectation and on higher moments of the number of orthogonal excitations of the Bose–Einstein condensate for low-energy states of (1.1).

Theorem 1.1

Let $V\in L^3({\mathbb{R}}^3)$ be pointwise nonnegative and spherically symmetric. Let ${\mathfrak{a}}_0>0$ denote the scattering length of V. Let $H_N$ be defined as in (1.1) with $0<\kappa <1/43$. Then, for every $\epsilon >0$, there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill |E_N-4\pi {\mathfrak{a}}_0{N}^{1+\kappa }|\le C{N}^{43\kappa +\epsilon }.\end{array}$$ 1.3

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

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

$$\begin{array}{c}\hfill ⟨{\psi }_N,{(H_N-E_N)}^2{\psi }_N⟩\le {\zeta }^2,\end{array}$$ 1.4

for a $\zeta >0$. Then, for every $\epsilon >0$ there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill ⟨U_N{\psi }_N,{\mathcal{N}}_{+}{U}_N{\psi }_N⟩\le C\left[\zeta ,+,{\zeta }^2,N^{13\kappa +\epsilon -1},+,N^{43\kappa +4\epsilon }\right]\end{array}$$ 1.5

for all $N\in \mathbb{N}$ large enough. If moreover ${\psi }_N=\chi (H_N\le E_N+\zeta ){\psi }_N$, then for all $k\in \mathbb{N}$ and all $\epsilon >0$ there exists $C>0$ such that

$$\begin{array}{c}\hfill ⟨U_N{\psi }_N,{\mathcal{N}}_{+}^k{U}_N{\psi }_N⟩\le C{\left[N^{20\kappa +\epsilon },{\zeta }^2,+,N^{44\kappa +2\epsilon }\right]}^k\end{array}$$ 1.6

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

The convergence $E_N/4\pi {\mathfrak{a}}_0{N}^{1+\kappa }\to 1$, as $N\to \infty$, has been first established, for Bose gases trapped by an external potential, in [19] (the choice $\kappa >0$ corresponds, in the terminology of [19], to the Thomas–Fermi limit).

It follows from (1.5) that the one-particle density matrix ${\gamma }_N={\mathrm{tr}}_{2,\dots ,N}|{\psi }_N⟩⟨{\psi }_N|$ associated with a normalized ${\psi }_N\in L_s^2({\mathrm{\Lambda }}^N)$ satisfying (1.4) is such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill 1-⟨{\phi }_0,{\gamma }_N{\phi }_0⟩=& \frac{1}{N}\left[N,-,⟨,{\psi }_N,,,a^{\ast },({\phi }_0),a,({\phi }_0),{\psi }_N,⟩\right]\hfill \\ \hfill =& \frac{1}{N}⟨U_N{\psi }_N,{\mathcal{N}}_{+}{U}_N{\psi }_N⟩\hfill \\ \hfill \le & C\left[\zeta ,N^{-1},+,{\zeta }^2,N^{13\kappa +\epsilon -2},+,N^{43\kappa +4\epsilon -1}\right]\hfill \end{array}\end{array}$$ 1.7

as $N\to \infty$. Here, we used the formula $U_N{a}^{\ast }({\phi }_0)a({\phi }_0)U_N=N-{\mathcal{N}}_{+}$; see (2.5). Equation (1.7) implies that low-energy states of (1.1) exhibit complete Bose–Einstein condensation, if $\kappa <1/43$.

We remark that the estimate (1.6) follows, in our analysis, from a stronger bound controlling not only the number but also the energy of the excitations of the condensate. As we will explain in Sect. 3, in order to estimate the energy of excitations in low-energy states, we first need to remove (at least part of) their correlations. If we choose, as we do in (1.6), ${\psi }_N\in L_s^2({\mathrm{\Lambda }}^N)$ with $\Vert {\psi }_N\Vert =1$ and ${\psi }_N=\chi (H_N\le E_N+\zeta ){\psi }_N$, we can introduce the corresponding renormalized excitation vector ${\xi }_N=e^B{U}_N{\psi }_N\in {\mathcal{F}}_{+}^{\le N}$, with the antisymmetric operator B defined as in (3.21) (the unitary operator $e^B$ will be referred to as a generalized Bogoliubov transformation). We will show in Sect. 6 that for every $k\in \mathbb{N}$, there exists $C>0$ such that

$$\begin{array}{c}\hfill ⟨{\xi }_N,({\mathcal{H}}_N+1){({\mathcal{N}}_{+}+1)}^{2k}{\xi }_N⟩\le C{\left[N^{20\kappa +\epsilon },{\zeta }^2,+,N^{44\kappa +2\epsilon }\right]}^{2k+1}\end{array}$$ 1.8

for all N large enough. Here ${\mathcal{H}}_N=\mathcal{K}+{\mathcal{V}}_N$, where

$$\begin{array}{c}\hfill \mathcal{K}=\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{a}_p^{\ast }{a}_p,\text{and}{\mathcal{V}}_N=\frac{1}{2N}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast },r\in {\mathrm{\Lambda }}^{\ast }:\\ r\ne -p,-q\end{array}}{N}^{\kappa }\widehat{V}(r/N^{1-\kappa })a_{p+r}^{\ast }{a}_q^{\ast }{a}_{q+r}{a}_p\end{array}$$ 1.9

are the kinetic and potential energy operators, restricted to ${\mathcal{F}}_{+}^{\le N}$. (Here, $\widehat{V}$ is the Fourier transform of the potential V, defined as in (2.4).) Equation (1.6) follows then from (1.8), because ${\mathcal{N}}_{+}$ commutes with ${\mathcal{H}}_N$, ${\mathcal{N}}_{+}\le \mathcal{K}\le {\mathcal{H}}_N$ and because conjugation with the generalized Bogoliubov transformation $e^B$ does not change the number of particles substantially; see Lemma 3.2 (for $k\in \mathbb{N}$ even, we also use simple interpolation).

In the Gross–Pitaevskii regime corresponding to $\kappa =0$ the convergence ${\gamma }_N\to |{\phi }_0⟩⟨{\phi }_0|$ has been first established in [16–18] and later, using a different approach, in [21].1 In this case (ie. $\kappa =0$), the bounds (1.3), (1.5) and (1.6) with $\epsilon =0$ (which are optimal in their N-dependence) have been shown in [4]. Previously, they have been established in [2], under the additional assumption of small potential. A simpler proof of the results of [2], extended also to systems of bosons trapped by an external potential, has been recently given in [20]. The result of [4] was used in [3] to determine the second order corrections to the ground state energy and the low-energy excitation spectrum of the Bose gas in the Gross–Pitaevskii regime. Note that our approach in the present paper could be easily extended to the case $\kappa =0$, leading to the same bounds obtained in [4]. We exclude the case $\kappa =0$ because we would have to modify certain definitions, making the notation more complicated (for example, the sets $P_H$ in (3.14) and $P_L$ in (4.2) would have to be defined in terms of cutoffs independent of N).

The methods of [16–18] can also be extended to show Bose–Einstein condensation for low-energy states of (1.1), for some $\kappa >0$. In fact, following the proof of [18, Theorem 5.1], it is possible to show that for a normalized ${\psi }_N\in L_s^2({\mathrm{\Lambda }}^N)$ with $\Vert {\psi }_N\Vert =1$ and such that $⟨{\psi }_N,H_N{\psi }_N⟩\le E_N+\zeta$, the expectation of the number of excitations is bounded by

$$\begin{array}{c}\hfill ⟨U_N{\psi }_N,{\mathcal{N}}_{+}{U}_N{\psi }_N⟩\le C\left[N^{\frac{15+20\kappa }{17}},+,\zeta \right]\end{array}$$ 1.10

which implies complete Bose–Einstein condensation for low-energy states, for all $\kappa <1/10$. For sufficiently small $\kappa >0$, Theorem 1.1 improves (1.10) because it gives a better rate2 (if $\kappa <15/711$) and because, through (1.6), it also provides (under stronger conditions on ${\psi }_N$) bounds for higher moments of the number of excitations ${\mathcal{N}}_{+}$.

In [10], in a slightly different setting, the authors obtain a bound of the form (1.6) for $k=1$, for the choice $\kappa =1/(55+1/3)$ (for normalized ${\psi }_N\in L_s^2({\mathrm{\Lambda }}^N)$ that satisfy $⟨{\psi }_N,H_N{\psi }_N⟩\le E_N+\zeta$). They use this result to show a lower bound on the ground state energy of the dilute Bose gas in the thermodynamic limit matching the prediction of Lee–Yang and Lee–Huang–Yang [13, 14].

After completion of our work, two more papers have appeared whose results are related with Theorem 1.1. Based on localization arguments from [8, 10], a bound for the expectation of ${\mathcal{N}}_{+}$ in low-energy states has been shown in [9], establishing Bose–Einstein condensation for all $\kappa <2/5$ (as pointed out there, using a refined analysis similar to that of [10], the range of $\kappa$ can be slightly improved). On the other hand, following an approach similar to [2], but with substantial simplifications (partly due to the fact that the author works in the grand canonical, rather than the canonical, ensemble), a new proof of Bose–Einstein condensation was obtained in [11], in the Gross–Pitaevskii regime, under the assumption of small potential. There is hope that the approach of [11] can be extended beyond the Gross–Pitaevskii regime, providing a simplified proof of Theorem 1.1, potentially allowing for larger values of $\kappa$.

The derivation of the bounds (1.5), (1.6), (1.8) is crucial to resolve the low-energy spectrum of the Hamiltonian (1.1). The extension of estimates on the ground state energy and on the excitation spectrum obtained in [3] for the Gross–Pitaevskii limit, to regimes with $\kappa >0$ small enough will be addressed in a separate paper [6], using the results of Theorem 1.1. With our techniques, it does not seem possible to obtain such precise information on the spectrum of (1.1) using only previously available bounds like (1.10).

Let us now briefly explain the strategy we use to prove Theorem 1.1. The first part of our analysis follows closely [4]. We start in Sect. 2 by introducing the excitation Hamiltonian ${\mathcal{L}}_N=U_N{H}_N{U}_N^{\ast }$, acting on the truncated Fock space ${\mathcal{F}}_{+}^{\le N}$; the result is given in (2.6), (2.7). The vacuum expectation $⟨\mathrm{\Omega },{\mathcal{L}}_N\mathrm{\Omega }⟩=N^{1+\kappa }\widehat{V}(0)/2$ is still very far from the correct ground state energy of ${\mathcal{L}}_N$ (and thus of $H_N$); the difference is of order $N^{1+\kappa }$. This is a consequence of the definition (1.2) of the unitary map $U_N$, whose action removes products of the condensate wave function ${\phi }_0$, leaving however all correlations among particles in the wave functions ${\alpha }_j\in L_{\perp }^2{(\mathrm{\Lambda })}^{{\otimes }_{s}j}$, $j=1,\dots ,N$.

To factor out correlations, we introduce in Sect. 3 a renormalized excitation Hamiltonian ${\mathcal{G}}_N=e^{-B}{\mathcal{L}}_N{e}^B$, defined through unitary conjugation of ${\mathcal{L}}_N$ with a generalized Bogoliubov transformation $e^B$. The antisymmetric operator $B:{\mathcal{F}}_{+}^{\le N}\to {\mathcal{F}}_{+}^{\le N}$ is quadratic in the modified creation and annihilation operators $b_p,b_p^{\ast }$ defined, for every momentum $p\in {\mathrm{\Lambda }}_{+}^{\ast }=2\pi {\mathbb{Z}}^3\backslash \{0\}$, in (2.8) ($b_p^{\ast }$ creates a particle with momentum p annihilating, at the same time, a particle with momentum zero; in other words, $b_p^{\ast }$ creates an excitation, moving a particle out of the condensate). The properties of ${\mathcal{G}}_N$ are listed in Prop. 3.3. In particular, Proposition 3.3 implies that to leading order, $⟨\mathrm{\Omega },{\mathcal{G}}_N\mathrm{\Omega }⟩\simeq 4\pi {\mathfrak{a}}_0{N}^{1+\kappa }$, if $\kappa$ is small enough.

Unfortunately, ${\mathcal{G}}_N$ is not coercive enough to prove directly that low-energy states exhibit condensation (in the sense that it is not clear how to estimate the difference between ${\mathcal{G}}_N$ and its vacuum expectation from below by the number of particle operator ${\mathcal{N}}_{+}$). For this reason, in Sect. 4, we define yet another renormalized excitation Hamiltonian ${\mathcal{J}}_N=e^{-A}{\mathcal{G}}_N{e}^A$, where now A is the antisymmetric operator (4.1), cubic in (modified) creation and annihilation operators (to be more precise, we only conjugate the main part of ${\mathcal{G}}_N$ with $e^A$; see (4.3)). Important properties of ${\mathcal{J}}_N$ are stated in Proposition 4.1. Up to negligible errors, the conjugation with $e^A$ completes the renormalization of quadratic and cubic terms; in (4.5), these terms have the same form they would have for particles interacting through a mean-field potential with Fourier transform $8\pi {\mathfrak{a}}_0{N}^{\kappa }\mathbf{1}(|p|<N^{\alpha })$, with a parameter $\alpha >0$ that will be chosen small enough, depending on $\kappa$ (in other words, the renormalization procedure allows us to replace, in all quadratic and cubic terms, the original interaction with Fourier transform $N^{-1+\kappa }\widehat{V}(p/N^{1-\kappa })$ decaying only for momenta $|p|>N^{1-\kappa }$, with a potential whose Fourier transform already decays on scales $N^{\alpha }\ll N^{1-\kappa }$).

The main problem with ${\mathcal{J}}_N$ is that its quartic terms (the restriction of the initial potential energy on the orthogonal complement of the condensate wave function) are still proportional to the local interaction with Fourier transform $N^{-1+\kappa }\widehat{V}(p/N^{1-\kappa })$.

One possibility to solve this problem is to neglect the original quartic terms (they are positive) and insert instead quartic terms proportional to the renormalized mean-field potential $8\pi {\mathfrak{a}}_0{N}^{\kappa }\mathbf{1}(|p|<N^{\alpha })$, so that Bose–Einstein condensation follows as it does for mean-field systems (see [22]). Since (with the notation $\check{\chi }$ for the inverse Fourier transform of the characteristic function on the ball of radius one)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{8\pi {\mathfrak{a}}_0{N}^{\kappa }}{N}\sum _{|r|<N^{\alpha }}{a}_{p+r}^{\ast }{a}_q^{\ast }{a}_{q+r}{a}_p& =8\pi {\mathfrak{a}}_0{N}^{3\alpha +\kappa -1}\int \check{\chi }(N^{\alpha }(x-y)){\check{a}}_x^{\ast }{\check{a}}_y^{\ast }{\check{a}}_y{\check{a}}_{x}dxdy\hfill \\ \hfill & \le C{N}^{3\alpha +\kappa -1}{\mathcal{N}}_{+}^2\hfill \end{array}\end{array}$$

and since we know from (1.10) that ${\mathcal{N}}_{+}\lesssim N^{\frac{15+20\kappa }{17}}$ in low-energy states, the insertion of the renormalized quartic terms produces an error that can be controlled by localization in the number of particles, if

$$\begin{array}{c}\hfill 3\alpha +\kappa -1+\frac{15+20\kappa }{17}=3\alpha +\frac{37}{17}\kappa -\frac{2}{17}<0\end{array}$$

This strategy was used in [4] to prove Bose–Einstein condensation with optimal rate in the Gross–Pitaevskii regime $\kappa =0$ (in this case, one can choose $\alpha =0$).

Here, we follow a different approach. We perform a last renormalization step, conjugating ${\mathcal{J}}_N$ through a unitary operator $e^D$, with D quartic in creation and annihilation operators. This leads to a new Hamiltonian ${\mathcal{M}}_N=e^{-D}{\mathcal{J}}_N{e}^D$ (in fact, it is more convenient to conjugate only the main part of ${\mathcal{J}}_N$, ignoring small contributions that can be controlled by other means; see (5.5)), where the original interaction $N^{-1+\kappa }\widehat{V}(p/N^{1-\kappa })$ is replaced by the mean-field potential $8\pi {\mathfrak{a}}_0{N}^{\kappa }\mathbf{1}(|p|<N^{\alpha })$ in all relevant terms.3 Condensation can then be shown as it is done for mean-field systems, with no need for localization. This is the main novelty of our analysis, compared with [4]. In Sect. 5, we define the final Hamiltonian ${\mathcal{M}}_N$ and in Proposition 5.1 we bound it from below. The proof of Proposition 5.1, which is technically the main part of our paper, is deferred to Sect. 7. In Sect. 6, we combine the results of the previous sections to conclude the proof of Theorem 1.1.

The results we prove with our new technique are stronger than what we would obtain using the approach of [4] in the sense that they allow for larger values of $\kappa$ and better rates. More importantly, we believe that the approach we propose here is more natural and that it leaves more space for extensions. In particular, with the final quartic renormalization step, we map the original Hamilton operator (1.1), with an interaction varying on momenta of order $N^{1-\kappa }$, into a new Hamiltonian having the same form, but now with an interaction restricted to momenta smaller than $N^{\alpha }$. If $\alpha <1-\kappa$, this leads to an effective regularization of the potential and it suggests that further improvements may be achieved by iteration; we plan to follow this strategy, which bears some similarities to the renormalization group analysis developed in [1], in future work.

In order to control errors arising from the quartic conjugation, it is important to use observables that were not employed in [4]. In particular, the expectation of the number of excitations with large momenta

$$\begin{array}{c}\hfill {\mathcal{N}}_{\ge N^{\gamma }}=\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }:|p|\ge N^{\gamma }}{a}_p^{\ast }{a}_p\end{array}$$

and of its powers ${\mathcal{N}}_{\ge N^{\gamma }}^2,{\mathcal{N}}_{\ge N^{\gamma }}^3$, as well as the expectation of products of the form ${\mathcal{K}}_L{\mathcal{N}}_{\ge N^{\gamma }}$ and ${\mathcal{K}}_L{\mathcal{N}}_{\ge N^{\gamma }}^2$, involving the kinetic energy operator restricted to low momenta ${\mathcal{K}}_L$, will play a crucial role in our analysis. It will therefore be important to establish bounds for the growth of these observables through all steps of the renormalization procedure (Lemmas 4.2, 4.3, 7.1, 7.2). In Sect. 6, an important step in the proof of Theorem 1.1 will consist in controlling the expectation of these observables on low-energy states of the renormalized Hamiltonian ${\mathcal{G}}_N$.

The Excitation Hamiltonian

We denote by $\mathcal{F}={⨁}_{n\ge 0}{L}^2{(\mathrm{\Lambda })}^{{\otimes }_{s}n}$ the bosonic Fock space over the one-particle space $L^2(\mathrm{\Lambda })$ and by $\mathrm{\Omega }=\{1,0,\dots \}$ the vacuum vector. We can define the number of particles operator $\mathcal{N}$ by setting ${(\mathcal{N}\psi )}^{(n)}=n{\psi }^{(n)}$ for all $\psi =\{{\psi }^{(0)},{\psi }^{(1)},\dots \}$ in a dense subspace of $\mathcal{F}$. For every one-particle wave function $g\in L^2(\mathrm{\Lambda })$, we define the creation operator $a^{\ast }(g)$ and its hermitian conjugate, the annihilation operator a(g), through

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {(a^{\ast }(g)\mathrm{\Psi })}^{(n)}(x_1,\dots ,x_n)& =\frac{1}{\sqrt{n}}\sum _{j=1}^{n}g(x_j){\mathrm{\Psi }}^{(n-1)}(x_1,\dots ,x_{j-1},x_{j+1},\dots ,x_n)\hfill \\ \hfill {(a(g)\mathrm{\Psi })}^{(n)}(x_1,\dots ,x_n)& =\sqrt{n+1}{\int }_{\mathrm{\Lambda }}\overline{g}(x){\mathrm{\Psi }}^{(n+1)}(x,x_1,\dots ,x_n)dx\hfill \end{array}\end{array}$$

Creation and annihilation operators are defined on the domain of ${\mathcal{N}}^{1/2}$, where they satisfy the bounds

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

and the canonical commutation relations

$$\begin{array}{c}\hfill [a(g),a^{\ast }(h)]=⟨g,h⟩,[a(g),a(h)]=[a^{\ast }(g),a^{\ast }(h)]=0\end{array}$$ 2.1

for all $g,h\in L^2(\mathrm{\Lambda })$ ($⟨.,.⟩$ denotes here the inner product on $L^2(\mathrm{\Lambda })$). For $p\in {\mathrm{\Lambda }}^{\ast }=2\pi {\mathbb{Z}}^3$, we define the plane wave ${\phi }_p\in L^2(\mathrm{\Lambda })$ through ${\phi }_p(x)=e^{-ip·x}$ for all $x\in \mathrm{\Lambda }$, and the operators $a_p^{\ast }=a({\phi }_p)$ and $a_p=a({\phi }_p)$ creating and, respectively, annihilating a particle with momentum p. It is sometimes convenient to switch to position space, introducing operator valued distributions ${\check{a}}_x,{\check{a}}_x^{\ast }$ such that

$$\begin{array}{c}\hfill a(f)={\int }_{\mathrm{\Lambda }}\overline{f}(x){\check{a}}_{x}dx,a^{\ast }(f)={\int }_{\mathrm{\Lambda }}f(x){\check{a}}_x^{\ast }dx\end{array}$$

In terms of creation and annihilation operators, the number of particles operator can be written as

$$\begin{array}{c}\hfill \mathcal{N}=\sum _{p\in {\mathrm{\Lambda }}^{\ast }}{a}_p^{\ast }{a}_p=\int a_x^{\ast }{a}_{x}dx\end{array}$$

We will describe excitations of the Bose–Einstein condensate on the truncated Fock space

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

constructed over the orthogonal complement $L_{\perp }^2(\mathrm{\Lambda })$ of the condensate wave function ${\phi }_0$. On ${\mathcal{F}}_{+}^{\le N}$, we denote the number of particles operator by ${\mathcal{N}}_{+}$. It is given by ${\mathcal{N}}_{+}={\sum }_{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{a}_p^{\ast }{a}_p$, where ${\mathrm{\Lambda }}_{+}^{\ast }={\mathrm{\Lambda }}^{\ast }\backslash \{0\}=2\pi {\mathbb{Z}}^3\backslash \{0\}$ is the momentum space for excitations. Given $\mathrm{\Theta }\ge 0$, we also introduce the restricted number of particles operators

$$\begin{array}{c}\hfill {\mathcal{N}}_{\ge \mathrm{\Theta }}=\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }:|p|\ge \mathrm{\Theta }}{a}_p^{\ast }{a}_p,\end{array}$$ 2.2

measuring the number of excitations with momentum larger or equal to $\mathrm{\Theta }$, and ${\mathcal{N}}_{<\mathrm{\Theta }}={\mathcal{N}}_{+}-{\mathcal{N}}_{\ge \mathrm{\Theta }}$.

Consider the operator $U_N:L_s^2({\mathrm{\Lambda }}^N)\to {\mathcal{F}}_{+}^{\le N}$ defined in (1.2). Identifying ${\psi }_N\in L_s^2({\mathrm{\Lambda }}^N)$ with the Fock space vector $\{0,\dots ,0,{\psi }_N,0,\dots \}$, we can also express $U_N$ in terms of creation and annihilation operators; we obtain

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

It is then easy to check that $U_N^{\ast }:{\mathcal{F}}_{+}^{\le N}\to L_s^2({\mathrm{\Lambda }}^N)$ 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 $U_N^{\ast }{U}_N=1$, ie. $U_N$ is unitary.

Using $U_N$, 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 ${\mathcal{L}}_N$, we first write the Hamiltonian (1.1) in momentum space, in terms of creation and annihilation operators. We find

$$\begin{array}{c}\hfill H_N=\sum _{p\in {\mathrm{\Lambda }}^{\ast }}{p}^2{a}_p^{\ast }{a}_p+\frac{1}{2N^{1-\kappa }}\sum _{p,q,r\in {\mathrm{\Lambda }}^{\ast }}\widehat{V}(r/N^{1-\kappa })a_{p+r}^{\ast }{a}_q^{\ast }{a}_p{a}_{q+r}\end{array}$$ 2.3

where

$$\begin{array}{c}\hfill \widehat{V}(k)={\int }_{{\mathbb{R}}^3}V(x)e^{-ik·x}dx\end{array}$$ 2.4

is the Fourier transform of V, defined for all $k\in {\mathbb{R}}^3$ (in fact, (1.1) is the restriction of (2.3) to the $N\in \mathbb{N}$-particle sector of the Fock space $\mathcal{F}$). We can now determine the excitation Hamiltonian ${\mathcal{L}}_N$ 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 $L_s^2({\mathrm{\Lambda }}^N)$ to itself). For any $p,q\in {\mathrm{\Lambda }}_{+}^{\ast }=2\pi {\mathbb{Z}}^3\backslash \{0\}$, we find (see [15]):

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill U_N{a}_0^{\ast }{a}_0{U}_N^{\ast }& =N-{\mathcal{N}}_{+}\hfill \\ \hfill U_N{a}_p^{\ast }{a}_0{U}_N^{\ast }& =a_p^{\ast }\sqrt{N-{\mathcal{N}}_{+}}\hfill \\ \hfill U_N{a}_0^{\ast }{a}_p{U}_N^{\ast }& =\sqrt{N-{\mathcal{N}}_{+}}{a}_p\hfill \\ \hfill U_N{a}_p^{\ast }{a}_q{U}_N^{\ast }& =a_p^{\ast }{a}_q\hfill \end{array}\end{array}$$ 2.5

We conclude that

$$\begin{array}{c}\hfill {\mathcal{L}}_N={\mathcal{L}}_N^{(0)}+{\mathcal{L}}_N^{(2)}+{\mathcal{L}}_N^{(3)}+{\mathcal{L}}_N^{(4)}\end{array}$$ 2.6

with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{L}}_N^{(0)}=& \frac{N-1}{2N}{N}^{\kappa }\widehat{V}(0)(N-{\mathcal{N}}_{+})+\frac{N^{\kappa }\widehat{V}(0)}{2N}{\mathcal{N}}_{+}(N-{\mathcal{N}}_{+})\hfill \\ \hfill {\mathcal{L}}_N^{(2)}=& \sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2{a}_p^{\ast }{a}_p+\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{N}^{\kappa }\widehat{V}(p/N^{1-\kappa })\left[b_p^{\ast },b_p,-,\frac{1}{N},a_p^{\ast },a_p\right]\hfill \\ \hfill & +\frac{1}{2}\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{N}^{\kappa }\widehat{V}(p/N^{1-\kappa })\left[b_p^{\ast },b_{-p}^{\ast },+,b_p,b_{-p}\right]\hfill \\ \hfill {\mathcal{L}}_N^{(3)}=& \frac{1}{\sqrt{N}}\sum _{p,q\in {\mathrm{\Lambda }}_{+}^{\ast }:p+q\ne 0}{N}^{\kappa }\widehat{V}(p/N^{1-\kappa })\left[b_{p+q}^{\ast },a_{-p}^{\ast },a_q,+,a_q^{\ast },a_{-p},b_{p+q}\right]\hfill \\ \hfill {\mathcal{L}}_N^{(4)}=& \frac{1}{2N}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast },r\in {\mathrm{\Lambda }}^{\ast }:\\ r\ne -p,-q\end{array}}{N}^{\kappa }\widehat{V}(r/N^{1-\kappa })a_{p+r}^{\ast }{a}_q^{\ast }{a}_p{a}_{q+r}\hfill \end{array}\end{array}$$ 2.7

where we introduced generalized creation and annihilation operators

$$\begin{array}{c}\hfill b_p^{\ast }=a_p^{\ast }\sqrt{\frac{N-{\mathcal{N}}_{+}}{N}},\text{and}{b}_p=\sqrt{\frac{N-{\mathcal{N}}_{+}}{N}}{a}_p\end{array}$$ 2.8

for all $p\in {\mathrm{\Lambda }}_{+}^{\ast }$. Observe that by (2.5),

$$\begin{array}{c}\hfill U_N^{\ast }{b}_p^{\ast }{U}_N=a_p^{\ast }\frac{a_0}{\sqrt{N}},U_N^{\ast }{b}_p{U}_N=\frac{a_0^{\ast }}{\sqrt{N}}{a}_p\end{array}$$

In other words, $b_p^{\ast }$ creates a particle with momentum $p\in {\mathrm{\Lambda }}_{+}^{\ast }$ but, at the same time, it annihilates a particle from the condensate; it creates an excitation, preserving the total number of particles in the system. 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 $b_p\simeq a_p$. Modified creation and annihilation operators satisfy the commutation relations

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [b_p,b_q^{\ast }]& =\left(1,-,\frac{{\mathcal{N}}_{+}}{N}\right){\delta }_{p,q}-\frac{1}{N}{a}_q^{\ast }{a}_p\hfill \\ \hfill [b_p,b_q]& =[b_p^{\ast },b_q^{\ast }]=0\hfill \end{array}\end{array}$$ 2.9

Furthermore, we find

$$\begin{array}{c}\hfill [b_p,a_q^{\ast }{a}_r]={\delta }_{\mathit{pq}}{b}_r,[b_p^{\ast },a_q^{\ast }{a}_r]=-{\delta }_{\mathit{pr}}{b}_q^{\ast }\end{array}$$ 2.10

for all $p,q,r\in {\mathrm{\Lambda }}_{+}^{\ast }$; 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

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

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 $\Vert b_p\Vert ,\Vert b_p^{\ast }{\Vert \le (N+1)}^{1/2}$.

We can also define modified operator valued distributions

$$\begin{array}{c}\hfill {\check{b}}_x=\sqrt{\frac{N-{\mathcal{N}}_{+}}{N}}{\check{a}}_x,\text{and}{\check{b}}_x^{\ast }={\check{a}}_x^{\ast }\sqrt{\frac{N-{\mathcal{N}}_{+}}{N}}\end{array}$$

in position space, for $x\in \mathrm{\Lambda }$. The commutation relations (2.9) take the form

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [{\check{b}}_x,{\check{b}}_y^{\ast }]& =\left(1,-,\frac{{\mathcal{N}}_{+}}{N}\right)\delta (x-y)-\frac{1}{N}{\check{a}}_y^{\ast }{\check{a}}_x\hfill \\ \hfill [{\check{b}}_x,{\check{b}}_y]& =[{\check{b}}_x^{\ast },{\check{b}}_y^{\ast }]=0\hfill \end{array}\end{array}$$

Moreover, (2.10) translates to

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [{\check{b}}_x,{\check{a}}_y^{\ast }{\check{a}}_z]& =\delta (x-y){\check{b}}_z,[{\check{b}}_x^{\ast },{\check{a}}_y^{\ast }{\check{a}}_z]=-\delta (x-z){\check{b}}_y^{\ast }\hfill \end{array}\end{array}$$

which also implies that $[{\check{b}}_x,{\mathcal{N}}_{+}]={\check{b}}_x$, $[{\check{b}}_x^{\ast },{\mathcal{N}}_{+}]=-{\check{b}}_x^{\ast }$.

Renormalized Excitation Hamiltonian

Conjugation with $U_N$ extracts, from the original quartic interaction in (2.3), some constant and some quadratic contributions, collected in ${\mathcal{L}}_N^{(0)}$ and ${\mathcal{L}}_N^{(2)}$ in (2.7). For bosons described by the Hamiltonian (1.1), this is not enough; there are still large contributions to the energy that are hidden in ${\mathcal{L}}_N^{(3)}$ and ${\mathcal{L}}_N^{(4)}$.

To extract the missing energy, we have to take into account correlations. To this end, we consider the ground state solution $f_{\ell }$ of the Neumann problem

$$\begin{array}{c}\hfill \left[-,\mathrm{\Delta },+,\frac{1}{2},V\right]f_{\ell }={\lambda }_{\ell }{f}_{\ell }\end{array}$$ 3.1

on the ball $|x|\le N^{1-\kappa }\ell$ (we omit the $N\in \mathbb{N}$-dependence in the notation for $f_{\ell }$ and for ${\lambda }_{\ell }$; notice that ${\lambda }_{\ell }$ scales as $N^{3\kappa -3}$), with the normalization $f_{\ell }(x)=1$ if $|x|=N^{1-\kappa }\ell$. By scaling, we observe that $f_{\ell }(N^{1-\kappa }.)$ satisfies the equation

$$\begin{array}{c}\hfill \left[-,\mathrm{\Delta },+,\frac{1}{2},N^{2-2\kappa },V,(N^{1-\kappa }x)\right]f_{\ell }(N^{1-\kappa }x)=N^{2-2\kappa }{\lambda }_{\ell }{f}_{\ell }(N^{1-\kappa }x)\end{array}$$

on the ball $|x|\le \ell$. From now on, we fix some $0<\ell <1/2$, so that the ball of radius $\ell$ is contained in the box $\mathrm{\Lambda }={[-1/2;1/2]}^3$. We then extend $f_{\ell }(N^{1-\kappa }.)$ to $\mathrm{\Lambda }$, by setting $f_N(x)=f_{\ell }(N^{1-\kappa }x)$, if $|x|\le \ell$ and $f_N(x)=1$ for $x\in \mathrm{\Lambda }$, with $|x|>\ell$. As a consequence,

$$\begin{array}{c}\hfill \left[-,\mathrm{\Delta },+,\frac{1}{2},N^{2-2\kappa },V,(N^{1-\kappa }.)\right]f_N=N^{2-2\kappa }{\lambda }_{\ell }{f}_N{\chi }_{\ell },\end{array}$$ 3.2

where ${\chi }_{\ell }$ denotes the characteristic function of the ball of radius $\ell$. The Fourier coefficients of the function $f_N$ are given by

$$\begin{array}{c}\hfill {\widehat{f}}_N(p)={\int }_{\mathrm{\Lambda }}{f}_{\ell }(N^{1-\kappa }x)e^{-ip·x}dx\end{array}$$ 3.3

for all $p\in {\mathrm{\Lambda }}^{\ast }$. Next, we define $w_{\ell }(x)=1-f_{\ell }(x)$ for $|x|\le N^{1-\kappa }\ell$ and $w_{\ell }(x)=0$ for all $|x|>N^{1-\kappa }\ell$. Its rescaled version $w_N:\mathrm{\Lambda }\to \mathbb{R}$ is defined through $w_N(x)=w_{\ell }(N^{1-\kappa }x)$ if $|x|\le \ell$ and $w_N(x)=0$ if $x\in \mathrm{\Lambda }$ with $|x|>\ell$. The Fourier coefficients of $w_N$ are given by

$$\begin{array}{c}\hfill {\widehat{w}}_N(p)={\int }_{\mathrm{\Lambda }}{w}_{\ell }(N^{1-\kappa }x)e^{-ip·x}dx=\frac{1}{N^{3-3\kappa }}{\widehat{w}}_{\ell }(p/N^{1-\kappa }),\end{array}$$

where

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

denotes the Fourier transform of the (compactly supported) function $w_{\ell }$. We find ${\widehat{f}}_N(p)={\delta }_{p,0}-N^{3\kappa -3}{\widehat{w}}_{\ell }(p/N^{1-\kappa })$. From (3.2), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& -p^2{\widehat{w}}_{\ell }(p/N^{1-\kappa })+\frac{N^{2-2\kappa }}{2}\sum _{q\in {\mathrm{\Lambda }}^{\ast }}\widehat{V}((p-q)/N^{1-\kappa }){\widehat{f}}_N(q)\hfill \\ \hfill & =N^{5-5\kappa }{\lambda }_{\ell }\sum _{q\in {\mathrm{\Lambda }}^{\ast }}{\widehat{\chi }}_{\ell }(p-q){\widehat{f}}_N(q).\hfill \end{array}\end{array}$$ 3.4

The next lemma summarizes important properties of the functions $w_{\ell }$ and $f_{\ell }$. Its proof can be found in [4, Appendix A] (replacing $N\in \mathbb{N}$ by $N^{1-\kappa }$ and noting that still $N^{1-\kappa }\ell \gg 1$ for $N\in \mathbb{N}$ sufficiently large and fixed $\ell \in (0;1/2)$).

Lemma 3.1

Let $V\in L^3({\mathbb{R}}^3)$ be nonnegative, compactly supported and spherically symmetric. Fix $\ell >0$ and let $f_{\ell }$ denote the solution of (3.1). For $N\in \mathbb{N}$ large enough, the following properties hold true.

• (i)
  We have

  $$\begin{array}{c}\hfill {\lambda }_{\ell }=\frac{3{\mathfrak{a}}_0}{N^{3-3\kappa }{\ell }^3}\left(1,+,\mathcal{O},(,{\mathfrak{a}}_0,/,\ell ,N^{1-\kappa },)\right).\end{array}$$ 3.5
• (ii)
  We have $0\le f_{\ell },w_{\ell }\le 1$. Moreover there exists a constant $C>0$ such that

  $$\begin{array}{c}\hfill \left|\int ,V,(x),f_{\ell },(x),d,x,-,8,\pi ,{\mathfrak{a}}_0\right|\le \frac{C{\mathfrak{a}}_0^2}{\ell N^{1-\kappa }}.\end{array}$$ 3.6
• (iii)
  There exists a constant $C>0$ such that

  $$\begin{array}{c}\hfill w_{\ell }(x)\le \frac{C}{|x|+1}\text{and}|\mathrm{\nabla }{w}_{\ell }(x)|\le \frac{C}{x^2+1}.\end{array}$$ 3.7

  for all $x\in {\mathbb{R}}^3$ and all $N\in \mathbb{N}$ large enough.
• (iv)
  There exists a constant $C>0$ such that

  $$\begin{array}{c}\hfill |{\widehat{w}}_N(p)|\le \frac{C}{N^{1-\kappa }{p}^2}\end{array}$$

  for all $p\in {\mathbb{R}}^3$ and all $N\in \mathbb{N}$ large enough (such that $N^{1-\kappa }\ge {\ell }^{-1}$).

We define $\eta :{\mathrm{\Lambda }}^{\ast }\to \mathbb{R}$ through

$$\begin{array}{c}\hfill {\eta }_p=-N{\widehat{w}}_N(p)=-\frac{N^{\kappa }}{N^{2-2\kappa }}{\widehat{\omega }}_{\ell }(p/N^{1-\kappa }).\end{array}$$ 3.8

In position space, this means that for $x\in \mathrm{\Lambda }$, we have

$$\begin{array}{c}\hfill \check{\eta }(x)=-N{w}_{\ell }(N^{1-\kappa }x),\end{array}$$ 3.9

so that we have in particular the $L^{\infty }$-bound

$$\begin{array}{c}\hfill \Vert \check{\eta }{\Vert }_{\infty }\le CN.\end{array}$$ 3.10

Lemma 3.1 also implies

$$\begin{array}{c}\hfill |{\eta }_p|\le \frac{C{N}^{\kappa }}{{|p|}^2}\end{array}$$ 3.11

for all $p\in {\mathrm{\Lambda }}_{+}^{\ast }=2\pi {\mathbb{Z}}^3\backslash \{0\}$, and for some constant $C>0$ independent of $N\in \mathbb{N}$ (for $N\in \mathbb{N}$ large enough). From (3.4), we find the relation

$$\begin{array}{c}\hfill \begin{array}{c}\hfill p^2{\eta }_p+\frac{1}{2}{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast {\widehat{f}}_N)(p)=N^{3-2\kappa }{\lambda }_{\ell }({\widehat{\chi }}_{\ell }\ast {\widehat{f}}_N)(p)\end{array}\end{array}$$ 3.12

or equivalently, expressing the r.h.s. through the coefficients ${\eta }_p$,

$$\begin{array}{c}\hfill \begin{array}{cc}& p^2{\eta }_p+\frac{1}{2}{N}^{\kappa }\widehat{V}(p/N^{1-\kappa })+\frac{1}{2N}\sum _{q\in {\mathrm{\Lambda }}^{\ast }}{N}^{\kappa }\widehat{V}((p-q)/N^{1-\kappa }){\eta }_q\hfill \\ \hfill & =N^{3-2\kappa }{\lambda }_{\ell }{\widehat{\chi }}_{\ell }(p)+N^{2-2\kappa }{\lambda }_{\ell }\sum _{q\in {\mathrm{\Lambda }}^{\ast }}{\widehat{\chi }}_{\ell }(p-q){\eta }_q.\hfill \end{array}\end{array}$$ 3.13

In our analysis, it is useful to restrict $\eta$ to high momenta. To this end, let $\alpha >0$ and

$$\begin{array}{c}\hfill P_H=\{p\in {\mathrm{\Lambda }}_{+}^{\ast }:|p|\ge N^{\alpha }\}.\end{array}$$ 3.14

We define ${\eta }_H\in {\ell }^2({\mathrm{\Lambda }}_{+}^{\ast })$ by

$$\begin{array}{c}\hfill {\eta }_H(p)={\eta }_p\chi (p\in P_H)={\eta }_p\chi (|p|\ge N^{\alpha }).\end{array}$$ 3.15

Equation (3.11) implies that

$$\begin{array}{c}\hfill \Vert {\eta }_H\Vert \le C{N}^{\kappa -\alpha /2}\end{array}$$ 3.16

and we assume from now on that $\alpha >2\kappa$ such that in particular

$$\begin{array}{c}\hfill \underset{N\to \infty }{lim}\Vert {\eta }_H\Vert =0.\end{array}$$ 3.17

Notice, on the other hand, that the $H^1$-norm of $\eta$ and ${\eta }_H$ diverge, as $N\to \infty$. From (3.9) and Lemma 3.1, part iii), we find

$$\begin{array}{c}\hfill \sum _{p\in P_H}{p}^2|{\eta }_p{|}^2\le \sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{p}^2|{\eta }_p{|}^2=\int {|\mathrm{\nabla }\check{\eta }(x)|}^{2}dx\le C{N}^{1+\kappa }\end{array}$$ 3.18

for all $N\in \mathbb{N}$ large enough. 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. (3.13) involves ${\eta }_0$). From Lemma 3.1, part iii), we obtain

$$\begin{array}{c}\hfill |{\eta }_0|\le N^{3\kappa -2}{\int }_{{\mathbb{R}}^3}{w}_{\ell }(x)dx\le C{N}^{\kappa }{\ell }^2\end{array}$$ 3.19

It will also be useful to have bounds for the function ${\check{\eta }}_H:\mathrm{\Lambda }\to \mathbb{R}$, having Fourier coefficients ${\eta }_H(p)$ as defined in (3.15). Writing ${\eta }_H(p)={\eta }_p-{\eta }_p\chi (|p|\le N^{\alpha })$, we obtain

$$\begin{array}{c}\hfill {\check{\eta }}_H(x)=\check{\eta }(x)-\sum _{\begin{array}{c}p\in {\mathrm{\Lambda }}^{\ast }:\\ |p|\le N^{\alpha }\end{array}}{\eta }_p{e}^{ip·x}=-N{w}_{\ell }(N^{1-\kappa }x)-\sum _{\begin{array}{c}p\in {\mathrm{\Lambda }}^{\ast }:\\ |p|\le N^{\alpha }\end{array}}{\eta }_p{e}^{ip·x}\end{array}$$

so that

$$\begin{array}{c}\hfill |{\check{\eta }}_H(x)|\le CN+C{N}^{\kappa }\sum _{\begin{array}{c}p\in {\mathrm{\Lambda }}^{\ast }:\\ |p|\le N^{\alpha }\end{array}}{|p|}^{-2}\le C(N+N^{\alpha +\kappa })\le C(N+N^{\alpha +\kappa })\end{array}$$ 3.20

for all $x\in \mathrm{\Lambda }$, if $N\in \mathbb{N}$ is large enough.

With the coefficients (3.15), we define the antisymmetric operator

$$\begin{array}{c}\hfill B=\frac{1}{2}\sum _{p\in P_H}\left({\eta }_p,b_p^{\ast },b_{-p}^{\ast },-,{\overline{\eta }}_p,b_{-p},b_p\right)\end{array}$$ 3.21

and the generalized Bogoliubov transformation $e^B:{\mathcal{F}}_{+}^{\le N}\to {\mathcal{F}}_{+}^{\le N}$. A first important observation is that conjugation with this unitary operator does not change the number of particles by too much. The proof of the following Lemma can be found in [7, Lemma 3.1] (a similar result has been previously established in [22]).

Lemma 3.2

Assume B is defined as in (3.21), with the coefficients ${\eta }_p$ as in (3.8), satisfying (3.17). For every $n\in \mathbb{N}$, there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill e^{-B}{({\mathcal{N}}_{+}+1)}^n{e}^B\le C{({\mathcal{N}}_{+}+1)}^n\end{array}$$ 3.22

as an operator inequality on ${\mathcal{F}}_{+}^{\le N}$. (The constant depends only on $\Vert {\eta }_H\Vert$ and on $n\in \mathbb{N}$.)

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

$$\begin{array}{c}\hfill {\mathcal{G}}_N=e^{-B}{\mathcal{L}}_N{e}^B=e^{-B}{U}_N{H}_N{U}_N^{\ast }{e}^B.\end{array}$$ 3.23

In the next propositions, we collect important properties of ${\mathcal{G}}_N$. Recall the notation ${\mathcal{H}}_N=\mathcal{K}+{\mathcal{V}}_N$, introduced in (1.9).

Proposition 3.3

Let $V\in L^3({\mathbb{R}}^3)$ be compactly supported, pointwise nonnegative and spherically symmetric. Let ${\mathcal{G}}_N$ be defined as in (3.23). Assume that the exponent $\alpha$ introduced in (3.14) is such that

$$\begin{array}{c}\hfill \alpha >6\kappa ,2\alpha +3\kappa <1\end{array}$$ 3.24

Then,

$$\begin{array}{c}\hfill {\mathcal{G}}_N=4\pi {\mathfrak{a}}_0{N}^{1+\kappa }+{\mathcal{H}}_N+{\theta }_{{\mathcal{G}}_N}\end{array}$$ 3.25

and there exists $C>0$ such that, for all $\delta >0$ and all $N\in \mathbb{N}$ large enough, we have

$$\begin{array}{c}\hfill \pm {\theta }_{{\mathcal{G}}_N}\le \delta {\mathcal{H}}_N+C{\delta }^{-1}{N}^{\alpha +2\kappa }{\mathcal{N}}_{+}+C{N}^{\alpha +2\kappa }\end{array}$$ 3.26

and the improved lower bound

$$\begin{array}{c}\hfill {\theta }_{{\mathcal{G}}_N}\ge -\delta {\mathcal{H}}_N-C{\delta }^{-1}{N}^{\kappa }{\mathcal{N}}_{+}-C{N}^{\alpha +2\kappa }.\end{array}$$ 3.27

Furthermore, for $\beta >0$, denote by ${\mathcal{G}}_N^{\text{eff}}$ the excitation Hamiltonian

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{G}}_N^{\text{eff}}=& 4\pi {\mathfrak{a}}_0{N}^{\kappa }(N-{\mathcal{N}}_{+})+[\widehat{V}(0)-4\pi {\mathfrak{a}}_0]N^{\kappa }{\mathcal{N}}_{+}\frac{(N-{\mathcal{N}}_{+})}{N}\hfill \\ \hfill & +N^{\kappa }\widehat{V}(0)\sum _{p\in P_H^c}{a}_p^{\ast }{a}_p(1-{\mathcal{N}}_{+}/N)+4\pi {\mathfrak{a}}_0{N}^{\kappa }\sum _{p\in P_H^c}[b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p}]\hfill \\ \hfill & +\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}p,q\in {\mathrm{\Lambda }}_{+}^{\ast }:|q|\le N^{\beta },\\ p+q\ne 0\end{array}}{N}^{\kappa }\widehat{V}(p/N^{1-\kappa })[b_{p+q}^{\ast }{a}_{-p}^{\ast }{a}_q+h.c.]+{\mathcal{H}}_N\hfill \end{array}\end{array}$$ 3.28

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

$$\begin{array}{c}\hfill \pm {\mathcal{E}}_{{\mathcal{G}}_N}\le C(N^{3\kappa -\alpha /2}+N^{\alpha +3\kappa /2-1/2}+N^{\kappa /2-\beta }){\mathcal{H}}_N+C{N}^{\alpha +2\kappa }\end{array}$$ 3.29

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

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

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \pm i[{\mathcal{N}}_{\ge c{N}^{\gamma }},{\mathcal{G}}_N],\pm i[{\mathcal{N}}_{<c{N}^{\gamma }},{\mathcal{G}}_N]& \le C(N^{\kappa +\alpha /2-\gamma }+N^{\kappa +\gamma /2})({\mathcal{H}}_N+1)\hfill \end{array}\end{array}$$ 3.30

for all $\alpha \ge \gamma >0$, $c>0$ fixed (independent of $N\in \mathbb{N}$) and $N\in \mathbb{N}$ large enough.

Finally, for every $k\in \mathbb{N}$, there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill \pm {\text{ad}}_{i{\mathcal{N}}_{+}}^{(k)}({\mathcal{G}}_N)=\pm [i{\mathcal{N}}_{+},\dots [i{\mathcal{N}}_{+},{\mathcal{G}}_N]\dots ]\le C{N}^{\kappa +\alpha /6}({\mathcal{H}}_N+1).\end{array}$$ 3.31

The proof of Proposition 3.3 is similar to the proof of [4, Prop. 4.2] and [3, Prop. 3.2], with the appropriate modifications dictated by the different scaling of the interaction. The main novelty in Proposition 3.3 is the bound (3.30) involving commutators of the restricted number of particles operator ${\mathcal{N}}_{\ge c{N}^{\gamma }}$. This can be obtained similarly to the bounds for ${\mathcal{E}}_{{\mathcal{G}}_N}$ and for $i[{\mathcal{N}}_{+},{\mathcal{G}}_N]$, because we have a full expansion of the operator ${\mathcal{G}}_N$ in a sum of terms whose commutators with ${\mathcal{N}}_{+}$ and with ${\mathcal{N}}_{\ge c{N}^{\gamma }}$ retains essentially the same form. In the version of this paper that is posted on the arXiv, we give a complete proof of Proposition 3.3 in “Appendix A”, adapting the arguments of [4, Prop. 4.2], [3, Prop. 3.2].

Cubic Renormalization

From Eq. (3.28), we observe that the cubic terms in ${\mathcal{G}}_N^{\text{eff}}$ still depend on the original interaction, which decays slowly in momentum (in contrast to the quadratic terms in the second line of (3.28), where the sum is now restricted to $P_H^c=\{p\in {\mathrm{\Lambda }}_{+}^{\ast }:|p|<N^{\alpha }\}$).

To renormalize the cubic terms in (3.28), we are going to conjugate ${\mathcal{G}}_N^{\text{eff}}$ with a unitary operator $e^A$, where the antisymmetric operator $A:{\mathcal{F}}_{+}^{\le N}\to {\mathcal{F}}_{+}^{\le N}$ is defined by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill A& =A_1-A_1^{\ast },\text{with}{A}_1=\frac{1}{\sqrt{N}}\sum _{r\in P_H,p\in P_L}{\eta }_r{b}_{r+p}^{\ast }{a}_{-r}^{\ast }{a}_p.\hfill \end{array}\end{array}$$ 4.1

The high-momentum set $P_H=\{p\in {\mathrm{\Lambda }}_{+}^{\ast }:|p|\ge N^{\alpha }\}$ is as in (3.14). The low-momentum set $P_L$ is defined by

$$\begin{array}{c}\hfill P_L=\{p\in {\mathrm{\Lambda }}_{+}^{\ast }:|p|\le N^{\beta }\}\end{array}$$ 4.2

with exponent $\beta >0$, that will be chosen as in (3.28).

Using the unitary operator $e^A$, we define ${\mathcal{J}}_N:{\mathcal{F}}_{+}^{\le N}\to {\mathcal{F}}_{+}^{\le N}$ by

$$\begin{array}{c}\hfill {\mathcal{J}}_N=e^{-A}{\mathcal{G}}_N^{\text{eff}}{e}^A.\end{array}$$ 4.3

Observe here that we only conjugate the main part ${\mathcal{G}}_N^{\text{eff}}$ of the renormalized excitation Hamiltonian ${\mathcal{G}}_N$; this makes the analysis a bit simpler (the difference ${\mathcal{G}}_N-{\mathcal{G}}_N^{\text{eff}}$ is small and can be estimated before applying the cubic conjugation).

The next proposition summarizes important properties of ${\mathcal{J}}_N$; it can be shown very similarly to [4, Prop. 5.2], of course with the appropriate changes of the scaling of the interaction. In the version of this paper that is posted on the arXiv, we give a complete proof of Proposition 4.1 in “Appendix B”, adapting the arguments of [4, Prop. 5.2].

Proposition 4.1

Suppose the exponents $\alpha$ and $\beta$ are such that

$$\begin{array}{c}\hfill i)\alpha >3\beta +2\kappa ,ii)3\alpha /2+2\kappa <1,iii)\alpha <5\beta ,iv)\beta >3\kappa /2,v)\beta <1/2\end{array}$$ 4.4

Let ${\mathcal{J}}_N$ be defined as in (4.3), let

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{J}}_N^{\text{eff}}=& 4\pi {\mathfrak{a}}_0{N}^{1+\kappa }-4\pi {\mathfrak{a}}_0{N}^{\kappa }{\mathcal{N}}_{+}^2/N+8\pi {\mathfrak{a}}_0{N}^{\kappa }\sum _{p\in P_H^c}[b_p^{\ast }{b}_p+\frac{1}{2}{b}_p^{\ast }{b}_{-p}^{\ast }+\frac{1}{2}{b}_p{b}_{-p}]\hfill \\ \hfill & +\frac{8\pi {\mathfrak{a}}_0{N}^{\kappa }}{\sqrt{N}}\sum _{\begin{array}{c}p\in P_H^c,q\in P_L:\\ p+q\ne 0\end{array}}[b_{p+q}^{\ast }{a}_{-p}^{\ast }{a}_q+h.c.]+{\mathcal{H}}_N,\hfill \end{array}\end{array}$$ 4.5

and set $\mu =max(3\alpha /2+2\kappa -1,3\kappa /2-\beta )$ ($\mu <0$ follows from (4.4)). Then, there exists a constant $C>0$ such that the self-adjoint operator ${\mathcal{E}}_{{\mathcal{J}}_N}={\mathcal{J}}_N-{\mathcal{J}}_N^{\text{eff}}$ satisfies the operator inequality

$$\begin{array}{c}\hfill \pm e^A{\mathcal{E}}_{{\mathcal{J}}_N}{e}^{-A}\le C(N^{-\beta /2}+N^{\mu })\mathcal{K}+C{N}^{\mu }{\mathcal{V}}_N+C{N}^{\mu -\kappa }{\mathcal{N}}_{+}+C{N}^{\alpha +2\kappa }(1+N^{\alpha +\beta /2-1})\end{array}$$ 4.6

in ${\mathcal{F}}_{+}^{\le N}$ for all $N\in \mathbb{N}$ sufficiently large.

The bounds for ${\mathcal{J}}_N$ given in Proposition 4.1 are still not enough to show Theorem 1.1. As we will discuss in the next section, the main problem is the quartic interaction term, contained in ${\mathcal{H}}_N$, which still depends on the singular interaction potential (in all other terms on the r.h.s. of (4.5), the singular potential has been replaced by the regular mean-field type potential, with Fourier transform $8\pi {\mathfrak{a}}_0{N}^{\kappa }{\mathbf{1}}_{P_H^c}(p)$, supported on momenta $|p|<N^{\alpha }$). To renormalize the quartic interaction, we will have to conjugate ${\mathcal{J}}_N^{\text{eff}}$ with yet another unitary operator, this time quartic in creation and annihilation operators. This last conjugation (which will be performed in the next section) will produce error terms. These errors will controlled in terms of the observables ${\mathcal{N}}_{+}$, $\mathcal{K}$ and ${\mathcal{V}}_N$ (as in (4.6)) but also, as we stressed at the end of Sect. 1, in terms of observables having the form ${\mathcal{N}}_{\ge N^{\gamma }}$ (the number of excitations having momentum larger or equal to $N^{\gamma }$), ${\mathcal{N}}_{\ge N^{\gamma }}^2$, ${\mathcal{N}}_{\ge N^{\gamma }}^3$, ${\mathcal{K}}_{\le N^{\gamma }}$ (the kinetic energy of excitations with momentum below $N^{\gamma }$), ${\mathcal{K}}_L{\mathcal{N}}_{\ge N^{\gamma }}$. For this reason, we need to control the action of $e^A$ on all these observables.

First of all, we bound the action of the cubic phase on the restricted number of particles operators ${\mathcal{N}}_{\ge \theta }={\sum }_{p\in {\mathrm{\Lambda }}_{+}^{\ast }:|p|\ge \theta }{a}_p^{\ast }{a}_p$. We will make use of the pull-through formula $a_p{\mathcal{N}}_{\ge \theta }=({\mathcal{N}}_{\ge \theta }+{\mathbf{1}}_{[\theta ,\infty )}(p))a_p$, which in particular implies that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \Vert {({\mathcal{N}}_{\ge \theta }+1)}^{1/2}{a}_p\xi \Vert & \le C\Vert a_p{({\mathcal{N}}_{\ge \theta }+1)}^{1/2}\xi \Vert ,\hfill \\ \hfill \Vert {({\mathcal{N}}_{\ge \theta }+1)}^{-1/2}{a}_p\xi \Vert & \le C\Vert a_p{({\mathcal{N}}_{\ge \theta }+1)}^{-1/2}\xi \Vert .\hfill \end{array}\end{array}$$ 4.7

Lemma 4.2

Assume the exponents $\alpha ,\beta$ satisfy (4.4) (in fact, here it is enough to assume that $\alpha >2\kappa$). Let $k\in {\mathbb{N}}_0$, $m=0,1,2$, $0<\gamma \le \alpha$, $c\ge 0$ (and $c<1$ if $\gamma =\alpha$). Then, there exists a constant $C>0$ such that the operator inequalities

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^{-sA}{({\mathcal{N}}_{+}+1)}^k{({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)}^m{e}^{\mathit{sA}}\le & C{({\mathcal{N}}_{+}+1)}^k{({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)}^m\hfill \end{array}\end{array}$$ 4.8

for all $s\in [-1;1]$ and all $N\in \mathbb{N}$.

Proof

The case $m=0$ follows from $m=1$. We start therefore with the case $m=1$. For $\xi \in {\mathcal{F}}_{+}^{\le N}$, we define the function ${\phi }_{\xi }:\mathbb{R}\to \mathbb{R}$ by

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

which has derivative

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\partial }_s{\phi }_{\xi }(s)& =2\mathrm{Re}⟨e^{\mathit{sA}}\xi ,{({\mathcal{N}}_{+}+1)}^k[{\mathcal{N}}_{\ge c{N}^{\gamma }},A_1]e^{\mathit{sA}}\xi ⟩\hfill \\ \hfill & +2\mathrm{Re}⟨e^{\mathit{sA}}\xi ,[{({\mathcal{N}}_{+}+1)}^k,A_1]({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)e^{\mathit{sA}}\xi ⟩,\hfill \end{array}\end{array}$$ 4.9

where $A_1$ as in (4.1). By the assumptions on $\gamma$ and c, we have $N^{\alpha }\ge N^{\alpha }-N^{\beta }\ge c{N}^{\gamma }$ for $N\in \mathbb{N}$ large enough. This implies in particular that

$$\begin{array}{c}\hfill [{\mathcal{N}}_{\ge c{N}^{\gamma }},b_{p+r}^{\ast }]=b_{p+r}^{\ast },[{\mathcal{N}}_{\ge c{N}^{\gamma }},a_{-r}^{\ast }]=a_{-r}^{\ast },[{\mathcal{N}}_{\ge c{N}^{\gamma }},a_p]=\chi (|p|\ge c{N}^{\gamma })a_p\end{array}$$

for $r\in P_H$ and $p\in P_L$, by (2.1) and (2.10). We then obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [{\mathcal{N}}_{\ge c{N}^{\gamma }},A_1]& =\frac{2}{\sqrt{N}}\sum _{\begin{array}{c}r\in P_H,p\in P_L\end{array}}{\eta }_r{b}_{r+p}^{\ast }{a}_{-r}^{\ast }{a}_p-\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}r\in P_H,p\in P_L,\\ |p|\ge c{N}^{\gamma }\end{array}}{\eta }_r{b}_{r+p}^{\ast }{a}_{-r}^{\ast }{a}_p\hfill \end{array}\end{array}$$ 4.10

as well as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [{({\mathcal{N}}_{+}+1)}^k,A_1]& =\frac{k}{\sqrt{N}}\sum _{r\in P_H,p\in P_L}{\eta }_r{b}_{r+p}^{\ast }{a}_{-r}^{\ast }{a}_p{({\mathcal{N}}_{+}+\mathrm{\Theta }({\mathcal{N}}_{+})+1)}^{k-1},\hfill \end{array}\end{array}$$ 4.11

for some function $\mathrm{\Theta }:\mathbb{N}\to (0;1)$ by the mean value theorem. Using the pull-through formula ${\mathcal{N}}_{+}{a}_p^{\ast }=a_p^{\ast }({\mathcal{N}}_{+}+1)$ and Cauchy–Schwarz, we estimate

$$\begin{array}{c}\hfill \begin{array}{cc}& |\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}r\in P_H,p\in P_L\end{array}}{\eta }_r⟨e^{\mathit{sA}}\xi ,{({\mathcal{N}}_{+}+1)}^k{b}_{r+p}^{\ast }{a}_{-r}^{\ast }{a}_p{e}^{\mathit{sA}}\xi ⟩|\hfill \\ \hfill & \le \frac{1}{\sqrt{N}}(\sum _{\begin{array}{c}r\in P_H,p\in P_L\end{array}}{\Vert {({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)}^{-1/2}{a}_{r+p}{a}_{-r}{({\mathcal{N}}_{+}+1)}^{k/2}{e}^{\mathit{sA}}\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill & \times (\sum _{\begin{array}{c}r\in P_H,p\in P_L\end{array}}{\eta }_r^2{\Vert {({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)}^{1/2}{a}_p{({\mathcal{N}}_{+}+1)}^{k/2}{e}^{\mathit{sA}}\xi \Vert }^2{)}^{1/2}\hfill \end{array}\end{array}$$

With the operator inequality ${\mathcal{N}}_{\ge c{N}^{\gamma }}\ge {\mathcal{N}}_{\ge N^{\alpha }}$ and with (4.7), we find that

$$\begin{array}{c}\hfill \begin{array}{cc}& |\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}r\in P_H,p\in P_L\end{array}}{\eta }_r⟨e^{\mathit{sA}}\xi ,{({\mathcal{N}}_{+}+1)}^k{b}_{r+p}^{\ast }{a}_{-r}^{\ast }{a}_p{e}^{\mathit{sA}}\xi ⟩|\hfill \\ \hfill & \le \frac{C}{\sqrt{N}}(\sum _{\begin{array}{c}r\in P_H,p\in P_L:|p+r|\ge c{N}^{\gamma }\end{array}}{\Vert a_{p+r}{({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)}^{-1/2}{a}_{-r}{({\mathcal{N}}_{+}+1)}^{k/2}{e}^{\mathit{sA}}\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill & \times \Vert {\eta }_H\Vert (\sum _{\begin{array}{c}p\in P_L\end{array}}{\Vert a_p{({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)}^{1/2}{({\mathcal{N}}_{+}+1)}^{k/2}{e}^{\mathit{sA}}\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill & \le \frac{C{N}^{\kappa -\alpha /2}}{\sqrt{N}}\Vert {({\mathcal{N}}_{\ge N^{\alpha }}+1)}^{1/2}{({\mathcal{N}}_{+}+1)}^{k/2}{e}^{\mathit{sA}}\xi \Vert \Vert {({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)}^{1/2}{({\mathcal{N}}_{+}+1)}^{(k+1)/2}{e}^{\mathit{sA}}\xi \Vert \hfill \\ \hfill & \le C{N}^{\kappa -\alpha /2}{\Vert {({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)}^{1/2}{({\mathcal{N}}_{+}+1)}^{k/2}{e}^{\mathit{sA}}\xi \Vert }^2=C{N}^{\kappa -\alpha /2}{\phi }_{\xi }(s).\hfill \end{array}\end{array}$$ 4.12

The same arguments show that

$$\begin{array}{c}\hfill \begin{array}{cc}& |\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}r\in P_H,p\in P_L,\\ |p|\ge c{N}^{\gamma }\end{array}}{\eta }_r⟨e^{\mathit{sA}}\xi ,{({\mathcal{N}}_{+}+1)}^k{b}_{r+p}^{\ast }{a}_{-r}^{\ast }{a}_p{e}^{\mathit{sA}}\xi ⟩|\hfill \\ \hfill & \le \frac{C}{\sqrt{N}}(\sum _{\begin{array}{c}r\in P_H,p\in P_L:|p+r|\ge c{N}^{\gamma }\end{array}}{\Vert a_{p+r}{({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)}^{-1/2}{a}_{-r}{({\mathcal{N}}_{+}+1)}^{k/2}{e}^{\mathit{sA}}\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill & \times \Vert {\eta }_H\Vert (\sum _{\begin{array}{c}p\in P_L\end{array}}{\Vert a_p{({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)}^{1/2}{({\mathcal{N}}_{+}+1)}^{k/2}{e}^{\mathit{sA}}\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill & \le C{N}^{\kappa -\alpha /2}{\phi }_{\xi }(s).\hfill \end{array}\end{array}$$ 4.13

Finally, we have that

$$\begin{array}{c}\hfill \begin{array}{cc}& |\frac{k}{\sqrt{N}}\sum _{r\in P_H,p\in P_L}{\eta }_r⟨e^{\mathit{sA}}\xi ,b_{r+p}^{\ast }{a}_{-r}^{\ast }{a}_p{({\mathcal{N}}_{+}+\mathrm{\Theta }({\mathcal{N}}_{+})+1)}^{k-1}({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)e^{\mathit{sA}}\xi ⟩|\hfill \\ \hfill & \le \frac{C}{\sqrt{N}}(\sum _{\begin{array}{c}r\in P_H,p\in P_L:|p+r|\ge c{N}^{\gamma }\end{array}}{\Vert a_{r+p}{a}_{-r}{({\mathcal{N}}_{+}+1)}^{(k-1)/2}{e}^{\mathit{sA}}\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill & \times (\sum _{\begin{array}{c}r\in P_H,p\in P_L\end{array}}{\eta }_r^2{\Vert a_p{({\mathcal{N}}_{+}+1)}^{(k-1)/2}({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)e^{\mathit{sA}}\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill & \le C{N}^{\kappa -\alpha /2}{\Vert {({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)}^{1/2}{({\mathcal{N}}_{+}+1)}^{k/2}{e}^{\mathit{sA}}\xi \Vert }^2=C{N}^{\kappa -\alpha /2}{\phi }_{\xi }(s).\hfill \end{array}\end{array}$$ 4.14

Recalling (4.9), (4.10) and that $\alpha \ge 2\kappa$, the bounds (4.12) to (4.14) show that

$$\begin{array}{c}\hfill {\partial }_s{\phi }_{\xi }(s)\le C{N}^{\kappa -\alpha /2}{\phi }_{\xi }(s)\le C{\phi }_{\xi }(s).\end{array}$$

Since the bounds are independent of $\xi \in {\mathcal{F}}_{+}^{\le N}$ and the same bounds hold true replacing A by $-A$ in the definition of ${\phi }_{\xi }$, the first inequality in (4.8) follows by Gronwall’s Lemma.

To prove (4.8) with $m=2$, we proceed similarly. Given $\xi \in {\mathcal{F}}_{+}^{\le N}$, we define the function ${\psi }_{\xi }:\mathbb{R}\to \mathbb{R}$ by

$$\begin{array}{c}\hfill {\psi }_{\xi }(s)=⟨\xi ,e^{-sA}{({\mathcal{N}}_{+}+1)}^k{({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)}^2{e}^{\mathit{sA}}\xi ⟩.\end{array}$$

Its derivative is equal to

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\partial }_s{\psi }_{\xi }(s)& =2\mathrm{Re}⟨e^{\mathit{sA}}\xi ,{({\mathcal{N}}_{+}+1)}^k[{({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)}^2,A_1]e^{\mathit{sA}}\xi ⟩\hfill \\ \hfill & +2\mathrm{Re}⟨e^{\mathit{sA}}\xi ,[{({\mathcal{N}}_{+}+1)}^k,A_1]{({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)}^2{e}^{\mathit{sA}}\xi ⟩\hfill \\ \hfill & =2\mathrm{Re}⟨e^{\mathit{sA}}\xi ,{({\mathcal{N}}_{+}+1)}^k[{\mathcal{N}}_{\ge c{N}^{\gamma }},[{\mathcal{N}}_{\ge c{N}^{\gamma }},A_1]]e^{\mathit{sA}}\xi ⟩\hfill \\ \hfill & +4\mathrm{Re}⟨e^{\mathit{sA}}\xi ,{({\mathcal{N}}_{+}+1)}^k[{\mathcal{N}}_{\ge c{N}^{\gamma }},A_1]({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)e^{\mathit{sA}}\xi ⟩\hfill \\ \hfill & +2\mathrm{Re}⟨e^{\mathit{sA}}\xi ,[{({\mathcal{N}}_{+}+1)}^k,A_1]{({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)}^2{e}^{\mathit{sA}}\xi ⟩.\hfill \end{array}\end{array}$$ 4.15

Comparing the contribution containing the double commutator in the last line on the r.h.s. of the last equation with (4.10) and using once again that $N^{\alpha }\ge N^{\alpha }-N^{\beta }\ge c{N}^{\gamma }$ for $N\in \mathbb{N}$ large enough, we observe that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [{\mathcal{N}}_{\ge c{N}^{\gamma }},[{\mathcal{N}}_{\ge c{N}^{\gamma }},A_1]]& =\frac{4}{\sqrt{N}}\sum _{\begin{array}{c}r\in P_H,p\in P_L\end{array}}{\eta }_r{b}_{r+p}^{\ast }{a}_{-r}^{\ast }{a}_p-\frac{3}{\sqrt{N}}\sum _{\begin{array}{c}r\in P_H,p\in P_L,\\ |p|\ge c{N}^{\gamma }\end{array}}{\eta }_r{b}_{r+p}^{\ast }{a}_{-r}^{\ast }{a}_p.\hfill \end{array}\end{array}$$ 4.16

Hence, the bounds (4.12) and (4.13) prove that

$$\begin{array}{c}\hfill |⟨e^{\mathit{sA}}\xi ,{({\mathcal{N}}_{+}+1)}^k[{\mathcal{N}}_{\ge c{N}^{\gamma }},[{\mathcal{N}}_{\ge c{N}^{\gamma }},A_1]]e^{\mathit{sA}}\xi ⟩|\le C{\phi }_{\xi }(s)\le C{\psi }_{\xi }(s).\end{array}$$

To bound the second contribution on the r.h.s. in (4.15), we recall (4.10) and we estimate

$$\begin{array}{c}\hfill \begin{array}{cc}& |\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}r\in P_H,p\in P_L\end{array}}{\eta }_r⟨e^{\mathit{sA}}\xi ,{({\mathcal{N}}_{+}+1)}^k{b}_{r+p}^{\ast }{a}_{-r}^{\ast }{a}_p({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)e^{\mathit{sA}}\xi ⟩|\hfill \\ & +|\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}r\in P_H,p\in P_L,\\ |p|\ge c{N}^{\gamma }\end{array}}{\eta }_r⟨e^{\mathit{sA}}\xi ,{({\mathcal{N}}_{+}+1)}^k{b}_{r+p}^{\ast }{a}_{-r}^{\ast }{a}_p({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)e^{\mathit{sA}}\xi ⟩|\hfill \\ & \le \frac{C}{\sqrt{N}}(\sum _{\begin{array}{c}r\in P_H,p\in P_L:|p+r|\ge c{N}^{\gamma }\end{array}}{\Vert a_{p+r}{a}_{-r}{({\mathcal{N}}_{+}+1)}^{k/2}{e}^{\mathit{sA}}\xi \Vert }^2{)}^{1/2}\hfill \\ & \times \Vert {\eta }_H\Vert (\sum _{\begin{array}{c}p\in P_L\end{array}}{\Vert a_p{({\mathcal{N}}_{+}+1)}^{k/2}({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)e^{\mathit{sA}}\xi \Vert }^2{)}^{1/2}\hfill \\ & \le C{N}^{\kappa -\alpha /2}{\Vert ({\mathcal{N}}_{\ge c{N}^{\gamma }}+1){({\mathcal{N}}_{+}+1)}^{k/2}{e}^{\mathit{sA}}\xi \Vert }^2=C{N}^{\kappa -\alpha /2}{\psi }_{\xi }(s)\hfill \end{array}\end{array}$$

Finally, the last contribution in (4.15) can be bounded as in (4.14), using (4.11). We have

$$\begin{array}{c}\hfill \begin{array}{cc}& |\frac{k}{\sqrt{N}}\sum _{r\in P_H,p\in P_L}{\eta }_r⟨e^{\mathit{sA}}\xi ,b_{r+p}^{\ast }{a}_{-r}^{\ast }{a}_p{({\mathcal{N}}_{+}+\mathrm{\Theta }({\mathcal{N}}_{+})+1)}^{k-1}{({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)}^2{e}^{\mathit{sA}}\xi ⟩|\hfill \\ \hfill & \le \frac{C}{\sqrt{N}}(\sum _{\begin{array}{c}r\in P_H,p\in P_L:|p+r|\ge c{N}^{\gamma }\end{array}}{\Vert a_{r+p}{a}_{-r}{({\mathcal{N}}_{+}+1)}^{k/2}{e}^{\mathit{sA}}\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill & \times (\sum _{\begin{array}{c}r\in P_H,p\in P_L\end{array}}{\eta }_r^2{\Vert a_p{({\mathcal{N}}_{+}+1)}^{(k-2)/2}{({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)}^2{e}^{\mathit{sA}}\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill & \le C{N}^{\kappa -\alpha /2}{\Vert ({\mathcal{N}}_{\ge c{N}^{\gamma }}+1){({\mathcal{N}}_{+}+1)}^{k/2}{e}^{\mathit{sA}}\xi \Vert }^2=C{N}^{\kappa -\alpha /2}{\psi }_{\xi }(s),\hfill \end{array}\end{array}$$

where, in the last step, we used that ${\mathcal{N}}_{\ge c{N}^{\gamma }}\le {\mathcal{N}}_{+}$. In conclusion, we have proved that

$$\begin{array}{c}\hfill {\partial }_s{\psi }_{\xi }(s)\le C{N}^{\kappa -\alpha /2}{\psi }_{\xi }(s)\le C{\psi }_{\xi }(s).\end{array}$$

Since the bounds are independent of $\xi \in {\mathcal{F}}_{+}^{\le N}$ and the same bounds hold true replacing $-A$ by A in the definition ${\psi }_{\xi }$, Gronwall’s lemma implies the last inequality in (4.8). $\square$

We denote the kinetic energy restricted to low momenta by

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathcal{K}}_{\le c{N}^{\gamma }}=\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }:|p|\le c{N}^{\gamma }}{p}^2{a}_p^{\ast }{a}_p.\end{array}\end{array}$$ 4.17

We will need the following estimates for the growth of the restricted kinetic energy.

Lemma 4.3

Assume the exponents $\alpha ,\beta$ satisfy (4.4) (here we only need $\alpha \ge 2\kappa$ and $\alpha >\beta$). Let $0<{\gamma }_1,{\gamma }_2\le \alpha$, and $c_1,c_2\ge 0$ (and also $c_j<1$, if ${\gamma }_j=\alpha$, for $j=1,2$). Then, there exists a constant $C>0$ such that the operator inequalities

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^{-sA}{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}}{e}^{\mathit{sA}}\le & {\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}}+N^{2\beta +2\kappa -\alpha -1}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^2,\hfill \\ \hfill e^{-sA}{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}}({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1)e^{\mathit{sA}}\le & {\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}}({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1)\hfill \\ \hfill & +N^{2\beta +2\kappa -\alpha -1}{({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1)}^2({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\hfill \\ \hfill \end{array}\end{array}$$ 4.18

for all $s\in [-1;1]$ and all $N\in \mathbb{N}$ sufficiently large.

Proof

Like the previous Lemma 4.2, this is an application of Gronwall’s lemma. Let us start to prove the first inequality in (4.18). Fix $\xi \in {\mathcal{F}}_{+}^{\le N}$ and define ${\phi }_{\xi }:\mathbb{R}\to \mathbb{R}$ by ${\phi }_{\xi }(s)=⟨\xi ,e^{-sA}{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}}{e}^{\mathit{sA}}\xi ⟩$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\partial }_s{\phi }_{\xi }(s)& =2\mathrm{Re}⟨\xi ,e^{-sA}[{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}},A_1]e^{\mathit{sA}}\xi ⟩.\hfill \\ \hfill \end{array}\end{array}$$

We notice first that

$$\begin{array}{c}\hfill [{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}},b_{p+r}^{\ast }]=[{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}},a_{-r}^{\ast }]=0\end{array}$$

if $r\in P_H$ and $p\in P_L$, because $|r|,|p+r|\ge N^{\alpha }-N^{\beta }>c_1{N}^{{\gamma }_1}$ for all $N\in \mathbb{N}$.

Using the commutation relations (2.1), we then compute

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}},A_1]& =-\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}r\in P_H,p\in P_L:|p|\le c_1{N}^{{\gamma }_1}\end{array}}{p}^2{\eta }_r{b}_{r+p}^{\ast }{a}_{-r}^{\ast }{a}_p.\hfill \end{array}\end{array}$$ 4.19

With (4.19) and $|p|\le N^{\beta }$ for $p\in P_L$, we then find that

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,e^{-sA}[{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}},A_1]e^{\mathit{sA}}\xi ⟩|\hfill \\ \hfill & \le \frac{C{N}^{\beta }}{\sqrt{N}}\sum _{\begin{array}{c}r\in P_H,p\in P_L:|p|\le c_1{N}^{{\gamma }_1}\end{array}}|p||{\eta }_r|\Vert a_{r+p}{a}_{-r}{e}^{\mathit{sA}}\xi \Vert \Vert a_p{e}^{\mathit{sA}}\xi \Vert \hfill \\ \hfill & \le \frac{C{N}^{\beta +\kappa -\alpha /2}}{\sqrt{N}}\Vert ({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)e^{\mathit{sA}}\xi \Vert \Vert {\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}}^{1/2}{e}^{\mathit{sA}}\xi \Vert .\hfill \end{array}\end{array}$$ 4.20

Finally, using Lemma 4.2 (with $c=\frac{1}{2}$, $\gamma =\alpha$ and $N\in \mathbb{N}$ sufficiently large), we conclude

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\partial }_s{\phi }_{\xi }(s)& \le C{N}^{\beta +\kappa -\alpha /2-1/2}\Vert ({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)e^{\mathit{sA}}\xi \Vert \Vert {\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}}^{1/2}{e}^{\mathit{sA}}\xi \Vert \hfill \\ \hfill & \le C{N}^{2\beta +2\kappa -\alpha -1}⟨\xi ,{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^2\xi ⟩+C{\phi }_{\xi }(s).\hfill \end{array}\end{array}$$

This proves the first inequality in (4.18), by Gronwall’s lemma.

Next, let us prove the second inequality in (4.18). We define ${\psi }_{\xi }:\mathbb{R}\to \mathbb{R}$ by

$$\begin{array}{c}\hfill {\psi }_{\xi }(s)=⟨\xi ,e^{-sA}{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}}({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1)e^{\mathit{sA}}\xi ⟩,\end{array}$$

and we compute

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\partial }_s{\psi }_{\xi }(s)& =2\mathrm{Re}⟨\xi ,e^{-sA}[{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}},A_1]({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1)e^{\mathit{sA}}\xi ⟩\hfill \\ \hfill & +2\mathrm{Re}⟨\xi ,e^{-sA}{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}}[{\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}},A_1]e^{\mathit{sA}}\xi ⟩.\hfill \end{array}\end{array}$$

First, we proceed as in (4.20) and obtain with (4.7) that

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,e^{-sA}[{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}},A_1]({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1)e^{\mathit{sA}}\xi ⟩|\hfill \\ \hfill & \le \frac{C{N}^{\beta }}{\sqrt{N}}\sum _{\begin{array}{c}r\in P_H,p\in P_L:\\ |p|\le c_1{N}^{{\gamma }_1}\end{array}}|p||{\eta }_r|\Vert a_{r+p}{({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1)}^{1/2}{a}_{-r}{e}^{\mathit{sA}}\xi \Vert \Vert a_p{({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1)}^{1/2}{e}^{\mathit{sA}}\xi \Vert \hfill \\ \hfill & \le \frac{C{N}^{\beta +\kappa -\alpha /2}}{\sqrt{N}}\Vert ({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1){({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}{e}^{\mathit{sD}}\xi \Vert \Vert {\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}}^{1/2}{({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1)}^{1/2}{e}^{\mathit{sA}}\xi \Vert .\hfill \end{array}\end{array}$$ 4.21

Equation (4.21) and Lemma 4.2 then imply

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,e^{-sA}[{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}},A_1]({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1)e^{\mathit{sA}}\xi ⟩|\hfill \\ \hfill & \le C{N}^{2\beta +2\kappa -\alpha -1}⟨\xi ,{({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1)}^2({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\xi ⟩+C{\psi }_{\xi }(s).\hfill \end{array}\end{array}$$ 4.22

Next, we recall the identity in (4.10) and that

$$\begin{array}{c}\hfill [{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}},b_{p+r}^{\ast }]=[{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}},a_{-r}^{\ast }]=0\end{array}$$

whenever $r\in P_H,p\in P_L$ and $N\in \mathbb{N}$, by assumption on $c_1$ and ${\gamma }_1$. We then estimate

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,e^{-sA}{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}}[{\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}},A_1]e^{\mathit{sA}}\xi ⟩|\hfill \\ \hfill & \le \frac{C}{\sqrt{N}}\sum _{\begin{array}{c}r\in P_H,p\in P_L,\\ v\in {\mathrm{\Lambda }}_{+}^{\ast }:|v|\le c_1{N}^{{\gamma }_1}\end{array}}{|v|}^2|{\eta }_r|\Vert a_{r+p}{({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1)}^{-1/2}{a}_{-r}{a}_v{e}^{\mathit{sD}}\xi \Vert \hfill \\ \hfill & \times \Vert a_p{({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1)}^{1/2}{a}_v{e}^{\mathit{sD}}\xi \Vert \hfill \\ \hfill & \le C{N}^{\kappa -\alpha /2}⟨e^{\mathit{sA}}\xi ,{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}}({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1)e^{\mathit{sA}}\xi ⟩\le C{\psi }_{\xi }(s).\hfill \end{array}\end{array}$$ 4.23

Hence, putting (4.22) and (4.23) together, we have proved that

$$\begin{array}{c}\hfill {\partial }_s{\psi }_{\xi }(s)\le C{N}^{2\beta +2\kappa -\alpha -1}⟨\xi ,{({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1)}^2({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\xi ⟩+C{\psi }_{\xi }(s).\end{array}$$

This implies the second bound in (4.18), by Gronwall’s lemma. $\square$

Next, we seek a bound for the growth of the potential energy operator. To this end, we first compute the commutator of ${\mathcal{V}}_N$ with the antisymmetric operator A. We introduce here the shorthand notation for the low-momentum part of the kinetic energy

$$\begin{array}{c}\hfill {\mathcal{K}}_L=\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }:|p|\le N^{\beta }}{p}^2{a}_p^{\ast }{a}_p=\sum _{p\in P_L}{p}^2{a}_p^{\ast }{a}_p.\end{array}$$ 4.24

Proposition 4.4

Assume the exponents $\alpha ,\beta$ satisfy (4.4). There exists a constant $C>0$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [{\mathcal{V}}_N,A]& =\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}_{+}^{\ast },p\in P_L:\\ p+u\ne 0\end{array}}{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast \eta /N)(u)[b_{p+u}^{\ast }{a}_{-u}^{\ast }{a}_p+h.c.]+{\mathcal{E}}_{[{\mathcal{V}}_N,A]}\hfill \end{array}\end{array}$$ 4.25

where the self-adjoint operator ${\mathcal{E}}_{[{\mathcal{V}}_N,A]}$ satisfies

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \pm {\mathcal{E}}_{[{\mathcal{V}}_N,A]}& \le \delta {\mathcal{V}}_N+{\delta }^{-1}C{N}^{\kappa -2\beta -1}{\mathcal{K}}_L({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)+{\delta }^{-1}C{N}^{2\alpha +3\kappa -2}{\mathcal{N}}_{+}\hfill \\ \hfill & +{\delta }^{-1}C{N}^{\kappa -1}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^2\hfill \end{array}\end{array}$$ 4.26

for all $\delta >0$ and for all $N\in \mathbb{N}$ sufficiently large.

Proof

From (4.1), we have

$$\begin{array}{c}\hfill [{\mathcal{V}}_N,A]=[{\mathcal{V}}_N,A_1]+\text{h.c.}\end{array}$$

Following [4, Prop. 8.1], we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [{\mathcal{V}}_N,A_1]+\text{h.c.}=& \frac{1}{\sqrt{N}}\sum _{u\in {\mathrm{\Lambda }}_{+}^{\ast },v\in P_L}^{\ast }{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast \eta /N)(u)b_{u+v}^{\ast }{a}_{-u}^{\ast }{a}_v\hfill \\ \hfill & +{\mathrm{\Theta }}_1+{\mathrm{\Theta }}_2+{\mathrm{\Theta }}_3+{\mathrm{\Theta }}_4+\text{h.c.},\hfill \end{array}\end{array}$$ 4.27

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathrm{\Theta }}_1& =-\frac{1}{N^{3/2}}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },v\in P_L,\\ r\in P_H^c\cup \{0\}\end{array}}^{\ast }{N}^{\kappa }\widehat{V}((u-r)/N^{1-\kappa }){\eta }_r{b}_{u+v}^{\ast }{a}_{-u}^{\ast }{a}_v,\hfill \\ \hfill {\mathrm{\Theta }}_2& =\frac{1}{N^{3/2}}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },p\in {\mathrm{\Lambda }}_{+}^{\ast },\\ r\in P_H,v\in P_L\end{array}}^{\ast }{N}^{\kappa }\widehat{V}(u/N^{1-\kappa }){\eta }_r{b}_{p+u}^{\ast }{a}_{v+r-u}^{\ast }{a}_{-r}^{\ast }{a}_p{a}_v,\hfill \\ \hfill {\mathrm{\Theta }}_3& =\frac{1}{N^{3/2}}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },p\in {\mathrm{\Lambda }}_{+}^{\ast },\\ r\in P_H,v\in P_L\end{array}}^{\ast }{N}^{\kappa }\widehat{V}(u/N^{1-\kappa }){\eta }_r{b}_{v+r}^{\ast }{a}_{p+u}^{\ast }{a}_{-r-u}^{\ast }{a}_p{a}_v,\hfill \\ \hfill {\mathrm{\Theta }}_4& =-\frac{1}{N^{3/2}}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },p\in {\mathrm{\Lambda }}_{+}^{\ast },\\ r\in P_H,v\in P_L\end{array}}^{\ast }{N}^{\kappa }\widehat{V}(u/N^{1-\kappa }){\eta }_r{b}_{v+r}^{\ast }{a}_{-r}^{\ast }{a}_{p+u}^{\ast }{a}_p{a}_{v+u}.\hfill \end{array}\end{array}$$ 4.28

Here and in the following, the notation ${\sum }^{\ast }$ indicates that we only sum over those momenta for which the arguments of the creation and annihilation operators are nonzero. The first term on the r.h.s. of (4.27) appears explicitly in (4.25), so let us estimate next the size of the operators ${\mathrm{\Theta }}_1$ to ${\mathrm{\Theta }}_4$, defined in (4.28). The bounds can be obtained similarly as in the proof of [4, Prop. 8.1].

Consider first ${\mathrm{\Theta }}_1$. For $\xi \in {\mathcal{F}}_{+}^{\le N}$, we switch to position space and find

$$\begin{array}{cc}\hfill |⟨\xi ,{\mathrm{\Theta }}_1\xi ⟩|& \le \frac{1}{N^{1/2}}\sum _{r\in P_H^c}|{\eta }_r|({\int }_{{\mathrm{\Lambda }}^2}dxdy{N}^{2-2\kappa }V(N^{1-\kappa }(x-y)){\Vert {\check{b}}_x{\check{a}}_y\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill & \times ({\int }_{{\mathrm{\Lambda }}^2}dxdy{N}^{2-2\kappa }V(N^{1-\kappa }(x-y))\Vert \sum _{v\in P_L}{e}^{\mathit{ivx}}{a}_v\xi {\Vert }^2{)}^{1/2}\hfill \\ \hfill & \le C{N}^{\alpha +3\kappa /2-1}\Vert {\mathcal{V}}_N^{1/2}\xi \Vert ({\int }_{\mathrm{\Lambda }}dx{e}^{i(v-v^{\prime })x}\sum _{v,v^{\prime }\in P_L}⟨\xi ,a_{v^{\prime }}^{\ast }{a}_v\xi ⟩{)}^{1/2}\hfill \\ \hfill & \le C{N}^{\alpha +3\kappa /2-1}\Vert {\mathcal{V}}_N^{1/2}\xi \Vert \Vert {\mathcal{N}}_{\le N^{\beta }}^{1/2}\xi \Vert .\hfill \end{array}$$ 4.29

The term ${\mathrm{\Theta }}_2$ on the r.h.s. of (4.28) can be controlled by

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,{\mathrm{\Theta }}_2\xi ⟩|\hfill \\ \hfill & =|\frac{1}{N^{1/2}}{\int }_{{\mathrm{\Lambda }}^2}dxdy{N}^{2-2\kappa }V(N^{1-\kappa }(x-y))\sum _{r\in P_H,v\in P_L}{e}^{\mathit{ivy}}{e}^{\mathit{iry}}{\eta }_r⟨\xi ,{\check{b}}_x^{\ast }{\check{a}}_y^{\ast }{a}_{-r}^{\ast }{\check{a}}_x{a}_v\xi ⟩|\hfill \\ \hfill & \le \frac{\Vert {\eta }_H\Vert }{N^{1/2}}[{\int }_{{\mathrm{\Lambda }}^2}dxdy{N}^{2-2\kappa }V(N^{1-\kappa }(x-y))\sum _{v\in P_L}{|v|}^{-2}{\Vert {\check{b}}_x{\check{a}}_y\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill & \times ({\int }_{{\mathrm{\Lambda }}^2}dxdy{N}^{2-2\kappa }V(N^{1-\kappa }(x-y))\sum _{v\in P_L}{|v|}^2{\Vert {({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}{\check{a}}_x{a}_v\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill & \le C{N}^{\beta /2+3\kappa /2-\alpha /2-1/2}\Vert {\mathcal{V}}_N^{1/2}\xi \Vert \Vert {\mathcal{K}}_L^{1/2}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$

In the last step, we used (4.7) to estimate

$$\begin{array}{cc}\hfill {\int }_{\mathrm{\Lambda }}dx{\Vert {({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}{\check{a}}_x\xi \Vert }^2& =\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{\Vert {({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}{a}_p\xi \Vert }^2\hfill \\ \hfill & \le C\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }}{\Vert a_p{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}\xi \Vert }^2\hfill \\ \hfill & =C\Vert {\mathcal{N}}_{+}^{1/2}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}{\xi \Vert }^2\hfill \end{array}$$ 4.30

for any $\xi \in {\mathcal{F}}_{+}^{\le N}$. The contributions ${\mathrm{\Theta }}_3$ and ${\mathrm{\Theta }}_4$ can be bounded similarly. We find

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,{\mathrm{\Theta }}_3\xi ⟩|\hfill \\ \hfill & =|\frac{1}{N^{1/2}}{\int }_{{\mathrm{\Lambda }}^2}dxdy{N}^{2-2\kappa }V(N^{1-\kappa }(x-y))\sum _{r\in P_H,v\in P_L}{e}^{-iry}{\eta }_r⟨\xi ,b_{v+r}^{\ast }{\check{a}}_x^{\ast }{\check{a}}_y^{\ast }{\check{a}}_x{a}_v\xi ⟩|\hfill \\ \hfill & \le \frac{C\Vert {\eta }_H\Vert }{N^{1/2}}({\int }_{{\mathrm{\Lambda }}^2}dxdy{N}^{2-2\kappa }V(N^{1-\kappa }(x-y))\sum _{v\in P_L}{|v|}^{-2}{\Vert {\check{a}}_x{\check{a}}_y\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill & \times ({\int }_{{\mathrm{\Lambda }}^2}dxdy{N}^{2-2\kappa }V(N^{1-\kappa }(x-y))\sum _{v\in P_L}{|v|}^2{\Vert {({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}{\check{a}}_x{a}_v\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill & \le C{N}^{\beta /2+3\kappa /2-\alpha /2-1/2}\Vert {\mathcal{V}}_N^{1/2}\xi \Vert \Vert {\mathcal{K}}_L^{1/2}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}\xi \Vert \hfill \end{array}\end{array}$$

as well as

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,{\mathrm{\Theta }}_4\xi ⟩|\hfill \\ \hfill & =|\frac{1}{N^{1/2}}{\int }_{{\mathrm{\Lambda }}^2}dxdy{N}^{2-2\kappa }V(N^{1-\kappa }(x-y))\sum _{r\in P_H,v\in P_L}{\eta }_r{e}^{-ivy}⟨\xi ,b_{v+r}^{\ast }{a}_{-r}^{\ast }{\check{a}}_x^{\ast }{\check{a}}_x{\check{a}}_y\xi ⟩|\hfill \\ \hfill & \le \frac{C\Vert {\eta }_H\Vert }{N^{1/2}}[{\int }_{{\mathrm{\Lambda }}^2}dxdy{N}^{2-2\kappa }V(N^{1-\kappa }(x-y))\sum _{v\in P_L}{\Vert {\check{a}}_x{\check{a}}_y\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill & \times [{\int }_{{\mathrm{\Lambda }}^2}dxdy{N}^{2-2\kappa }V(N^{1-\kappa }(x-y))\sum _{r\in P_H,v\in P_L}{\Vert {\check{a}}_x{a}_{v+r}{a}_{-r}\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill & \le C{N}^{3\beta /2+3\kappa /2-\alpha /2-1/2}\Vert {\mathcal{V}}_N^{1/2}\xi \Vert \Vert ({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\xi \Vert .\hfill \end{array}\end{array}$$

Summarizing (using $\alpha >3\beta +2\kappa$) we proved that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \pm \sum _{i=1}^4({\mathrm{\Theta }}_i+\text{h.c.})& \le \delta {\mathcal{V}}_N+{\delta }^{-1}C{N}^{2\alpha +3\kappa -2}{\mathcal{N}}_{+}+{\delta }^{-1}C{N}^{\kappa -2\beta -1}{\mathcal{K}}_L({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\hfill \\ \hfill & +{\delta }^{-1}C{N}^{\kappa -1}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^2\hfill \end{array}\end{array}$$ 4.31

for any $\delta >0$. Setting ${\mathcal{E}}_{[{\mathcal{V}}_N,A]}={\sum }_{i=1}^4({\mathrm{\Theta }}_i+\text{h.c.})$, this proves the claim. $\square$

From Proposition 4.4, we immediately get a bound for the action of $e^A$ on ${\mathcal{V}}_N$.

Corollary 4.5

Assume the exponents $\alpha ,\beta$ satisfy (4.4). Then, there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^{-sA}{\mathcal{V}}_N{e}^{\mathit{sA}}& \le C{\mathcal{V}}_N+C(N^{\kappa }+N^{2\alpha +3\kappa -2})({\mathcal{N}}_{+}+1)\hfill \\ \hfill & +C{N}^{\kappa -2\beta -1}{\mathcal{K}}_L({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)+C{N}^{\kappa -3\beta -2}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^3.\hfill \end{array}\end{array}$$ 4.32

for all $s\in [-1;1]$ and $N\in \mathbb{N}$ large enough.

Proof

We apply Gronwall’s lemma. Given $\xi \in {\mathcal{F}}_{+}^{\le N}$, we define ${\phi }_{\xi }(s)=⟨\xi ,e^{-sA}{\mathcal{V}}_N{e}^{\mathit{sA}}\xi ⟩$ and compute its derivative s.t.

$$\begin{array}{c}\hfill {\partial }_s{\phi }_{\xi }(s)=⟨\xi ,e^{-sA}[{\mathcal{V}}_N,A]e^{\mathit{sA}}\xi ⟩.\end{array}$$

Hence, we can apply (4.25) and estimate

$$\begin{array}{c}\hfill \begin{array}{cc}& |\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}_{+}^{\ast },v\in P_L:\\ v+u\ne 0\end{array}}{N}^{\kappa }⟨e^{\mathit{sA}}\xi ,(\widehat{V}(./N^{1-\kappa })\ast \eta /N)(u)b_{v+u}^{\ast }{a}_{-u}^{\ast }{a}_v{e}^{\mathit{sA}}\xi ⟩|\hfill \\ \hfill & \le \frac{N^{\kappa /2}{\Vert \check{\eta }\Vert }_{\infty }}{N}({\int }_{{\mathrm{\Lambda }}^2}dxdy{N}^{2-2\kappa }V(N^{1-\kappa }(x-y)){\Vert {\check{a}}_x{\check{a}}_y{e}^{\mathit{sA}}\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill & \times ({\int }_{{\mathrm{\Lambda }}^2}dxdy{N}^{3-3\kappa }V(N^{1-\kappa }(x-y))\Vert \sum _{v\in P_L}{e}^{\mathit{ivx}}{a}_v{e}^{\mathit{sA}}{\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill & \le C{N}^{\kappa /2}\Vert {\mathcal{V}}_N^{1/2}{e}^{\mathit{sA}}\xi \Vert \Vert {\mathcal{N}}_{\le N^{\beta }}{e}^{\mathit{sA}}\xi \Vert \le C{N}^{\kappa }⟨\xi ,e^{-sA}{\mathcal{N}}_{+}{e}^{\mathit{sA}}\xi ⟩+C{\phi }_{\xi }(s).\hfill \end{array}\end{array}$$

Here, we used (3.10), which shows that $\Vert \check{\eta }{\Vert }_{\infty }\le CN$. Using Lemma 4.2, this simplifies to

$$\begin{array}{c}\hfill \begin{array}{cc}& |\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}_{+}^{\ast },v\in P_L:\\ v+u\ne 0\end{array}}{N}^{\kappa }⟨e^{\mathit{sD}}\xi ,(\widehat{V}(./N^{1-\kappa })\ast \eta /N)(u)b_{u+v}^{\ast }{a}_{-u}^{\ast }{a}_v{e}^{\mathit{sD}}\xi ⟩|\hfill \\ & \le C{\phi }_{\xi }(s)+C{N}^{\kappa }⟨\xi ,({\mathcal{N}}_{+}+1)\xi ⟩.\hfill \end{array}\end{array}$$ 4.33

Together with (4.25), the bound (4.26) (choosing $\delta =1$) and an application of Lemma 4.2 as well as of Lemma 4.3, the claim follows from Gronwall’s lemma.

$\square$

Quartic Renormalization

To explain why the bounds for ${\mathcal{J}}_N$ obtained in Prop. 4.1 are not enough to show Theorem 1.1, we introduce, for $r\in {\mathrm{\Lambda }}_{+}^{\ast }$, the operators

$$\begin{array}{c}\hfill c_r^{\ast }=\frac{1}{\sqrt{N}}\sum _{\begin{array}{c}v\in {\mathrm{\Lambda }}_{+}^{\ast }:v\ne -r,\\ v\in P_L,v+r\in P_L^c\end{array}}{a}_{v+r}^{\ast }{a}_v,e_r^{\ast }=\frac{1}{2\sqrt{N}}\sum _{\begin{array}{c}v\in {\mathrm{\Lambda }}_{+}^{\ast }:v\ne -r,\\ v\in P_L,v+r\in P_L\end{array}}{a}_{v+r}^{\ast }{a}_v.\end{array}$$ 5.1

We denote the adjoints of $c_r^{\ast }$ and $e_r^{\ast }$ by $c_r$ and $e_r$, respectively. Notice in particular that $e_r^{\ast }=e_{-r}$ for all $r\in {\mathrm{\Lambda }}_{+}^{\ast }$. A straightforward computation shows that

$$\begin{array}{c}\hfill \begin{array}{cc}& \frac{8\pi {\mathfrak{a}}_0{N}^{\kappa }}{\sqrt{N}}\sum _{\begin{array}{c}p\in P_H^c,q\in P_L:\\ p+q\ne 0\end{array}}[b_{p+q}^{\ast }{a}_{-p}^{\ast }{a}_q+\text{h.c.}]\hfill \\ & =8\pi {\mathfrak{a}}_0{N}^{\kappa }\sum _{p\in P_H^c}[b_{-p}^{\ast }{e}_{-p}+e_{-p}^{\ast }{b}_{-p}+b_{-p}^{\ast }{e}_p^{\ast }+e_p{b}_{-p}+b_{-p}^{\ast }{c}_p^{\ast }+c_p{b}_{-p}].\hfill \end{array}\end{array}$$ 5.2

Together with (4.5), this suggests to bound the Hamiltonian ${\mathcal{J}}_N$ from below by completing the square in the operators $g_r^{\ast }:=b_r^{\ast }+c_r^{\ast }+e_r^{\ast }$ and $g_r:=b_r+c_r+e_r$, for $r\in P_H^c\subset {\mathrm{\Lambda }}_{+}^{\ast }$. A better look at (4.5) reveals, however, that several terms that are needed to complete the square are still hidden in the energy ${\mathcal{H}}_N$. Since these terms are not small, we need to extract them from ${\mathcal{H}}_N$ by conjugation with a unitary operator $e^D$, with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D& =D_1-D_1^{\ast },\text{where}{D}_1=\frac{1}{2N}\sum _{r\in P_H,p,q\in P_L}{\eta }_r{a}_{p+r}^{\ast }{a}_{q-r}^{\ast }{a}_p{a}_q.\hfill \end{array}\end{array}$$ 5.3

Since $[D,{\mathcal{N}}_{+}]=0$, we have the identity

$$\begin{array}{c}\hfill e^{-sD}{({\mathcal{N}}_{+}+1)}^k{e}^{\mathit{sD}}={({\mathcal{N}}_{+}+1)}^k\end{array}$$ 5.4

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

Using $e^D$, we define the final excitation Hamiltonian

$$\begin{array}{c}\hfill {\mathcal{M}}_N=e^{-D}{\mathcal{J}}_N^{\text{eff}}{e}^D.\end{array}$$ 5.5

The next proposition provides an important lower bound for ${\mathcal{M}}_N$. Its proof is given in Sect. 7.

Proposition 5.1

Suppose the exponents $\alpha$ (in the definition of the set $P_H$ in (3.14)) and $\beta$ (in the definition of the set $P_L$ in (4.2)) are such that

$$\begin{array}{c}\hfill i)\alpha >3\beta +2\kappa ,ii)1>\alpha +\beta +2\kappa ,iii)5\beta >\alpha ,iv)\beta >3\kappa ,v)1/2>\beta ,\end{array}$$ 5.6

Set $\gamma =min(\alpha ,1-\alpha -\kappa )$ ($\gamma >0$ from (5.6)) and let $m_0\in \mathbb{R}$ be s.t. $m_0\beta =\alpha$. Let $V\in L^3({\mathbb{R}}^3)$ be compactly supported, pointwise nonnegative and spherically symmetric. Then, ${\mathcal{M}}_N$, as defined as in (5.5), is bounded from below by

$$\begin{array}{c}\hfill {\mathcal{M}}_N\ge 4\pi {\mathfrak{a}}_0{N}^{1+\kappa }+\frac{1}{4}\mathcal{K}+{\mathcal{E}}_{{\mathcal{M}}_N}\end{array}$$ 5.7

for a self-adjoint operator ${\mathcal{E}}_{{\mathcal{M}}_N}$ satisfying

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^A{e}^D{\mathcal{E}}_{{\mathcal{M}}_N}& e^{-D}{e}^{-A}\hfill \\ \hfill \ge & -C{N}^{-\beta }\mathcal{K}-C{N}^{-\beta -\kappa }{\mathcal{V}}_N-C{N}^{\beta +2\kappa -1}\mathcal{K}{\mathcal{N}}_{\ge N^{\beta }}\hfill \\ \hfill & -C{N}^{\alpha +\beta +2\kappa -1}\mathcal{K}{\mathcal{N}}_{\ge N^{\lfloor m_0\rfloor \beta }}\hfill \\ \hfill & -C\sum _{j=3}^{2\lfloor m_0\rfloor -1}{N}^{j\beta /2+\beta /2+2\kappa -1}\mathcal{K}{\mathcal{N}}_{\ge \frac{1}{2}{N}^{j\beta /2}}-C{N}^{3\alpha +\kappa }\hfill \end{array}\end{array}$$ 5.8

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

Proof of Theorem 1.1

For $\epsilon >0$ sufficiently small, we define

$$\begin{array}{c}\hfill \alpha =14\kappa +4\epsilon ,\beta =4\kappa +\epsilon .\end{array}$$ 6.1

The choice $\kappa <1/43$ guarantees, if $\epsilon >0$ is small enough, that all conditions in (5.6) (and thus also in (3.24) and (4.4)) are satisfied.

From (3.25) and (3.26), we obtain the upper bound

$$\begin{array}{c}\hfill E_N\le 4\pi {\mathfrak{a}}_0{N}^{1+\kappa }+C{N}^{16\kappa +4\epsilon }\end{array}$$ 6.2

for the ground state energy of $H_N$. From (3.25) and (3.27), on the other hand, we obtain

$$\begin{array}{c}\hfill {\mathcal{H}}_N\le 2({\mathcal{G}}_N-4\pi {\mathfrak{a}}_0{N}^{1+\kappa })+C{N}^{\kappa }{\mathcal{N}}_{+}+C{N}^{16\kappa +4\epsilon }\end{array}$$

With (6.2) and setting ${\mathcal{G}}_N^{\prime }={\mathcal{G}}_N-E_N$, we deduce that

$$\begin{array}{c}\hfill {\mathcal{H}}_N\le 2{\mathcal{G}}_N^{\prime }+C{N}^{\kappa }{\mathcal{N}}_{+}+C{N}^{16\kappa +4\epsilon }\end{array}$$ 6.3

Next, we prove (1.5). From (3.29) and (6.3) we arrive at

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{G}}_N& ={\mathcal{G}}_N^{\text{eff}}+{\mathcal{E}}_{{\mathcal{G}}_N}\ge {\mathcal{G}}_N^{\text{eff}}-C{N}^{-(7\kappa +2\epsilon )/2}{\mathcal{G}}_N^{\prime }-C{N}^{-(5\kappa +2\epsilon )/2}{\mathcal{N}}_{+}-C{N}^{16\kappa +4\epsilon }\hfill \end{array}\end{array}$$

Writing ${\mathcal{G}}_{\text{eff}}=e^A{\mathcal{J}}_N{e}^{-A}$ and recalling that $\kappa <1/43$ (and that $\epsilon >0$ is small enough), Prop. 4.1 and (6.3) imply that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{G}}_N\ge & e^A{\mathcal{J}}_N^{\text{eff}}{e}^{-A}+e^A{\mathcal{E}}_{{\mathcal{J}}_N}{e}^{-A}-C{N}^{-(7\kappa +2\epsilon )/2}{\mathcal{G}}_N^{\prime }-C{N}^{-(5\kappa +2\epsilon )/2}{\mathcal{N}}_{+}-C{N}^{16\kappa +4\epsilon }\hfill \\ \hfill \ge & e^A{\mathcal{J}}_N^{\text{eff}}{e}^{-A}-C{N}^{-(5\kappa +2\epsilon )/2}{\mathcal{G}}_N^{\prime }-C{N}^{-(3\kappa +2\epsilon )/2}{\mathcal{N}}_{+}-C{N}^{16\kappa +4\epsilon }\hfill \end{array}\end{array}$$

Inserting ${\mathcal{J}}_{\text{eff}}=e^D{\mathcal{M}}_N{e}^{-D}$ and applying Prop. 5.1, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{G}}_N\ge & 4\pi {\mathfrak{a}}_0{N}^{1+\kappa }+\frac{1}{4}{e}^A{e}^D\mathcal{K}{e}^{-D}{e}^{-A}+e^A{e}^D{\mathcal{E}}_{{\mathcal{M}}_N}{e}^{-D}{e}^{-A}\hfill \\ \hfill & -C{N}^{-(5\kappa +2\epsilon )/2}{\mathcal{G}}_N^{\prime }-C{N}^{-(3\kappa +2\epsilon )/2}{\mathcal{N}}_{+}-C{N}^{16\kappa +4\epsilon }\hfill \end{array}\end{array}$$ 6.4

With $\mathcal{K}\ge {(2\pi )}^2{\mathcal{N}}_{+}$ and Lemma 4.2 (with $m=0$ and $k=1$) we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^A{e}^D\mathcal{K}{e}^{-D}{e}^{-A}& \ge {(2\pi )}^2{e}^A{e}^D{\mathcal{N}}_{+}{e}^{-D}{e}^{-A}={(2\pi )}^2{e}^A{\mathcal{N}}_{+}{e}^{-A}\ge c{\mathcal{N}}_{+}\hfill \end{array}\end{array}$$ 6.5

for a constant $c>0$ small enough (but independent of N). If N is large enough, we conclude (using also the upper bound (6.2)), that

$$\begin{array}{c}\hfill {\mathcal{N}}_{+}\le C{\mathcal{G}}_N^{\prime }-C{e}^A{e}^D{\mathcal{E}}_{{\mathcal{M}}_N}{e}^{-D}{e}^{-A}+C{N}^{16\kappa +4\epsilon }\end{array}$$ 6.6

To bound the error term $e^A{e}^D{\mathcal{E}}_{{\mathcal{M}}_N}{e}^{-D}{e}^{-A}$, we need (according to (5.8)) to control observables of the form $N^{-1}\mathcal{K}{\mathcal{N}}_{\ge c{N}^{\gamma }}$. To this end, we observe, first of all, that, by Cauchy–Schwarz and by (6.3),

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill N^{-1}\mathcal{K}{\mathcal{N}}_{\ge c{N}^{\gamma }}\le & {\delta }^{-1}{N}^{\kappa -2\gamma }\mathcal{K}+\delta N^{2\gamma -\kappa -2}\mathcal{K}{\mathcal{N}}_{\ge c{N}^{\gamma }}^2\hfill \\ \hfill \le & {\delta }^{-1}{N}^{\kappa -2\gamma }\mathcal{K}+2\delta N^{2\gamma -\kappa -2}{\mathcal{N}}_{\ge c{N}^{\gamma }}{\mathcal{G}}_N^{\prime }{\mathcal{N}}_{\ge c{N}^{\gamma }}+C\delta N^{-1}\mathcal{K}{\mathcal{N}}_{\ge c{N}^{\gamma }}.\hfill \end{array}\end{array}$$ 6.7

Choosing $\delta >0$ sufficiently small, we thus have

$$\begin{array}{c}\hfill \begin{array}{cc}& N^{-1}\mathcal{K}{\mathcal{N}}_{\ge c{N}^{\gamma }}\le C{N}^{\kappa -2\gamma }\mathcal{K}+C{N}^{2\gamma -\kappa -2}{\mathcal{N}}_{\ge c{N}^{\gamma }}{\mathcal{G}}_N^{\prime }{\mathcal{N}}_{\ge c{N}^{\gamma }}.\hfill \end{array}\end{array}$$ 6.8

We write

$$\begin{array}{c}\hfill {\mathcal{N}}_{\ge c{N}^{\gamma }}{\mathcal{G}}_N^{\prime }{\mathcal{N}}_{\ge c{N}^{\gamma }}={\mathcal{N}}_{\ge c{N}^{\gamma }}^2{\mathcal{G}}_N^{\prime }+{\mathcal{N}}_{\ge c{N}^{\gamma }}[{\mathcal{G}}_N^{\prime },{\mathcal{N}}_{\ge c{N}^{\gamma }}].\end{array}$$ 6.9

Using (6.3) (similarly as we did in (6.7)) and ${\mathcal{N}}_{\ge c{N}^{\gamma }}\le N$, ${\mathcal{N}}_{\ge c{N}^{\gamma }}\le C{N}^{-2\gamma }\mathcal{K}$, we can bound the expectation of the first term on the r.h.s. of the last equation, for an arbitrary $\xi \in {\mathcal{F}}_{+}^{\le N}$, by

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,{\mathcal{N}}_{\ge c{N}^{\gamma }}^2{\mathcal{G}}_N^{\prime }\xi ⟩|\hfill \\ \hfill & \le {⟨\xi ,{\mathcal{N}}_{\ge c{N}^{\gamma }}^3\xi ⟩}^{1/2}{⟨\xi ,{\mathcal{G}}_N^{\prime }{\mathcal{N}}_{\ge c{N}^{\gamma }}{\mathcal{G}}_N^{\prime }\xi ⟩}^{1/2}\hfill \\ \hfill & \le C{N}^{1/2-\gamma }{⟨\xi ,\mathcal{K}{\mathcal{N}}_{\ge c{N}^{\gamma }}^2\xi ⟩}^{1/2}{⟨\xi ,{\mathcal{G}}_N^{{}^{\prime }2}\xi ⟩}^{1/2}\hfill \\ \hfill & \le C{N}^{1/2-\gamma }{⟨\xi ,{\mathcal{G}}_N^{{}^{\prime }2}\xi ⟩}^{1/2}{⟨\xi ,{\mathcal{N}}_{\ge c{N}^{\gamma }}{\mathcal{G}}_N^{\prime }{\mathcal{N}}_{\ge c{N}^{\gamma }}\xi ⟩}^{1/2}\hfill \\ \hfill & +C{N}^{1+\kappa /2-2\gamma }{⟨\xi ,{\mathcal{G}}_N^{{}^{\prime }2}\xi ⟩}^{1/2}{⟨\xi ,\mathcal{K}{\mathcal{N}}_{\ge c{N}^{\gamma }}\xi ⟩}^{1/2}\hfill \\ \hfill & \le \delta ⟨\xi ,{\mathcal{N}}_{\ge c{N}^{\gamma }}{\mathcal{G}}_N^{\prime }{\mathcal{N}}_{\ge c{N}^{\gamma }}\xi ⟩+C{\delta }^{-1}{N}^{1-2\gamma }⟨\xi ,{\mathcal{G}}_N^{{}^{\prime }2}\xi ⟩\hfill \\ \hfill & +C\delta N^{1+\kappa -2\gamma }{⟨\xi ,\mathcal{K}{\mathcal{N}}_{\ge c{N}^{\gamma }}\xi ⟩}^{1/2}.\hfill \end{array}\end{array}$$ 6.10

On the other hand, to estimate the commutator term in Eq. (6.9), we notice that $\mathcal{A}:={({\mathcal{H}}_N+1)}^{-1/2}i[{\mathcal{G}}_N^{\prime },{\mathcal{N}}_{\ge c{N}^{\gamma }}]{({\mathcal{H}}_N+1)}^{-1/2}$ is a bounded, self-adjoint operator with $\Vert \mathcal{A}\Vert \le C{N}^{\kappa +\alpha /2-\gamma }+C{N}^{\kappa +\gamma /2}$, by (3.30). Setting $\mu =max(\alpha ,3\gamma )$, this implies, with (6.3),

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,{\mathcal{N}}_{\ge c{N}^{\gamma }}[{\mathcal{G}}_N^{\prime },{\mathcal{N}}_{\ge c{N}^{\gamma }}]\xi ⟩|\hfill \\ \hfill & \le \delta ⟨\xi ,{\mathcal{N}}_{\ge c{N}^{\gamma }}({\mathcal{H}}_N+1){\mathcal{N}}_{\ge c{N}^{\gamma }}\xi ⟩+C{\delta }^{-1}{N}^{2\kappa -2\gamma +\mu }⟨\xi ,({\mathcal{H}}_N+1)\xi ⟩\hfill \\ \hfill & \le 2\delta ⟨\xi ,{\mathcal{N}}_{\ge c{N}^{\gamma }}{\mathcal{G}}_N^{\prime }{\mathcal{N}}_{\ge c{N}^{\gamma }}\xi ⟩+C\delta N^{1+\kappa -2\gamma }⟨\xi ,\mathcal{K}{\mathcal{N}}_{\ge c{N}^{\gamma }}\xi ⟩\hfill \\ \hfill & +C{\delta }^{-1}{N}^{3\kappa -2\gamma +\mu }⟨\xi ,{\mathcal{N}}_{+}\xi ⟩+C{\delta }^{-1}{N}^{3\kappa +\alpha -2\gamma +\mu }{\Vert \xi \Vert }^2\hfill \end{array}\end{array}$$ 6.11

for all $\xi \in {\mathcal{F}}_{+}^{\le N}$. Plugging (6.10) and (6.11) into (6.9), we find that, for sufficiently small $\delta >0$,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{N}}_{\ge c{N}^{\gamma }}{\mathcal{G}}_N^{\prime }{\mathcal{N}}_{\ge c{N}^{\gamma }}\le & C\delta N^{1+\kappa -2\gamma }\mathcal{K}{\mathcal{N}}_{\ge c{N}^{\gamma }}+C{\delta }^{-1}{N}^{1-2\gamma }{\mathcal{G}}_N^{{}^{\prime }2}\hfill \\ \hfill & +C{\delta }^{-1}{N}^{3\kappa -2\gamma +\mu }{\mathcal{N}}_{+}+C{\delta }^{-1}{N}^{3\kappa -2\gamma +\mu +\alpha }\hfill \end{array}\end{array}$$ 6.12

Inserting into (6.8) and choosing $\delta >0$ small enough, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& N^{-1}\mathcal{K}{\mathcal{N}}_{\ge c{N}^{\gamma }}\le C{N}^{\kappa -2\gamma }\mathcal{K}+C{N}^{-\kappa -1}{\mathcal{G}}_N^{{}^{\prime }2}+C{N}^{2\kappa +\mu -2}{\mathcal{N}}_{+}+C{N}^{2\kappa +\mu +\alpha -2}\hfill \end{array}\end{array}$$ 6.13

Applying (6.13) to the r.h.s. of (5.8) we find, using also (6.3), (6.1), and the choice $\kappa <1/43$,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^A{e}^D{\mathcal{E}}_{{\mathcal{M}}_N}{e}^{-D}{e}^{-A}& \ge -C{N}^{-\epsilon }{\mathcal{N}}_{+}-C{N}^{-(\kappa +\epsilon )}{\mathcal{G}}_N^{\prime }-C{N}^{13\kappa +3\epsilon -1}{\mathcal{G}}_N^{{}^{\prime }2}-C{N}^{43\kappa +12\epsilon }\hfill \end{array}\end{array}$$ 6.14

Inserting the last equation into (6.6) and using (6.2), we conclude that for N large enough,

$$\begin{array}{c}\hfill {\mathcal{N}}_{+}\le C{\mathcal{G}}_N^{\prime }+C{N}^{13\kappa +3\epsilon -1}{\mathcal{G}}_N^{{}^{\prime }2}+C{N}^{43\kappa +12\epsilon }\end{array}$$

For ${\psi }_N\in L_s^2({\mathrm{\Lambda }}^N)$ with $\Vert {\psi }_N\Vert =1$ and $⟨{\psi }_N,{(H_N-E_N)}^2{\psi }_N⟩\le {\zeta }^2$, the corresponding excitation vector ${\xi }_N=e^B{U}_N{\psi }_N$ is such that $⟨{\xi }_N,{\mathcal{G}}_N^{{}^{\prime }2}{\xi }_N⟩\le {\zeta }^2$ and thus

$$\begin{array}{c}\hfill ⟨{\xi }_N,{\mathcal{N}}_{+}{\xi }_N⟩\le C\left[\zeta ,+,{\zeta }^2,N^{13\kappa +3\epsilon -1},+,N^{43\kappa +12\epsilon }\right]\end{array}$$

which proves (1.5), using Lemma 3.2. From (6.3), we obtain also

$$\begin{array}{c}\hfill ⟨{\xi }_N,{\mathcal{H}}_N{\xi }_N⟩\le C\left[\zeta ,N^{\kappa },+,{\zeta }^2,N^{14\kappa +3\epsilon -1},+,N^{44\kappa +12\epsilon }\right],\end{array}$$ 6.15

an estimate that will be needed to arrive at (1.6).

Evaluating (6.14) on a normalized ground state ${\xi }_N$ of ${\mathcal{G}}_N$ and inserting the result in (6.4) we also deduce that

$$\begin{array}{c}\hfill E_N\ge 4\pi {\mathfrak{a}}_0{N}^{1+\kappa }-C{N}^{43\kappa +12\epsilon }\end{array}$$

Together with the upper bound (6.2), this concludes the proof of (1.3).

We still have to show (1.6) for $k>0$. To this end, we will prove the stronger bound (1.8); Eq. (1.6) follows then immediately from ${\mathcal{N}}_{+}\le {\mathcal{H}}_N$ and by Lemma 3.2. We denote by $Q_{\zeta }$ the spectral subspace of ${\mathcal{G}}_N$ associated with energies below $E_N+\zeta$. We use induction to show that for all $k\in \mathbb{N}$, there exists a constant $C>0$ (depending on k) such that

$$\begin{array}{c}\hfill \underset{\xi \in Q_{\zeta }}{sup}\frac{⟨\xi ,({\mathcal{H}}_N+1){({\mathcal{N}}_{+}+1)}^{2k}\xi ⟩}{{\Vert \xi \Vert }^2}\le C{\left[N^{44\kappa +12\epsilon },+,{\zeta }^2,N^{20\kappa +5\epsilon }\right]}^{2k+1}\end{array}$$ 6.16

for all $k\in \mathbb{N}$. This proves (1.8) and thus, with the bound ${\mathcal{N}}_{+}\le {\mathcal{H}}_N$ and with Lemma 3.2, also (1.6). The case $k=0$ follows from (6.15). From now on, we assume (6.16) to hold true, and we prove the same bound, with k replaced by $(k+1)$ (and with a new constant C). To this end, we start by observing that combining (6.3) and (6.6),

$$\begin{array}{c}\hfill {\mathcal{H}}_N+1\le C{N}^{\kappa }{\mathcal{G}}_N^{\prime }-C{N}^{\kappa }{e}^A{e}^D{\mathcal{E}}_{{\mathcal{M}}_N}{e}^{-D}{e}^{-A}+C{N}^{17\kappa +4\epsilon }\end{array}$$

Hence,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {({\mathcal{N}}_{+}+1)}^{2(k+1)}({\mathcal{H}}_N+1)& ={({\mathcal{N}}_{+}+1)}^{k+1}({\mathcal{H}}_N+1){({\mathcal{N}}_{+}+1)}^{k+1}\hfill \\ \hfill & \le C{N}^{\kappa }{({\mathcal{N}}_{+}+1)}^{k+1}{\mathcal{G}}_N^{\prime }{({\mathcal{N}}_{+}+1)}^{k+1}\hfill \\ \hfill & -C{N}^{\kappa }{({\mathcal{N}}_{+}+1)}^{k+1}{e}^A{e}^D{\mathcal{E}}_{{\mathcal{M}}_N}{e}^{-D}{e}^{-A}{({\mathcal{N}}_{+}+1)}^{k+1}\hfill \\ \hfill & +C{N}^{17\kappa +4\epsilon }{({\mathcal{N}}_{+}+1)}^{2(k+1)}\hfill \end{array}\end{array}$$ 6.17

We estimate the first term on the r.h.s. by

$$\begin{array}{c}\hfill \begin{array}{cc}& N^{\kappa }{({\mathcal{N}}_{+}+1)}^{k+1}{\mathcal{G}}_N^{\prime }{({\mathcal{N}}_{+}+1)}^{k+1}\hfill \\ \hfill & \le N^{\kappa }{({\mathcal{N}}_{+}+1)}^{2(k+1)}{\mathcal{G}}_N^{\prime }+N^{\kappa }{({\mathcal{N}}_{+}+1)}^{k+1}[{\mathcal{G}}_N^{\prime },{({\mathcal{N}}_{+}+1)}^{k+1}]\hfill \\ \hfill & =N^{\kappa }{({\mathcal{N}}_{+}+1)}^{2(k+1)}{\mathcal{G}}_N^{\prime }\hfill \\ \hfill & +N^{\kappa }\sum _{j=1}^{k+1}\left(\genfrac{}{}{0pt}{}{k+1}{j}\right){({\mathcal{N}}_{+}+1)}^{k+1}{\text{ad}}_{{\mathcal{N}}_{+}}^{(j)}({\mathcal{G}}_N){({\mathcal{N}}_{+}+1)}^{k+1-j}\hfill \end{array}\end{array}$$

By Cauchy–Schwarz, we find

$$\begin{array}{c}\hfill \begin{array}{cc}& N^{\kappa }{({\mathcal{N}}_{+}+1)}^{k+1}{\mathcal{G}}_N^{\prime }{({\mathcal{N}}_{+}+1)}^{k+1}\hfill \\ \hfill & \le N^{\kappa }{({\mathcal{N}}_{+}+1)}^{2(k+1)}+N^{\kappa }{\mathcal{G}}_N^{\prime }{({\mathcal{N}}_{+}+1)}^{2(k+1)}{\mathcal{G}}_N^{\prime }\hfill \\ \hfill & +N^{\kappa }\sum _{j=1}^{k+1}\left(\genfrac{}{}{0pt}{}{k+1}{j}\right){({\mathcal{N}}_{+}+1)}^{k+1}{\text{ad}}_{{\mathcal{N}}_{+}}^{(j)}({\mathcal{G}}_N){({\mathcal{N}}_{+}+1)}^{k+1-j}\hfill \end{array}\end{array}$$

With ${({\mathcal{N}}_{+}+1)}^{2(k+1)}\le {({\mathcal{N}}_{+}+1)}^{2k+1}({\mathcal{H}}_N+1)$ and with the estimate

$$\begin{array}{c}\hfill \Vert {({\mathcal{H}}_N+1)}^{-1/2}{\text{ad}}_{{\mathcal{N}}_{+}}^{(j)}({\mathcal{G}}_N){({\mathcal{H}}_N+1)}^{-1/2}\Vert \le C{N}^{7\kappa /3+2\epsilon /3}\end{array}$$ 6.18

from (3.31) we obtain, using again Cauchy–Schwarz,

$$\begin{array}{c}\hfill \begin{array}{cc}& N^{\kappa }⟨\xi ,{({\mathcal{N}}_{+}+1)}^{k+1}{\mathcal{G}}_N^{\prime }{({\mathcal{N}}_{+}+1)}^{k+1}\xi ⟩\hfill \\ \hfill & \le C\left[N^{\kappa },{\zeta }^2,+,N^{7\kappa /3+2\epsilon /3}\right]{\Vert \xi \Vert }^2\hfill \\ \hfill & \times {\left[\underset{\xi \in Q_{\zeta }}{sup},\frac{⟨\xi ,{({\mathcal{N}}_{+}+1)}^{2(k+1)}({\mathcal{H}}_N+1)\xi ⟩}{{\Vert \xi \Vert }^2}\right]}^{1/2}\hfill \\ \hfill & {\left[\underset{\xi \in Q_{\zeta }}{sup},\frac{⟨\xi ,{({\mathcal{N}}_{+}+1)}^{2k}({\mathcal{H}}_N+1)\xi ⟩}{{\Vert \xi \Vert }^2}\right]}^{1/2}\hfill \end{array}\end{array}$$

for every $\xi \in Q_{\zeta }$. Hence, for any $\delta >0$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}& N^{\kappa }\frac{⟨\xi ,{({\mathcal{N}}_{+}+1)}^{k+1}{\mathcal{G}}_N^{\prime }{({\mathcal{N}}_{+}+1)}^{k+1}\xi ⟩}{{\Vert \xi \Vert }^2}\hfill \\ \hfill & \le \delta \underset{\xi \in Q_{\zeta }}{sup}\frac{⟨\xi ,{({\mathcal{N}}_{+}+1)}^{2(k+1)}({\mathcal{H}}_N+1)\xi ⟩}{{\Vert \xi \Vert }^2}\hfill \\ \hfill & +C{\delta }^{-1}{\left[N^{\kappa },{\zeta }^2,+,N^{7\kappa /3+2\epsilon /3}\right]}^2\underset{\xi \in Q_{\zeta }}{sup}\frac{⟨\xi ,{({\mathcal{N}}_{+}+1)}^{2k}({\mathcal{H}}_N+1)\xi ⟩}{{\Vert \xi \Vert }^2}\hfill \end{array}\end{array}$$ 6.19

To bound the contribution proportional to $e^A{e}^D{\mathcal{E}}_{{\mathcal{M}}_N}{e}^{-D}{e}^{-A}$ on the r.h.s. of (6.17), we have to control, according to (6.8), terms of the form

$$\begin{array}{c}\hfill \begin{array}{cc}& {({\mathcal{N}}_{+}+1)}^{k+1}{\mathcal{N}}_{\ge c{N}^{\gamma }}{\mathcal{G}}_N^{\prime }{\mathcal{N}}_{\ge c{N}^{\gamma }}{({\mathcal{N}}_{+}+1)}^{k+1}\hfill \\ \hfill & ={({({\mathcal{N}}_{+}+1)}^{k+1}{\mathcal{N}}_{\ge c{N}^{\gamma }})}^2{\mathcal{G}}_N^{\prime }+{({\mathcal{N}}_{+}+1)}^{k+1}{\mathcal{N}}_{\ge c{N}^{\gamma }}\left[{\mathcal{G}}_N^{\prime },,,{({\mathcal{N}}_{+}+1)}^{k+1},{\mathcal{N}}_{\ge c{N}^{\gamma }}\right]\hfill \\ \hfill & =:\text{A}+\text{B}\hfill \end{array}\end{array}$$

For an arbitrary $\xi \in Q_{\zeta }$, we can bound the expectation of $\text{A}$ by Cauchy–Schwarz as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{⟨\xi ,\text{A}\xi ⟩}{{\Vert \xi \Vert }^2}& \le \frac{⟨\xi ,{({({\mathcal{N}}_{+}+1)}^{k+1}{\mathcal{N}}_{\ge c{N}^{\gamma }})}^2\xi ⟩}{{\Vert \xi \Vert }^2}+\frac{⟨{\mathcal{G}}_N^{\prime }\xi ,{({({\mathcal{N}}_{+}+1)}^{k+1}{\mathcal{N}}_{\ge c{N}^{\gamma }})}^2{\mathcal{G}}_N^{\prime }\xi ⟩}{{\Vert \xi \Vert }^2}\hfill \\ \hfill & \le N^2(1+{\zeta }^2)\underset{\xi \in Q_{\zeta }}{sup}\frac{⟨\xi ,{({\mathcal{N}}_{+}+1)}^{2k}{\mathcal{N}}_{\ge c{N}^{\gamma }}^2\xi ⟩}{{\Vert \xi \Vert }^2}\hfill \\ \hfill & \le N^{2-2\gamma }(1+{\zeta }^2)\underset{\xi \in Q_{\zeta }}{sup}\frac{⟨\xi ,{({\mathcal{N}}_{+}+1)}^{2k+1}\mathcal{K}\xi ⟩}{{\Vert \xi \Vert }^2}\hfill \\ \hfill & \le N^{2-2\gamma }(1+{\zeta }^2){\left[\underset{\xi \in Q_{\zeta }}{sup},\frac{⟨\xi ,{({\mathcal{N}}_{+}+1)}^{2k}\mathcal{K}\xi ⟩}{{\Vert \xi \Vert }^2}\right]}^{1/2}\hfill \\ \hfill & \times {\left[\underset{\xi \in Q_{\zeta }}{sup},\frac{⟨\xi ,{({\mathcal{N}}_{+}+1)}^{2(k+1)}\mathcal{K}\xi ⟩}{{\Vert \xi \Vert }^2}\right]}^{1/2}\hfill \end{array}\end{array}$$ 6.20

As for the term $\text{B}$, we can write

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \text{B}=& {({\mathcal{N}}_{+}+1)}^{k+1}{\mathcal{N}}_{\ge c{N}^{\gamma }}^2\left[{\mathcal{G}}_N^{\prime },,,{({\mathcal{N}}_{+}+1)}^{k+1}\right]\hfill \\ \hfill & +{({\mathcal{N}}_{+}+1)}^{k+1}{\mathcal{N}}_{\ge c{N}^{\gamma }}\left[{\mathcal{G}}_N^{\prime },,,{\mathcal{N}}_{\ge c{N}^{\gamma }}\right]{({\mathcal{N}}_{+}+1)}^{k+1}\hfill \\ \hfill =& \sum _{j=1}^{k+1}\left(\genfrac{}{}{0pt}{}{k+1}{j}\right){({\mathcal{N}}_{+}+1)}^{k+1}{\mathcal{N}}_{\ge c{N}^{\gamma }}^2{\text{ad}}_{{\mathcal{N}}_{+}}^{(j)}({\mathcal{G}}_N^{\prime }){({\mathcal{N}}_{+}+1)}^{k+1-j}\hfill \\ \hfill & +{({\mathcal{N}}_{+}+1)}^{k+1}{\mathcal{N}}_{\ge c{N}^{\gamma }}\left[{\mathcal{G}}_N^{\prime },,,{\mathcal{N}}_{\ge c{N}^{\gamma }}\right]{({\mathcal{N}}_{+}+1)}^{k+1}\hfill \end{array}\end{array}$$

From (6.18) and using (3.30) to estimate

$$\begin{array}{c}\hfill \Vert {({\mathcal{H}}_N+1)}^{-1/2}[{\mathcal{N}}_{\ge c{N}^{\gamma }},{\mathcal{G}}_N]{({\mathcal{H}}_N+1)}^{-1/2}\Vert \le C{N}^{8\kappa +2\epsilon -\gamma }+C{N}^{\kappa +\gamma /2},\end{array}$$

we obtain for every $\xi \in Q_{\zeta }$ that

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,\text{B}\xi ⟩|\hfill \\ \hfill & \le C{N}^{7\kappa /3+2\epsilon /3}\Vert {({\mathcal{H}}_N+1)}^{1/2}{\mathcal{N}}_{\ge c{N}^{\gamma }}^2{({\mathcal{N}}_{+}+1)}^{k+1}\xi \Vert \Vert {({\mathcal{H}}_N+1)}^{1/2}{({\mathcal{N}}_{+}+1)}^k\xi \Vert \hfill \\ \hfill & +C{N}^{8\kappa +2\epsilon -\gamma }\Vert {({\mathcal{H}}_N+1)}^{1/2}{\mathcal{N}}_{\ge c{N}^{\gamma }}{({\mathcal{N}}_{+}+1)}^{k+1}\xi \Vert \Vert {({\mathcal{H}}_N+1)}^{1/2}{({\mathcal{N}}_{+}+1)}^{k+1}\xi \Vert \hfill \\ \hfill & +C{N}^{\kappa +\gamma /2}\Vert {({\mathcal{H}}_N+1)}^{1/2}{\mathcal{N}}_{\ge c{N}^{\gamma }}{({\mathcal{N}}_{+}+1)}^{k+1}\xi \Vert \Vert {({\mathcal{H}}_N+1)}^{1/2}{({\mathcal{N}}_{+}+1)}^{k+1}\xi \Vert .\hfill \end{array}\end{array}$$

Applying the bounds ${\mathcal{N}}_{+}\le N$, ${\mathcal{N}}_{\ge c{N}^{\gamma }}\le C{N}^{-2\gamma }\mathcal{K}$ and (6.3) yields on the one hand

$$\begin{array}{c}\hfill \begin{array}{cc}& \Vert {({\mathcal{H}}_N+1)}^{1/2}{\mathcal{N}}_{\ge c{N}^{\gamma }}{({\mathcal{N}}_{+}+1)}^{k+1}\xi \Vert \Vert {({\mathcal{H}}_N+1)}^{1/2}{({\mathcal{N}}_{+}+1)}^{k+1}\xi \Vert \hfill \\ \hfill & \le C\Vert {\mathcal{G}}_N^{\prime }{\mathcal{N}}_{\ge c{N}^{\gamma }}{({\mathcal{N}}_{+}+1)}^{k+1}\xi \Vert \Vert {({\mathcal{H}}_N+1)}^{1/2}{({\mathcal{N}}_{+}+1)}^{k+1}\xi \Vert \hfill \\ \hfill & +C{N}^{1+\kappa /2-\gamma }{\Vert {({\mathcal{H}}_N+1)}^{1/2}{({\mathcal{N}}_{+}+1)}^{k+1}\xi \Vert }^2\hfill \\ \hfill & \le \delta ⟨\xi ,{({\mathcal{N}}_{+}+1)}^{k+1}{\mathcal{N}}_{\ge c{N}^{\gamma }}{\mathcal{G}}_N^{\prime }{\mathcal{N}}_{\ge c{N}^{\gamma }}{({\mathcal{N}}_{+}+1)}^{k+1}\xi ⟩\hfill \\ \hfill & +C({\delta }^{-1}+N^{1+\kappa /2-\gamma }){\Vert {({\mathcal{H}}_N+1)}^{1/2}{({\mathcal{N}}_{+}+1)}^{k+1}\xi \Vert }^2\hfill \end{array}\end{array}$$

for any $\delta >0$. Since $8\kappa +2\epsilon -\gamma \le 1+\kappa /2-\gamma$ and $\kappa +\gamma /2\le 1+\kappa /2-\gamma$ for all $\gamma \le \alpha$ if $\kappa <1/43$, this implies with the choice $\delta =\frac{1}{4}{(N^{8\kappa +2\epsilon -\gamma }+N^{\kappa +\gamma /2})}^{-1}$ that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,\text{B}\xi ⟩|\le & C{N}^{7\kappa /3+2\epsilon /3}\Vert {({\mathcal{H}}_N+1)}^{1/2}{\mathcal{N}}_{\ge c{N}^{\gamma }}^2{({\mathcal{N}}_{+}+1)}^{k+1}\xi \Vert \Vert {({\mathcal{H}}_N+1)}^{1/2}{({\mathcal{N}}_{+}+1)}^k\xi \Vert \hfill \\ \hfill & +C(N^{1+17\kappa /2+2\epsilon -\gamma }+N^{1+3\kappa /2-\gamma /2}){\Vert {({\mathcal{H}}_N+1)}^{1/2}{({\mathcal{N}}_{+}+1)}^{k+1}\xi \Vert }^2\hfill \\ \hfill & +\frac{1}{4}⟨\xi ,{({\mathcal{N}}_{+}+1)}^{k+1}{\mathcal{N}}_{\ge c{N}^{\gamma }}{\mathcal{G}}_N^{\prime }{\mathcal{N}}_{\ge c{N}^{\gamma }}{({\mathcal{N}}_{+}+1)}^{k+1}\xi ⟩.\hfill \end{array}\end{array}$$ 6.21

On the other hand, we can estimate

$$\begin{array}{c}\hfill \begin{array}{cc}& \Vert {({\mathcal{H}}_N+1)}^{1/2}{\mathcal{N}}_{\ge c{N}^{\gamma }}^2{({\mathcal{N}}_{+}+1)}^{k+1}\xi \Vert \hfill \\ \hfill & {\le N\Vert (\mathcal{K}+1)}^{1/2}{\mathcal{N}}_{\ge c{N}^{\gamma }}{({\mathcal{N}}_{+}+1)}^{k+1}\xi \Vert +\Vert {\mathcal{V}}_N^{1/2}{\mathcal{N}}_{\ge c{N}^{\gamma }}^2{({\mathcal{N}}_{+}+1)}^{k+1}\xi \Vert .\hfill \end{array}\end{array}$$ 6.22

Expressing ${\mathcal{V}}_N$ in position space, we find, with $\varphi ={\mathcal{N}}_{\ge c{N}^{\gamma }}{({\mathcal{N}}_{+}+1)}^{k+1}\xi$,

$$\begin{array}{c}\hfill \Vert {\mathcal{V}}_N^{1/2}{\mathcal{N}}_{\ge c{N}^{\gamma }}{\varphi \Vert }^2=\int dxdy{N}^{2-2\kappa }V(N^{1-\kappa }(x-y)){\Vert {\check{a}}_x{\check{a}}_y{\mathcal{N}}_{\ge c{N}^{\gamma }}\varphi \Vert }^2\end{array}$$ 6.23

We have

$$\begin{array}{c}\hfill {\check{a}}_x{\mathcal{N}}_{\ge c{N}^{\gamma }}=({\mathcal{N}}_{\ge c{N}^{\gamma }}+1){\check{a}}_x-a({\check{\chi }}_x)\end{array}$$

where

$$\begin{array}{c}\hfill {\check{\chi }}_x(y)=\check{\chi }(y-x)=\sum _{p\in {\mathrm{\Lambda }}_{+}^{\ast }:|p|\le c{N}^{\gamma }}{e}^{ip·(x-y)}\end{array}$$

is such that $\Vert {\check{\chi }}_x\Vert =\Vert \chi \Vert \le C{N}^{3\gamma /2}$. Hence, we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \Vert {\check{a}}_x{\check{a}}_y& {\mathcal{N}}_{\ge c{N}^{\gamma }}\varphi \Vert \le N\Vert {\check{a}}_x{\check{a}}_y\varphi \Vert +N^{1/2}\Vert {\check{\chi }}_x\Vert \Vert {\check{a}}_y\varphi \Vert +N^{1/2}\Vert {\check{\chi }}_y\Vert \Vert {\check{a}}_x\varphi \Vert .\hfill \end{array}\end{array}$$

Inserting in (6.23), we find

$$\begin{array}{c}\hfill \Vert {\mathcal{V}}_N^{1/2}{\mathcal{N}}_{\ge c{N}^{\gamma }}{\varphi \Vert }^2\le C{N}^2\Vert {\mathcal{V}}_N^{1/2}{\varphi \Vert }^2+C{N}^{3\gamma +\kappa }{\Vert {\mathcal{N}}_{+}^{1/2}\varphi \Vert }^2.\end{array}$$

From (6.22), we conclude that

$$\begin{array}{c}\hfill \Vert {({\mathcal{H}}_N+1)}^{1/2}{\mathcal{N}}_{\ge c{N}^{\gamma }}^2{\mathcal{N}}_{+}^{k+1}\xi \Vert \le N\Vert {({\mathcal{H}}_N+1)}^{1/2}{\mathcal{N}}_{\ge c{N}^{\gamma }}{({\mathcal{N}}_{+}+1)}^{k+1}\xi \Vert \end{array}$$

for all $\gamma \le \alpha =14\kappa +4\epsilon$, if $\kappa <1/43$. Using now similar arguments as before (6.21), we conclude that together with (6.21), we have

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,\text{B}\xi ⟩|\hfill \\ \hfill & \le \frac{1}{2}⟨\xi ,{({\mathcal{N}}_{+}+1)}^{k+1}{\mathcal{N}}_{\ge c{N}^{\gamma }}{\mathcal{G}}_N^{\prime }{\mathcal{N}}_{\ge c{N}^{\gamma }}{({\mathcal{N}}_{+}+1)}^{k+1}\xi ⟩\hfill \\ \hfill & +C{N}^{2+10\kappa /3+2\epsilon /3-\gamma }\Vert {({\mathcal{H}}_N+1)}^{1/2}{({\mathcal{N}}_{+}+1)}^{k+1}\xi \Vert \Vert {({\mathcal{H}}_N+1)}^{1/2}{({\mathcal{N}}_{+}+1)}^k\xi \Vert \hfill \\ \hfill & +C{N}^{2+14\kappa /3+4\epsilon /3}{\Vert {({\mathcal{H}}_N+1)}^{1/2}{({\mathcal{N}}_{+}+1)}^k\xi \Vert }^2\hfill \\ \hfill & +C(N^{1+17\kappa /2+2\epsilon -2\gamma }+N^{1+3\kappa /2-\gamma /2}){\Vert {({\mathcal{H}}_N+1)}^{1/2}{({\mathcal{N}}_{+}+1)}^{k+1}\xi \Vert }^2\hfill \\ \hfill \end{array}\end{array}$$

Combining this with (6.20), we arrive at

$$\begin{array}{c}\hfill \begin{array}{cc}& \frac{⟨\xi ,{({\mathcal{N}}_{+}+1)}^{k+1}{\mathcal{N}}_{\ge c{N}^{\gamma }}{\mathcal{G}}_N^{\prime }{\mathcal{N}}_{\ge c{N}^{\gamma }}{({\mathcal{N}}_{+}+1)}^{k+1}\xi ⟩}{{\Vert \xi \Vert }^2}\hfill \\ \hfill & \le \left[N^{2-2\gamma },{\zeta }^2,+,N^{2+10\kappa /3+2\epsilon /3-\gamma }\right]{\left[\underset{\xi \in Q_{\zeta }}{sup},\frac{⟨\xi ,{({\mathcal{N}}_{+}+1)}^{2k}({\mathcal{H}}_N+1)\xi ⟩}{{\Vert \xi \Vert }^2}\right]}^{1/2}\hfill \\ \hfill & \times {\left[\underset{\xi \in Q_{\zeta }}{sup},\frac{⟨\xi ,{({\mathcal{N}}_{+}+1)}^{2(k+1)}({\mathcal{H}}_N+1)\xi ⟩}{{\Vert \xi \Vert }^2}\right]}^{1/2}\hfill \\ \hfill & +C{N}^{2+14\kappa /3+4\epsilon /3}\left[\underset{\xi \in Q_{\zeta }}{sup},\frac{⟨\xi ,{({\mathcal{N}}_{+}+1)}^{2k}({\mathcal{H}}_N+1)\xi ⟩}{{\Vert \xi \Vert }^2}\right]\hfill \\ \hfill & +C(N^{1+17\kappa /2+2\epsilon -2\gamma }+N^{1+3\kappa /2-\gamma /2})\left[\underset{\xi \in Q_{\zeta }}{sup},\frac{⟨\xi ,{({\mathcal{N}}_{+}+1)}^{2(k+1)}({\mathcal{H}}_N+1)\xi ⟩}{{\Vert \xi \Vert }^2}\right]\hfill \end{array}\end{array}$$

for all $\xi \in Q_z$. With (6.8), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& \frac{N^{-1}⟨\xi ,{({\mathcal{N}}_{+}+1)}^{k+1}\mathcal{K}{\mathcal{N}}_{\ge c{N}^{\gamma }}{({\mathcal{N}}_{+}+1)}^{k+1}\xi ⟩}{{\Vert \xi \Vert }^2}\hfill \\ \hfill & \le C{N}^{\kappa -2\gamma }\frac{⟨\xi ,{({\mathcal{N}}_{+}+1)}^{k+1}\mathcal{K}{({\mathcal{N}}_{+}+1)}^{k+1}\xi ⟩}{{\Vert \xi \Vert }^2}\hfill \\ \hfill & +C\left[N^{-\kappa },{\zeta }^2,+,N^{\gamma +7\kappa /3+2\epsilon /3}\right]{\left[\underset{\xi \in Q_{\zeta }}{sup},\frac{⟨\xi ,{({\mathcal{N}}_{+}+1)}^{2k}({\mathcal{H}}_N+1)\xi ⟩}{{\Vert \xi \Vert }^2}\right]}^{1/2}\hfill \\ \hfill & \times {\left[\underset{\xi \in Q_{\zeta }}{sup},\frac{⟨\xi ,{({\mathcal{N}}_{+}+1)}^{2(k+1)}({\mathcal{H}}_N+1)\xi ⟩}{{\Vert \xi \Vert }^2}\right]}^{1/2}\hfill \\ \hfill & +C{N}^{2\gamma +11\kappa /3+4\epsilon /3}\left[\underset{\xi \in Q_{\zeta }}{sup},\frac{⟨\xi ,{({\mathcal{N}}_{+}+1)}^{2k}({\mathcal{H}}_N+1)\xi ⟩}{{\Vert \xi \Vert }^2}\right]\hfill \\ \hfill & +C(N^{15\kappa /2+2\epsilon -1}+N^{\kappa /2+3\gamma /2-1})\left[\underset{\xi \in Q_{\zeta }}{sup},\frac{⟨\xi ,{({\mathcal{N}}_{+}+1)}^{2(k+1)}({\mathcal{H}}_N+1)\xi ⟩}{{\Vert \xi \Vert }^2}\right].\hfill \end{array}\end{array}$$

Applying this bound to (5.8) and recalling that $\kappa <1/43$, we conclude that

$$\begin{array}{c}\hfill \begin{array}{cc}& \frac{N^{\kappa }⟨\xi ,{({\mathcal{N}}_{+}+1)}^{k+1}{e}^A{e}^D{\mathcal{E}}_{{\mathcal{M}}_N}{e}^{-D}{e}^{-A}{({\mathcal{N}}_{+}+1)}^{k+1}\xi ⟩}{{\Vert \xi \Vert }^2}\hfill \\ \hfill & \ge -C{N}^{-\epsilon }\left[\underset{\xi \in Q_{\zeta }}{sup},\frac{⟨\xi ,({\mathcal{H}}_N+1){({\mathcal{N}}_{+}+1)}^{2(k+1)}\xi ⟩}{{\Vert \xi \Vert }^2}\right]\hfill \\ \hfill & -C\left[N^{20\kappa +5\epsilon },{\zeta }^2,+,N^{44\kappa +12\epsilon }\right]{\left[\underset{\xi \in Q_{\zeta }}{sup},\frac{⟨\xi ,{({\mathcal{N}}_{+}+1)}^{2k}({\mathcal{H}}_N+1)\xi ⟩}{{\Vert \xi \Vert }^2}\right]}^{1/2}\hfill \\ \hfill & \times {\left[\underset{\xi \in Q_{\zeta }}{sup},\frac{⟨\xi ,{({\mathcal{N}}_{+}+1)}^{2(k+1)}({\mathcal{H}}_N+1)\xi ⟩}{{\Vert \xi \Vert }^2}\right]}^{1/2}.\hfill \end{array}\end{array}$$

Therefore, for any $\delta >0$, we find (if N is large enough)

$$\begin{array}{c}\hfill \begin{array}{cc}& \frac{N^{\kappa }⟨\xi ,{({\mathcal{N}}_{+}+1)}^{k+1}{e}^A{e}^D{\mathcal{E}}_{{\mathcal{M}}_N}{e}^{-D}{e}^{-A}{({\mathcal{N}}_{+}+1)}^{k+1}\xi ⟩}{{\Vert \xi \Vert }^2}\hfill \\ \hfill & \ge -\delta \underset{\xi \in Q_{\zeta }}{sup}\frac{⟨\xi ,({\mathcal{H}}_N+1){({\mathcal{N}}_{+}+1)}^{2(k+1)}\xi ⟩}{{\Vert \xi \Vert }^2}\hfill \\ \hfill & -C{\delta }^{-1}{\left[N^{20\kappa +5\epsilon },{\zeta }^2,+,N^{44\kappa +12\epsilon }\right]}^2\underset{\xi \in Q_{\zeta }}{sup}\frac{⟨\xi ,({\mathcal{H}}_N+1){({\mathcal{N}}_{+}+1)}^{2k}\xi ⟩}{{\Vert \xi \Vert }^2}.\hfill \end{array}\end{array}$$

From the last bound, (6.19) and (6.17), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& \frac{⟨\xi ,{({\mathcal{N}}_{+}+1)}^{2(k+1)}({\mathcal{H}}_N+1)\xi ⟩}{{\Vert \xi \Vert }^2}\hfill \\ \hfill & \le \delta \underset{\xi \in Q_{\zeta }}{sup}\frac{⟨\xi ,{({\mathcal{N}}_{+}+1)}^{2(k+1)}({\mathcal{H}}_N+1)\xi ⟩}{{\Vert \xi \Vert }^2}\hfill \\ \hfill & +C{\delta }^{-1}{\left[N^{20\kappa +5\epsilon },{\zeta }^2,+,N^{44\kappa +12\epsilon }\right]}^2\underset{\xi \in Q_{\zeta }}{sup}\frac{⟨\xi .,{({\mathcal{N}}_{+}+1)}^{2k}({\mathcal{H}}_N+1)\xi ⟩}{{\Vert \xi \Vert }^2}\hfill \end{array}\end{array}$$

for any $\xi \in Q_{\zeta }$. Taking the supremum over all $\xi \in Q_{\zeta }$, and choosing $\delta >0$ small enough, we arrive at

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \underset{\xi \in Q_{\zeta }}{sup}& \frac{⟨\xi ,{({\mathcal{N}}_{+}+1)}^{2(k+1)}({\mathcal{H}}_N+1)\xi ⟩}{{\Vert \xi \Vert }^2}\hfill \\ \hfill & \le C{\left[N^{20\kappa +5\epsilon },{\zeta }^2,+,N^{44\kappa +12\epsilon }\right]}^2\underset{\xi \in Q_{\zeta }}{sup}\frac{⟨\xi ,{({\mathcal{N}}_{+}+1)}^{2k}({\mathcal{H}}_N+1)\xi ⟩}{{\Vert \xi \Vert }^2}\hfill \\ \hfill & \le C{\left[N^{20\kappa +5\epsilon },{\zeta }^2,+,N^{44\kappa +12\epsilon }\right]}^{2k+1}\hfill \end{array}\end{array}$$

by the induction assumption. $\square$

Analysis of ${\mathcal{M}}_N$

This section is devoted to the proof of Proposition 5.1. In Sect. 7.1 we establish bounds on the growth of the number of excitations and of their energy with respect to the action of $e^D$, with the quartic operator $D=D_1-D_1^{\ast }$ with

$$\begin{array}{c}\hfill D_1=\frac{1}{2N}\sum _{r\in P_H,p,q\in P_L}{\eta }_r{a}_{p+r}^{\ast }{a}_{q-r}^{\ast }{a}_p{a}_q\end{array}$$ 7.1

as defined in (5.3). In Sect. 7.2, we compute the different parts of the excitation Hamiltonian ${\mathcal{M}}_N$, introduced in (5.5). Finally, in Sect. 7.3, we conclude the proof of Proposition 5.1.

Growth of Number and Energy of Excitations

The first lemma of this section controls the growth of the number of excitations with high momentum.

Lemma 7.1

Assume the exponents $\alpha ,\beta$ satisfy (5.6). Let $k\in {\mathbb{N}}_0$, $m=1,2,3$, $0<\gamma \le \alpha$ and $c>0$ ($c<1$ if $\gamma =\alpha$). Then, there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^{-sD}{({\mathcal{N}}_{+}+1)}^k{({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)}^m{e}^{\mathit{sD}}\le & C{({\mathcal{N}}_{+}+1)}^k{({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)}^m,\hfill \end{array}\end{array}$$ 7.2

for all $s\in [-1;1]$ and all $N\in \mathbb{N}$ large enough.

Proof

Since $[{\mathcal{N}}_{+},{\mathcal{N}}_{\ge c{N}^{\gamma }}]=0$ and $[{\mathcal{N}}_{+},D]=0$, it is enough to prove the lemma for $k=0$. We consider first $m=1$. For $\xi \in {\mathcal{F}}_{+}^{\le N}$, we define the function ${\phi }_{\xi }:\mathbb{R}\to \mathbb{R}$ by

$$\begin{array}{c}\hfill {\phi }_{\xi }(s)=⟨\xi ,e^{-sD}({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)e^{\mathit{sD}}\xi ⟩\end{array}$$

so that differentiating yields

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\partial }_s{\phi }_{\xi }(s)& =2\mathrm{Re}⟨e^{\mathit{sD}}\xi ,[{\mathcal{N}}_{\ge c{N}^{\gamma }},D_1]e^{\mathit{sD}}\xi ⟩\hfill \end{array}\end{array}$$ 7.3

with $D_1$ as in (7.1). By assumption, $N^{\alpha }\ge N^{\alpha }-N^{\beta }\ge c{N}^{\gamma }$ for sufficiently large $N\in \mathbb{N}$. This implies that

$$\begin{array}{c}\hfill [{\mathcal{N}}_{\ge c{N}^{\gamma }},a_{p+r}^{\ast }]=a_{p+r}^{\ast },[{\mathcal{N}}_{\ge c{N}^{\gamma }},a_{q-r}^{\ast }]=a_{q-r}^{\ast }\end{array}$$

for $r\in P_H$ and $p,q\in P_L$, by (2.1) and (2.10). We then compute

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [{\mathcal{N}}_{\ge c{N}^{\gamma }},D_1]& =\frac{1}{N}\sum _{\begin{array}{c}r\in P_H,p,q\in P_L\end{array}}{\eta }_r{a}_{p+r}^{\ast }{a}_{q-r}^{\ast }{a}_p{a}_q-\frac{1}{N}\sum _{\begin{array}{c}r\in P_H,p,q\in P_L,\\ |p|\ge c{N}^{\gamma }\end{array}}{\eta }_r{a}_{p+r}^{\ast }{a}_{q-r}^{\ast }{a}_p{a}_q.\hfill \end{array}\end{array}$$ 7.4

and apply Cauchy–Schwarz to obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |{\partial }_s{\phi }_{\xi }(s)|& \le \frac{C}{N}(\sum _{\begin{array}{c}r\in P_H,p,q\in P_L,\\ |p+r|\ge c{N}^{\gamma },|q-r|\ge c{N}^{\gamma }\end{array}}{\Vert a_{p+r}{({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)}^{-1/2}{a}_{q-r}{e}^{\mathit{sD}}\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill & \times \Vert {\eta }_H\Vert (\sum _{\begin{array}{c}p,q\in P_L\end{array}}{\Vert a_p{({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)}^{1/2}{a}_q{e}^{\mathit{sD}}\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill & \le C{N}^{\kappa +3\beta /2-\alpha /2}{\phi }_{\xi }(s)\le C{\phi }_{\xi }(s).\hfill \end{array}\end{array}$$ 7.5

Since the bound is independent of $\xi \in {\mathcal{F}}_{+}^{\le N}$ and it also holds true if we replace D by $-D$ in the definition of ${\phi }_{\xi }$, this proves (7.2), for $m=1$.

For $m=3$, we define

$$\begin{array}{c}\hfill {\psi }_{\xi }(s)=⟨\xi ,e^{-sD}{({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)}^3{e}^{\mathit{sD}}\xi ⟩\end{array}$$

with derivative

$$\begin{array}{c}\hfill {\partial }_s{\psi }_{\xi }(s)=2\mathrm{Re}⟨e^{\mathit{sD}}\xi ,[{({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)}^3,D_1]e^{\mathit{sD}}\xi ⟩\end{array}$$

We have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [{({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)}^3,D_1]=& 3({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)[{\mathcal{N}}_{\ge c{N}^{\gamma }},D_1]({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)\hfill \\ \hfill & +[{\mathcal{N}}_{\ge c{N}^{\gamma }},[{\mathcal{N}}_{\ge c{N}^{\gamma }},[{\mathcal{N}}_{\ge c{N}^{\gamma }},D_1]]].\hfill \end{array}\end{array}$$ 7.6

The contribution of the first term on the r.h.s. of (7.6) can be controlled as in (7.5) (replacing $e^{\mathit{sD}}\xi$ with $({\mathcal{N}}_{\ge c{N}^{\gamma }}+1)e^{\mathit{sD}}\xi$). With (7.4) and using again that $N^{\alpha }\ge N^{\alpha }-N^{\beta }\ge c{N}^{\gamma }$, we obtain that

$$\begin{array}{c}\hfill \begin{array}{cc}& [{\mathcal{N}}_{\ge c{N}^{\gamma }},[{\mathcal{N}}_{\ge c{N}^{\gamma }},[{\mathcal{N}}_{\ge c{N}^{\gamma }},D_1]]]\hfill \\ & =\frac{4}{N}\sum _{\begin{array}{c}r\in P_H,p,q\in P_L\end{array}}{\eta }_r{a}_{p+r}^{\ast }{a}_{q-r}^{\ast }{a}_p{a}_q-\frac{7}{N}\sum _{\begin{array}{c}r\in P_H,p,q\in P_L,\\ |p|\ge c{N}^{\gamma }\end{array}}{\eta }_r{a}_{p+r}^{\ast }{a}_{q-r}^{\ast }{a}_p{a}_q\hfill \\ & +\frac{3}{N}\sum _{\begin{array}{c}r\in P_H,p,q\in P_L,\\ |p|,|q|\ge c{N}^{\gamma }\end{array}}{\eta }_r{a}_{p+r}^{\ast }{a}_{q-r}^{\ast }{a}_p{a}_q.\hfill \end{array}\end{array}$$

All these contributions can be controlled like those in (7.4). We conclude that

$$\begin{array}{c}\hfill |{\partial }_s{\psi }_{\xi }(s)|\le C{\psi }_{\xi }(s)\end{array}$$

This proves (7.2) with $m=3$. The case $m=2$ follows by operator monotonicity of the function $x\mapsto x^{2/3}$. $\square$

Next, we prove bounds for the growth of the low-momentum part of the kinetic energy, defined as in (4.17).

Lemma 7.2

Assume the exponents $\alpha ,\beta$ satisfy (5.6). Let $0<{\gamma }_1,{\gamma }_2\le \alpha$, $c_1,c_2\ge 0$ (and $c_j\le 1$ if ${\gamma }_j=\alpha$, for $j=1,2$). Then, there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^{-sD}{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}}{e}^{\mathit{sD}}\le & {\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}}+N^{2\beta -1}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^2,\hfill \\ \hfill e^{-sD}{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}}({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1)e^{\mathit{sD}}\le & {\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}}({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1)\hfill \\ \hfill & +N^{2\beta -1}{({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1)}^2({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\hfill \\ \hfill \end{array}\end{array}$$ 7.7

for all $s\in [-1;1]$ and all $N\in \mathbb{N}$ sufficiently large.

Proof

Fix $\xi \in {\mathcal{F}}_{+}^{\le N}$ and define ${\phi }_{\xi }:\mathbb{R}\to \mathbb{R}$ by ${\phi }_{\xi }(s)=⟨\xi ,e^{-sD}{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}}{e}^{\mathit{sD}}\xi ⟩$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\partial }_s{\phi }_{\xi }(s)& =2\mathrm{Re}⟨\xi ,e^{-sD}[{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}},D_1]e^{\mathit{sD}}\xi ⟩.\hfill \\ \hfill \end{array}\end{array}$$

We notice that

$$\begin{array}{c}\hfill [{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}},a_{p+r}^{\ast }]=[{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}},a_{q-r}^{\ast }]=0\end{array}$$

if $r\in P_H$ and $p,q\in P_L$, because $|r|,|p+r|,|q-r|\ge N^{\alpha }-N^{\beta }>c_1{N}^{{\gamma }_1}$ for $N\in \mathbb{N}$ large enough.

Using (2.1), we then compute

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}},D_1]& =-\frac{1}{N}\sum _{\begin{array}{c}r\in P_H,p,q\in P_L:|p|\le c_1{N}^{{\gamma }_1}\end{array}}{p}^2{\eta }_r{a}_{p+r}^{\ast }{a}_{q-r}^{\ast }{a}_p{a}_q.\hfill \end{array}\end{array}$$ 7.8

and, using that $|p|\le N^{\beta }$ for $p\in P_L$, we obtain with Cauchy–Schwarz

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,e^{-sD}[{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}},D_1]e^{\mathit{sD}}\xi ⟩|\hfill \\ \hfill & \le \frac{C{N}^{\beta }}{N}\sum _{\begin{array}{c}r\in P_H,p,q\in P_L:|p|\le c_1{N}^{{\gamma }_1}\end{array}}|p||{\eta }_r|\Vert a_{r+p}{a}_{q-r}{e}^{\mathit{sD}}\xi \Vert \Vert a_p{a}_q{e}^{\mathit{sD}}\xi \Vert \hfill \\ \hfill & \le C{N}^{5\beta /2+\kappa -\alpha /2-1/2}\Vert ({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)e^{\mathit{sD}}\xi \Vert \Vert {\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}}^{1/2}{e}^{\mathit{sD}}\xi \Vert .\hfill \end{array}\end{array}$$ 7.9

With Lemma 7.1 choosing $c=\frac{1}{2}$ and $\gamma =\alpha$, this implies for $N\in \mathbb{N}$ large enough that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\partial }_s{\phi }_{\xi }(s)& \le C{N}^{5\beta /2+\kappa -\alpha /2-1/2}\Vert ({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)e^{\mathit{sD}}\xi \Vert \Vert {\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}}^{1/2}{e}^{\mathit{sD}}\xi \Vert \hfill \\ \hfill & \le C{N}^{2\beta -1}⟨\xi ,{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^2\xi ⟩+C{\phi }_{\xi }(s).\hfill \end{array}\end{array}$$

This proves the first inequality in (7.7), by Gronwall’s lemma and $\alpha >3\beta +2\kappa \ge 0$.

Next, let us prove the second inequality in (7.7). We define ${\psi }_{\xi }:\mathbb{R}\to \mathbb{R}$ by

$$\begin{array}{c}\hfill {\psi }_{\xi }(s)=⟨\xi ,e^{-sD}{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}}({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1)e^{\mathit{sD}}\xi ⟩,\end{array}$$

and we compute

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\partial }_s{\psi }_{\xi }(s)& =2\mathrm{Re}⟨\xi ,e^{-sD}[{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}},D_1]({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1)e^{\mathit{sD}}\xi ⟩\hfill \\ \hfill & +2\mathrm{Re}⟨\xi ,e^{-sD}{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}}[{\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}},D_1]e^{\mathit{sD}}\xi ⟩.\hfill \end{array}\end{array}$$

First, we proceed as in (7.9) and obtain with (4.7) that

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,e^{-sD}[{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}},D_1]({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1)e^{\mathit{sD}}\xi ⟩|\hfill \\ \hfill & \le \frac{C{N}^{\beta }}{N}\sum _{\begin{array}{c}r\in P_H,p,q\in P_L:\\ |p|\le c_1{N}^{{\gamma }_1}\end{array}}|p||{\eta }_r|\Vert a_{r+p}{a}_{q-r}{({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1)}^{1/2}{e}^{\mathit{sD}}\xi \Vert \hfill \\ \hfill & \times \Vert a_q{a}_p{({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1)}^{1/2}{e}^{\mathit{sD}}\xi \Vert \hfill \\ \hfill & \le C{N}^{5\beta /2+\kappa -\alpha /2-1/2}\Vert ({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1){({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}{e}^{\mathit{sD}}\xi \Vert \hfill \\ \hfill & \times \Vert {\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}}^{1/2}{({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1)}^{1/2}{e}^{\mathit{sD}}\xi \Vert .\hfill \end{array}\end{array}$$

Here, we used in the last step that $[a_{q-r},{\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}]=a_{q-r}$ for $r\in P_H$, $q\in P_L$ and that ${\mathcal{N}}_{c_2{N}^{{\gamma }_2}}\ge {\mathcal{N}}_{N^{\alpha }-N^{\beta }}$ for $N\in \mathbb{N}$ large enough. The last bound and Lemma 7.1 imply that

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,e^{-sD}[{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}},D_1]({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1)e^{\mathit{sD}}\xi ⟩|\hfill \\ \hfill & \le C{N}^{2\beta -1}⟨\xi ,{({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1)}^2({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\xi ⟩+C{\psi }_{\xi }(s).\hfill \end{array}\end{array}$$ 7.10

Next, we recall the identity (7.4) and that

$$\begin{array}{c}\hfill [{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}},a_{p+r}^{\ast }]=[{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}},a_{q-r}^{\ast }]=0\end{array}$$

whenever $r\in P_H,p,q\in P_L$ and $N\in \mathbb{N}$ is sufficiently large. We then obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,e^{-sD}{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}}[{\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}},D_1]e^{\mathit{sD}}\xi ⟩|\hfill \\ \hfill & \le \frac{C}{N}\sum _{\begin{array}{c}r\in P_H,p,q\in P_L,\\ v\in {\mathrm{\Lambda }}_{+}^{\ast }:|v|\le c_1{N}^{{\gamma }_1}\end{array}}{|v|}^2|{\eta }_r|\Vert a_{r+p}{({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1)}^{-1/2}{a}_{q-r}{a}_v{e}^{\mathit{sD}}\xi \Vert \hfill \\ \hfill & \times \Vert a_p{a}_q{({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1)}^{1/2}{a}_v{e}^{\mathit{sD}}\xi \Vert \hfill \\ \hfill & \le C{N}^{3\beta /2+\kappa -\alpha /2}⟨e^{\mathit{sD}}\xi ,{\mathcal{K}}_{\le c_1{N}^{{\gamma }_1}}({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1)e^{\mathit{sD}}\xi ⟩\le C{\psi }_{\xi }(s).\hfill \end{array}\end{array}$$ 7.11

Hence, putting (7.10) and (7.11) together, we have proved that

$$\begin{array}{c}\hfill {\partial }_s{\psi }_{\xi }(s)\le C{N}^{2\beta -1}⟨\xi ,{({\mathcal{N}}_{\ge c_2{N}^{{\gamma }_2}}+1)}^2({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\xi ⟩+C{\psi }_{\xi }(s),\end{array}$$

which implies the second bound in (7.7), by Gronwall’s lemma. $\square$

It will also be important to control the potential energy operator, restricted to low momenta. We define

$$\begin{array}{c}\hfill {\mathcal{V}}_{N,L}=\frac{1}{2N}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },p,q\in {\mathrm{\Lambda }}_{+}^{\ast }:\\ p+u,q+u,p,q\in P_L\end{array}}{N}^{\kappa }\widehat{V}(u/N^{1-\kappa })a_{p+u}^{\ast }{a}_q^{\ast }{a}_p{a}_{q+u}.\end{array}$$ 7.12

Notice that ${\mathcal{V}}_{N,L}={\mathcal{V}}_{N,L}^{\ast }$ by symmetry of the momentum restrictions. To calculate $e^D{\mathcal{V}}_{N,L}{e}^{-D}$, we will use the next lemma, which will also be useful in the next subsections.

Lemma 7.3

Assume the exponents $\alpha ,\beta$ satisfy (5.6). Let $F={(F_p)}_{p\in {\mathrm{\Lambda }}_{+}^{\ast }}\in {\ell }^{\infty }({\mathrm{\Lambda }}_{+}^{\ast })$ and define

$$\begin{array}{c}\hfill Z=\frac{1}{2N}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },p,q\in {\mathrm{\Lambda }}_{+}^{\ast }:\\ p+u,q+u,p,q\in P_L\end{array}}{F}_u{a}_{p+u}^{\ast }{a}_q^{\ast }{a}_p{a}_{q+u}\end{array}$$ 7.13

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

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \pm (e^{-sD}Z{e}^{\mathit{sD}}-Z)& \le {C\Vert F\Vert }_{\infty }{N}^{\beta -1}{\mathcal{K}}_L({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)+C{\Vert F\Vert }_{\infty }{N}^{3\beta -2}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^3\hfill \end{array}\end{array}$$ 7.14

for all $s\in [-1;1]$, and for all $N\in \mathbb{N}$ sufficiently large.

Proof

Given $\xi \in {\mathcal{F}}_{+}^{\le N}$, we define ${\phi }_{\xi }:\mathbb{R}\to \mathbb{R}$ by

$$\begin{array}{c}\hfill {\phi }_{\xi }(s)=⟨\xi ,e^{-sD}\text{Z}{e}^{\mathit{sD}}\xi ⟩,\end{array}$$

which has derivative

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\partial }_s{\phi }_{\xi }(s)& =2\mathrm{Re}⟨\xi ,e^{-sD}[\text{Z},D_1]e^{\mathit{sD}}\xi ⟩.\hfill \end{array}\end{array}$$

By assumption, we have $\alpha >3\beta +2\kappa$ so that $|r|,|v+r|,|w-r|\ge N^{\alpha }-N^{\beta }>N^{\beta }$ if $r\in P_H$ and $v,w\in P_L$, for sufficiently large $N\in \mathbb{N}$. This implies in particular that

$$\begin{array}{c}\hfill [a_p{a}_{q+u},a_{v+r}^{\ast }{a}_{w-r}^{\ast }]=0\end{array}$$

whenever $q+u,p\in P_L$ and $r\in P_H$, $v,w\in P_L$. As a consequence, we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [\text{Z},D_1]& =-\frac{1}{2N^2}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },r\in P_H,v,w\in P_L:\\ w-u,v+u\in P_L\end{array}}{F}_u{\eta }_r{a}_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_{w-u}{a}_{v+u}\hfill \\ \hfill & -\frac{1}{N^2}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },r\in P_H,v,w,p\in P_L:\\ p+u,v+u\in P_L\end{array}}{F}_u{\eta }_r{a}_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_{p+u}^{\ast }{a}_w{a}_{v+u}{a}_p.\hfill \end{array}\end{array}$$ 7.15

With (4.7) and $N^{\alpha }-N^{\beta }>\frac{1}{2}{N}^{\alpha }$ for $N\in \mathbb{N}$ large enough, we can bound

$$\begin{array}{c}\hfill \begin{array}{cc}& |\frac{1}{N^2}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },r\in P_H,v,w\in P_L:\\ w-u,v+u\in P_L\end{array}}{F}_u{\eta }_r⟨e^{\mathit{sD}}\xi ,a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_{w-u}{a}_{v+u}{e}^{\mathit{sD}}\xi ⟩|\hfill \\ & \le \frac{{C\Vert F\Vert }_{\infty }}{N^2}(\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },r\in P_H,v,w\in P_L:\\ w-u,v+u\in P_L\end{array}}{|v+u|}^{-2}{\Vert a_{v+r}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{-1/2}{a}_{w-r}{e}^{\mathit{sD}}\xi \Vert }^2{)}^{1/2}\hfill \\ & \times (\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },r\in P_H,v,w\in P_L:\\ w-u,v+u\in P_L\end{array}}{\eta }_r^2{|v+u|}^2{\Vert a_{w-u}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}{a}_{v+u}{e}^{\mathit{sD}}\xi \Vert }^2{)}^{1/2}\hfill \\ & \le {C\Vert F\Vert }_{\infty }{N}^{7\beta /2+\kappa -\alpha /2-3/2}\Vert {({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}{e}^{\mathit{sD}}\xi \Vert \Vert {\mathcal{K}}_L^{1/2}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}{e}^{\mathit{sD}}\xi \Vert .\hfill \end{array}\end{array}$$

and

$$\begin{array}{c}\hfill \begin{array}{cc}& |\frac{1}{N^2}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },r\in P_H,v,w,p\in P_L:\\ p+u,v+u\in P_L\end{array}}{F}_u{\eta }_r⟨e^{\mathit{sD}}\xi ,a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_{p+u}^{\ast }{a}_w{a}_{v+u}{a}_p{e}^{\mathit{sD}}\xi ⟩|\hfill \\ & \le \frac{{C\Vert F\Vert }_{\infty }}{N^2}(\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },r\in P_H,v,w,p\in P_L:\\ p+u,v+u\in P_L\end{array}}{|p+u|}^2{|p|}^{-2}\hfill \\ & \times \Vert a_{v+r}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{-1/2}{a}_{w-r}{a}_{p+u}{e}^{\mathit{sD}}{\xi \Vert }^2{)}^{1/2}\hfill \\ & \times (\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },r\in P_H,v,w,p\in P_L:\\ p+u,v+u\in P_L\end{array}}{\eta }_r^2{|p|}^2{|p+u|}^{-2}\hfill \\ & \times \Vert a_w{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}{a}_{v+u}{a}_p{e}^{\mathit{sD}}{\xi \Vert }^2{)}^{1/2}\hfill \\ & \le {C\Vert F\Vert }_{\infty }{N}^{5\beta /2+\kappa -\alpha /2-1}⟨\xi ,e^{-sD}{\mathcal{K}}_L({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)e^{\mathit{sD}}\xi ⟩.\hfill \end{array}\end{array}$$

Lemmas 7.1, 7.2 and the assumption $\alpha >3\beta +2\kappa \ge 0$ implies

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \pm {\partial }_s{\phi }_s(\xi )& \le {C\Vert F\Vert }_{\infty }{N}^{\beta -1}⟨\xi ,{\mathcal{K}}_L({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\xi ⟩\hfill \\ \hfill & +{C\Vert F\Vert }_{\infty }{N}^{3\beta -2}⟨\xi ,{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^3\xi ⟩.\hfill \end{array}\end{array}$$

Hence, integrating the last equation from zero to $s\in [-1;1]$ proves the lemma.

$\square$

With ${sup}_{p\in {\mathrm{\Lambda }}^{\ast }}|N^{\kappa }\widehat{V}(p/N^{1-\kappa })|\le C{N}^{\kappa }$, we obtain immediately the following result.

Corollary 7.4

Assume the exponents $\alpha ,\beta$ satisfy (5.6). Then, there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \pm (e^{-sD}{\mathcal{V}}_{N,L}{e}^{\mathit{sD}}-{\mathcal{V}}_{N,L})& \le C{N}^{\beta +\kappa -1}{\mathcal{K}}_L({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\hfill \\ \hfill & +C{N}^{3\beta +\kappa -2}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^3\hfill \end{array}\end{array}$$

for all $s\in [-1;1]$, and for all $N\in \mathbb{N}$ sufficiently large.

We also need rough bounds for the conjugation of the full potential energy operator ${\mathcal{V}}_N$. To this end, we will make use of the following estimate for the commutator of ${\mathcal{V}}_N$ with $D=D_1-D_1^{\ast }$, with $D_1$ defined in (7.1).

Proposition 7.5

Assume the exponents $\alpha ,\beta$ satisfy (5.6). Then,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [{\mathcal{V}}_N,D]& =\frac{1}{2N}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}_{+}^{\ast },p,q\in P_L:\\ p+u,q-u\ne 0\end{array}}{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast \eta /N)(u)(a_{p+u}^{\ast }{a}_{q-u}^{\ast }{a}_p{a}_q+h.c.)\hfill \\ \hfill & +{\mathcal{E}}_{[{\mathcal{V}}_N,D]}\hfill \end{array}\end{array}$$ 7.16

and there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \pm {\mathcal{E}}_{[{\mathcal{V}}_N,D]}& \le \delta {\mathcal{V}}_N+C{N}^{\alpha +\kappa -1}{\mathcal{V}}_N+C{N}^{\alpha +\kappa -1}{\mathcal{V}}_{N,L}\hfill \\ \hfill & +{\delta }^{-1}C{N}^{\beta +\kappa -1}{\mathcal{K}}_L({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)+{\delta }^{-1}C{N}^{3\beta +\kappa -1}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^2\hfill \end{array}\end{array}$$ 7.17

for all $\delta >0$ and for all $N\in \mathbb{N}$ sufficiently large.

Proof

We have

$$\begin{array}{c}\hfill [{\mathcal{V}}_N,D]=[{\mathcal{V}}_N,D_1]+\text{h.c.}\end{array}$$

To compute the commutator $[{\mathcal{V}}_N,D_1]$, we compute first of all that

$$\begin{array}{c}\hfill \begin{array}{cc}& [a_{p+u}^{\ast }{a}_q^{\ast }{a}_p{a}_{q+u},a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w]\hfill \\ \hfill & =a_{p+u}^{\ast }{a}_q^{\ast }{a}_{q+u}{a}_{w-r}^{\ast }{a}_v{a}_w{\delta }_{p,v+r}+a_{p+u}^{\ast }{a}_q^{\ast }{a}_p{a}_{w-r}^{\ast }{a}_v{a}_w{\delta }_{q+u,v+r}\hfill \\ \hfill & +a_{p+u}^{\ast }{a}_q^{\ast }{a}_{v+r}^{\ast }{a}_{q+u}{a}_v{a}_w{\delta }_{p,w-r}+a_{p+u}^{\ast }{a}_q^{\ast }{a}_{v+r}^{\ast }{a}_p{a}_v{a}_w{\delta }_{q+u,w-r}\hfill \\ \hfill & -a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_q^{\ast }{a}_w{a}_p{a}_{q+u}{\delta }_{p+u,v}-a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_{p+u}^{\ast }{a}_w{a}_p{a}_{q+u}{\delta }_{q,v}\hfill \\ \hfill & -a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_q^{\ast }{a}_p{a}_{q+u}{\delta }_{p+u,w}-a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_{p+u}^{\ast }{a}_p{a}_{q+u}{\delta }_{q,w}.\hfill \end{array}\end{array}$$

Putting the terms in the first and last line on the r.h.s. into normal order, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [{\mathcal{V}}_N,D_1]+\text{h.c.}=& \frac{1}{2N}\sum _{u\in {\mathrm{\Lambda }}^{\ast },v,w\in P_L}^{\ast }{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast \eta /N)(u)a_{v+u}^{\ast }{a}_{w-u}^{\ast }{a}_v{a}_w\hfill \\ \hfill & +{\mathrm{\Phi }}_1+{\mathrm{\Phi }}_2+{\mathrm{\Phi }}_3+{\mathrm{\Phi }}_4+\text{h.c.},\hfill \end{array}\end{array}$$ 7.18

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathrm{\Phi }}_1& =-\frac{1}{2N^2}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },v,w\in P_L,\\ r\in P_H^c\cup \{0\}\end{array}}^{\ast }{N}^{\kappa }\widehat{V}((u-r)/N^{1-\kappa }){\eta }_r{a}_{v+u}^{\ast }{a}_{w-u}^{\ast }{a}_v{a}_w,\hfill \\ \hfill {\mathrm{\Phi }}_2& =-\frac{1}{2N^2}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },r\in P_H,\\ v,w\in P_L\end{array}}^{\ast }{N}^{\kappa }\widehat{V}(u/N^{1-\kappa }){\eta }_r{a}_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_{w-u}{a}_{v+u},\hfill \\ \hfill {\mathrm{\Phi }}_3& =\frac{1}{N^2}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },q\in {\mathrm{\Lambda }}_{+}^{\ast },\\ r\in P_H,v,w\in P_L\end{array}}^{\ast }{N}^{\kappa }\widehat{V}(u/N^{1-\kappa }){\eta }_r{a}_{w-r+u}^{\ast }{a}_{v+r}^{\ast }{a}_q^{\ast }{a}_{q+u}{a}_v{a}_w,\hfill \\ \hfill {\mathrm{\Phi }}_4& =-\frac{1}{N^2}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },q\in {\mathrm{\Lambda }}_{+}^{\ast },\\ r\in P_H,v,w\in P_L\end{array}}^{\ast }{N}^{\kappa }\widehat{V}(u/N^{1-\kappa }){\eta }_r{a}_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_q^{\ast }{a}_w{a}_{v-u}{a}_{q+u}.\hfill \end{array}\end{array}$$ 7.19

The first term on the r.h.s. in (7.18) appears explicitly in (7.16). Hence, let us estimate the size of the operators ${\mathrm{\Phi }}_1$ to ${\mathrm{\Phi }}_4$, defined in (7.19).

Starting with ${\mathrm{\Phi }}_1$, we switch to position space and find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\mathrm{\Phi }}_1\xi ⟩|\le & \frac{1}{N}\sum _{r\in P_H^c\cup \{0\}}|{\eta }_r|({\int }_{{\mathrm{\Lambda }}^2}dxdy{N}^{2-2\kappa }V(N^{1-\kappa }(x-y)){\Vert {\check{b}}_x{\check{a}}_y\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill & \times ({\int }_{{\mathrm{\Lambda }}^2}dxdy{N}^{2-2\kappa }V(N^{1-\kappa }(x-y))\Vert \sum _{w,v\in P_L}{e}^{ivx+iwy}{a}_v{a}_w\xi {\Vert }^2{)}^{1/2}\hfill \\ \hfill \le & C{N}^{\alpha +\kappa -1}\Vert {\mathcal{V}}_N^{1/2}\xi \Vert \Vert {\mathcal{V}}_{N,L}^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$ 7.20

The term ${\mathrm{\Phi }}_2$ on the r.h.s. of (7.19) can be controlled by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\mathrm{\Phi }}_2\xi ⟩|& =|\frac{1}{N}{\int }_{{\mathrm{\Lambda }}^2}dxdy{N}^{2-2\kappa }V(N^{1-\kappa }(x-y))\hfill \\ \hfill & \times \sum _{\begin{array}{c}r\in P_H,\\ v,w\in P_L\end{array}}^{\ast }{e}^{-iwx}{e}^{-ivy}{\eta }_r⟨\xi ,a_{v+r}^{\ast }{a}_{w-r}^{\ast }{\check{a}}_x{\check{a}}_y\xi ⟩|\hfill \\ \hfill & \le \frac{C{N}^{3\beta }\Vert {\eta }_H\Vert }{N}({\int }_{{\mathrm{\Lambda }}^2}dxdy{N}^{2-2\kappa }V(N^{1-\kappa }(x-y)){\Vert {\check{a}}_x{\check{a}}_y\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill & \times ({\int }_{{\mathrm{\Lambda }}^2}dxdy{N}^{2-2\kappa }V(N^{1-\kappa }(x-y))\sum _{r\in P_H,v,w\in P_L}{\Vert a_{v+r}{a}_{w-r}\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill & \le C{N}^{9\beta /2+3\kappa /2-\alpha /2-3/2}\Vert {\mathcal{V}}_N^{1/2}\xi \Vert \Vert ({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\xi \Vert .\hfill \end{array}\end{array}$$

Finally, the contributions ${\mathrm{\Phi }}_3$ and ${\mathrm{\Phi }}_4$ can be bounded as follows. We obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\mathrm{\Phi }}_3\xi ⟩|\le & \frac{1}{N}{\int }_{{\mathrm{\Lambda }}^2}dxdy{N}^{2-2\kappa }V(N^{1-\kappa }(x-y))\sum _{\begin{array}{c}r\in P_H,\\ v,w\in P_L\end{array}}|{\eta }_r||⟨\xi ,a_{v+r}^{\ast }{\check{a}}_x^{\ast }{\check{a}}_y^{\ast }{\check{a}}_y{a}_v{a}_w\xi ⟩|\hfill \\ \hfill \le & \frac{C{N}^{3\beta /2}\Vert {\eta }_H\Vert }{N}({\int }_{{\mathrm{\Lambda }}^2}dxdy{N}^{2-2\kappa }V(N^{1-\kappa }(x-y))\sum _{v\in P_L}{|v|}^{-2}{\Vert {\check{a}}_x{\check{a}}_y\xi \Vert }^2{)}^{1/2}\hfill \\ & \times (N^{\kappa -1}{\int }_{\mathrm{\Lambda }}dx\sum _{v,w\in P_L}{|v|}^2{\Vert {({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}{\check{a}}_x{a}_w{a}_v\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill \le & C{N}^{2\beta +3\kappa /2-\alpha /2-1/2}\Vert {\mathcal{V}}_N^{1/2}\xi \Vert \Vert {\mathcal{K}}_L^{1/2}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}\xi \Vert \hfill \end{array}\end{array}$$

as well as

$$\begin{array}{cc}\hfill |⟨\xi ,{\mathrm{\Phi }}_4\xi ⟩|\le & \frac{1}{N}{\int }_{{\mathrm{\Lambda }}^2}dxdy{N}^{2-2\kappa }V(N^{1-\kappa }(x-y))\hfill \\ & \times \sum _{r\in P_H,v,w\in P_L}|{\eta }_r||⟨\xi ,a_{v+r}^{\ast }{a}_{w-r}^{\ast }{\check{a}}_y^{\ast }{a}_w{\check{a}}_x{\check{a}}_y\xi ⟩|\hfill \\ & \le \frac{C{N}^{3\beta /2}\Vert {\eta }_H\Vert }{N}[{\int }_{{\mathrm{\Lambda }}^2}dxdy{N}^{2-2\kappa }V(N^{1-\kappa }(x-y)){\Vert {\check{a}}_x{\check{a}}_y\xi \Vert }^2{)}^{1/2}\hfill \\ & \times (N^{\kappa -1}{\int }_{\mathrm{\Lambda }}dy\sum _{\begin{array}{c}r\in P_H,\\ v,w\in P_L\end{array}}{\Vert {\check{a}}_y{a}_{v+r}{a}_{w-r}{({\mathcal{N}}_{+}+1)}^{1/2}\xi \Vert }^2{)}^{1/2}\hfill \\ & \le C{N}^{3\beta +3\kappa /2-\alpha /2-1/2}\Vert {\mathcal{V}}_N^{1/2}\xi \Vert \Vert ({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\xi \Vert .\hfill \end{array}$$

In conclusion, the previous bounds imply with the assumption (5.6) (in particular, since $\alpha >3\beta +2\kappa$ and $3\beta -2<0$) that

$$\begin{array}{c}\hfill \begin{array}{cc}& \pm ({\mathrm{\Phi }}_1+{\mathrm{\Phi }}_2+{\mathrm{\Phi }}_3+{\mathrm{\Phi }}_4+\text{h.c.})\hfill \\ \hfill & \le \delta {\mathcal{V}}_N+C{N}^{\alpha +\kappa -1}{\mathcal{V}}_N+C{N}^{\alpha +\kappa -1}{\mathcal{V}}_{N,L}+{\delta }^{-1}C{N}^{\beta +\kappa -1}{\mathcal{K}}_L({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\hfill \\ \hfill & +{\delta }^{-1}C{N}^{3\beta +\kappa -1}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^2\hfill \end{array}\end{array}$$ 7.21

holds true in ${\mathcal{F}}_{+}^{\le N}$ for any $\delta >0$. This concludes the proof. $\square$

With Proposition 7.5, we obtain a bound for the growth of ${\mathcal{V}}_N$.

Corollary 7.6

Assume the exponents $\alpha ,\beta$ satisfy (5.6). Then, there exists a constant $C>0$ such that the operator inequality

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^{-sD}{\mathcal{V}}_N{e}^{\mathit{sD}}& \le C{\mathcal{V}}_N+C{\mathcal{V}}_{N,L}+C{N}^{\beta +\kappa -1}{\mathcal{K}}_L({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\hfill \\ \hfill & +C{N}^{3\beta +\kappa }({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1).\hfill \end{array}\end{array}$$

for all $s\in [-1;1]$ and for all $N\in \mathbb{N}$ sufficiently large.

Proof

We apply Gronwall’s lemma. Given a normalized vector $\xi \in {\mathcal{F}}_{+}^{\le N}$, we define ${\phi }_{\xi }(s)=⟨\xi ,e^{-sD}{\mathcal{V}}_N{e}^{\mathit{sD}}\xi ⟩$ and compute its derivative s.t.

$$\begin{array}{c}\hfill {\partial }_s{\phi }_{\xi }(s)=⟨\xi ,e^{-sD}[{\mathcal{V}}_N,D]e^{\mathit{sD}}\xi ⟩.\end{array}$$

Hence, we can apply (7.16) and estimate

$$\begin{array}{c}\hfill \begin{array}{cc}& |\frac{1}{2N}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}_{+}^{\ast },v,w\in P_L:\\ v+u,w-u\ne 0\end{array}}{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast \eta /N)(u)⟨e^{\mathit{sD}}\xi ,a_{v+u}^{\ast }{a}_{w-u}^{\ast }{a}_v{a}_w{e}^{\mathit{sD}}\xi ⟩|\hfill \\ \hfill & \le \frac{\Vert \check{\eta }{\Vert }_{\infty }}{N}({\int }_{{\mathrm{\Lambda }}^2}dxdy{N}^{2-2\kappa }V(N^{1-\kappa }(x-y)){\Vert {\check{a}}_x{\check{a}}_y{e}^{\mathit{sD}}\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill & \times ({\int }_{{\mathrm{\Lambda }}^2}dxdy{N}^{2-2\kappa }V(N^{1-\kappa }(x-y))\Vert \sum _{v,w\in P_L}{e}^{ivx+iwy}{a}_v{a}_w{e}^{\mathit{sD}}\xi {\Vert }^2{)}^{1/2}\hfill \\ \hfill & \le C\Vert {\mathcal{V}}_N^{1/2}{e}^{\mathit{sD}}\xi \Vert \Vert {\mathcal{V}}_{N,L}^{1/2}{e}^{\mathit{sD}}\xi \Vert \le C{\phi }_{\xi }(s)+C⟨\xi ,e^{-sD}{\mathcal{V}}_{N,L}{e}^{\mathit{sD}}\xi ⟩.\hfill \end{array}\end{array}$$ 7.22

Here, we used (3.10), which shows that $\Vert \check{\eta }{\Vert }_{\infty }\le CN$. Using Corollary 7.4 (recalling that $\alpha >3\beta +2\kappa$ and $2\beta \le 1$) and ${\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}\le N$ in ${\mathcal{F}}_{+}^{\le N}$, this simplifies to

$$\begin{array}{c}\hfill \begin{array}{cc}& |\frac{1}{2N}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}_{+}^{\ast },v,w\in P_L:\\ v+u,w-u\ne 0\end{array}}{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast \eta /N)(u)⟨e^{\mathit{sD}}\xi ,a_{v+u}^{\ast }{a}_{w-u}^{\ast }{a}_v{a}_w{e}^{\mathit{sD}}\xi ⟩|\hfill \\ & \le C{\phi }_{\xi }(s)+C⟨\xi ,{\mathcal{V}}_{N,L}\xi ⟩+C{N}^{\beta +\kappa -1}⟨\xi ,{\mathcal{K}}_L({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\xi ⟩\hfill \\ & +C{N}^{3\beta +\kappa }⟨\xi ,({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\xi ⟩.\hfill \end{array}\end{array}$$

Together with (7.16), the bound (7.17) (choosing $\delta =1$) and an application of Lemma 7.1 and of Lemma 7.2, the claim follows now from Gronwall’s lemma.

$\square$

Finally, we need control for the growth of the full kinetic energy operator $\mathcal{K}$. To this end, we need to estimate its commutator with D.

Proposition 7.7

Assume the exponents $\alpha ,\beta$ satisfy (5.6). Let $m_0\in \mathbb{R}$ be such that $m_0\beta =\alpha$ (from (5.6) it follows that $3<m_0<5$). Then,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [\mathcal{K},D]& =-\frac{1}{2N}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },p,q\in P_L:\\ p+u,q-u\ne 0\end{array}}{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast {\widehat{f}}_N)(u)(a_{p+u}^{\ast }{a}_{q-u}^{\ast }{a}_p{a}_q+h.c.)\hfill \\ \hfill & +{\mathcal{E}}_{[\mathcal{K},D]},\hfill \end{array}\end{array}$$ 7.23

where the self-adjoint operator ${\mathcal{E}}_{[\mathcal{K},D]}$ satisfies

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \pm {\mathcal{E}}_{[\mathcal{K},D]}\le & C{N}^{5\beta /4+\kappa }{\mathcal{K}}_{\le 2N^{3\beta /2}}+\delta \mathcal{K}\hfill \\ \hfill & +C{\delta }^{-1}\sum _{j=3}^{2\lfloor m_0\rfloor -1}{N}^{j\beta /2+3\beta /2+2\kappa -1}{\mathcal{K}}_L({\mathcal{N}}_{\ge \frac{1}{2}{N}^{j\beta /2}}+1)\hfill \\ \hfill & +C{\delta }^{-1}{N}^{\alpha +\beta +2\kappa -1}{\mathcal{K}}_L({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\lfloor m_0\rfloor \beta }}+1)+C\hfill \\ \hfill \end{array}\end{array}$$ 7.24

for all $\delta >0$ and for all $N\in \mathbb{N}$ sufficiently large.

Proof

Using that $[\mathcal{K},D]=[\mathcal{K},D_1]+\text{h.c.}$, a straight forward computation shows that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [\mathcal{K},D_1]+\text{h.c.}& =-\frac{1}{2N}\sum _{\begin{array}{c}r\in {\mathrm{\Lambda }}^{\ast },v,w\in P_L:\\ v+r,w-r\ne 0\end{array}}{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast {\widehat{f}}_N)(r)a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w\hfill \\ \hfill & +{\mathrm{\Sigma }}_1+{\mathrm{\Sigma }}_2+{\mathrm{\Sigma }}_3+\text{h.c.},\hfill \end{array}\end{array}$$ 7.25

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathrm{\Sigma }}_1=& \frac{1}{2N}\sum _{\begin{array}{c}r\in P_H^c\cup \{0\},v,w\in P_L:\\ v+r,w-r\ne 0\end{array}}{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast {\widehat{f}}_N)(r)a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w,\hfill \\ \hfill {\mathrm{\Sigma }}_2=& \frac{1}{2N}\sum _{\begin{array}{c}r\in P_H,v,w\in P_L:\\ v+r,w-r\ne 0\end{array}}{N}^{3-2\kappa }{\lambda }_{\ell }({\widehat{\chi }}_{\ell }\ast {\widehat{f}}_N)(r)a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w,\hfill \\ \hfill {\mathrm{\Sigma }}_3=& \frac{2}{N}\sum _{\begin{array}{c}r\in P_H,v,w\in P_L:\\ v+r,w-r\ne 0\end{array}}r·v{\eta }_r{a}_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w.\hfill \end{array}\end{array}$$ 7.26

Let us estimate the size of the operators ${\mathrm{\Sigma }}_1,{\mathrm{\Sigma }}_2$ and ${\mathrm{\Sigma }}_3$. Using $|(\widehat{V}(./N^{1-\kappa })\ast {\widehat{f}}_N)(r)|\le C$, we control the operator ${\mathrm{\Sigma }}_1$ by

$$\begin{array}{cc}\hfill |⟨\xi ,{\mathrm{\Sigma }}_1\xi ⟩|& =|\frac{1}{2N}\sum _{\begin{array}{c}r\in P_H^c\cup \{0\},v,w\in P_L:\\ v+r,w-r\ne 0\end{array}}{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast {\widehat{f}}_N)(r)⟨\xi ,b_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w\xi ⟩|\hfill \\ \hfill & \le \frac{C{N}^{\kappa }}{N}\sum _{\begin{array}{c}r\in {\mathrm{\Lambda }}^{\ast },v,w\in P_L:|r|\le N^{3\beta /2},\\ v+r,w-r\ne 0\end{array}}\Vert a_{w-r}{a}_{v+r}\xi \Vert \Vert a_v{a}_w\xi \Vert \hfill \\ \hfill & +\frac{C{N}^{\kappa }}{N}\sum _{j=3}^{2\lfloor m_0\rfloor -1}\hfill \\ \hfill & \sum _{\begin{array}{c}r\in P_H^c\cup \{0\},v,w\in P_L:\\ N^{j\beta /2}\le |r|\le N^{(j+1)\beta /2},\\ v+r,w-r\ne 0\end{array}}\Vert a_{w-r}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{j\beta /2}}+1)}^{-1/2}{a}_{v+r}\xi \Vert \Vert a_v{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{j\beta /2}}+1)}^{1/2}{a}_w\xi \Vert \hfill \\ \hfill & +\frac{C{N}^{\kappa }}{N}\hfill \\ \hfill & \sum _{\begin{array}{c}r\in P_H^c\cup \{0\},v,w\in P_L:\\ N^{\lfloor m_0\rfloor \beta }\le |r|\le N^{\alpha },\\ v+r,w-r\ne 0\end{array}}\Vert a_{w-r}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\lfloor m_0\rfloor \beta }}+1)}^{-1/2}{a}_{v+r}\xi \Vert \Vert a_v{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\lfloor m_0\rfloor \beta }}+1)}^{1/2}{a}_w\xi \Vert .\hfill \\ \hfill \end{array}$$ 7.27

By Cauchy–Schwarz, the first term on the r.h.s. of (7.27) can be controlled by

$$\begin{array}{c}\hfill \frac{C{N}^{\kappa }}{N}\sum _{\begin{array}{c}r\in {\mathrm{\Lambda }}^{\ast },v,w\in P_L:|r|\le N^{3\beta /2},\\ v+r,w-r\ne 0\end{array}}\Vert a_{w-r}{a}_{v+r}\xi \Vert \Vert a_v{a}_w\xi \Vert \le C{N}^{5\beta /4+\kappa }⟨\xi ,{\mathcal{K}}_{\le 2N^{3\beta /2}}\xi ⟩.\end{array}$$

The second contribution on the r.h.s. of (7.27) can be bounded by

$$\begin{array}{c}\hfill \begin{array}{cc}& \frac{C{N}^{\kappa }}{N}\sum _{j=3}^{2\lfloor m_0\rfloor -1}\sum _{\begin{array}{c}r\in P_H^c\cup \{0\},v,w\in P_L:\\ N^{j\beta /2}\le |r|\le N^{(j+1)\beta /2},\\ v+r,w-r\ne 0\end{array}}\Vert a_{w-r}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{j\beta /2}}+1)}^{-1/2}{a}_{v+r}\xi \Vert \Vert a_w{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{j\beta /2}}+1)}^{1/2}{a}_v\xi \Vert \hfill \\ \hfill & \le C\sum _{j=3}^{2\lfloor m_0\rfloor -1}{N}^{j\beta /4+3\beta /4+\kappa -1/2}\Vert {\mathcal{K}}^{1/2}\xi \Vert \Vert {\mathcal{K}}_L^{1/2}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{j\beta /2}}+1)}^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$ 7.28

Similarly, we find that

$$\begin{array}{c}\hfill \begin{array}{cc}& \frac{C{N}^{\kappa }}{N}\sum _{\begin{array}{c}r\in P_H^c\cup \{0\},v,w\in P_L:\\ N^{\lfloor m_0\rfloor \beta }\le |r|\le N^{\alpha },\\ v+r,w-r\ne 0\end{array}}\Vert a_{w-r}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\lfloor m_0\rfloor \beta }}+1)}^{-1/2}{a}_{v+r}\xi \Vert \Vert a_w{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\lfloor m_0\rfloor \beta }}+1)}^{1/2}{a}_v\xi \Vert \hfill \\ \hfill & \le C{N}^{\alpha /2+\beta /2+\kappa -1/2}\Vert {\mathcal{K}}^{1/2}\xi \Vert \Vert {\mathcal{K}}_L^{1/2}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\lfloor m_0\rfloor \beta }}+1)}^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$ 7.29

In summary, the previous three bounds imply that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \pm {\mathrm{\Sigma }}_1\le & C{N}^{5\beta /4+\kappa }{\mathcal{K}}_{\le 2N^{3\beta /2}}+\delta \mathcal{K}+C{\delta }^{-1}{N}^{\alpha +\beta +2\kappa -1}{\mathcal{K}}_L({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\lfloor m_0\rfloor \beta }}+1)\hfill \\ \hfill & +C{\delta }^{-1}\sum _{j=3}^{2\lfloor m_0\rfloor -1}{N}^{j\beta /2+3\beta /2+2\kappa -1}{\mathcal{K}}_L({\mathcal{N}}_{\ge \frac{1}{2}{N}^{j\beta /2}}+1)\hfill \end{array}\end{array}$$ 7.30

for some constant $C>0$ and all $\delta >0$.

Next, let us switch to ${\mathrm{\Sigma }}_2$ and ${\mathrm{\Sigma }}_3$, defined in (7.26). Since $({\widehat{\chi }}_{\ell }\ast {\widehat{f}}_N)(r)={\widehat{\chi }}_{\ell }(r)+N^{-1}{\eta }_r$, with

$$\begin{array}{c}\hfill {\widehat{\chi }}_{\ell }(r)=\frac{4\pi }{{|r|}^2}\left(\frac{sin(\ell |r|)}{|r|},-,\ell ,cos,(\ell |r|)\right)\end{array}$$

we find

$$\begin{array}{c}\hfill |({\widehat{\chi }}_{\ell }\ast {\widehat{f}}_N)(r){|\le C|r|}^{-2}\end{array}$$

This, together with Lemma 3.1(i), Cauchy–Schwarz and $\alpha >3\beta +2\kappa$, implies that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\mathrm{\Sigma }}_2\xi ⟩|\le & \frac{C{N}^{\kappa }}{N}\sum _{\begin{array}{c}r\in P_H,v,w\in P_L:\\ v+r,w-r\ne 0\end{array}}{|r|}^{-2}\Vert a_{v+r}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{-1/2}{a}_{w-r}\xi \Vert \hfill \\ \hfill & \times \Vert a_v{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}{a}_w\xi \Vert \hfill \\ \hfill \le & C{N}^{-\beta -1/2}\Vert {({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}\xi \Vert \Vert {\mathcal{K}}_L^{1/2}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}\xi \Vert .\hfill \end{array}\end{array}$$ 7.31

Similarly, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\mathrm{\Sigma }}_3\xi ⟩|\le & \frac{C}{N}\sum _{r\in P_H,v,w\in P_L}|r||v||{\eta }_r|\Vert a_{v+r}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{-1/2}{a}_{w-r}\xi \Vert \hfill \\ \hfill & \times \Vert a_v{a}_w{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}\xi \Vert \hfill \\ \hfill \le & C{N}^{-1/2}\Vert {\mathcal{K}}^{1/2}\xi \Vert \Vert {\mathcal{K}}_L^{1/2}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}\xi \Vert ,\hfill \end{array}\end{array}$$ 7.32

where we used that $|r|/|v+r|\le 2$ for $r\in P_H$, $v\in P_L$ and $N\in \mathbb{N}$ large enough. Combining (7.30), (7.31) and (7.32) and defining ${\mathcal{E}}_{[\mathcal{K},D]}={\sum }_{i=1}^3({\mathrm{\Sigma }}_i+\text{h.c.})$ proves the claim. $\square$

Corollary 7.8

Assume the exponents $\alpha ,\beta$ satisfy (5.6). Let $m_0\in \mathbb{R}$ be such that $m_0\beta =\alpha$ ($3<m_0<5$ from (5.6)). Then, there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}& e^{-sD}\mathcal{K}{e}^{\mathit{sD}}\hfill \\ \hfill & \le C\mathcal{K}+C{\mathcal{V}}_N+C{\mathcal{V}}_{N,L}+C{N}^{5\beta /4+\kappa }{\mathcal{K}}_{\le N^{3\beta /2}}\hfill \\ \hfill & +C\sum _{j=3}^{2\lfloor m_0\rfloor -1}{N}^{j\beta /2+3\beta /2+2\kappa -1}[{\mathcal{K}}_L+N^{2\beta }({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)]({\mathcal{N}}_{\ge \frac{1}{2}{N}^{j\beta /2}}+1)\hfill \\ \hfill & +C{N}^{\alpha +\beta +2\kappa -1}[{\mathcal{K}}_L+N^{2\beta }({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)]({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\lfloor m_0\rfloor \beta }}+1)+C{N}^{13\beta /4+\kappa }\hfill \end{array}\end{array}$$ 7.33

for all $s\in [-1;1]$ and for all $N\in \mathbb{N}$ sufficiently large.

Proof

Given $\xi \in {\mathcal{F}}_{+}^{\le N}$, we define ${\phi }_{\xi }(s)=⟨\xi ,e^{-sD}\mathcal{K}{e}^{\mathit{sD}}\xi ⟩$. Differentiation yields

$$\begin{array}{c}\hfill {\partial }_s{\phi }_{\xi }(s)=⟨\xi ,e^{-sD}[\mathcal{K},D]e^{\mathit{sD}}\xi ⟩,\end{array}$$

s.t., to bound the derivative of ${\phi }_{\xi }$, we can apply Proposition 7.7. Arguing exactly as in (7.22), we obtain with ${sup}_{x\in \mathrm{\Lambda }}|f_N(x)|\le 1$ the operator inequality

$$\begin{array}{c}\hfill \begin{array}{cc}& \pm \frac{1}{2N}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}_{+}^{\ast },v,w\in P_L:\\ v+u,w-u\ne 0\end{array}}{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast {\widehat{f}}_N)(u)a_{v+u}^{\ast }{a}_{w-u}^{\ast }{a}_v{a}_w\le C{\mathcal{V}}_N+C{\mathcal{V}}_{N,L}.\hfill \end{array}\end{array}$$

Now, the claim follows from the bound (7.24) (choosing $\delta =1$), the previous bound and an application of Corollaries 7.6, 7.4, Lemmas 7.1, 7.2 and the operator bound ${\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}\le 4N^{-2\alpha }\mathcal{K}$, by Gronwall’s Lemma. $\square$

Action of Quartic Renormalization on Excitation Hamiltonian

We compute now the main contributions to ${\mathcal{M}}_N=e^{-D}{\mathcal{J}}_N^{\text{eff}}{e}^D$. From (4.5) and recalling that $[{\mathcal{N}}_{+},D]=0$, we can decompose

$$\begin{array}{c}\hfill {\mathcal{M}}_N=4\pi {\mathfrak{a}}_0{N}^{1+\kappa }-4\pi {\mathfrak{a}}_0{N}^{\kappa -1}{\mathcal{N}}_{+}^2/N+{\mathcal{M}}_N^{(2)}+{\mathcal{M}}_N^{(3)}+{\mathcal{M}}_N^{(4)}\end{array}$$ 7.34

where the operators ${\mathcal{M}}_N^{(i)},i=2,3,4,$ are defined by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{M}}_N^{(2)}=& 8\pi {\mathfrak{a}}_0{N}^{\kappa }\sum _{p\in P_H^c}{e}^{-D}{b}_p^{\ast }{b}_p{e}^D+4\pi {\mathfrak{a}}_0{N}^{\kappa }\sum _{p\in P_H^c}{e}^{-D}[b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p}]e^D\hfill \\ \hfill {\mathcal{M}}_N^{(3)}=& \frac{8\pi {\mathfrak{a}}_0{N}^{\kappa }}{\sqrt{N}}\sum _{\begin{array}{c}p\in P_H^c,q\in P_L:\\ p+q\ne 0\end{array}}{e}^{-D}[b_{p+q}^{\ast }{a}_{-p}^{\ast }{a}_q+\text{h.c.}]e^D,\hfill \\ \hfill {\mathcal{M}}_N^{(4)}=& e^{-D}{\mathcal{H}}_N{e}^D=e^{-D}\mathcal{K}{e}^D+e^{-D}{\mathcal{V}}_N{e}^D.\hfill \end{array}\end{array}$$ 7.35

Analysis of ${\mathcal{M}}_N^{(2)}$

In this section, we determine the main contributions to ${\mathcal{M}}_N^{(2)}$, defined in (7.35) by

$$\begin{array}{c}\hfill {\mathcal{M}}_N^{(2)}=8\pi {\mathfrak{a}}_0{N}^{\kappa }\sum _{p\in P_H^c}{e}^{-D}{b}_p^{\ast }{b}_p{e}^D+4\pi {\mathfrak{a}}_0{N}^{\kappa }\sum _{p\in P_H^c}{e}^{-D}[b_p^{\ast }{b}_{-p}^{\ast }+b_p{b}_{-p}]e^D\end{array}$$ 7.36

The main result of this section is the following proposition.

Proposition 7.9

Assume the exponents $\alpha ,\beta$ satisfy (5.6). Then

$$\begin{array}{c}\hfill {\mathcal{M}}_N^{(2)}=8\pi {\mathfrak{a}}_0{N}^{\kappa }\sum _{p\in P_H^c}[b_p^{\ast }{b}_p+\frac{1}{2}{b}_p^{\ast }{b}_{-p}^{\ast }+\frac{1}{2}{b}_p{b}_{-p}]+{\mathcal{E}}_{{\mathcal{M}}_N}^{(2)}\end{array}$$ 7.37

and there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \pm e^A{e}^D{\mathcal{E}}_{{\mathcal{M}}_N}^{(2)}{e}^{-D}{e}^{-A}& \le C{N}^{-\beta -2\kappa }\mathcal{K}+C{N}^{\kappa }\hfill \end{array}\end{array}$$ 7.38

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

Proof

We start with the identity

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathcal{M}}_N^{(2)}-8\pi {\mathfrak{a}}_0{N}^{\kappa }\sum _{p\in P_H^c}[b_p^{\ast }{b}_p+\frac{1}{2}{b}_p^{\ast }{b}_{-p}^{\ast }+\frac{1}{2}{b}_p{b}_{-p}]\hfill \\ \hfill & =8\pi {\mathfrak{a}}_0{N}^{\kappa }{\int }_0^{1}dt\sum _{p\in P_H^c}{e}^{-tD}[b_p^{\ast }{b}_p+\frac{1}{2}{b}_p^{\ast }{b}_{-p}^{\ast }+\frac{1}{2}{b}_p{b}_{-p},D_1]e^{\mathit{tD}}+\text{h.c.}\hfill \end{array}\end{array}$$ 7.39

and a straight-forward computation shows that

$$\begin{array}{c}\hfill \begin{array}{cc}& [b_p^{\ast }{b}_p+\frac{1}{2}{b}_p^{\ast }{b}_{-p}^{\ast }+\frac{1}{2}{b}_p{b}_{-p},a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_w{a}_v]\hfill \\ \hfill & =b_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{b}_w({\delta }_{p,v+r}+{\delta }_{p,w-r}-{\delta }_{p,v}-{\delta }_{p,w})\hfill \\ \hfill & -\frac{1}{2}{b}_{v+r}^{\ast }{b}_{w-r}^{\ast }({\delta }_{p,w}{\delta }_{-p,v}+{\delta }_{-p,w}{\delta }_{p,v})+\frac{1}{2}{b}_v{b}_w({\delta }_{p,w-r}{\delta }_{-p,v+r}+{\delta }_{-p,w-r}{\delta }_{p,v+r})\hfill \\ \hfill & -\frac{1}{2}{b}_{v+r}^{\ast }{b}_{w-r}^{\ast }(a_{-p}^{\ast }{a}_w{\delta }_{p,v}+a_p^{\ast }{a}_w{\delta }_{-p,v}+a_{-p}^{\ast }{a}_v{\delta }_{p,w}+a_p^{\ast }{a}_v{\delta }_{-p,w})\hfill \\ \hfill & +\frac{1}{2}(a_{w-r}^{\ast }{a}_{-p}{\delta }_{p,v+r}+a_{v+r}^{\ast }{a}_{-p}{\delta }_{p,w-r}+a_{w-r}^{\ast }{a}_p{\delta }_{-p,v+r}+a_{v+r}^{\ast }{a}_p{\delta }_{-p,w-r})b_v{b}_w.\hfill \end{array}\end{array}$$

As a consequence, we find that

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathcal{M}}_N^{(2)}-8\pi {\mathfrak{a}}_0{N}^{\kappa }\sum _{p\in P_H^c}[b_p^{\ast }{b}_p+\frac{1}{2}{b}_p^{\ast }{b}_{-p}^{\ast }+\frac{1}{2}{b}_p{b}_{-p}]={\int }_0^{1}dt{e}^{-tD}\sum _{j=1}^5({\text{V}}_j+\text{h.c.})e^{\mathit{tD}},\hfill \end{array}\end{array}$$ 7.40

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\text{V}}_1& =-\frac{8\pi {\mathfrak{a}}_0{N}^{\kappa }}{2N}\sum _{\begin{array}{c}r\in P_H,v\in P_L\end{array}}{\eta }_r{b}_{v+r}^{\ast }{b}_{-v-r}^{\ast },\hfill \\ \hfill {\text{V}}_2& =\frac{8\pi {\mathfrak{a}}_0{N}^{\kappa }}{2N}\sum _{\begin{array}{c}r\in P_H,v\in P_L:\\ v+r\in P_H^c,v+r\ne 0\end{array}}{\eta }_r{b}_v{b}_{-v},\hfill \\ \hfill {\text{V}}_3& =\frac{8\pi {\mathfrak{a}}_0{N}^{\kappa }}{2N}\sum _{\begin{array}{c}r\in P_H,v,w\in P_L:\\ v+r,w-r\ne 0\end{array}}{\eta }_r(-2+{\chi }_{\{r+v\in P_H^c\}}+{\chi }_{\{w-r\in P_H^c\}})b_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{b}_w,\hfill \\ \hfill {\text{V}}_4& =-\frac{8\pi {\mathfrak{a}}_0{N}^{\kappa }}{N}\sum _{\begin{array}{c}r\in P_H,v,w\in P_L:\\ v+r,w-r\ne 0\end{array}}{\eta }_r{b}_{v+r}^{\ast }{b}_{w-r}^{\ast }{a}_{-v}^{\ast }{a}_w,\hfill \\ \hfill {\text{V}}_5& =\frac{8\pi {\mathfrak{a}}_0{N}^{\kappa }}{N}\sum _{\begin{array}{c}r\in P_H,v,w\in P_L:\\ r-w\in P_H^c,v+r,w-r\ne 0\end{array}}{\eta }_r{a}_{v+r}^{\ast }{a}_{r-w}{b}_v{b}_w.\hfill \\ \hfill \end{array}\end{array}$$ 7.41

Here, ${\chi }_{\{p\in S\}}$ denotes as usual the characteristic function for the set $S\subset {\mathrm{\Lambda }}_{+}^{\ast }$, evaluated at $p\in {\mathrm{\Lambda }}_{+}^{\ast }$. Let us briefly explain how to bound the different contributions ${\text{V}}_1$ to ${\text{V}}_5$, defined in (7.41). Using Cauchy–Schwarz, the first two contributions are bounded by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \pm ({\text{V}}_1+{\text{V}}_2)& \le C{N}^{2\kappa +3\beta -\alpha /2-1}({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)+C{N}^{2\kappa +3\beta /2-1}({\mathcal{K}}_L+1)\hfill \end{array}\end{array}$$

where, for ${\text{V}}_2$, we used that $v+r\in P_H^c$ implies that $|r|\le N^{\alpha }+N^{\beta }$ and furthermore that ${\sum }_{N^{\alpha }\le |r|\le N^{\alpha }+N^{\beta }}|{\eta }_r|\le N^{\kappa +\beta }$. The contributions ${\text{V}}_3$ to ${\text{V}}_5$, on the other hand, can be controlled by

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,({\text{V}}_3+{\text{V}}_4+{\text{V}}_5)\xi ⟩|\hfill \\ & \le \frac{C{N}^{\kappa }}{N}\sum _{\begin{array}{c}r\in P_H,v,w\in P_L:\\ v+r,w-r\ne 0\end{array}}|{\eta }_r|\Vert a_{v+r}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{-1/2}{a}_{w-r}\xi \Vert \Vert a_v{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}{a}_w\xi \Vert \hfill \\ & +\frac{C{N}^{\kappa }}{N}\sum _{\begin{array}{c}r\in P_H,v,w\in P_L:\\ v+r,w-r\ne 0\end{array}}|{\eta }_r|\Vert a_{v+r}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{-1/2}{a}_{w-r}{a}_w\xi \Vert \Vert a_v{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}\xi \Vert \hfill \\ & +\frac{C{N}^{\kappa }}{N}\sum _{\begin{array}{c}r\in P_H,v,w\in P_L:\\ v+r,w-r\ne 0\end{array}}|{\eta }_r|\Vert a_{v+r}\xi \Vert \Vert a_v{a}_w{a}_{w-r}\xi \Vert \hfill \\ & \le C{N}^{2\kappa +3\beta /2-\alpha /2}⟨\xi ,({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\xi ⟩\le C{N}^{\kappa }⟨\xi ,({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\xi ⟩\hfill \end{array}\end{array}$$

for any $\xi \in {\mathcal{F}}_{+}^{\le N}$. In conclusion (since $2\kappa +3\beta -\alpha /2-1<\kappa$ from (5.6)), we have proved that

$$\begin{array}{c}\hfill \pm \sum _{j=1}^5({\text{V}}_j+\text{h.c.})\le C{N}^{2\kappa +3\beta /2-1}{\mathcal{K}}_L+C{N}^{\kappa }({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1).\end{array}$$

Now, applying this bound together with (7.40), Lemmas 4.2, 4.3, 7.1, 7.2 and the operator inequality ${\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}\le 4N^{-2\alpha }\mathcal{K}$ proves the claim. $\square$

Analysis of ${\mathcal{M}}_N^{(3)}$

In this section, we determine the main contributions to ${\mathcal{M}}_N^{(3)}$, defined in (7.35) by

$$\begin{array}{c}\hfill {\mathcal{M}}_N^{(3)}=\frac{8\pi {\mathfrak{a}}_0{N}^{\kappa }}{\sqrt{N}}\sum _{\begin{array}{c}p\in P_H^c,q\in P_L:\\ p+q\ne 0\end{array}}{e}^{-D}(b_{p+q}^{\ast }{a}_{-p}^{\ast }{a}_q+\text{h.c.})e^D.\end{array}$$ 7.42

Proposition 7.10

Assume the exponents $\alpha ,\beta$ satisfy (5.6). Then, we have that

$$\begin{array}{c}\hfill {\mathcal{M}}_N^{(3)}=\frac{8\pi {\mathfrak{a}}_0{N}^{\kappa }}{\sqrt{N}}\sum _{\begin{array}{c}p\in P_H^c,q\in P_L:\\ p+q\ne 0\end{array}}(b_{p+q}^{\ast }{a}_{-p}^{\ast }{a}_q+h.c.)+{\mathcal{E}}_{{\mathcal{M}}_N}^{(3)}\end{array}$$ 7.43

and there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}& \pm e^A{e}^D{\mathcal{E}}_{{\mathcal{M}}_N}^{(3)}{e}^{-D}{e}^{-A}\hfill \\ \hfill & \le C{N}^{-\beta }\mathcal{K}+C{N}^{\alpha +\beta /2+2\kappa -1}\mathcal{K}({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)+C{N}^{\alpha +\beta /2+2\kappa }\hfill \end{array}\end{array}$$ 7.44

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

Proof

Let us define the operator $\text{Y}:{\mathcal{F}}_{+}^{\le N}\to {\mathcal{F}}_{+}^{\le N}$ by

$$\begin{array}{c}\hfill \text{Y}=\frac{8\pi {\mathfrak{a}}_0{N}^{\kappa }}{\sqrt{N}}\sum _{\begin{array}{c}p\in P_H^c,q\in P_L:\\ p+q\ne 0\end{array}}(b_{p+q}^{\ast }{a}_{-p}^{\ast }{a}_q+\text{h.c.}),\end{array}$$ 7.45

so that ${\mathcal{M}}_N^{(3)}=e^{-D}\text{Y}{e}^D$. We recall the definition (7.1) and observe that

$$\begin{array}{c}\hfill e^{-D}\text{Y}{e}^D-\text{Y}={\int }_0^{1}ds{e}^{-sD}[\text{Y},D_1]e^{\mathit{sD}}+\text{h.c.}.\end{array}$$ 7.46

This implies that it is enough to control the commutator $[\text{Y},D_1]$ after conjugation with $e^{\mathit{tD}}$, for any $t\in [-1;1]$. Note that, if $p\in P_H^c,q\in P_L,r\in P_H$ and $v,w\in P_L$, we have $|v+r|\ge N^{\alpha }-N^{\beta }>\frac{1}{2}{N}^{\alpha }>N^{\beta }$ s.t. $[a_{-p}^{\ast }{a}_q,a_{v+r}^{\ast }{a}_{w-r}^{\ast }]=0$, for $N\in \mathbb{N}$ large enough. Then, a lengthy but straightforward calculation shows that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [b_{p+q}^{\ast }{a}_{-p}^{\ast }{a}_q,a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w]& =-b_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_q({\delta }_{-p,w}{\delta }_{p+q,v}+{\delta }_{-p,v}{\delta }_{p+q,w})\hfill \\ \hfill & -b_{p+q}^{\ast }{a}_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_q(a_w{\delta }_{-p,v}+a_v{\delta }_{-p,w})\hfill \\ \hfill & -b_{-p}^{\ast }{a}_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_q(a_w{\delta }_{p+q,v}+a_v{\delta }_{p+q,w})\hfill \end{array}\end{array}$$

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [a_q^{\ast }{a}_{-p}{b}_{p+q},a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w]& =a_q^{\ast }{a}_v{b}_w{\delta }_{-p,w-r}{\delta }_{p+q,v+r}+a_q^{\ast }{a}_v{b}_w{\delta }_{-p,v+r}{\delta }_{p+q,w-r}\hfill \\ \hfill & +a_q^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w{b}_{p+q}{\delta }_{-p,v+r}+a_q^{\ast }{a}_{v+r}^{\ast }{a}_v{a}_w{b}_{p+q}{\delta }_{-p,w-r}\hfill \\ \hfill & -a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_w{a}_{-p}{b}_{p+q}{\delta }_{q,v}-a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_{-p}{b}_{p+q}{\delta }_{q,w}\hfill \\ \hfill & +a_q^{\ast }{a}_{w-r}^{\ast }{a}_{-p}{a}_v{b}_w{\delta }_{p+q,v+r}+a_q^{\ast }{a}_{v+r}^{\ast }{a}_{-p}{a}_v{b}_w{\delta }_{p+q,w-r}.\hfill \end{array}\end{array}$$

As a consequence, we conclude that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [\text{Y},D_1]+\text{h.c.}& =\sum _{i=1}^6({\mathrm{\Psi }}_i+\text{h.c.}),\hfill \end{array}\end{array}$$ 7.47

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathrm{\Psi }}_1& =-\frac{8\pi {\mathfrak{a}}_0{N}^{\kappa }}{N^{3/2}}\sum _{\begin{array}{c}r\in P_H,v,w\in P_L:\\ v+w\in P_L\end{array}}^{\ast }{\eta }_r{b}_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_{v+w},\hfill \\ \hfill {\mathrm{\Psi }}_2& =\frac{8\pi {\mathfrak{a}}_0{N}^{\kappa }}{N^{3/2}}\sum _{\begin{array}{c}r\in P_H,v,w\in P_L:\\ v+r,r-w\in P_H^c,v+w\in P_L\end{array}}^{\ast }{\eta }_r{a}_{v+w}^{\ast }{a}_v{b}_w,\hfill \\ \hfill {\mathrm{\Psi }}_3& =-\frac{16\pi {\mathfrak{a}}_0{N}^{\kappa }}{N^{3/2}}\sum _{r\in P_H,q,v,w\in P_L}^{\ast }{\eta }_r{b}_{q-v}^{\ast }{a}_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_q{a}_w,\hfill \\ \hfill {\mathrm{\Psi }}_4& =\frac{8\pi {\mathfrak{a}}_0{N}^{\kappa }}{N^{3/2}}\sum _{\begin{array}{c}r\in P_H,q,v,w\in P_L:\\ v+r\in P_H^c\end{array}}^{\ast }{\eta }_r{a}_q^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w{b}_{q-v-r},\hfill \\ \hfill {\mathrm{\Psi }}_5& =\frac{8\pi {\mathfrak{a}}_0{N}^{\kappa }}{N^{3/2}}\sum _{\begin{array}{c}r\in P_H,q,v,w\in P_L:\\ v+r-q\in P_H^c\end{array}}^{\ast }{\eta }_r{a}_q^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w{b}_{q-v-r},\hfill \\ \hfill {\mathrm{\Psi }}_6& =-\frac{8\pi {\mathfrak{a}}_0{N}^{\kappa }}{N^{3/2}}\sum _{p\in P_H^c,r\in P_H,v,w\in P_L}^{\ast }{\eta }_r{a}_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_w{a}_{-p}{b}_{p+v}.\hfill \end{array}\end{array}$$ 7.48

Let us explain how to control the operators ${\mathrm{\Psi }}_1$ to ${\mathrm{\Psi }}_6$, defined in (7.48). We start with ${\mathrm{\Psi }}_1$. Given $\xi \in {\mathcal{F}}_{+}^{\le N}$, we find that

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,{\mathrm{\Psi }}_1\xi ⟩|\hfill \\ \hfill & =|\frac{8\pi {\mathfrak{a}}_0{N}^{\kappa }}{N^{3/2}}\sum _{r\in P_H,v,w\in P_L}^{\ast }{\eta }_r⟨\xi ,b_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_{v+w}\xi ⟩|\hfill \\ \hfill & \le \frac{C{N}^{\kappa }}{N^{3/2}}\sum _{r\in P_H,v,w\in P_L}^{\ast }|{\eta }_r|\Vert {({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{-1/2}{a}_{v+r}{a}_{w-r}\xi \Vert \Vert {({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}{a}_{v+w}\xi \Vert \hfill \\ \hfill & \le C{N}^{3\beta +2\kappa -\alpha /2-1}⟨\xi ,({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\xi ⟩\le C{N}^{3\beta /2+\kappa -1}⟨\xi ,({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\xi ⟩.\hfill \end{array}\end{array}$$

The contribution ${\mathrm{\Psi }}_2$ can be bounded by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\mathrm{\Psi }}_2\xi ⟩|& =|\frac{8\pi {\mathfrak{a}}_0{N}^{\kappa }}{N^{3/2}}\sum _{\begin{array}{c}r\in P_H,v,w\in P_L:\\ v+r\in P_H^c\end{array}}^{\ast }{\eta }_r⟨\xi a_{v+w}^{\ast }{a}_v{b}_w\xi ⟩|\hfill \\ \hfill & \le C{N}^{\beta /2+\kappa -1}⟨\xi ,{\mathcal{K}}_{\le 2N^{\beta }}\xi ⟩\sum _{\begin{array}{c}{N}^{\alpha }\le |r|\le N^{\alpha }+N^{\beta }\end{array}}|{\eta }_r|\le C{N}^{3\beta /2+2\kappa -1}⟨\xi ,{\mathcal{K}}_{\le 2N^{\beta }}\xi ⟩.\hfill \end{array}\end{array}$$

Notice here, that we used that $|r|\le N^{\alpha }+N^{\beta }$ if $r+v\in P_H^c$ and $v\in P_L$. Next, we apply as usual Cauchy–Schwarz to estimate the terms ${\mathrm{\Psi }}_3$ to ${\mathrm{\Psi }}_5$ by

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,{\mathrm{\Psi }}_3\xi ⟩+⟨\xi ,{\mathrm{\Psi }}_4\xi ⟩+⟨\xi ,{\mathrm{\Psi }}_5\xi ⟩|\hfill \\ \hfill & \le C{N}^{3\beta +2\kappa -\alpha /2}⟨\xi ,({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\xi ⟩\le C{N}^{3\beta /2+\kappa }⟨\xi ,({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\xi ⟩\hfill \end{array}\end{array}$$

for all $\alpha >3\beta +2\kappa$. Finally, the term ${\mathrm{\Psi }}_6$ can be controlled by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\mathrm{\Psi }}_6\xi ⟩|& =|\frac{8\pi {\mathfrak{a}}_0{N}^{\kappa }}{N^{3/2}}\sum _{p\in P_H^c,r\in P_H,v,w\in P_L}^{\ast }{\eta }_r⟨\xi ,a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_w{a}_{-p}{b}_{p+v}\xi ⟩|\hfill \\ \hfill & \le C{N}^{\kappa -3/2}\sum _{p\in P_H^c,r\in P_H,v,w\in P_L}^{\ast }{|w|}^{-1}\Vert {({\mathcal{N}}_{\ge N^{\alpha }/2}+1)}^{-1/2}{a}_{v+r}{a}_{w-r}\xi \Vert \hfill \\ \hfill & \times |w||{\eta }_r|\Vert a_w{a}_{-p}{b}_{p+v}{({\mathcal{N}}_{\ge N^{\alpha }/2}+1)}^{1/2}\xi \Vert \hfill \\ \hfill & \le C{N}^{\alpha +\beta /2+2\kappa -1}⟨\xi ,{\mathcal{K}}_L({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\xi ⟩+C{N}^{\alpha +\beta /2+2\kappa }⟨\xi ,({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\xi ⟩.\hfill \end{array}\end{array}$$

In conclusion, the previous estimates show that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \pm [\sum _{i=1}^6({\mathrm{\Psi }}_i+\text{h.c.})]& \le C{N}^{3\beta /2+2\kappa -1}{\mathcal{K}}_{\le 2N^{\beta }}+C{N}^{\alpha +\beta /2+2\kappa -1}{\mathcal{K}}_L({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\hfill \\ \hfill & +C{N}^{\alpha +\beta /2+2\kappa }({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1),\hfill \end{array}\end{array}$$

so that, together with (7.46) and (7.47), an application of the Lemmas 4.2, 4.3, 7.1, 7.2 and the operator bound ${\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}\le 4N^{-2\alpha }\mathcal{K}$ proves the claim. $\square$

Analysis of ${\mathcal{M}}_N^{(4)}$

In this section, we determine the main contributions to ${\mathcal{M}}_N^{(4)}=e^{-D}{\mathcal{H}}_N{e}^D$, defined in (7.35). To this end, we start with the observation that

$$\begin{array}{c}\hfill {\mathcal{M}}_N^{(4)}={\mathcal{H}}_N+{\int }_0^{1}ds{e}^{-sD}([\mathcal{K},D_1]+[{\mathcal{V}}_N,D_1])e^{\mathit{sD}}+\text{h.c.},\end{array}$$ 7.49

with $D_1$ defined in (7.1). By Propositions 7.5 and 7.7, this implies that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{M}}_N^{(4)}& ={\mathcal{H}}_N-\frac{N^{\kappa }}{2N}\sum _{\begin{array}{c}r\in {\mathrm{\Lambda }}^{\ast },v,w\in P_L:\\ v+r,w-r\ne 0\end{array}}{\int }_0^{1}ds\widehat{V}(r/N^{1-\kappa })e^{-sD}(a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w+\text{h.c.})e^{\mathit{sD}}\hfill \\ \hfill & +{\int }_0^{1}ds{e}^{-sD}({\mathcal{E}}_{[\mathcal{K},D]}+{\mathcal{E}}_{[{\mathcal{V}}_N,D]})e^{\mathit{sD}},\hfill \end{array}\end{array}$$ 7.50

where we used that $\widehat{V}(·/N^{1-\kappa })\ast ({\widehat{f}}_N-\eta /N)(r)=\widehat{V}(·/N^{1-\kappa })(r)$ for all $r\in {\mathrm{\Lambda }}_{+}^{\ast }$. Moreover, the operators ${\mathcal{E}}_{[{\mathcal{V}}_N,D]}$ and ${\mathcal{E}}_{[\mathcal{K},D]}$ are explicitly given by

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathcal{E}}_{[{\mathcal{V}}_N,D]}=\sum _{i=1}^4({\mathrm{\Phi }}_i+\text{h.c.}),{\mathcal{E}}_{[\mathcal{K},D]}=\sum _{j=1}^3({\mathrm{\Sigma }}_j+\text{h.c.})\end{array}\end{array}$$ 7.51

where we recall the definitions (7.19) and (7.26). Let us analyze the different contributions in (7.50), separately. We start with the second term on the r.h.s. of (7.50).

Proposition 7.11

Assume the exponents $\alpha ,\beta$ satisfy (5.6). Then, we have

$$\begin{array}{c}\hfill \begin{array}{cc}& \frac{1}{2N}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },p,q\in P_L:\\ p+u,q-u\ne 0\end{array}}{N}^{\kappa }\widehat{V}(r/N^{1-\kappa })e^{-sD}(a_{p+u}^{\ast }{a}_{q-u}^{\ast }{a}_p{a}_q+h.c.)e^{\mathit{sD}}\hfill \\ \hfill & =\frac{1}{2N}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },p,q\in P_L:\\ p+u,q-u\ne 0\end{array}}{N}^{\kappa }\widehat{V}(r/N^{1-\kappa })(a_{p+u}^{\ast }{a}_{q-u}^{\ast }{a}_p{a}_q+h.c.)\hfill \\ \hfill & +\frac{s}{N}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },v,w\in P_L:\\ v+u,w-u\in P_L\end{array}}^{\ast }{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast \eta /N)(u)a_{v+u}^{\ast }{a}_{w-u}^{\ast }{a}_v{a}_w+{\mathcal{E}}_1(s)+{\mathcal{E}}_2(s)\hfill \end{array}\end{array}$$ 7.52

and there exists a constant $C>0$ s.t. ${\mathcal{E}}_1(s)$ and ${\mathcal{E}}_2(s)$ satisfy

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \pm {\mathcal{E}}_1(s)& \le C(N^{\alpha +\beta +2\kappa -1}+N^{-3\beta -3\kappa })\mathcal{K}+C{N}^{2\beta +\kappa },\hfill \\ \hfill \pm {\mathcal{E}}_2(s)& \le C{N}^{\beta +\kappa -1}{\mathcal{K}}_L({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)+C(N^{-\beta -\kappa }+C{N}^{3\beta /2+\kappa /2-1}){\int }_0^{s}dt{e}^{-tD}{\mathcal{V}}_N{e}^{\mathit{tD}}\hfill \\ \hfill & +C{N}^{2\beta +2\kappa -1}{\int }_0^{s}dt{e}^{-tD}{\mathcal{K}}_{\le 2N^{\beta }}({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)e^{\mathit{tD}},\hfill \end{array}\end{array}$$ 7.53

for all $\delta >0$, $s\in [-1;1]$ and for all $N\in \mathbb{N}$ sufficiently large.

Proof

For definiteness, let us denote by $\text{W}:{\mathcal{F}}_{+}^{\le N}\to {\mathcal{F}}_{+}^{\le N}$ the operator

$$\begin{array}{c}\hfill \text{W}=\frac{1}{2N}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },p,q\in P_L:\\ p+u,q-u\ne 0\end{array}}{N}^{\kappa }\widehat{V}(u/N^{1-\kappa })(a_{p+u}^{\ast }{a}_{q-u}^{\ast }{a}_p{a}_q+\text{h.c.})\end{array}$$ 7.54

and consider the identity

$$\begin{array}{c}\hfill \begin{array}{cc}& e^{-sD}\text{W}{e}^{\mathit{sD}}-\text{W}\hfill \\ & ={\int }_0^{s}dt{e}^{-tD}[\text{W},D_1]e^{\mathit{tD}}+\text{h.c.}\hfill \\ & =\frac{1}{2N}{\int }_0^{s}dt\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },p,q\in P_L:\\ p+u,q-u\ne 0\end{array}}{N}^{\kappa }\widehat{V}(r/N^{1-\kappa })e^{-tD}[(a_{p+u}^{\ast }{a}_{q-u}^{\ast }{a}_p{a}_q+\text{h.c.}),D_1]e^{\mathit{tD}}+\text{h.c.}\hfill \end{array}\end{array}$$ 7.55

Now, observe that

$$\begin{array}{c}\hfill [a_p,a_{v+r}^{\ast }]=[a_q,a_{v+r}^{\ast }]=[a_p,a_{w-r}^{\ast }]=[a_q,a_{w-r}^{\ast }]=0\end{array}$$

for all $p,q\in P_L$ and $r\in P_H$, $v,w\in P_L$ and $N\in \mathbb{N}$ sufficiently large. Then, proceeding as in the proof of Proposition 7.5, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& [a_{p+u}^{\ast }{a}_{q-u}^{\ast }{a}_p{a}_q,a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w]\hfill \\ \hfill & =-a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_{q-u}^{\ast }{a}_w{a}_p{a}_q{\delta }_{p+u,v}-a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_{p+u}^{\ast }{a}_w{a}_p{a}_q{\delta }_{q-u,v}\hfill \\ \hfill & -a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_{q-u}^{\ast }{a}_p{a}_q{\delta }_{p+u,w}-a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_{p+u}^{\ast }{a}_p{a}_q{\delta }_{q-u,w}.\hfill \end{array}\end{array}$$ 7.56

and

$$\begin{array}{c}\hfill \begin{array}{cc}& [a_p^{\ast }{a}_q^{\ast }{a}_{p-u}{a}_{q+u},a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w]\hfill \\ \hfill & =a_p^{\ast }{a}_q^{\ast }{a}_{q+u}{a}_{w-r}^{\ast }{a}_v{a}_w{\delta }_{p-u,v+r}+a_p^{\ast }{a}_q^{\ast }{a}_{p-u}{a}_{w-r}^{\ast }{a}_v{a}_w{\delta }_{q+u,v+r}\hfill \\ \hfill & +a_p^{\ast }{a}_q^{\ast }{a}_{v+r}^{\ast }{a}_{q+u}{a}_v{a}_w{\delta }_{p-u,w-r}+a_p^{\ast }{a}_q^{\ast }{a}_{v+r}^{\ast }{a}_{p-u}{a}_v{a}_w{\delta }_{q+u,w-r}\hfill \\ \hfill & -a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_q^{\ast }{a}_w{a}_{p-u}{a}_{q+u}{\delta }_{p,v}-a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_p^{\ast }{a}_w{a}_{p-u}{a}_{q+u}{\delta }_{q,v}\hfill \\ \hfill & -a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_q^{\ast }{a}_{p-u}{a}_{q+u}{\delta }_{p,w}-a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_p^{\ast }{a}_{p-u}{a}_{q+u}{\delta }_{q,w}.\hfill \end{array}\end{array}$$ 7.57

Combining the last two identities and putting non-normally ordered contributions into normal order, we find that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [\text{W},D_1]+\text{h.c.}=& \frac{1}{N}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },v,w\in P_L:\\ v+u,w-u\in P_L\end{array}}^{\ast }{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast \eta /N)(u)a_{v+u}^{\ast }{a}_{w-u}^{\ast }{a}_v{a}_w\hfill \\ & +\sum _{j=1}^6({\zeta }_j+\text{h.c.}),\hfill \end{array}\end{array}$$ 7.58

where

$$\begin{array}{cc}\hfill {\zeta }_1& =-\frac{1}{2N^2}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },v,w\in P_L:\\ v+u,w-u\in P_L,\\ r\in P_H^c\cup \{0\}\end{array}}^{\ast }{N}^{\kappa }\widehat{V}((u-r)/N^{1-\kappa }){\eta }_r{a}_{v+u}^{\ast }{a}_{w-u}^{\ast }{a}_v{a}_w,\hfill \\ \hfill {\zeta }_2& =-\frac{1}{2N^2}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },r\in P_H,\\ v,w\in P_L:\\ w-u,v+u\in P_L\end{array}}^{\ast }{N}^{\kappa }\widehat{V}(u/N^{1-\kappa }){\eta }_r{a}_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_{w-u}{a}_{v+u},\hfill \\ \hfill {\zeta }_3& =-\frac{1}{2N^2}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },r\in P_H,\\ v,w\in P_L\end{array}}^{\ast }{N}^{\kappa }\widehat{V}(u/N^{1-\kappa }){\eta }_r{a}_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_{w-u}{a}_{v+u},\hfill \\ \hfill {\zeta }_4& =-\frac{1}{N^2}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },r\in P_H,\\ v,w,q\in P_L:\\ v-u\in P_L\end{array}}^{\ast }{N}^{\kappa }\widehat{V}(u/N^{1-\kappa }){\eta }_r{a}_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_{q-u}^{\ast }{a}_w{a}_{v-u}{a}_q,\hfill \\ \hfill {\zeta }_5& =\frac{1}{N^2}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },r\in P_H,\\ v,w,q\in P_L:\\ v+r+u\in P_L\end{array}}^{\ast }{N}^{\kappa }\widehat{V}(u/N^{1-\kappa }){\eta }_r{a}_{v+r+u}^{\ast }{a}_q^{\ast }{a}_{w-r}^{\ast }{a}_{q+u}{a}_v{a}_w,\hfill \\ \hfill {\zeta }_6& =-\frac{1}{N^2}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },r\in P_H,\\ v,w,q\in P_L\end{array}}^{\ast }{N}^{\kappa }\widehat{V}(u/N^{1-\kappa }){\eta }_r{a}_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_q^{\ast }{a}_w{a}_{v-u}{a}_{q+u}.\hfill \\ \hfill \end{array}$$ 7.59

Let us briefly explain how to control the operators ${\zeta }_1$ to ${\zeta }_6$, defined in (7.2.3).

Noting that $v+u\in P_L$ implies $|u|\le 2N^{\beta }$ whenever $v\in P_L$, the first two contributions ${\zeta }_1$ and ${\zeta }_2$ in (7.2.3) can be controlled by

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,{\zeta }_1\xi ⟩|+|⟨\xi ,{\zeta }_2\xi ⟩|\hfill \\ \hfill & \le \frac{C{N}^{\kappa }}{2N^2}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },v,w\in P_L:\\ v+u,w-u\in P_L,\\ r\in P_H^c\cup \{0\}\end{array}}^{\ast }|{\eta }_r|\frac{|w-u|}{|v|}\Vert a_{v+u}{a}_{w-u}\xi \Vert \frac{|v|}{|w-u|}\Vert a_v{a}_w\xi \Vert \hfill \\ \hfill & +\frac{C{N}^{\kappa }}{2N^2}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },r\in P_H,\\ v,w\in P_L:\\ w-u,v+u\in P_L\end{array}}^{\ast }|{\eta }_r|\Vert a_{v+r}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{-1/2}{a}_{w-r}\xi \Vert \hfill \\ \hfill & \times \Vert a_{w-u}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}{a}_{v+u}\xi \Vert \hfill \\ \hfill & \le C{N}^{\alpha +\beta +2\kappa -1}⟨\xi ,{\mathcal{K}}_{\le 2N^{\beta }}\xi ⟩+N^{7\beta /2+2\kappa -\alpha /2-1}⟨\xi ,({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\xi ⟩\hfill \\ \hfill & +N^{7\beta /2+2\kappa -\alpha /2-2}⟨\xi ,{\mathcal{K}}_{\le 2N^{\beta }}({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\xi ⟩\hfill \\ \hfill & \le C{N}^{\alpha +\beta +2\kappa -1}⟨\xi ,{\mathcal{K}}_{\le 2N^{\beta }}\xi ⟩+C{N}^{2\beta +\kappa -1}⟨\xi ,({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\xi ⟩.\hfill \end{array}\end{array}$$ 7.60

By switching to position space, the term ${\zeta }_3$ can be bounded by

$$\begin{array}{cc}& |⟨\xi ,{\zeta }_3\xi ⟩|\hfill \\ \hfill & \le C{N}^{3\beta /2+\kappa -\alpha /2-1}({\int }_{{\mathrm{\Lambda }}^2}dxdy{N}^{2-2\kappa }V(N^{1-\kappa }(x-y)){\Vert {\check{a}}_x{\check{a}}_y\xi \Vert }^2{)}^{1/2}\hfill \\ \hfill & \times ({\int }_{{\mathrm{\Lambda }}^2}dxdy{N}^{2-2\kappa }V(N^{1-\kappa }(x-y))\sum _{r\in P_H,w\in P_L}\Vert \sum _{v\in P_L}{e}^{\mathit{ivx}}{a}_{v+r}{a}_{w-r}\xi {\Vert }^2{)}^{1/2}\hfill \\ \hfill & \le C{N}^{3\beta /2+\kappa -\alpha /2-1}\Vert {\mathcal{V}}_N^{1/2}\xi \Vert (N^{\kappa -1}{\int }_{\mathrm{\Lambda }}dx\sum _{r\in P_H,w\in P_L}\Vert \sum _{v\in P_L}{e}^{\mathit{ivx}}{a}_{v+r}{a}_{w-r}\xi {\Vert }^2{)}^{1/2}\hfill \\ \hfill & \le C{N}^{3\beta /2+\kappa /2-1}⟨\xi ,{\mathcal{V}}_N\xi ⟩+C{N}^{3\beta /2+\kappa /2}.\hfill \end{array}$$

We proceed similarly as above for the terms ${\zeta }_4$ and ${\zeta }_5$ which yields

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,{\zeta }_4\xi ⟩|+|⟨\xi ,{\zeta }_5\xi ⟩|\hfill \\ \hfill & \le \frac{C{N}^{\kappa }}{N^2}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },r\in P_H,\\ v,w,q\in P_L:v-u\in P_L\end{array}}^{\ast }{|q|}^{-1}|q-u|\Vert a_{v+r}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{-1/2}{a}_{w-r}{a}_{q-u}\xi \Vert \hfill \\ \hfill & \times |{\eta }_r{||q||q-u|}^{-1}\Vert a_w{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}{a}_{v-u}{a}_q\xi \Vert \hfill \\ \hfill & +\frac{C{N}^{\kappa }}{N^2}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },r\in P_H,\\ v,w,q\in P_L:\\ v+r+u\in P_L\end{array}}^{\ast }{(|q||v|}^{-1}\Vert a_{v+r+u}{a}_q{a}_{w-r}\xi \Vert )(|{\eta }_r{||q|}^{-1}|v|\Vert a_{q+u}{a}_v{a}_w\xi \Vert )\hfill \\ \hfill & \le C{N}^{5\beta /2+2\kappa -\alpha /2-1}⟨\xi ,{\mathcal{K}}_{\le 3N^{\beta }}({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\xi ⟩\hfill \\ \hfill & \le C{N}^{\beta +\kappa -1}⟨\xi ,{\mathcal{K}}_{\le 3N^{\beta }}({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\xi ⟩,\hfill \end{array}\end{array}$$ 7.61

where, for ${\zeta }_5$, we used that $v+r+u\in P_L$ implies that $|u|\ge \frac{3}{4}{N}^{\alpha }$, and thus $|q+u|\ge \frac{1}{2}{N}^{\alpha }$, whenever $v,q\in P_L$, $r\in P_H$ and $N\in \mathbb{N}$ sufficiently large (otherwise $|v+r+u|\ge \frac{1}{4}{N}^{\alpha }-N^{\beta }>N^{\beta }$ for large enough $N\in \mathbb{N}$). Finally, ${\zeta }_6$ can be controlled by

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,{\zeta }_6\xi ⟩|\hfill \\ \hfill & =|\frac{1}{N}\sum _{\begin{array}{c}r\in P_H,\\ v,w,q\in P_L\end{array}}^{\ast }{\int }_{{\mathrm{\Lambda }}^2}{N}^{2-2\kappa }V(N^{1-\kappa }(x-y))e^{-ivx-iqy}{\eta }_r⟨\xi ,a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_q^{\ast }{a}_w{\check{a}}_x{\check{a}}_y\xi ⟩|\hfill \\ \hfill & \le C{N}^{\beta /2+\kappa -\alpha /2-1/2}\Vert {\mathcal{V}}_N^{1/2}\xi \Vert (N^{\kappa -1}{\int }_{\mathrm{\Lambda }}dx\sum _{\begin{array}{c}r\in P_H,\\ w,q\in P_L\end{array}}^{\ast }|q|\Vert \sum _{v\in P_L}{e}^{-ivx}{a}_{v+r}{a}_{w-r}{a}_q\xi {\Vert }^2{)}^{1/2}\hfill \\ \hfill & \le C{N}^{\beta /2+\kappa /2-1/2}\Vert {\mathcal{V}}_N^{1/2}\xi \Vert \Vert {\mathcal{K}}_L^{1/2}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}\xi \Vert \hfill \end{array}\end{array}$$

In summary, the previous estimates show that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \pm \sum _{j=1}^6({\zeta }_j+\text{h.c.})& \le \delta {\mathcal{V}}_N+C{N}^{3\beta /2+\kappa /2-1}{\mathcal{V}}_N+C{N}^{\alpha +\beta +2\kappa -1}{\mathcal{K}}_{\le 2N^{\beta }}+C{N}^{2\beta +\kappa }\hfill \\ \hfill & +C(1+{\delta }^{-1})N^{\beta +\kappa -1}{\mathcal{K}}_{\le 3N^{\beta }}({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\hfill \end{array}\end{array}$$ 7.62

for all $\delta >0$. On the other hand, by Lemma 7.3, we also know that

$$\begin{array}{c}\hfill \begin{array}{cc}& \pm [\frac{1}{N}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },v,w\in P_L:\\ v+u,w-u\in P_L\end{array}}^{\ast }{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast \eta /N)(u){\int }_0^{s}dt{e}^{-tD}{a}_{v+u}^{\ast }{a}_{w-u}^{\ast }{a}_v{a}_w{e}^{\mathit{tD}}\hfill \\ & -\frac{s}{N}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },v,w\in P_L:\\ v+u,w-u\in P_L\end{array}}^{\ast }{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast \eta /N)(u)a_{v+u}^{\ast }{a}_{w-u}^{\ast }{a}_v{a}_w]\hfill \\ & \le C{N}^{-3\beta -3\kappa }\mathcal{K}+C{N}^{3\beta +\kappa -2}+C{N}^{\beta +\kappa -1}{\mathcal{K}}_L({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1).\hfill \end{array}\end{array}$$ 7.63

Now, going back to (7.55), the bounds (7.62) and (7.63) imply that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^{-sD}\text{W}{e}^{\mathit{sD}}& =\text{W}+\frac{s}{N}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },v,w\in P_L:\\ v+u,w-u\in P_L\end{array}}^{\ast }{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast \eta /N)(u)a_{v+u}^{\ast }{a}_{w-u}^{\ast }{a}_v{a}_w\hfill \\ & +{\mathcal{E}}_1(s)+{\mathcal{E}}_2(s,\delta ),\hfill \end{array}\end{array}$$ 7.64

where the self-adjoint operators ${\mathcal{E}}_1(s)$ and ${\mathcal{E}}_2(s)$ are bounded by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \pm {\mathcal{E}}_1(s)& \le C(N^{\alpha +\beta +2\kappa -1}+N^{-3\beta -3\kappa })\mathcal{K}+C{N}^{2\beta +\kappa },\hfill \end{array}\end{array}$$

as well as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \pm {\mathcal{E}}_2(s,\delta )& \le C{N}^{\beta +\kappa -1}{\mathcal{K}}_L({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)+C(\delta +C{N}^{3\beta /2+\kappa /2-1}){\int }_0^{s}dt{e}^{-tD}{\mathcal{V}}_N{e}^{\mathit{tD}}\hfill \\ \hfill & +C(1+{\delta }^{-1})N^{\beta +\kappa -1}{\int }_0^{s}dt{e}^{-tD}{\mathcal{K}}_{\le 2N^{\beta }}({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)e^{\mathit{tD}},\hfill \end{array}\end{array}$$

for all $\delta >0$ and uniformly in $s\in [-1;1]$. Defining ${\mathcal{E}}_2(s)={\mathcal{E}}_2(s,N^{-\beta -\kappa })$, this concludes the proof. $\square$

Equipped with Proposition 7.11, we go back to (7.50) and conclude that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{M}}_N^{(4)}& \ge {\mathcal{H}}_N-\frac{1}{2N}\sum _{\begin{array}{c}r\in {\mathrm{\Lambda }}^{\ast },v,w\in P_L:\\ v+r,w-r\ne 0\end{array}}\widehat{V}(r/N^{1-\kappa })(a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w+\text{h.c.})\hfill \\ & -\frac{1}{2N}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },v,w\in P_L:\\ v+u,w-u\in P_L\end{array}}^{\ast }{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast \eta /N)(u)a_{v+u}^{\ast }{a}_{w-u}^{\ast }{a}_v{a}_w\hfill \\ & -\frac{1}{8}\mathcal{K}-C{N}^{2\beta +\kappa }+{\int }_0^{1}ds{\mathcal{E}}_2(s)+{\int }_0^{1}ds{e}^{-sD}({\mathcal{E}}_{[{\mathcal{V}}_N,D]}+{\mathcal{E}}_{[\mathcal{K},D]})e^{\mathit{sD}},\hfill \end{array}\end{array}$$ 7.65

for all $\alpha \ge 3\beta +2\kappa \ge 0$ with $\alpha +\beta +2\kappa -1<0$, $0\le \kappa <\beta$ and $N\in \mathbb{N}$ large enough.

Next, let us analyse the error terms related to ${\mathcal{E}}_2(s)$ and ${\mathcal{E}}_{[{\mathcal{V}}_N,D]}$ further. The bounds (7.53) and (7.21) (with $\delta =c{N}^{-\beta -\kappa }$ for a sufficiently small $c>0$; this choice guarantees that we can extract the term ${\mathcal{V}}_{N,L}$ in (7.66), with an error that can be absorbed in $\mathcal{K}$) imply, together with Lemmas 7.1, 7.2, Corollaries 7.4 and 7.6 and with the assumption (5.6) on the exponents $\alpha ,\beta$, that

$$\begin{array}{c}\hfill \begin{array}{cc}& {\int }_0^{1}ds(e^D{\mathcal{E}}_2(s)e^{-D}+e^{(1-s)D}{\mathcal{E}}_{[{\mathcal{V}}_N,D]}{e}^{-(1-s)D})\hfill \\ \hfill & \ge -C{N}^{2\beta +2\kappa -1}{\mathcal{K}}_L({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)-\tilde{C}{N}^{-\beta -\kappa }({\mathcal{V}}_N+{\mathcal{V}}_{N,L})-C{N}^{2\beta }({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\hfill \\ \hfill & -C{N}^{4\beta +2\kappa -1}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^2\hfill \end{array}\end{array}$$

for all $N\in \mathbb{N}$ large enough and for an arbitrarily small constant $\tilde{C}>0$. With Corollary 7.4 and (7.65), we conclude that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{M}}_N^{(4)}& \ge {\mathcal{H}}_N-\frac{1}{2N}\sum _{\begin{array}{c}r\in {\mathrm{\Lambda }}^{\ast },v,w\in P_L:\\ v+r,w-r\ne 0\end{array}}\widehat{V}(r/N^{1-\kappa })(a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w+\text{h.c.})\hfill \\ & -\frac{1}{2N}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },v,w\in P_L:\\ v+u,w-u\in P_L\end{array}}^{\ast }{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast \eta /N)(u)a_{v+u}^{\ast }{a}_{w-u}^{\ast }{a}_v{a}_w\hfill \\ & -\frac{1}{4}\mathcal{K}-C{N}^{2\beta +\kappa }-\tilde{C}{N}^{-\beta -\kappa }{\mathcal{V}}_{N,L}+{\int }_0^{1}ds{e}^{-sD}{\mathcal{E}}_{[\mathcal{K},D]}{e}^{\mathit{sD}}+{\mathcal{E}}_{{\mathcal{M}}_N}^{(41)},\hfill \end{array}\end{array}$$ 7.66

where the error ${\mathcal{E}}_{{\mathcal{M}}_N}^{(41)}$ is such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^D{\mathcal{E}}_{{\mathcal{M}}_N}^{(41)}{e}^{-D}& \ge -C{N}^{2\beta +2\kappa -1}{\mathcal{K}}_L({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)-C{N}^{-\beta -\kappa }{\mathcal{V}}_N\hfill \\ \hfill & -C{N}^{2\beta }{\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}-C{N}^{4\beta +2\kappa -1}{\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}^2\hfill \end{array}\end{array}$$

Applying Lemmas 4.2, 4.3 and Corollary 4.5, we deduce with the operator inequality ${\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}\le 4N^{-2\alpha }\mathcal{K}$ that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^A{e}^D{\mathcal{E}}_{{\mathcal{M}}_N}^{(41)}{e}^{-D}{e}^{-A}& \ge -C{N}^{-\beta }\mathcal{K}-C{N}^{-\beta -\kappa }{\mathcal{V}}_N-C{N}^{2\beta +2\kappa -1}\hfill \\ \hfill & -C{N}^{2\beta +2\kappa -1}\mathcal{K}{\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}\hfill \end{array}\end{array}$$ 7.67

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

Now, we switch to the contribution containing the operator ${\mathcal{E}}_{[\mathcal{K},D]}$ on the r.h.s. of the lower bound (7.66). We recall once again that

$$\begin{array}{c}\hfill {\int }_0^{1}ds{e}^{-sD}{\mathcal{E}}_{[\mathcal{K},D]}{e}^{\mathit{sD}}={\int }_0^{1}ds\sum _{j=1}^3{e}^{-sD}({\mathrm{\Sigma }}_j+\text{h.c.})e^{\mathit{sD}},\end{array}$$

where the operators ${\mathrm{\Sigma }}_1,{\mathrm{\Sigma }}_2$ and ${\mathrm{\Sigma }}_3$ were defined in (7.26). It turns out that ${\mathrm{\Sigma }}_2$ and ${\mathrm{\Sigma }}_3$ are negligible errors while ${\mathrm{\Sigma }}_1$ still contains an important contribution of leading order. We start with the analysis of the contribution related to ${\mathrm{\Sigma }}_1$.

Proposition 7.12

Assume the exponents $\alpha ,\beta$ satisfy (5.6). Then, we have that

$$\begin{array}{c}\hfill \begin{array}{cc}& \frac{1}{2N}\sum _{\begin{array}{c}u\in P_H^c\cup \{0\},p,q\in P_L:\\ p+u,q-u\ne 0\end{array}}{N}^{\kappa }(\widehat{V}(/N^{1-\kappa })\ast {\widehat{f}}_N)(u)e^{-sD}(a_{p+u}^{\ast }{a}_{q-u}^{\ast }{a}_p{a}_q+h.c.)e^{\mathit{sD}}\hfill \\ \hfill & =\frac{1}{2N}\sum _{\begin{array}{c}u\in P_H^c\cup \{0\},p,q\in P_L:\\ p+u,q-u\ne 0\end{array}}{N}^{\kappa }(\widehat{V}(/N^{1-\kappa })\ast {\widehat{f}}_N)(u)(a_{p+u}^{\ast }{a}_{q-u}^{\ast }{a}_p{a}_q+h.c.)+{\mathcal{E}}_3(s)\hfill \end{array}\end{array}$$ 7.68

and there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}& \pm e^A{e}^D{\mathcal{E}}_3(s)e^{-D}{e}^{-A}\hfill \\ \hfill & \le C{N}^{\alpha +\beta +2\kappa -1}\mathcal{K}+C{N}^{\alpha +\beta +2\kappa -1}\mathcal{K}{\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+C{N}^{4\beta +2\kappa }+C{N}^{\alpha +3\beta +2\kappa -1}\hfill \end{array}\end{array}$$ 7.69

for all $s\in [-1;1]$ and for all $N\in \mathbb{N}$ sufficiently large.

Proof

We proceed as in Proposition 7.11 and recall ${\mathrm{\Sigma }}_1:{\mathcal{F}}_{+}^{\le N}\to {\mathcal{F}}_{+}^{\le N}$ to be

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathrm{\Sigma }}_1=\frac{1}{2N}\sum _{\begin{array}{c}u\in P_H^c\cup \{0\},p,q\in P_L:\\ p+u,q-u\ne 0\end{array}}{N}^{\kappa }(\widehat{V}(/N^{1-\kappa })\ast {\widehat{f}}_N)(u)(a_{p+u}^{\ast }{a}_{q-u}^{\ast }{a}_p{a}_q+\text{h.c.}).\hfill \end{array}\end{array}$$

We then have

$$\begin{array}{c}\hfill \begin{array}{cc}& e^{-sD}{\mathrm{\Sigma }}_1{e}^{\mathit{sD}}-{\mathrm{\Sigma }}_1={\int }_0^{s}dt{e}^{-tD}[{\mathrm{\Sigma }}_1,D_1]e^{\mathit{tD}}+\text{h.c.}\hfill \end{array}\end{array}$$ 7.70

Similarly as in (7.58) and (7.2.3), we find that

$$\begin{array}{c}\hfill [{\mathrm{\Sigma }}_1,D_1]+\text{h.c.}=\sum _{i=1}^8({\mathrm{\Gamma }}_i+\text{h.c.}),\end{array}$$ 7.71

where

$$\begin{array}{cc}\hfill {\mathrm{\Gamma }}_1& =\frac{1}{N^2}\sum _{\begin{array}{c}u\in P_H^c\cup \{0\},r\in P_H,v,w\in P_L:\\ v+u+r,w-u-r\in P_L\end{array}}^{\ast }{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast {\widehat{f}}_N)(u){\eta }_r{a}_{v+u+r}^{\ast }{a}_{w-u-r}^{\ast }{a}_v{a}_w,\hfill \\ \hfill {\mathrm{\Gamma }}_2& =-\frac{1}{2N^2}\sum _{\begin{array}{c}u\in P_H^c\cup \{0\},r\in P_H,\\ v,w\in P_L:\\ w-u,v+u\in P_L\end{array}}^{\ast }{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast {\widehat{f}}_N)(u){\eta }_r{a}_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_{w-u}{a}_{v+u},\hfill \\ \hfill {\mathrm{\Gamma }}_3& =-\frac{1}{2N^2}\sum _{\begin{array}{c}u\in P_H^c\cup \{0\},r\in P_H,\\ v,w\in P_L\end{array}}^{\ast }{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast {\widehat{f}}_N)(u){\eta }_r{a}_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_{w-u}{a}_{v+u},\hfill \\ \hfill {\mathrm{\Gamma }}_4& =-\frac{1}{N^2}\sum _{\begin{array}{c}u\in P_H^c\cup \{0\},r\in P_H,\\ v,w,q\in P_L:\\ v-u\in P_L\end{array}}^{\ast }{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast {\widehat{f}}_N)(u){\eta }_r{a}_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_{q-u}^{\ast }{a}_w{a}_{v-u}{a}_q,\hfill \\ \hfill {\mathrm{\Gamma }}_5& =\frac{1}{N^2}\sum _{\begin{array}{c}u\in P_H^c\cup \{0\},r\in P_H,\\ v,w,q\in P_L:\\ v+r+u\in P_L\end{array}}^{\ast }{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast {\widehat{f}}_N)(u){\eta }_r{a}_{v+r+u}^{\ast }{a}_q^{\ast }{a}_{w-r}^{\ast }{a}_{q+u}{a}_v{a}_w,\hfill \\ \hfill {\mathrm{\Gamma }}_6& =-\frac{1}{N^2}\sum _{\begin{array}{c}u\in P_H^c\cup \{0\},r\in P_H,\\ v,w,q\in P_L\end{array}}^{\ast }{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast {\widehat{f}}_N)(u){\eta }_r{a}_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_q^{\ast }{a}_w{a}_{v-u}{a}_{q+u}.\hfill \end{array}$$

The operators ${\mathrm{\Gamma }}_1$ to ${\mathrm{\Gamma }}_6$ can be bounded similarly as in the proof of Proposition 7.11. Let us start with ${\mathrm{\Gamma }}_1$. Applying as usual Cauchy-Schwarz implies that

$$\begin{array}{c}\hfill \begin{array}{cc}& |⟨\xi ,{\mathrm{\Gamma }}_1\xi ⟩|\hfill \\ & \le \frac{C{N}^{\kappa }}{N^2}\sum _{\begin{array}{c}u\in P_H^c\cup \{0\},r\in P_H,v,w\in P_L:\\ v+u+r,w-u-r\in P_L\end{array}}^{\ast }{(|v|}^{-1}\Vert a_{v+u+r}{a}_{w-u-r}\xi \Vert )(|{\eta }_r||v|\Vert a_v{a}_w\xi \Vert )\hfill \\ & \le C{N}^{\alpha /2+5\beta /2+2\kappa -1/2}\Vert \xi \Vert \Vert {\mathcal{K}}_L^{1/2}\xi \Vert \le C{N}^{\alpha +\beta +2\kappa -1}⟨\xi ,{\mathcal{K}}_L\xi ⟩+C{N}^{4\beta +2\kappa }{\Vert \xi \Vert }^2\hfill \end{array}\end{array}$$

where we used that $v+u+r\in P_L$ implies $|u|\ge N^{\alpha }-3N^{\beta }$ and $|r|\le N^{\alpha }+3N^{\beta }$ whenever $u\in P_H^c,r\in P_H$ and $v\in P_L$ (otherwise $|u+r+v|\ge |r|-|u|-N^{\beta }\ge 2N^{\beta }>N^{\beta }$ if either $|u|\le N^{\alpha }-3N^{\beta }$ or $|r|\ge N^{\alpha }+3N^{\beta }$, in contradiction to $u+r+v\in P_L$) for $N\in \mathbb{N}$ sufficiently large. Notice in addition that ${\sum }_{N^{\alpha }-3N^{\beta }\le |u|\le N^{\alpha }}\le C{N}^{2\alpha +\beta }$.

The term ${\mathrm{\Gamma }}_2$ can be estimated exactly as the term ${\zeta }_2$ in (7.60), that is

$$\begin{array}{c}\hfill |⟨\xi ,{\mathrm{\Gamma }}_2\xi ⟩|\le C{N}^{\alpha +\beta +2\kappa -1}⟨\xi ,{\mathcal{K}}_{\le 2N^{\beta }}\xi ⟩+C{N}^{2\beta +\kappa -1}⟨\xi ,({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\xi ⟩.\end{array}$$

The contribution ${\mathrm{\Gamma }}_3$ can be controlled by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\mathrm{\Gamma }}_3\xi ⟩|& \le \frac{C{N}^{\kappa }}{2N^2}\sum _{\begin{array}{c}u\in P_H^c\cup \{0\},r\in P_H,\\ v,w\in P_L\end{array}}^{\ast }|{\eta }_r|\Vert a_{v+r}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{-1/2}{a}_{w-r}\xi \Vert \hfill \\ & \Vert a_{w-u}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}{a}_{v+u}\xi \Vert \hfill \\ & \le C{N}^{\alpha +3\beta +2\kappa -1}⟨\xi ,({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\xi ⟩.\hfill \end{array}\end{array}$$

The terms ${\mathrm{\Gamma }}_4$ and ${\mathrm{\Gamma }}_5$ can be bounded exactly as in (7.61). We find

$$\begin{array}{c}\hfill |⟨\xi ,{\mathrm{\Gamma }}_4\xi ⟩|+|⟨\xi ,{\mathrm{\Gamma }}_5\xi ⟩|\le C{N}^{\beta +\kappa -1}⟨\xi ,{\mathcal{K}}_{\le 2N^{\beta }}({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\xi ⟩,\end{array}$$

Finally, the last contribution ${\mathrm{\Gamma }}_6$ is bounded by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |⟨\xi ,{\mathrm{\Gamma }}_6\xi ⟩|& \le \frac{C{N}^{\kappa }}{N^2}\sum _{\begin{array}{c}u\in P_H^c\cup \{0\},r\in P_H,\\ v,w,q\in P_L\end{array}}^{\ast }{(|q||w|}^{-1}\Vert a_{v+r}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{-1/2}{a}_{w-r}{a}_q\xi \Vert )\hfill \\ & \times (|{\eta }_r{||w||q|}^{-1}\Vert a_{v-u}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}{a}_w{a}_{q+u}\xi \Vert )\hfill \\ & \le C{N}^{\alpha +\beta +2\kappa -1}⟨\xi ,{\mathcal{K}}_L({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\xi ⟩.\hfill \end{array}\end{array}$$

In conclusion, the above estimates imply that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \pm \sum _{i=1}^6({\mathrm{\Gamma }}_i+\text{h.c.})& \le C{N}^{\alpha +\beta +2\kappa -1}{\mathcal{K}}_{\le 2N^{\beta }}+C{N}^{\alpha +\beta +2\kappa -1}{\mathcal{K}}_{\le 2N^{\beta }}({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)\hfill \\ \hfill & +C{N}^{\alpha +3\beta +2\kappa -1}({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)+C{N}^{4\beta +2\kappa }\hfill \end{array}\end{array}$$

for all $\alpha >3\beta +2\kappa \ge 0$ and for all $N\in \mathbb{N}$ sufficiently large. Combining this estimate with the identites (7.70) and (7.71), and applying Lemmas 4.2, 4.3, 7.1 as well as Lemma 7.2 together with the operator inequality ${\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}\le 4N^{-2\alpha }\mathcal{K}$ proves the proposition. $\square$

Applying Proposition 7.12 to the lower bound (7.66) and defining ${\mathcal{E}}_{{\mathcal{M}}_N}^{(42)}={\int }_0^{1}ds{\mathcal{E}}_3(s)$ with $\mathcal{E}(s)$ from Proposition 7.12, we conclude that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{M}}_N^{(4)}& \ge {\mathcal{H}}_N-\frac{1}{2N}\sum _{\begin{array}{c}r\in {\mathrm{\Lambda }}^{\ast },v,w\in P_L:\\ v+r,w-r\ne 0\end{array}}\widehat{V}(r/N^{1-\kappa })(a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w+\text{h.c.})\hfill \\ & -\frac{1}{2N}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },v,w\in P_L:\\ v+u,w-u\in P_L\end{array}}^{\ast }{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast \eta /N)(u)a_{v+u}^{\ast }{a}_{w-u}^{\ast }{a}_v{a}_w\hfill \\ & +\frac{1}{2N}\sum _{\begin{array}{c}u\in P_H^c\cup \{0\},p,q\in P_L:\\ p+u,q-u\ne 0\end{array}}{N}^{\kappa }(\widehat{V}(/N^{1-\kappa })\ast {\widehat{f}}_N)(u)(a_{p+u}^{\ast }{a}_{q-u}^{\ast }{a}_p{a}_q+h.c.)\hfill \\ & -\frac{1}{4}\mathcal{K}-C{N}^{-\beta -\kappa }{\mathcal{V}}_{N,L}+{\mathcal{E}}_{{\mathcal{M}}_N}^{(41)}+{\mathcal{E}}_{{\mathcal{M}}_N}^{(42)}+{\int }_0^{1}ds{e}^{-sD}({\mathrm{\Sigma }}_2+{\mathrm{\Sigma }}_3+\text{h.c.})e^{\mathit{sD}},\hfill \end{array}\end{array}$$ 7.72

where ${\mathcal{E}}_{{\mathcal{M}}_N}^{(41)}$ satisfies the lower bound (7.67), ${\mathcal{E}}_{{\mathcal{M}}_N}^{(42)}$ satisfies the bound (7.69) and where the operators ${\mathrm{\Sigma }}_2$ and ${\mathrm{\Sigma }}_3$ were defined in (7.26).

Let us finally estimate the size of the error in the last line of (7.72), involving the two operators ${\mathrm{\Sigma }}_2$ and ${\mathrm{\Sigma }}_3$. Using the estimate (7.31) together with Lemmas 4.2, 4.3, 7.1 and 7.2, we find for ${\mathcal{E}}_{{\mathcal{M}}_N}^{(43)}={\int }_0^{1}ds{e}^{-sD}({\mathrm{\Sigma }}_2+\text{h.c.})e^{\mathit{sD}}$

$$\begin{array}{c}\hfill \begin{array}{c}\hfill e^A{e}^D{\mathcal{E}}_{{\mathcal{M}}_N}^{(43)}{e}^{-D}{e}^{-A}\ge -C{N}^{-\beta -1}\mathcal{K}{\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}-C{N}^{-5\beta -4\kappa }\mathcal{K}-C{N}^{\beta }.\end{array}\end{array}$$ 7.73

Finally, consider the operator ${\mathcal{E}}_{{\mathcal{M}}_N}^{(44)}={\int }_0^{1}ds{e}^{-sD}({\mathrm{\Sigma }}_3+\text{h.c.})e^{\mathit{sD}}$, with ${\mathrm{\Sigma }}_3$ defined in (7.26). Let $m_0\in \mathbb{R}$ be such that $m_0\beta =\alpha$ (in particular, $\lfloor m_0\rfloor \ge 3$). Here, we use the bound (7.32) to find first of all that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{E}}_{{\mathcal{M}}_N}^{(44)}\ge & -{\int }_0^{1}ds\Vert {\mathcal{K}}^{1/2}{e}^{\mathit{sD}}\xi \Vert (N^{-1/2}\Vert {\mathcal{K}}_L^{1/2}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}\xi \Vert \hfill \\ \hfill & +N^{\beta -1}\Vert {({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{3/2}\xi \Vert )\hfill \end{array}\end{array}$$

for any $\xi \in {\mathcal{F}}_{+}^{\le N}$ with $\Vert \xi \Vert =1$. Notice that we applied once again Lemmas 7.1 and 7.2 in the second factor. With Corollary 7.8, the first factor is bounded by

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathcal{E}}_{{\mathcal{M}}_N}^{(44)}\hfill \\ \hfill & \ge -C(\Vert {\mathcal{K}}^{1/2}\xi \Vert +\Vert {\mathcal{V}}_N^{1/2}\xi \Vert +\Vert {\mathcal{V}}_{N,L}^{1/2}\xi \Vert +N^{5\beta /8+\kappa /2}\Vert {\mathcal{K}}_{\le N^{3\beta /2}}^{1/2}\xi \Vert \hfill \\ \hfill & +N^{-1/2}\Vert {\mathcal{K}}_L^{1/2}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}\xi \Vert +N^{3\beta /2+\kappa /2}\hfill \\ \hfill & +\sum _{j=3}^{2\lfloor m_0\rfloor -1}{N}^{j\beta /4+3\beta /4+\kappa -1/2}[\Vert {\mathcal{K}}_L^{1/2}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{j\beta /2}}+1)}^{1/2}\xi \Vert \hfill \\ \hfill & +N^{\beta }\Vert {({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{j\beta /2}}+1)}^{1/2}\xi \Vert ]\hfill \\ \hfill & +N^{\alpha /2+\beta /2+\kappa -1/2}[\Vert {\mathcal{K}}_L^{1/2}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\lfloor m_0\rfloor \beta }}+1)}^{1/2}\xi \Vert \hfill \\ \hfill & +N^{\beta }\Vert {({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\lfloor m_0\rfloor \beta }}+1)}^{1/2}\xi \Vert ])\hfill \\ \hfill & \times (N^{-1/2}\Vert {\mathcal{K}}_L^{1/2}{({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{1/2}\xi \Vert +N^{\beta -1}\Vert {({\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}+1)}^{3/2}\xi \Vert )\hfill \end{array}\end{array}$$

for all exponents $\alpha ,\beta$ satisfying (5.6) and $N\in \mathbb{N}$ sufficiently large. It follows that

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathcal{E}}_{{\mathcal{M}}_N}^{(44)}\ge {\mathcal{E}}_{{\mathcal{M}}_N}^{(441)}+{\mathcal{E}}_{{\mathcal{M}}_N}^{(442)}+{\mathcal{E}}_{{\mathcal{M}}_N}^{(443)},\end{array}\end{array}$$ 7.74

where

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathcal{E}}_{{\mathcal{M}}_N}^{(441)}=-\frac{1}{8}\mathcal{K}-\tilde{C}{N}^{-\alpha }{\mathcal{V}}_{N,L}-C{N}^{3\beta +\kappa },{\mathcal{E}}_{{\mathcal{M}}_N}^{(442)}=N^{-\alpha }{\mathcal{V}}_N\end{array}\end{array}$$ 7.75

with an arbitrarily small constant $\tilde{C}>0$ and where after an additional application of Lemmas 4.2, 4.3, 7.1 and 7.2 together with the operator bound ${\mathcal{N}}_{\ge \mathrm{\Theta }}\le {\mathrm{\Theta }}^{-2}\mathcal{K}$, the error ${\mathcal{E}}_{{\mathcal{M}}_N}^{(443)}$ is such that

$$\begin{array}{c}\hfill \begin{array}{cc}& e^A{e}^D{\mathcal{E}}_{{\mathcal{M}}_N}^{(443)}{e}^{-D}{e}^{-A}\hfill \\ \hfill & \ge -C{N}^{\alpha +\beta +2\kappa -1}\mathcal{K}-C{N}^{\alpha -1}\mathcal{K}{\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}-C{N}^{\alpha +3\beta +2\kappa -1}\hfill \\ \hfill & -C\sum _{j=3}^{2\lfloor m_0\rfloor -1}{N}^{j\beta /2+\beta /2+2\kappa -1}\mathcal{K}{\mathcal{N}}_{\ge \frac{1}{2}{N}^{j\beta /2}}-C{N}^{\alpha +\beta +2\kappa -1}\mathcal{K}{\mathcal{N}}_{\ge \frac{1}{2}{N}^{\lfloor m_0\rfloor \beta }}\hfill \end{array}\end{array}$$ 7.76

for all exponents $\alpha ,\beta$ satisfying (5.6) and $N\in \mathbb{N}$ sufficiently large.

Choosing $\tilde{C}>0$ sufficiently large (but independently of $N\in \mathbb{N}$) and arguing as right before (7.66), we deduce that

$$\begin{array}{c}\hfill \begin{array}{cc}& e^A(\tilde{C}{N}^{-\alpha }{e}^D{\mathcal{V}}_{N,L}{e}^{-D}+e^D{\mathcal{E}}_{{\mathcal{M}}_N}^{(442)}{e}^{-D})e^{-A}\hfill \\ \hfill & \ge -C{N}^{-\alpha }{\mathcal{V}}_N-C{N}^{-3\beta -\kappa }{\mathcal{N}}_{+}-C{N}^{-2\beta -\kappa -1}\mathcal{K}{\mathcal{N}}_{\ge \frac{1}{2}{N}^{\alpha }}\hfill \end{array}\end{array}$$ 7.77

for all $\alpha ,\beta$ satisfying (5.6) and $N\in \mathbb{N}$ sufficiently large. This follows through another application of Corollaries 4.5, 7.4 and 7.6, together with Lemmas 4.2, 4.3, 7.1 and 7.2. We summarize these bounds in the following corollary.

Corollary 7.13

Let $m_0\in \mathbb{R}$ be such that $m_0\beta =\alpha$ and let ${\mathcal{M}}_N^{(4)}$ be defined as in (7.35). For every $\tilde{C}>0$, there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{M}}_N^{(4)}& \ge \frac{1}{2}\mathcal{K}+{\mathcal{V}}_N-\frac{1}{2N}\sum _{\begin{array}{c}r\in {\mathrm{\Lambda }}^{\ast },v,w\in P_L:\\ v+r,w-r\ne 0\end{array}}\widehat{V}(r/N^{1-\kappa })(a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w+\text{h.c.})\hfill \\ & -\frac{1}{2N}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },v,w\in P_L:\\ v+u,w-u\in P_L\end{array}}^{\ast }{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast \eta /N)(u)a_{v+u}^{\ast }{a}_{w-u}^{\ast }{a}_v{a}_w\hfill \\ & +\frac{1}{2N}\sum _{\begin{array}{c}u\in P_H^c\cup \{0\},p,q\in P_L:\\ p+u,q-u\ne 0\end{array}}{N}^{\kappa }(\widehat{V}(/N^{1-\kappa })\ast {\widehat{f}}_N)(u)(a_{p+u}^{\ast }{a}_{q-u}^{\ast }{a}_p{a}_q+h.c.)\hfill \\ & -\tilde{C}{N}^{-\beta -\kappa }{\mathcal{V}}_{N,L}+{\mathcal{E}}_{{\mathcal{M}}_N}^{(4)}\hfill \end{array}\end{array}$$ 7.78

where

$$\begin{array}{c}\hfill \begin{array}{cc}& e^A{e}^D{\mathcal{E}}_{{\mathcal{M}}_N}^{(4)}{e}^{-D}{e}^{-A}\hfill \\ \hfill & \ge -C{N}^{-\beta }\mathcal{K}-C{N}^{-\beta -\kappa }{\mathcal{V}}_N-C{N}^{\alpha +\beta +2\kappa -1}\mathcal{K}{\mathcal{N}}_{\ge \frac{1}{2}{N}^{\lfloor m_0\rfloor \beta }}\hfill \\ \hfill & -C\sum _{j=3}^{2\lfloor m_0\rfloor -1}{N}^{j\beta /2+\beta /2+2\kappa -1}\mathcal{K}{\mathcal{N}}_{\ge \frac{1}{2}{N}^{j\beta /2}}-C{N}^{2\beta +\kappa }\hfill \end{array}\end{array}$$ 7.79

for all exponents $\alpha ,\beta$ satisfying (5.6) and for all $N\in \mathbb{N}$ sufficiently large.

Proof

The proof follows from defining ${\mathcal{E}}_{{\mathcal{M}}_N}^{(4)}={\sum }_{j=1}^3{\mathcal{E}}_{{\mathcal{M}}_N}^{(4j)}+{\sum }_{j=1}^3{\mathcal{E}}_{{\mathcal{M}}_N}^{(44j)}$ and combining (7.67), (7.72), (7.69), (7.73), (7.74), (7.75), (7.77), (7.76) and the operator bound ${\mathcal{N}}_{+}\le {(2\pi )}^{-2}\mathcal{K}$ in ${\mathcal{F}}_{+}^{\le N}$. $\square$

Proof of Proposition 5.1

Recall from (7.34) the decomposition

$$\begin{array}{c}\hfill {\mathcal{M}}_N=4\pi {\mathfrak{a}}_0{N}^{1+\kappa }-4\pi {\mathfrak{a}}_0{N}^{\kappa -1}{\mathcal{N}}_{+}^2/N+{\mathcal{M}}_N^{(2)}+{\mathcal{M}}_N^{(3)}+{\mathcal{M}}_N^{(4)}\end{array}$$

Collecting the results of Propositions 7.9, 7.10 and Corollary 7.13, we deduce that

$$\begin{array}{cc}\hfill {\mathcal{M}}_N& \ge 4\pi {\mathfrak{a}}_0{N}^{1+\kappa }-4\pi {\mathfrak{a}}_0{N}^{\kappa -1}{\mathcal{N}}_{+}^2+8\pi {\mathfrak{a}}_0{N}^{\kappa }\sum _{p\in P_H^c}[b_p^{\ast }{b}_p+\frac{1}{2}{b}_p^{\ast }{b}_{-p}^{\ast }+\frac{1}{2}{b}_p{b}_{-p}]\hfill \\ \hfill & +\frac{8\pi {\mathfrak{a}}_0{N}^{\kappa }}{\sqrt{N}}\sum _{\begin{array}{c}p\in P_H^c,q\in P_L:\\ p+q\ne 0\end{array}}[b_{-p}^{\ast }{a}_{p+q}^{\ast }{a}_q+\text{h.c.}]+\frac{1}{2}\mathcal{K}\hfill \\ \hfill & +{\mathcal{V}}_N-\frac{1}{2N}\sum _{\begin{array}{c}r\in {\mathrm{\Lambda }}^{\ast },v,w\in P_L:\\ v+r,w-r\ne 0\end{array}}\widehat{V}(r/N^{1-\kappa })(a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w+\text{h.c.})\hfill \\ \hfill & -\frac{1}{2N}\sum _{\begin{array}{c}r\in {\mathrm{\Lambda }}^{\ast },v,w\in P_L:\\ v+r,w-r\in P_L\end{array}}^{\ast }{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast \eta /N)(r)a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w\hfill \\ \hfill & +\frac{1}{2N}\sum _{\begin{array}{c}r\in P_H^c\cup \{0\},v,w\in P_L:\\ v+r,w-r\ne 0\end{array}}{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast {\widehat{f}}_N)(r)(a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w+\text{h.c.})\hfill \\ \hfill & -\tilde{C}{N}^{-\beta -\kappa }{\mathcal{V}}_{N,L}+{\mathcal{E}}_{{\mathcal{M}}_N}^{\prime },\hfill \end{array}$$ 7.80

where ${\mathcal{E}}_{{\mathcal{M}}_N}^{\prime }$ satisfies the lower bound

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^A{e}^D{\mathcal{E}}_{{\mathcal{M}}_N}^{\prime }{e}^{-D}{e}^{-A}& \ge -C{N}^{-\beta }\mathcal{K}-C{N}^{-\beta -\kappa }{\mathcal{V}}_N-C{N}^{\alpha +\beta +2\kappa -1}\mathcal{K}{\mathcal{N}}_{\ge \frac{1}{2}{N}^{\lfloor m_0\rfloor \beta }}\hfill \\ \hfill & -C\sum _{j=3}^{2\lfloor m_0\rfloor -1}{N}^{j\beta /2+\beta /2+2\kappa -1}\mathcal{K}{\mathcal{N}}_{\ge \frac{1}{2}{N}^{j\beta /2}}-C{N}^{\alpha +\beta /2+2\kappa }\hfill \end{array}\end{array}$$ 7.81

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

We combine next the terms on the third, fourth and fifth lines in (7.3). We first notice that

$$\begin{array}{cc}& \frac{1}{2N}\sum _{\begin{array}{c}r\in {\mathrm{\Lambda }}^{\ast },v,w\in P_L:\\ v+r,w-r\ne 0\end{array}}\widehat{V}(r/N^{1-\kappa })(a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w+a_v^{\ast }{a}_w^{\ast }{a}_{w-r}{a}_{v+r})\hfill \\ & =\frac{1}{2N}\sum _{\begin{array}{c}r\in {\mathrm{\Lambda }}^{\ast },v,w\in {\mathrm{\Lambda }}_{+}^{\ast }:\\ v,w\in P_L,\\ v+r,w-r\ne 0\end{array}}\widehat{V}(r/N^{1-\kappa })a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w\hfill \\ & +\frac{1}{2N}\sum _{\begin{array}{c}r\in {\mathrm{\Lambda }}^{\ast },v,w\in {\mathrm{\Lambda }}_{+}^{\ast }:\\ v+r,w-r\in P_L\end{array}}\widehat{V}(r/N^{1-\kappa })a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w\hfill \\ & =\frac{1}{2N}\sum _{\begin{array}{c}r\in {\mathrm{\Lambda }}^{\ast },v,w\in {\mathrm{\Lambda }}_{+}^{\ast }:\\ (v,w)\in P_L^2\text{or}(v+r,w-r)\in P_L^2\end{array}}^{\ast }\widehat{V}(r/N^{1-\kappa })a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w\hfill \\ & +\frac{1}{2N}\sum _{\begin{array}{c}r\in {\mathrm{\Lambda }}^{\ast },v,w\in {\mathrm{\Lambda }}_{+}^{\ast }:\\ (v,w,v+r,w-r)\in P_L^4\end{array}}^{\ast }\widehat{V}(r/N^{1-\kappa })a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w\hfill \end{array}$$ 7.82

Arguing in the same way for the contribution on the fifth line in (7.3), using that $({\widehat{f}}_N-\eta /N)(p)={\delta }_{p,0}$ for all $p\in {\mathrm{\Lambda }}_{+}^{\ast }$, and using that $v\in P_L$ and $v+r\in P_L$ implies in particular that $r\in P_H^c$, we therefore obtain that

$$\begin{array}{cc}& {\mathcal{V}}_N-\frac{1}{2N}\sum _{\begin{array}{c}r\in {\mathrm{\Lambda }}^{\ast },v,w\in {\mathrm{\Lambda }}_{+}^{\ast }:\\ (v,w)\in P_L^2\text{or}(v+r,w-r)\in P_L^2\end{array}}^{\ast }\widehat{V}(r/N^{1-\kappa })a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w\hfill \\ \hfill & -\frac{1}{2N}\sum _{\begin{array}{c}r\in {\mathrm{\Lambda }}^{\ast },v,w\in {\mathrm{\Lambda }}_{+}^{\ast }:\\ (v,w,v+r,w-r)\in P_L^4\end{array}}\widehat{V}(r/N^{1-\kappa })a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w\hfill \\ \hfill & -\frac{1}{2N}\sum _{\begin{array}{c}r\in {\mathrm{\Lambda }}^{\ast },v,w\in P_L:\\ v+r,w-r\in P_L\end{array}}^{\ast }{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast \eta /N)(r)a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w\hfill \\ \hfill & +\frac{1}{2N}\sum _{\begin{array}{c}r\in P_H^c\cup \{0\},v,w\in P_L:\\ v+r,w-r\ne 0\end{array}}{N}^{\kappa }(\widehat{V}(/N^{1-\kappa })\ast {\widehat{f}}_N)(r)(a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w+\text{h.c.})\hfill \\ \hfill & ={\mathcal{V}}_N-\frac{1}{2N}\sum _{\begin{array}{c}r\in {\mathrm{\Lambda }}^{\ast },v,w\in {\mathrm{\Lambda }}_{+}^{\ast }:\\ (v,w)\in P_L^2\text{or}(v+r,w-r)\in P_L^2\end{array}}^{\ast }\widehat{V}(r/N^{1-\kappa })a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w\hfill \\ \hfill & +\frac{1}{2N}\sum _{\begin{array}{c}r\in P_H^c\cup \{0\},v,w\in P_L:\\ (v,w)\in P_L^2\text{or}(v+r,w-r)\in P_L^2\end{array}}^{\ast }{N}^{\kappa }(\widehat{V}(/N^{1-\kappa })\ast {\widehat{f}}_N)(r)a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w.\hfill \\ \hfill \end{array}$$ 7.83

Now, notice furthermore that

$$\begin{array}{cc}& {\mathcal{V}}_N-\frac{1}{2N}\sum _{\begin{array}{c}r\in {\mathrm{\Lambda }}^{\ast },v,w\in {\mathrm{\Lambda }}_{+}^{\ast }:\\ (v,w)\in P_L^2\text{or}(v+r,w-r)\in P_L^2\end{array}}^{\ast }\widehat{V}(r/N^{1-\kappa })a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w\hfill \\ \hfill & =\frac{1}{2N}\sum _{\begin{array}{c}r\in {\mathrm{\Lambda }}^{\ast },v,w\in {\mathrm{\Lambda }}_{+}^{\ast }:\\ (v,w)\in {(P_L^2)}^c\text{and}\\ (v+r,w-r)\in {(P_L^2)}^c\end{array}}^{\ast }\widehat{V}(r/N^{1-\kappa })a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w,\hfill \end{array}$$

such that, after switching to position space, the pointwise positivity $V\ge 0$ implies

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathcal{V}}_N-\frac{1}{2N}\sum _{\begin{array}{c}r\in {\mathrm{\Lambda }}^{\ast },v,w\in {\mathrm{\Lambda }}_{+}^{\ast }:\\ (v,w)\in P_L^2\text{or}(v+r,w-r)\in P_L^2\end{array}}^{\ast }\widehat{V}(r/N^{1-\kappa })a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w\hfill \\ & ={\int }_{{\mathrm{\Lambda }}^2}dxdy{N}^{2-2\kappa }V(N^{1-\kappa }(x-y))\hfill \\ & \times [a^{\ast }({({\check{\chi }}_{P_L^c})}_x)a^{\ast }({({\check{\chi }}_{P_L^c})}_y)+a^{\ast }({({\check{\chi }}_{P_L})}_x)a^{\ast }({({\check{\chi }}_{P_L^c})}_y)+a^{\ast }({({\check{\chi }}_{P_L^c})}_x)a^{\ast }({({\check{\chi }}_{P_L})}_y)]\hfill \\ & \times [a({({\check{\chi }}_{P_L^c})}_x)a({({\check{\chi }}_{P_L^c})}_y)+a({({\check{\chi }}_{P_L})}_x)a({({\check{\chi }}_{P_L^c})}_y)+a({({\check{\chi }}_{P_L^c})}_x)a({({\check{\chi }}_{P_L})}_y)]\hfill \\ & \ge 0.\hfill \end{array}\end{array}$$ 7.84

Here, we used that ${\mathrm{\Lambda }}_{+}^{\ast }=P_L\cup P_L^c$ and we denote by ${\check{\chi }}_S$ the distribution which has Fourier transform ${\chi }_S$, the characteristic function of the set $S\subset {\mathrm{\Lambda }}_{+}^{\ast }$.

Combining (7.3), (7.3), (7.3) and (7.84), it follows that

$$\begin{array}{cc}\hfill {\mathcal{M}}_N& \ge 4\pi {\mathfrak{a}}_0{N}^{1+\kappa }-4\pi {\mathfrak{a}}_0{N}^{\kappa -1}{\mathcal{N}}_{+}^2+8\pi {\mathfrak{a}}_0{N}^{\kappa }\sum _{p\in P_H^c}[b_p^{\ast }{b}_p+\frac{1}{2}{b}_p^{\ast }{b}_{-p}^{\ast }+\frac{1}{2}{b}_p{b}_{-p}]\hfill \\ \hfill & +\frac{8\pi {\mathfrak{a}}_0{N}^{\kappa }}{\sqrt{N}}\sum _{\begin{array}{c}p\in P_H^c,q\in P_L:\\ p+q\ne 0\end{array}}[b_{-p}^{\ast }{a}_{p+q}^{\ast }{a}_q+\text{h.c.}]+\frac{1}{2}\mathcal{K}\hfill \\ \hfill & +\frac{1}{2N}\sum _{\begin{array}{c}r\in P_H^c\cup \{0\},v,w\in P_L:\\ (v,w)\in P_L^2\text{or}(v+r,w-r)\in P_L^2\end{array}}^{\ast }{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast {\widehat{f}}_N)(r)a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w\hfill \\ \hfill & -\tilde{C}{N}^{-\beta -\kappa }{\mathcal{V}}_{N,L}+{\mathcal{E}}_{{\mathcal{M}}_N}^{\prime }\hfill \end{array}$$ 7.85

Using Lemma 3.1, part ii), we have $(\widehat{V}(./N^{1-\kappa })\ast {\widehat{f}}_N)(0)=8\pi {\mathfrak{a}}_0+\mathcal{O}(N^{\kappa -1})$. This implies

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{M}}_N& \ge 4\pi {\mathfrak{a}}_0{N}^{1+\kappa }+8\pi {\mathfrak{a}}_0{N}^{\kappa }\sum _{p\in P_H^c}[b_p^{\ast }{b}_p+\frac{1}{2}{b}_p^{\ast }{b}_{-p}^{\ast }+\frac{1}{2}{b}_p{b}_{-p}]\hfill \\ & +\frac{8\pi {\mathfrak{a}}_0{N}^{\kappa }}{\sqrt{N}}\sum _{\begin{array}{c}p\in P_H^c,q\in P_L:\\ p+q\ne 0\end{array}}[b_{-p}^{\ast }{a}_{p+q}^{\ast }{a}_q+\text{h.c.}]+\frac{1}{2}\mathcal{K}\hfill \\ & +\frac{1}{2N}\sum _{\begin{array}{c}r\in P_H^c,v,w\in P_L:\\ (v,w)\in P_L^2\text{or}(v+r,w-r)\in P_L^2\end{array}}^{\ast }{N}^{\kappa }(\widehat{V}(./N^{1-\kappa })\ast {\widehat{f}}_N)(r)a_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w\hfill \\ & -\tilde{C}{N}^{-\beta -\kappa }{\mathcal{V}}_{N,L}+{\mathcal{E}}_{{\mathcal{M}}_N}^{\prime \prime },\hfill \end{array}\end{array}$$ 7.86

where, by (7.81) and Lemmas 4.2 and 7.1,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^A{e}^D{\mathcal{E}}_{{\mathcal{M}}_N}^{\prime \prime }{e}^{-D}{e}^{-A}& \ge -C{N}^{-\beta }\mathcal{K}-C{N}^{-\beta -\kappa }{\mathcal{V}}_N-C{N}^{\alpha +\beta +2\kappa -1}\mathcal{K}{\mathcal{N}}_{\ge \frac{1}{2}{N}^{\lfloor m_0\rfloor \beta }}\hfill \\ \hfill & -C\sum _{j=3}^{2\lfloor m_0\rfloor -1}{N}^{j\beta /2+\beta /2+2\kappa -1}\mathcal{K}{\mathcal{N}}_{\ge \frac{1}{2}{N}^{j\beta /2}}-C{N}^{\alpha +\beta /2+2\kappa }\hfill \end{array}\end{array}$$ 7.87

Similarly, for $r\in P_H^c$, we know that

$$\begin{array}{c}\hfill |(\widehat{V}(./N^{1-\kappa })\ast {\widehat{f}}_N)(r)-8\pi {\mathfrak{a}}_0|\le C{N}^{\alpha +\kappa -1}.\end{array}$$

Therefore, proceeding exactly as between (7.27) and (7.30), with $(\widehat{V}(./N^{1-\kappa })\ast {\widehat{f}}_N)(r)$ replaced by $|(\widehat{V}(/N^{1-\kappa })\ast {\widehat{f}}_N)(r)-8\pi {\mathfrak{a}}_0|$, we deduce that

$$\begin{array}{cc}\hfill {\mathcal{M}}_N& \ge 4\pi {\mathfrak{a}}_0{N}^{1+\kappa }+\frac{1}{2}\mathcal{K}+8\pi {\mathfrak{a}}_0{N}^{\kappa }\sum _{p\in P_H^c}[b_p^{\ast }{b}_p+\frac{1}{2}{b}_p^{\ast }{b}_{-p}^{\ast }+\frac{1}{2}{b}_p{b}_{-p}]\hfill \\ & +\frac{8\pi {\mathfrak{a}}_0{N}^{\kappa }}{\sqrt{N}}\sum _{\begin{array}{c}p\in P_H^c,q\in P_L:\\ p+q\ne 0\end{array}}[b_{-p}^{\ast }{a}_{p+q}^{\ast }{a}_q+\text{h.c.}]\hfill \\ & +\frac{4\pi {\mathfrak{a}}_0{N}^{\kappa }}{N}\sum _{\begin{array}{c}r\in P_H^c,v,w\in P_L:\\ (v,w)\in P_L^2\\ \text{or}(v+r,w-r)\in P_L^2\end{array}}^{\ast }{a}_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w-\tilde{C}{N}^{-\beta -\kappa }{\mathcal{V}}_{N,L}+{\mathcal{E}}_{{\mathcal{M}}_N}^{\prime \prime \prime },\hfill \end{array}$$ 7.88

with ${\mathcal{E}}_{{\mathcal{M}}_N}^{\prime \prime \prime }$ satisfying the same bound (7.87) as ${\mathcal{E}}_{{\mathcal{M}}_N}^{\prime \prime }$. Here we used Lemmas 4.2, 4.3, 7.1 and 7.2, as well as the assumption (5.6).

Finally, recalling the definition (5.1) and the identity (5.2), we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{M}}_N& \ge 4\pi {\mathfrak{a}}_0{N}^{1+\kappa }+\frac{1}{2}\mathcal{K}+8\pi {\mathfrak{a}}_0{N}^{\kappa }\sum _{p\in P_H^c}[b_p^{\ast }{b}_p+\frac{1}{2}{b}_p^{\ast }{b}_{-p}^{\ast }+\frac{1}{2}{b}_p{b}_{-p}]\hfill \\ \hfill & +8\pi {\mathfrak{a}}_0{N}^{\kappa }\sum _{p\in P_H^c}[b_{-p}^{\ast }{e}_{-p}+e_{-p}^{\ast }{b}_{-p}+b_{-p}^{\ast }{e}_p^{\ast }+e_p{b}_{-p}+b_{-p}^{\ast }{c}_p^{\ast }+c_p{b}_{-p}]\hfill \\ \hfill & +\frac{4\pi {\mathfrak{a}}_0{N}^{\kappa }}{N}\sum _{\begin{array}{c}r\in P_H^c,v,w\in P_L:\\ (v,w)\in P_L^2\\ \text{or}(v+r,w-r)\in P_L^2\end{array}}^{\ast }{a}_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w-\tilde{C}{N}^{-\beta -\kappa }{\mathcal{V}}_{N,L}+{\mathcal{E}}_{{\mathcal{M}}_N}^{\prime \prime \prime }.\hfill \end{array}\end{array}$$ 7.89

To express also the first term in the third line of (7.89) in terms of the modified creation and annihilation fields defined in (5.1), we first observe that

$$\begin{array}{c}\hfill \begin{array}{cc}& \frac{4\pi {\mathfrak{a}}_0{N}^{\kappa }}{N}\sum _{\begin{array}{c}r\in P_H^c,v,w\in P_L:\\ (v,w)\in P_L^2\\ \text{or}(v+r,w-r)\in P_L^2\end{array}}^{\ast }{a}_{v+r}^{\ast }{a}_{w-r}^{\ast }{a}_v{a}_w\hfill \\ & =\frac{4\pi {\mathfrak{a}}_0{N}^{\kappa }}{N}\sum _{r\in P_H^c}\sum _{\begin{array}{c}v,w\in P_L:\\ (v,w)\in P_L^2\\ \text{or}(v+r,w-r)\in P_L^2\end{array}}^{\ast }{a}_{v+r}^{\ast }{a}_v{a}_{w-r}^{\ast }{a}_w-\frac{4\pi {\mathfrak{a}}_0{N}^{\kappa }}{N}\sum _{\begin{array}{c}r\in P_H^c,v\in P_L:\\ (v,v+r)\in P_L^2\end{array}}^{\ast }{a}_{v+r}^{\ast }{a}_{v+r}\hfill \\ & \ge \frac{4\pi {\mathfrak{a}}_0{N}^{\kappa }}{N}\sum _{r\in P_H^c}\sum _{\begin{array}{c}v,w\in P_L:\\ (v,w)\in P_L^2\\ \text{or}(v+r,w-r)\in P_L^2\end{array}}^{\ast }{a}_{v+r}^{\ast }{a}_v{a}_{w-r}^{\ast }{a}_w-C{N}^{3\beta +\kappa -1}{\mathcal{N}}_{+}-C.\hfill \end{array}\end{array}$$

Then, for a fixed $r\in P_H^c$, we have that

$$\begin{array}{c}\hfill \{(v,w)\in {\mathrm{\Lambda }}_{+}^{\ast }\times {\mathrm{\Lambda }}_{+}^{\ast }:(v,w)\in P_L^2\text{or}(v+r,w-r)\in P_L^2\}=\bigcup _{j=1}^7{S}_j,\end{array}$$

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill S_1& =\{(v,w)\in {\mathrm{\Lambda }}_{+}^{\ast }\times {\mathrm{\Lambda }}_{+}^{\ast }:v\in P_L,w\in P_L,v+r\in P_L,w-r\in P_L\},\hfill \\ \hfill S_2& =\{(v,w)\in {\mathrm{\Lambda }}_{+}^{\ast }\times {\mathrm{\Lambda }}_{+}^{\ast }:v\in P_L,w\in P_L,v+r\in P_L,w-r\in P_L^c\},\hfill \\ \hfill S_3& =\{(v,w)\in {\mathrm{\Lambda }}_{+}^{\ast }\times {\mathrm{\Lambda }}_{+}^{\ast }:v\in P_L,w\in P_L,v+r\in P_L^c,w-r\in P_L\},\hfill \\ \hfill S_4& =\{(v,w)\in {\mathrm{\Lambda }}_{+}^{\ast }\times {\mathrm{\Lambda }}_{+}^{\ast }:v\in P_L,w\in P_L,v+r\in P_L^c,w-r\in P_L^c\},\hfill \\ \hfill S_5& =\{(v,w)\in {\mathrm{\Lambda }}_{+}^{\ast }\times {\mathrm{\Lambda }}_{+}^{\ast }:v\in P_L^c,w\in P_L,v+r\in P_L,w-r\in P_L\},\hfill \\ \hfill S_6& =\{(v,w)\in {\mathrm{\Lambda }}_{+}^{\ast }\times {\mathrm{\Lambda }}_{+}^{\ast }:v\in P_L,w\in P_L^c,v+r\in P_L,w-r\in P_L\},\hfill \\ \hfill S_7& =\{(v,w)\in {\mathrm{\Lambda }}_{+}^{\ast }\times {\mathrm{\Lambda }}_{+}^{\ast }:v\in P_L^c,w\in P_L^c,v+r\in P_L,w-r\in P_L\}.\hfill \end{array}\end{array}$$

In particular, the union ${\bigcup }_{j=1}^7{S}_j$ is a disjoint union. As a consequence, we find that

$$\begin{array}{c}\hfill \begin{array}{cc}& \frac{4\pi {\mathfrak{a}}_0{N}^{\kappa }}{N}\sum _{r\in P_H^c}\sum _{\begin{array}{c}v,w\in P_L:\\ (v,w)\in P_L^2\\ \text{or}(v+r,w-r)\in P_L^2\end{array}}^{\ast }{a}_{v+r}^{\ast }{a}_v{a}_{w-r}^{\ast }{a}_w\hfill \\ \hfill & =8\pi {\mathfrak{a}}_0{N}^{\kappa }\sum _{r\in P_H^c}[e_r^{\ast }{c}_{-r}^{\ast }+c_{-r}{e}_r+\frac{1}{2}{d}_r^{\ast }{e}_{-r}^{\ast }+\frac{1}{2}{e}_{-r}{e}_r+\frac{1}{2}{c}_r^{\ast }{c}_{-r}^{\ast }+\frac{1}{2}{c}_{-r}{c}_r]\hfill \\ \hfill & +8\pi {\mathfrak{a}}_0{N}^{\kappa }\sum _{r\in P_H^c}[e_r^{\ast }{e}_r+c_r^{\ast }{e}_r+e_r^{\ast }{c}_r].\hfill \end{array}\end{array}$$

Inserting in (7.88), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{M}}_N& \ge 4\pi {\mathfrak{a}}_0{N}^{1+\kappa }+\frac{1}{2}\mathcal{K}+8\pi {\mathfrak{a}}_0{N}^{\kappa }\sum _{r\in P_H^c}(b_r^{\ast }+c_r^{\ast }+e_r^{\ast })(b_r+c_r+e_r)\hfill \\ \hfill & +4\pi {\mathfrak{a}}_0{N}^{\kappa }\sum _{r\in P_H^c}[(b_r^{\ast }+c_r^{\ast }+e_r^{\ast })(b_{-r}^{\ast }+c_{-r}^{\ast }+e_{-r}^{\ast })+\text{h.c.}]\hfill \\ \hfill & -8\pi {\mathfrak{a}}_0{N}^{\kappa }\sum _{r\in P_H^c}\left[c_r^{\ast },c_r,+,b_r^{\ast },c_r,+,c_r^{\ast },b_r\right]-\tilde{C}{N}^{-\beta -\kappa }{\mathcal{V}}_{N,L}+{\mathcal{E}}_{{\mathcal{M}}_N}^{\prime \prime \prime }\hfill \end{array}\end{array}$$ 7.90

with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^A{e}^D{\mathcal{E}}_{{\mathcal{M}}_N}^{\prime \prime \prime }{e}^{-D}{e}^{-A}& \ge -C{N}^{-\beta }\mathcal{K}-C{N}^{-\beta -\kappa }{\mathcal{V}}_N-C{N}^{\alpha +\beta +2\kappa -1}\mathcal{K}{\mathcal{N}}_{\ge \frac{1}{2}{N}^{\lfloor m_0\rfloor \beta }}\hfill \\ \hfill & -C\sum _{j=3}^{2\lfloor m_0\rfloor -1}{N}^{j\beta /2+\beta /2+2\kappa -1}\mathcal{K}{\mathcal{N}}_{\ge \frac{1}{2}{N}^{j\beta /2}}-C{N}^{\alpha +\beta /2+2\kappa }\hfill \end{array}\end{array}$$

Let us now estimate the remaining terms on the last line of (7.90). For $\xi \in {\mathcal{F}}_{+}^{\le N}$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}& |8\pi {\mathfrak{a}}_0{N}^{\kappa }\sum _{r\in P_H^c}⟨\xi ,c_r^{\ast }{c}_r\xi ⟩|\hfill \\ \hfill & \le \frac{C{N}^{\kappa }}{N}\sum _{\begin{array}{c}r\in P_H^c,v,w\in P_L:\\ v\in P_L,r+v\in P_L^c,\\ w\in P_L,w+r\in P_L^c\end{array}}^{\ast }{(|w||v|}^{-1}\Vert a_{r+v}{a}_w{\xi \Vert )(|v||w|}^{-1}\Vert a_v{a}_{w+r}\xi \Vert )\hfill \\ \hfill & \le C{N}^{\beta +\kappa -1}⟨\xi ,{\mathcal{K}}_L({\mathcal{N}}_{\ge N^{\beta }}+1)\xi ⟩,\hfill \end{array}\end{array}$$ 7.91

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |8\pi {\mathfrak{a}}_0{N}^{\kappa }\sum _{r\in P_H^c}⟨\xi ,(b_r^{\ast }{c}_r+c_r^{\ast }{b}_r)\xi ⟩|& \le \frac{1}{4}\sum _{r\in P_H^c}⟨\xi ,b_r^{\ast }{b}_r\xi ⟩+C{N}^{2\kappa }\sum _{r\in P_H^c}⟨\xi ,c_r^{\ast }{c}_r\xi ⟩\hfill \\ \hfill & \le \frac{1}{4}\mathcal{K}+C{N}^{\beta +2\kappa -1}⟨\xi ,{\mathcal{K}}_L({\mathcal{N}}_{\ge N^{\beta }}+1)\xi ⟩,\hfill \end{array}\end{array}$$ 7.92

Similarly, we can bound

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill N^{-\beta -\kappa }⟨\xi ,{\mathcal{V}}_{N,L}\xi ⟩\le & C{N}^{-\beta -1}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },p,q\in {\mathrm{\Lambda }}_{+}^{\ast }:\\ p+u,q+u,p,q\in P_L\end{array}}\Vert a_{p+u}{a}_q\xi \Vert \Vert a_p{a}_{q+u}\xi \Vert \hfill \\ \hfill \le & C{N}^{-\beta -1}\sum _{\begin{array}{c}u\in {\mathrm{\Lambda }}^{\ast },p,q\in {\mathrm{\Lambda }}_{+}^{\ast }:\\ p+u,q+u,p,q\in P_L\end{array}}\frac{{|q|}^2}{{|p|}^2}{\Vert a_{p+u}{a}_q\xi \Vert }^2\hfill \\ \hfill \le & C{N}^{-1}\Vert {\mathcal{K}}^{1/2}{\mathcal{N}}_{+}^{1/2}{\xi \Vert }^2\le C{\Vert {\mathcal{K}}^{1/2}\xi \Vert }^2\hfill \end{array}\end{array}$$

Thus, choosing the constant $\tilde{C}>0$ small enough and applying Lemmas 7.2, 4.3 and 4.2 to the r.h.s. of (7.91) and to the second term on the r.h.s. of (7.92), we conclude that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{M}}_N& \ge 4\pi {\mathfrak{a}}_0{N}^{1+\kappa }+\frac{1}{4}\mathcal{K}+8\pi {\mathfrak{a}}_0{N}^{\kappa }\sum _{r\in P_H^c}(b_r^{\ast }+c_r^{\ast }+e_r^{\ast })(b_r+c_r+e_r)\hfill \\ \hfill & +4\pi {\mathfrak{a}}_0{N}^{\kappa }\sum _{r\in P_H^c}[(b_r^{\ast }+c_r^{\ast }+e_r^{\ast })(b_{-r}^{\ast }+c_{-r}^{\ast }+e_{-r}^{\ast })+\text{h.c.}]+{\mathcal{E}}_{{\mathcal{M}}_N}^{\prime \prime \prime \prime }\hfill \end{array}\end{array}$$ 7.93

where ${\mathcal{E}}_{{\mathcal{M}}_N}^{\prime \prime \prime \prime }$ is such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^A{e}^D{\mathcal{E}}_{{\mathcal{M}}_N}^{\prime \prime \prime \prime }{e}^{-A}{e}^{-D}\ge & -C{N}^{-\beta }\mathcal{K}-C{N}^{-\beta -\kappa }{\mathcal{V}}_N\hfill \\ \hfill & -C{N}^{\beta +2\kappa -1}\mathcal{K}{\mathcal{N}}_{\ge N^{\beta }}-C{N}^{\alpha +\beta +2\kappa -1}\mathcal{K}{\mathcal{N}}_{\ge \frac{1}{2}{N}^{\lfloor m_0\rfloor \beta }}\hfill \\ \hfill & -C\sum _{j=3}^{2\lfloor m_0\rfloor -1}{N}^{j\beta /2+\beta /2+2\kappa -1}\mathcal{K}{\mathcal{N}}_{\ge \frac{1}{2}{N}^{j\beta /2}}-C{N}^{\alpha +\beta /2+2\kappa }\hfill \end{array}\end{array}$$ 7.94

We introduce the operators

$$\begin{array}{c}\hfill g_r^{\ast }=b_r^{\ast }+c_r^{\ast }+e_r^{\ast },g_r=b_r+c_r+e_r.\end{array}$$

With the algebraic identity

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \sum _{r\in P_H^c}[g_r^{\ast }{g}_r+\frac{1}{2}{g}_r^{\ast }{g}_{-r}^{\ast }+\frac{1}{2}{g}_{-r}{g}_r]& =\frac{1}{2}\sum _{r\in P_H^c}(g_r^{\ast }+g_{-r})(g_r+g_{-r}^{\ast })-\frac{1}{2}\sum _{r\in P_H^c}[g_r,g_r^{\ast }],\hfill \end{array}\end{array}$$

we conclude that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{M}}_N& \ge 4\pi {\mathfrak{a}}_0{N}^{1+\kappa }+\frac{1}{4}\mathcal{K}-4\pi {\mathfrak{a}}_0{N}^{\kappa }\sum _{r\in P_H^c}[g_r,g_r^{\ast }]+{\mathcal{E}}_{{\mathcal{M}}_N}^{\prime \prime \prime \prime }\hfill \end{array}\end{array}$$

Since

$$\begin{array}{c}\hfill \begin{array}{c}\hfill [b_r,c_r^{\ast }]=[b_r,e_r^{\ast }]=[c_r,b_r^{\ast }]=[e_r,b_r^{\ast }]=[c_r,e_r^{\ast }]=[e_r,c_r^{\ast }]=0,\end{array}\end{array}$$

we obtain that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [g_r,g_r^{\ast }]& =\frac{N-{\mathcal{N}}_{+}}{N}-\frac{1}{N}{a}_r^{\ast }{a}_r+\frac{1}{N}\sum _{\begin{array}{c}v\in {\mathrm{\Lambda }}_{+}^{\ast }:v\in P_L,\\ v+r\in P_L^c\end{array}}{a}_v^{\ast }{a}_v-\frac{1}{N}\sum _{\begin{array}{c}v\in {\mathrm{\Lambda }}_{+}^{\ast }:v\in P_L,\\ v+r\in P_L^c\end{array}}{a}_{v+r}^{\ast }{a}_{v+r}\hfill \\ \hfill & +\frac{1}{4N}\sum _{\begin{array}{c}v\in {\mathrm{\Lambda }}_{+}^{\ast }:v\in P_L,\\ v+r\in P_L\end{array}}{a}_v^{\ast }{a}_v-\frac{1}{4N}\sum _{\begin{array}{c}v\in {\mathrm{\Lambda }}_{+}^{\ast }:v\in P_L,\\ v+r\in P_L\end{array}}{a}_{v+r}^{\ast }{a}_{v+r}.\hfill \end{array}\end{array}$$

A straightforward computation then shows that

$$\begin{array}{c}\hfill -4\pi {\mathfrak{a}}_0{N}^{\kappa }\sum _{p\in P_H^c}[g_r,g_r^{\ast }]\ge -C{N}^{3\alpha +\kappa }(1-{\mathcal{N}}_{+}/N)-C{N}^{3\alpha +\kappa }{\mathcal{N}}_{+}/N\ge -C{N}^{3\alpha +\kappa }.\end{array}$$

Thus

$$\begin{array}{c}\hfill {\mathcal{M}}_N\ge 4\pi {\mathfrak{a}}_0{N}^{1+\kappa }+\frac{1}{4}\mathcal{K}+{\mathcal{E}}_{{\mathcal{M}}_N}\end{array}$$

where ${\mathcal{E}}_{{\mathcal{M}}_N}$ satisfies

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^A{e}^D{\mathcal{E}}_{{\mathcal{M}}_N}{e}^{-A}{e}^{-D}\ge & -C{N}^{-\beta }\mathcal{K}-C{N}^{-\beta -\kappa }{\mathcal{V}}_N\hfill \\ \hfill & -C{N}^{\beta +2\kappa -1}\mathcal{K}{\mathcal{N}}_{\ge N^{\beta }}-C{N}^{\alpha +\beta +2\kappa -1}\mathcal{K}{\mathcal{N}}_{\ge \frac{1}{2}{N}^{\lfloor m_0\rfloor \beta }}\hfill \\ \hfill & -C\sum _{j=3}^{2\lfloor m_0\rfloor -1}{N}^{j\beta /2+\beta /2+2\kappa -1}\mathcal{K}{\mathcal{N}}_{\ge \frac{1}{2}{N}^{j\beta /2}}-C{N}^{3\alpha +\kappa }\hfill \end{array}\end{array}$$

This concludes the proof of Proposition 5.1. $\square$

Funding

Open access funding provided by University of Zurich. Funding was provided by H2020 European Research Council (Grant No. ERC - ADG 2018 - Project 834782 CLaQS), Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (Grant Nos. NCCR SwissMAP and Projekt 200020_172623).