2 Setting and trial state

To show Proposition 1.3, we find it convenient to work with rescaled variables. We consider the transformation $x_j \to x_j / L$ and, motivated by the choice $L = {\tilde {\rho }}^{-\gamma }$ in Proposition 1.3, we set $N = {\tilde {\rho }}^{1-3\gamma }$ (we will look for trial states with expected number of particles close to N to make sure that (1.8) holds true). It follows that the Hamiltonian (1.5) is unitarily equivalent to the operator $L^{-2} \mathcal {H}_N = {\tilde {\rho }}^{2\gamma } \mathcal {H}_N$ , with $\mathcal {H}_N$ acting on the Fock space $\mathcal {F} (\Lambda )$ defined over the unit box $\Lambda = \Lambda _1 = [-1/2; 1/2 ]^3$ (with periodic boundary conditions) so that $(\mathcal {H}_N \Psi )^{(n)} = \mathcal {H}_N^{(n)} \Psi ^{(n)}$ , with

$$ \begin{align*} \mathcal{H}_N^{(n)} = \sum_{j=1}^n -\Delta_{x_j} + \sum_{1 \leq i,j \leq n} N^{2-2\kappa} V (N^{1-\kappa} (x_i - x_j)) \end{align*} $$

and $\kappa = (2\gamma - 1)/ (3\gamma -1)$ . The assumption $\gamma> 1$ in Proposition 1.3 allows us to restrict our attention to $\kappa \in (1/2; 2/3)$ .

For any momentum $p \in \Lambda ^* = 2\pi \mathbb {Z}^3$ , we introduce on the Fock space $\mathcal {F} (\Lambda ) = \bigoplus _{n \geq 0} L^2_s (\Lambda ^{n})$ the operators $a_p^*, a_p$ , creating and, respectively, annihilating a particle with momentum p. Creation and annihilation operators satisfy the canonical commutation relations

(2.1) $$ \begin{align} \left[ a_p , a_q^* \right] = \delta_{pq} , \qquad \left[ a_p , a_q \right] = \left[ a^*_p, a_q^* \right] = 0. \end{align} $$

On $\mathcal {F}(\Lambda )$ , we define the number of particles operator $\mathcal {N} = \sum _{p \in \Lambda ^*} a_p^* a_p$ . Expressed in terms of creation and annihilation operators, the Hamiltonian $\mathcal {H}_N$ takes the form

(2.2) $$ \begin{align} \mathcal{H}_N=\sum_{p\in\Lambda^*}p^2a_p^*a_p+\frac{1}{2N^{1-\kappa}} \sum_{p,q,r\in \Lambda^*} \widehat{V} (r/N^{1-\kappa}) \, a_{p+r}^*a_q^*a_{q+r}a_{p}. \end{align} $$

We now construct our trial state. To generate a condensate, we use a Weyl operator

(2.3) $$ \begin{align} W_{N_0}=\exp \big[ \sqrt{N_0} a_0^* - \sqrt{N_0}a_0 \big] \end{align} $$

with a parameter $N_0$ to be specified later on. While $W_{N_0}$ leaves $a_p, a_p^*$ invariant, for all $p \in \Lambda ^* \backslash \{ 0 \}$ , it produces shifts of $a_0, a_0^*$ ; in other words,

(2.4) $$ \begin{align} W_{N_0}^* a_0 \, W_{N_0} = a_0 + \sqrt{N_0} , \qquad W_{N_0}^* a_0^* \, W_{N_0} = a_0^* + \sqrt{N_0}. \end{align} $$

When acting on the vacuum vector $\Omega = \{ 1, 0, \dots \}$ , (2.3) generates a coherent state in the zero-momentum mode $\varphi _0 (x) \equiv 1$ , with expected number of particles $N_0$ .

It turns out, however, that the coherent state does not approximate the ground state energy, not even to leading order. To get closer to the ground state energy, it is crucial to add correlations among particles. To this end, we fix $0 < \ell < 1/2$ and we consider the lowest energy solution $f_{\ell }$ of the Neumann problem

(2.5) $$ \begin{align} \left[ -\Delta + \frac{1}{2} V \right] f_{\ell} = \lambda_{\ell} f_{\ell} \end{align} $$

on the ball $|x| \leq N^{1-\kappa }\ell $ , with the normalisation $f_\ell (x) = 1$ if $|x| =N^{1-\kappa } \ell $ . Furthermore, by rescaling, we define $f_{N,\ell } (x) := f_\ell \big ( N^{1-\kappa }x\big )$ for $|x| \leq \ell $ . We extend $f_{N,\ell }$ to a function on $\Lambda $ , by fixing $f_{N,\ell } (x) = 1$ , for all $x \in \Lambda $ , with $|x|> \ell $ . Then

(2.6) $$ \begin{align} \left[ -\Delta + \frac12 N^{2-2\kappa} V ( N^{1-\kappa}x ) \right] f_{N,\ell} (x) = N^{2-2\kappa} \lambda_\ell f_{N,\ell} (x) \chi_\ell(x) \end{align} $$

for all $x \in \Lambda $ , where $\chi _\ell $ denotes the characteristic function of the ball of radius $\ell $ . We denote by $ \widehat {f}_{N,\ell } (p)$ the Fourier coefficients of the function $f_{N,\ell }$ , for $p \in \Lambda ^*$ . We also define $w_\ell (x) = 1 - f_\ell (x)$ (with $w_\ell (x) = 0$ for $|x|> N^{1-\kappa } \ell $ ) and its rescaled version $w_{N,\ell } : \Lambda \to \mathbb {R}$ through $w_{N,\ell } (x) = w_\ell (N^{1-\kappa } x) = 1 - f_{N,\ell } (x)$ . The Fourier coefficients of $w_{N,\ell }$ are given by

$$ \begin{align*} \widehat{w}_{N,\ell} (p) = \int_{\Lambda} w_\ell ( N^{1-\kappa} x) e^{-i p \cdot x} dx =\frac1{N^{3-3\kappa}}\, \widehat{w}_\ell \big(p/N^{1-\kappa}\big) \end{align*} $$

where $\widehat {w}_\ell (k)$ denotes the Fourier transform of the (compactly supported) function $w_\ell $ . Some important properties of the solution of the eigenvalue problem (2.5) are summarised in the following lemma, whose proof can be found in [Reference Boccato, Brennecke, Cenatiempo and Schlein3, Appendix A] (replacing $N\in \mathbb {N}$ by $N^{1-\kappa }$ ).

We consider the coefficients $\eta : \Lambda ^* \to \mathbb {R}$ defined through

(2.7) $$ \begin{align} \eta_p = -N \widehat{w}_{N,\ell} (p) = - \frac{N^\kappa}{N^{2-2\kappa}} \,\widehat{w}_\ell(p/N^{1-\kappa}). \end{align} $$

Lemma 2.1 implies that

(2.8) $$ \begin{align} |\eta_p| \leq \frac{C N^\kappa}{p^2} \end{align} $$

for all $p \in \Lambda _+^*=2\pi \mathbb {Z}^3 \backslash \{0\}$ and for some constant $C>0$ independent of $N\in \mathbb {N}$ (for $N\in \mathbb {N}$ large enough). From (2.6), we find the relation

(2.9) $$ \begin{align} p^2 \eta_p + \frac{N^\kappa}{2}\widehat{V} (p/N^{1-\kappa}) + \frac{1}{2N} \sum_{q \in \Lambda^*} N^\kappa\widehat{V} ((p-q) /N^{1-\kappa})\eta_q = N^{3-2\kappa} \lambda_{\ell} (\widehat{\chi}_\ell * \widehat{f}_{N,\ell}) (p). \end{align} $$

From Lemma 2.1, part iii), we also obtain

(2.10) $$ \begin{align} |\eta_0| \leq N^{3-2\kappa} \int_{\mathbb{R}^3} w_\ell (x) dx \leq C N^\kappa . \end{align} $$

The coefficients $\eta _p$ will be used to model, through a Bogoliubov transformation, short-distance correlations among particles. To reach this goal, it is enough to act on momenta $|p| \gg N^{\kappa /2}$ . On low momenta, the Bogoliubov transformation is needed to diagonalise the (renormalised) quadratic part of the Hamiltonian. For $\varepsilon> 0$ small enough, we therefore define the set

(2.11) $$ \begin{align} P_L \; = \Big\{ p \in \Lambda^*_+ : |p| \leq N^{\kappa/2+\varepsilon} \Big\}\, \end{align} $$

of low momenta. We will denote its complement by $P_L^c = \Lambda _+^* \backslash P_L$ . For $p \in \Lambda ^*_+$ we set

$$ \begin{align*} \nu_p = \tau_p \chi(p \in P_L) + \eta_p \chi(p \in P_L^c) \end{align*} $$

with $\eta _p$ defined in (2.7), $\tau _p \in \mathbb {R}$ defined by

(2.12) $$ \begin{align} \tanh(2\tau_p) = - \frac{8 \pi \mathfrak{a} N^\kappa}{p^2 +8 \pi \mathfrak{a} N^\kappa}\, \end{align} $$

and $\chi (p \in S)$ denoting the indicator function of the set S. With these coefficients, we define the Bogoliubov transformation

(2.13) $$ \begin{align} T_\nu \; = \exp \bigg(\;\frac 12 \sum_{p \in \Lambda^*_+} \nu_p \big(a^*_p a^*_{-p} - \mbox{h.c.} \big) \;\bigg). \end{align} $$

For any $p \neq 0$ we have

(2.14) $$ \begin{align} T^*_\nu a_p T_\nu = \gamma_p a_p + \sigma_p a^*_{-p} \end{align} $$

with the notation $\gamma _p = \cosh (\nu _p)$ and $\sigma _p = \sinh (\nu _p)$ .

With the Weyl operator (2.3) and the Bogoliubov transformation (2.13), we obtain the ‘squeezed’ coherent state $\widetilde {\Psi }_N = W_{N_0} T_\nu \Omega $ . Choosing $N_0$ so that $N = N_0 + \| \sigma \|^2$ , one can show that this trial state has approximately N particles and, to leading order, the correct ground state energy. However, as observed in [Reference Erdős, Schlein and Yau9] (for a similar trial state) and later in [Reference Napiórkowski, Reuvers and Solovej17], the energy of the quasi-free state $\widetilde {\Psi }_N$ does not match the second-order correction in (1.3). To prove Proposition 1.3, we therefore need to modify the trial state. We do so by replacing the vacuum $\Omega $ in the definition of $\widetilde {\Psi }_N$ by the normalised Fock space vector $\xi _\nu / \| \xi _\nu \|$ , with $\xi _\nu = e^{A_\nu } \Omega $ and the cubic phase

(2.15) $$ \begin{align} \begin{aligned} A_\nu =\frac 1 {\sqrt {N}} \sum_{\substack{r\in P_H,v \in P_S:\\r+v\in P_H}} \nu_r \sinh (\nu_v) \,a^*_{r+v} a^*_{-r} a^*_{-v} \Theta_{r,v} = \frac 1 {\sqrt {N}} \sum_{\substack{r\in P_H,v \in P_S:\\r+v\in P_H}} \eta_r \sigma_v \,a^*_{r+v} a^*_{-r} a^*_{-v} \Theta_{r,v}. \end{aligned}\end{align} $$

Here, we introduced the momentum sets

(2.16) $$ \begin{align} P_H &= \{ p \in \Lambda_+^* : |p|> N^{1-\kappa-\varepsilon} \}, \nonumber\\ P_S &=\Big\{p\in \Lambda^*_+: N^{\kappa/2-\varepsilon} \leq |p|\leq N^{\kappa/2 + \varepsilon}\Big\}. \end{align} $$

Notice that $P_S \subset P_L$ . On the other hand, to make sure that $P_H \cap P_L = \emptyset $ , from now on we will require that $\varepsilon> 0$ is so small that $3\kappa - 2 + 4 \varepsilon < 0$ . Moreover, in (2.15) we included, for every $r \in P_H$ and every $v \in P_S$ , the cutoff

$$ \begin{align*} \Theta_{r,v}&= \prod_{s \in P_H}\!\Big[ 1-\chi(\mathcal{N}_s>0)\chi(\mathcal{N}_{-s+v}>0)\Big] \nonumber\\ & \quad\times \prod_{w \in P_S}\! \Big[ 1-\chi(\mathcal{N}_w>0)\chi(\mathcal{N}_{r-w}+\mathcal{N}_{-r-v-w}>0) \Big] \end{align*} $$

where $\mathcal {N}_p = a^*_p a_p$ and $\chi (t>0)$ is the indicator function of the set $\{ t>0 \}$ .

Let us briefly discuss the action of the cutoff $\Theta _{r,v}$ . To understand its role in the computation of $e^{A_\nu } \Omega $ , we observe that, for every integer $m \geq 2$ , $r_1, \dots , r_m \in P_H$ , $v_1, \dots , v_m \in P_S$ , with $r_1 + v_1, \dots , r_m +v_m \in P_H$ , we find

$$ \begin{align*} \Theta_{r_m,v_m} &a^*_{r_{m-1}+v_{m-1}} a^*_{-r_{m-1}}a^*_{-v_{m-1}}\dots a^*_{r_1+v_1}a^*_{-r_1}a^*_{-v_1}\Omega=\\ &= \prod_{i,j=1}^{m-1}\prod_{\substack{p_\ell\in\{-r_\ell,r_\ell+v_\ell\},\\\ell=i,j,m}}\delta_{p_i\neq -p_j+v_m}\delta_{-p_m+v_i\neq p_j} \\ &\hspace{4cm} \times a^*_{r_{m-1}+v_{m-1}}a^*_{-r_{m-1}}a^*_{-v_{m-1}}\dots a^*_{r_1+v_1}a^*_{-r_1}a^*_{-v_1}\Omega. \end{align*} $$

The choice $i=j$ in the product on the second line introduces restrictions of the form $v_m \neq v_i$ and $p_m \neq p_i$ where $p_\ell \in \{-r_\ell ,r_\ell +v_\ell \}$ for $\ell = m,i$ , for all $i \in \{1, \dots , m-1 \}$ (the condition $p_i \neq -p_i + v_m$ , on the other hand, is trivially satisfied due to the assumption $p_i\in P_H,v_m\in P_S$ ). For $m\geq 3$ , the cutoff $\Theta _{r_m,v_m}$ implements additional restrictions involving three indices of the form $-p_i+v_j \neq p_k$ with $p_\ell \in \{-r_\ell ,r_\ell +v_\ell \}, \ell =i,j,k$ where $i,j,k = 1,\dots ,m, i\neq j\neq k$ , so that exactly one of the three indices is m. We conclude that, for any $m\geq 2$ ,

(2.17) $$ \begin{align} A_\nu^m\Omega = &\frac1{N^{m/2}}\sum_{\substack{r_1\in P_H,v_1\in P_S:\\r_1+v_1\in P_H}}\dots\sum_{\substack{r_m\in P_H,v_m\in P_S:\nonumber\\r_m+v_m\in P_H}}\prod_{i=1}^m \eta_{r_i}\sigma_{v_i}\nonumber\\ &\hspace{2cm} \times \theta(\{r_j,v_j\}_{j=1}^m)a^*_{r_m+v_m}a^*_{-r_m}a^*_{-v_m}\dots a^*_{r_1+v_1}a^*_{-r_1}a^*_{v_1}\Omega \end{align} $$

where

(2.18) $$ \begin{align} \theta\big( \{r_j, v_j \}_{j=1}^{m} \big) = \prod_{\substack{i,j,k=1\\j\neq k}}^m\prod_{\substack{p_i\in\{-r_i,r_i+v_i\}\\ p_k\in\{-r_k,r_k+v_k\}}}\delta_{-p_i+v_j\neq p_k}. \end{align} $$

To illustrate the reason for the introduction of the cutoff, let us compute the norm $\| \xi _\nu \|$ of the vector $\xi _\nu = e^{A_\nu } \Omega $ . With (2.17), we find

(2.19) $$ \begin{align} \| \xi_\nu\|^2 &= \sum_{m\geq 0}\frac{1}{(m!)^2} \| (A_\nu)^m \Omega \|^2 \nonumber\\ &= \sum_{m\geq 0}\frac1{(m!)^2}\frac1{N^m} \sum_{\substack{v_1, {\tilde{v}}_1 \in P_S \\ r_1, {\tilde{r}}_1 \in P_H: \\ r_1 + v_1 ,\, {\tilde{r}}_1 + {\tilde{v}}_1 \in P_H}} \cdots \sum_{\substack{v_m, {\tilde{v}}_m \in P_S \\ r_m, {\tilde{r}}_m \in P_H: \\ r_m + v_m ,\, {\tilde{r}}_m + {\tilde{v}}_m \in P_H}} \theta\big(\{{\tilde{r}}_j,{\tilde{v}}_j\}_{j=1}^m\big) \theta\big(\{r_j,v_j\}_{j=1}^m\big) \nonumber\\ & \hspace{1cm}\times\prod_{i=1}^m \eta_{r_i}\eta_{{\tilde{r}}_i} \sigma_{v_i} \sigma_{{\tilde{v}}_i} {\bigl\langle a^*_{r_m+v_m}a^*_{-r_m}a^*_{-v_m}\dots a^*_{-v_1}\Omega, a^*_{\tilde{r}_m+\tilde{v}_m}a^*_{-\tilde{r}_m}a^*_{-\tilde{v}_m} \dots a^*_{-\tilde{v}_1}\Omega\bigr\rangle}. \end{align} $$

Clearly, for the expectation on the last line not to vanish, all creation and annihilation operators with momenta in $P_S$ must be contracted among themselves. Since, on the support of $\theta (\{ r_j, v_j \}_{j=1}^m)$ , $v_i \neq v_j$ for all $i\neq j$ (and, similarly, ${\tilde {v}}_i \neq {\tilde {v}}_j$ for all $i\neq j$ on the support of $\theta (\{ \widetilde {r}_j, \widetilde {v}_j \}_{j=1}^m)$ ), we have $(m!)$ identical contributions arising from this pairing. We end up with

(2.20) $$ \begin{align} & \| \xi_\nu\|^2 =\sum_{m\geq 0}\frac1{m!}\frac1{N^m} \sum_{\substack{v_1 \in P_S, r_1, {\tilde{r}}_1 \in P_H: \\ r_1 + v_1 ,\, {\tilde{r}}_1 + v_1 \in P_H}} \cdots \sum_{\substack{v_m \in P_S, r_m, {\tilde{r}}_m \in P_H: \\ r_m + v_m ,\, {\tilde{r}}_m + v_m \in P_H}} \hskip -0.5cm \theta\big( \{r_j, v_j \}_{j=1}^{m} \big) \theta\big( \{ {\tilde{r}}_j, v_j \}_{j=1}^{m} \big) \nonumber\\ & \hspace{4.5cm}\times \prod_{i=1}^m \eta_{r_i}\eta_{{\tilde{r}}_i} \sigma^2_{v_i} {\bigl\langle\Omega,A_{r_1,v_1}\dots A_{r_m,v_m} A^*_{\tilde{r}_m,v_m} \dots A^*_{\tilde{r}_1,v_1} \Omega\bigr\rangle} \end{align} $$

where we have introduced the notation $A_{r_i,v_i} = a_{r_i+v_i} a_{-r_i}$ .

It is now important to observe that, because of the presence of the cutoffs, the annihilation operators in $A_{r_j, v_j}$ must be contracted with the creation operators in $A_{\widetilde {r}_j, v_j}$ . In fact, if this was not the case, we would have $-r_j = -\widetilde {r}_\ell $ or $-r_j = \widetilde {r}_\ell + v_\ell $ and $r_j + v_j = -\widetilde {r}_k$ or $r_j + v_j = \widetilde {r}_k + v_k$ , with at least one of the two indices $\ell , k$ different from j. This would imply one of the four relations $\widetilde {r}_\ell + v_j = -\widetilde {r}_k$ , $\widetilde {r}_\ell + v_j = \widetilde {r}_k + v_k$ , $ \widetilde {r}_\ell + v_\ell = \widetilde {r}_k + v_j$ , $-\widetilde {r}_\ell - v_\ell + v_j = \widetilde {r}_k + v_k$ , all of which are forbidden by the cutoff $\theta \big ( \{ \widetilde {r}_j, \widetilde {v}_j \}_{j=1}^{m} \big )$ . We conclude that

(2.21) $$ \begin{align} {\bigl\langle\Omega,A_{r_1,v_1}\dots A_{r_m,v_m} A^*_{\tilde{r}_m,v_m} \dots A^*_{\tilde{r}_1,v_1} \Omega\bigr\rangle} &\theta\big( \{r_j, v_j \}_{j=1}^{m} \big) \theta\big( \{ {\tilde{r}}_j, {\tilde{v}}_j \}_{j=1}^{m} \big) \nonumber\\ &\quad = \prod_{i=1}^m \Big(\delta_{{\tilde{r}}_i, r_i} + \delta_{-{\tilde{r}}_i, r_i+v_i}\Big) \theta\big( \{r_j, v_j \}_{j=1}^{m} \big)\, \end{align} $$

(after identification of the momenta, the second cutoff becomes superfluous). From (2.20), we obtain

(2.22) $$ \begin{align} \|\xi_\nu \|^2 &= \sum_{m\geq 0}\frac1{m!}\frac1{N^{m}} \sum_{\substack{ v_1 \in P_S, r_1 \in P_H : \\ r_1 +v_1 \in P_H } } \cdots \sum_{\substack{ v_m \in P_S, r_m \in P_H : \\ r_m +v_m \in P_H } } \theta\big(\{r_j, v_j\}_{j=1}^{m}\big) \, \prod_{i=1}^{m}\eta_{r_i} \big( \eta_{r_i} +\eta_{r_i+v_i} \big) \sigma_{v_i}^2\nonumber\\ &= \sum_{m\geq 0}\frac1{2^m m!}\frac1{N^{m}} \sum_{\substack{ v_1 \in P_S, r_1 \in P_H : \\ r_1 +v_1 \in P_H } } \cdots \sum_{\substack{ v_m \in P_S, r_m \in P_H : \\ r_m +v_m \in P_H } } \theta\big(\{r_j, v_j\}_{j=1}^{m}\big) \, \prod_{i=1}^{m} \big( \eta_{r_i} +\eta_{r_i+v_i} \big)^2 \sigma_{v_i}^2 \end{align} $$

where we used the invariance of $\theta $ , with respect to $-r_i \to r_i + v_i$ . The cutoffs have been used first to exclude coinciding momenta in $v_1, \dots , v_m$ and in $\widetilde {v}_1, \dots , \widetilde {v}_m$ (which implies that, up to permutations, the pairing of the momenta in $P_S$ is unique) and then in (2.21) to make sure that annihilation operators in $A_{r_j, v_j}$ can only be contracted with the creation operators in $A^*_{\widetilde {r}_j, v_j}$ . This substantially simplifies computations. Similar simplifications will arise in the computation of the energy of our trial state.

Apart from the formula (2.22) for the norm $\| \xi _\nu \|^2$ , we will also need bounds on the expectation, in the state $\xi _\nu / \| \xi _\nu \|$ , of the number of particles operator $\mathcal {N}$ , of $\mathcal {N}^2$ , of the kinetic energy operator $\mathcal {K}$ and of the product $\mathcal {K} \mathcal {N}$ . These bounds are collected in the next proposition, whose proof will be discussed in Section 5.

Proposition 2.2. Let $\xi _\nu =e^{A_\nu }\Omega $ with $e^{A_\nu }$ defined in (2.15) with $\kappa \in (1/2 ;2/3)$ and $\varepsilon> 0$ such that $3\kappa -2 + 4\varepsilon < 0$ . Then, under the assumptions of Theorem 1.1, we have

(2.23) $$ \begin{align} \frac{\langle \xi_\nu, \mathcal{N}^{\,j} \xi_\nu \rangle}{\| \xi_\nu \|^2} & \leq C N^{(9\kappa/2-2+\varepsilon)j} \end{align} $$

and

(2.24) $$ \begin{align} \frac{ {\bigl\langle\xi_\nu,\mathcal{K} \mathcal{N}^{j-1} \xi_\nu\bigr\rangle}}{\|\xi_\nu\|^2} \leq C N^{5\kappa/2}N^{(9\kappa/2-2+\varepsilon)(j-1)} \end{align} $$

for $j=1,2$ .

Using the Weyl operator $W_{N_0}$ from (2.3), the Bogoliubov transformation $T_\nu $ defined in (2.13) and the cubic phase $A_\nu $ introduced in (2.15) (or, equivalently, the vector $\xi _\nu = e^{A_\nu } \Omega $ ), we can now define our trial state

(2.25) $$ \begin{align} \Psi_N = \frac{W_{N_0} T_\nu e^{A_\nu} \Omega}{\| W_{N_0} T_\nu e^{A_\nu} \Omega \|} = W_{N_0} T_\nu \frac{\xi_\nu}{\| \xi_\nu \|} . \end{align} $$

Here, we choose $N_0> 0$ such that

(2.26) $$ \begin{align} N = N_0 + \| \sigma_L \|^2 \end{align} $$

where $\sigma _L$ denotes the restriction to the set $P_L$ of the coefficients $\sigma _p = \sinh (\nu _p)$ , with $\nu _p$ defining the Bogoliubov transformation $T_\nu $ ; see (2.13).

Let us briefly compare our trial state with the one of [Reference Yau and Yin20]. In both approaches, the condensate is perturbed with operators creating double and triple excitations, the latter having two particles with high momenta and one particle with low momentum. Moreover, similarly as in [Reference Yau and Yin20], we impose cutoffs making sure that each low momentum appears only once. In contrast to [Reference Yau and Yin20], we also impose cutoffs on high momenta. Moreover, we have a clearer separation between creation of pairs (obtained through the Bogoliubov transformation $T_\nu $ ) and creation of triples. Finally, in our approach, we create triple excitations through the action of $e^{A_\nu }$ on the vacuum; the algebraic structure of the exponential makes the analysis and the combinatorics much simpler.

As shown in the next proposition, the choice (2.26) of $N_0$ guarantees that $\Psi _N$ has the expected number of particles.

Proposition 2.3. Let $\Psi _N$ be defined in (2.25) with the parameter $N_0$ appearing in (2.3) defined by (2.26). Let $\kappa \in (1/2; 2/3)$ and $\varepsilon> 0$ so that $3\kappa -2 + 4\varepsilon < 0$ . Then

(2.27) $$ \begin{align} \langle \Psi_N, \mathcal{N} \Psi_N \rangle \geq N ,\qquad \langle \Psi_N, \mathcal{N}^2 \Psi_N \rangle \leq C N^2 \end{align} $$

for all N large enough.

To prove Proposition 2.3 (and later to show other properties of the trial state $\Psi _N$ ) in the next lemma we collect some bounds for norms of the coefficients appearing in the definition of $A_\nu $ in (2.15). We denote here by $\eta _L, \eta _{L^c}, \eta _S, \eta _H$ the restriction of $\eta : \Lambda ^* \to \mathbb {R}$ to the set $P_L, P_L^c, P_S$ and, respectively, $P_H$ . Similarly, we define $\gamma _L, \gamma _{L^c}, \gamma _H, \gamma _S$ and $\sigma _L, \sigma _{L^c}, \sigma _H, \sigma _S$ .

Lemma 2.4. We have

$$ \begin{align*} &\| \eta_{L^c} \|^2 \leq C N^{3 \kappa/2 -\varepsilon} , \quad && \| \eta_{L^c} \|^2_{H^1} \leq C N^{1+\kappa}, \quad && \|\eta_{L^c} \|_\infty \leq C N^{-2\varepsilon} \\[0.2cm] &\| \eta_H \|^2 \leq C N^{3\kappa-1 +\varepsilon} , \quad && \| \eta_H \|^2_{H^1} \leq C N^{1+\kappa} , \quad && \|\eta_H \|_\infty \leq C N^{3\kappa-2 +2\varepsilon} . \end{align*} $$

In particular, this implies that $\|\gamma _H\|_\infty , \|\sigma _H\|_\infty \leq C$ . Moreover, we have

$$ \begin{align*} \|\gamma_L \|^2_\infty , \|\sigma_L \|^2_\infty \leq C N^{\kappa/2} , \qquad \quad \| \gamma_L \sigma_L \|_1 \leq C N^{3\kappa/2 +\varepsilon} \end{align*} $$

and

$$ \begin{align*} \| \gamma_L \|^2 \leq C N^{3\kappa/2 + 3\varepsilon}, \qquad \| \sigma_L \|^2 \leq C N^{3\kappa/2} \, , \qquad \quad \| \sigma_L \|^2_{H^1} \leq C N^{5\kappa/2 +\varepsilon}. \end{align*} $$

Finally, we observe that

$$ \begin{align*} \| \sigma_S\|^2 \leq C N^{3\kappa/2}, \qquad \| \sigma_S\|^2_{H^1} \leq C N^{5\kappa/2+\varepsilon}, \qquad \| \gamma_S \|_\infty^2, \| \sigma_S \|^2_\infty \leq C N^\varepsilon . \end{align*} $$

Proof. The bounds for $\|\eta _{L^c}\|$ , $\| \eta _H\|$ , $\|\eta _{L^c}\|_\infty $ and $\| \eta _H\|_\infty $ follow from (2.8). On the other hand, with the notation $\check {\eta } (x) = - N w_\ell ( N^{1-\kappa } x)$ for the function on $\Lambda $ with Fourier coefficients $\eta _p$ , we find from Lemma 2.1, part iii),

$$ \begin{align*} \| \eta_{L^c} \|_{H^1}^2 \leq C \sum_{p \in \Lambda^*} p^2 |\eta_p|^2 \leq C \int |\nabla \check{\eta} (x)|^2 dx \leq C N^{1+ \kappa} \int_{\mathbb{R}^3} \frac{1}{(|x|^2+1)^2}\, dx \leq C N^{1+\kappa} . \end{align*} $$

To show bounds for $\sigma _L, \gamma _L$ we observe that, with (2.12) and $\gamma _p^2 = 1 + \sigma _p^2$ , we obtain

(2.28) $$ \begin{align} \sigma^2_p = \frac{p^2 + 8\pi \frak{a} N^\kappa - \sqrt{|p|^4 + 16 \pi \frak{a} N^\kappa p^2}}{2 \sqrt{|p|^4 + 16 \pi \frak{a} N^\kappa p^2}} , \qquad \sigma_p \gamma_p = \frac{-8\pi \frak{a} N^\kappa}{2 \sqrt{|p|^4 + 16 \pi \frak{a} N^\kappa p^2}}. \end{align} $$

Recalling that $P_L = \{ p \in \Lambda ^*_+ : |p| \leq N^{\kappa /2+\varepsilon } \}$ , we find

(2.29) $$ \begin{align} \| \sigma_L\|^2_\infty \leq \mathop{\mathrm{sup}}\limits_{p \in P_L : |p| \leq N^{\kappa/2}} C \frac{N^{\kappa/2}}{|p|} + \mathop{\mathrm{sup}}\limits_{p \in P_L : |p|\geq N^{\kappa/2}} C \frac{N^{2\kappa}}{|p|^4} \leq C N^{\kappa /2}. \end{align} $$

Moreover, by definition of $P_S$ we get

$$ \begin{align*} \|\sigma_S\|_\infty^2\leq \mathop{\mathrm{sup}}\limits_{N^{\kappa/2-\varepsilon}\leq|p|\leq N^{\kappa/2}} C \frac{N^{\kappa/2}}{|p|} + \mathop{\mathrm{sup}}\limits_{p \in P_L: |p|\geq N^{\kappa/2}} C \frac{N^{2\kappa}}{|p|^4} \leq C N^\varepsilon. \end{align*} $$

Using again $\gamma ^2_p= 1+ \sigma ^2_p$ , we find $\| \gamma _L\|_\infty \leq C N^{\kappa /2}$ and $\|\gamma _S\|_\infty \leq C N^\varepsilon $ . Similarly, we obtain

(2.30) $$ \begin{align} \| \sigma_L\|^2 \leq C \sum_{p \in P_L : |p| \leq N^{\kappa/2}} \frac{N^{\kappa/2}}{|p|} + C \sum_{p \in P_L : |p|> N^{\kappa/2}} \frac{N^{2\kappa}}{|p|^4} \leq C N^{3\kappa/2} \end{align} $$

and thus $\|\gamma _L\|^2 \leq C N^{3\kappa /2 + 3 \varepsilon }$ . Moreover, we have

(2.31) $$ \begin{align} \| \sigma_L\|^2_{H_1} \leq C \sum_{p \in P_L : |p| \leq N^{\kappa/2}} |p| N^{\kappa/2} + C \sum_{p \in P_L : |p| \geq N^{\kappa/2}} \frac{N^{2\kappa} }{p^2} \leq C N^{5\kappa/2 + \varepsilon}. \end{align} $$

The bound for $\| \sigma _L \gamma _L\|_1$ is proved similarly, using the expression for $\gamma _p \sigma _p$ in (2.28). Finally, we note that the estimates (2.30) and (2.31) do not improve when we consider the restriction of $\sigma _p$ to $P_S\subset P_L$ ; hence, $\| \sigma _S\|^2 \leq C N^{3\kappa /2}$ and $\| \sigma _S\|^2_{H^1} \leq C N^{5\kappa /2+\varepsilon }$ .

We can now return to the proof of Proposition 2.3.

Proof Proof of Proposition 2.3.

From (2.4) and (2.14) we get

(2.32) $$ \begin{align} T_\nu^* W_{N_0}^* \mathcal{N} W_{N_0} T_\nu =\; &N_0+\sum_{p\in \Lambda_+^*}\sigma_p^2+\sqrt{N_0}(a_0+a^*_0)+ a^*_0 a_0 \nonumber\\ & + \sum_{p\in \Lambda^*_+}\big[ (\sigma_p^2+\gamma_p^2)a_p^*a_p + \gamma_p \sigma_p (a_p a_{-p} + \mbox{h.c.}) \big] . \end{align} $$

By definition of $\xi _\nu $ , $a_0 \xi _\nu = 0$ and $\langle \xi _\nu , a_p a_{-p} \xi _\nu \rangle = 0$ for every $p \in \Lambda ^*$ , as well as the definition $N = N_0 + \| \sigma _L \|^2$ , we obtain that

(2.33) $$ \begin{align} \langle \Psi_N, \mathcal{N} \Psi_N \rangle = \frac{\langle \xi_\nu , T_\nu^* W_{N_0}^* \mathcal{N} W_{N_0} T_\nu \xi_\nu \rangle}{\| \xi_\nu \|^2} = N +\sum_{p\in P_L^c}\sigma_p^2 + \sum_{p\in P_S\cup P_H} (\sigma_p^2+\gamma_p^2)\frac{\langle \xi_\nu,a_p^*a_p \xi_\nu\rangle}{\|\xi_\nu\|^2} . \end{align} $$

This immediately implies that $\langle \Psi _N, \mathcal {N} \Psi _N \rangle \geq N$ .

With $a_0 \xi _\nu = 0$ and the assumption $3\kappa - 2 + 4 \varepsilon < 0$ , (2.32) also implies that

$$ \begin{align*} \langle \xi_\nu , T_\nu^* W_{N_0}^* \mathcal{N}^2 W_{N_0} T_\nu \xi_\nu \rangle \leq \; &C N^2 + C \sum_{p,q \in \Lambda^*_+} (\sigma_p^2 + \gamma_p^2) (\sigma_q^2 + \gamma_q^2) \langle \xi_\nu, a_p^* a_p a_q^* a_q \xi_\nu \rangle \\ &+ C \sum_{p,q \in \Lambda^*_+} \gamma_p \sigma_p \gamma_q \sigma_q \langle \xi_\nu, (a_p a_{-p} + a^*_p a_{-p}^*) (a_q a_{-q} + a_q^* a_{-q}^*) \xi_\nu \rangle. \end{align*} $$

Since $\xi _\nu $ is a superposition of states with $3m$ particles with momenta in $P_H \cup P_S$ , we obtain, writing $a_p a_{-p} a_q^* a_{-q}^* = a_q^* a_p a_{-q}^* a_{-p} +(\delta _{p,q}+ \delta _{-p, q})( a_{p}^* a_{p} +1) + \delta _{p,q} a^*_{-p} a_{-p}$ and similarly for $a_p^* a_{-p}^* a_q a_{-q}$ ,

$$ \begin{align*} \langle \xi_\nu , T_\nu^* W_{N_0}^* \mathcal{N}^2 W_{N_0} T_\nu \xi_\nu \rangle \leq \; &CN^2 + C N^{2\varepsilon} \langle \xi_\nu , (\mathcal{N}+1)^2 \xi_\nu \rangle + C \sum_{p \in P_H \cup P_S} \gamma_p^2 \sigma_p^2 \langle \xi_\nu, a_p^* a_p \xi_\nu \rangle \\ & + C \sum_{p \in \Lambda^*_+} \gamma_p^2 \sigma_p^2 \| \xi_\nu \|^2 + C \hspace{-.3cm} \sum_{p,q \in P_H\cup P_S} \hspace{-.3cm} \gamma_p \sigma_p \gamma_q \sigma_q \langle \xi_\nu, a^*_p a_q a^*_{-p} a_{-q} \xi_\nu \rangle . \end{align*} $$

Estimating the last term through

$$ \begin{align*} \Big| \sum_{p,q \in P_H \cup P_S} \gamma_p &\sigma_p \gamma_q \sigma_q \langle \xi_\nu, a^*_p a_q a^*_{-p} a_{-q} \xi_\nu \rangle \Big| \\ \leq \; & \sum_{p,q \in P_H \cup P_S} |\gamma_p | | \sigma_p | |\gamma_q | | \sigma_q | \, \| a^* _q a_p \xi_\nu \| \| a^*_{-p} a_{-q} \xi_\nu \| \\ \leq \; & \sum_{p,q \in P_H \cup P_S} | \gamma_p | |\sigma_p | |\gamma_q | |\sigma_q | \, \big[ \| a_q a_p \xi_\nu \| + \| a_p \xi_\nu \| \big] \big[ \| a_{-p} a_{-q} \xi_\nu \| + \| a_{-q} \xi_\nu \| \big] \\ \leq \; & C \| \gamma_{S \cup H} \|_\infty^2 \| \sigma_{S \cup H} \|_\infty^2 \| (\mathcal{N}+1) \xi_\nu \|^2 + C \| \gamma_{S \cup H} \|_\infty^2 \| \sigma_{S \cup H} \|^2 \| \mathcal{N}^{1/2} \xi_\nu \|^2\\ &+ C \| \gamma_{S \cup H} \|_\infty^2 \| \sigma_{S \cup H} \|_\infty \| \sigma_{S \cup H} \| \| (\mathcal{N}+1) \xi_\nu \| \| \mathcal{N}^{1/2} \xi_\nu \|, \end{align*} $$

we conclude with the bounds in Lemma 2.4 and in Proposition 2.2 that, for $3\kappa -2 + 4\varepsilon < 0,$

$$ \begin{align*} \langle \xi_\nu , T_\nu^* W_{N_0}^* \mathcal{N}^2 W_{N_0} T_\nu \xi_\nu \rangle \leq \;& C N^2 + C N^{2\varepsilon} \langle \xi_\nu, \mathcal{N}^2 \xi_\nu \rangle + C N^\varepsilon \| \sigma \|^2 \langle \xi_\nu, \mathcal{N} \xi_\nu \rangle + C \| \gamma \|_\infty^2 \| \sigma \|^2 \\ \leq \; & C N^2 + C N^{9\kappa -4 + 4\varepsilon} + C N^{6\kappa -2 +2\varepsilon} + C N^{2\kappa} \leq C N^2. \\[-36pt] \end{align*} $$

The next theorem, whose proof occupies the rest of the article, determines the energy of $\Psi _N$ and, combined with Proposition 2.3, allows us to conclude the proof of Proposition 1.3.

Theorem 2.5. Let $\mathcal {H}_N$ and $\Psi _N \in \mathcal {F}$ be defined as in (2.2) and (2.25), respectively, and let $E_N^\Psi = \langle \Psi _N, \mathcal {H}_N \Psi _N \rangle $ . Let $\kappa \in (1/2; 2/3)$ , $\varepsilon> 0$ so small that $3\kappa -2 + 4\varepsilon < 0$ . Then, under the assumption of Theorem 1.1, we have

(2.34) $$ \begin{align} E_N^\Psi \leq 4 \pi \mathfrak{a} N^{1+\kappa} \bigg(1+\frac{128}{15\sqrt{\pi}}(\mathfrak{a}^3N^{3\kappa-2})^{1/2}\bigg)+ C N^{5\kappa/2} \max \{ N^{-\varepsilon} \hspace{-.1cm} , N^{9\kappa-5 +6\varepsilon} \hspace{-.1cm} ,N^{21\kappa/4-3+3\varepsilon}\} \end{align} $$

for all N large enough.

3 Energy of the trial state

In this section we prove Theorem 2.5. With (2.25) and introducing the notations

(3.1) $$ \begin{align} \mathcal{G}_N = T^*_\nu \mathcal{L}_N T_\nu, \quad \text{with} \quad \mathcal{L}_N = W^*_{N_0} \mathcal{H}_N W_{N_0}\, \end{align} $$

we write

$$ \begin{align*} E^{\Psi}_N = \langle \Psi_N, \mathcal{H}_N \Psi_N \rangle = \frac{\langle \xi_\nu , \mathcal{G}_N \xi_\nu \rangle}{\| \xi_\nu \|^2 } \end{align*} $$

with $\xi _\nu = e^{A_\nu } \Omega $ , as defined before (2.15). With (2.2) and recalling from (2.4) that

(3.2) $$ \begin{align} W^*_{N_0} a_p W_{N_0}=a_p+\sqrt{N_0}\,\delta_{p,0} \end{align} $$

we obtain $\mathcal {L}_N= \mathcal {L}^{(0)}_N +\mathcal {L}^{(1)}_N+ \mathcal {L}^{(2)}_N + \mathcal {L}^{(3)}_N + \mathcal {L}^{(4)}_N $ , with

(3.3) $$ \begin{align} \mathcal{L}^{(0)}_N=\;&\frac {N_0^2} {2} \,N^{\kappa-1} \widehat{V}(0)\nonumber\\ \mathcal{L}^{(1)}_N=\; & N_0^{3/2}\, N^{\kappa-1}\widehat{V}(0)(a_0+\mbox{h.c.})\nonumber\\ \mathcal{L}^{(2)}_N=\;&\sum_{p\in \Lambda^*}p^2a^*_pa_p + \frac{N_0}{N} \sum_{p\in \Lambda^*} N^\kappa \big( \widehat{V}(p/N^{1-\kappa}) + \widehat V(0) \big) a_p^*a_p \nonumber\\ & \hskip 3cm + \frac{N_0}{2 N}\sum_{p\in \Lambda^*}N^\kappa \widehat{V}(p/N^{1-\kappa})(a_p^*a_{-p}^*+ a_p a_{-p})\nonumber\\ \mathcal{L}^{(3)}_N=\; & \frac{\sqrt{N_0}}N \sum_{p,r\in \Lambda^*} N^{\kappa} \widehat{V}(r/N^{1-\kappa})(a^*_pa^*_ra_{p+r}+ a^*_{p+r} a_r a_p )\nonumber \\ \mathcal{L}^{(4)}_N=\; &\frac1{2N}\sum_{p,q,r \in \Lambda^*} N^{\kappa} \widehat{V}(r/N^{1-\kappa})a^*_{p+r} a^*_q a_{p}a_{q+r}. \end{align} $$

To compute $\mathcal {G}_N$ , we have to conjugate the operators in (3.3) with the Bogoliubov transformation $T_\nu $ . The result is described in the following proposition, whose proof will be discussed in Section 4.

Proposition 3.1. Let

(3.4) $$ \begin{align} \mathcal{K} = \sum_{p \in \Lambda^*_+} p^2 a^*_p a_p, \qquad \mathcal{V}_N^{(H)}= \frac{1}{2 N}\sum_{\substack{r\in \Lambda^*,\, p,q \in P_H:\\ p+r,q+r\in P_H}}N^\kappa \widehat{V}(r/N^{1-\kappa}) a_{p+r}^*a_q^*a_{p}a_{q+r} \, \end{align} $$

and (recalling from (2.26) that $N_0 = N - \| \sigma _L \|^2$ )

(3.5) $$ \begin{align} \mathcal{C}_N =\; \frac{\sqrt{N_0}}{ N} \sum_{\substack{p,r \in P_H\\ p+r \in P_S}} N^\kappa \widehat V(r/N^{1-\kappa})\, \sigma_{p+r} \gamma_p\gamma_r \,(a^*_{p+r} a^*_{-p} a^*_{-r} + \mbox{h.c.} ). \end{align} $$

Moreover, let

(3.6) $$ \begin{align} C_{\mathcal{G}_N} =\; & \frac{N^{1+\kappa}}{2} \widehat{V}(0) +\sum_{p\in {\Lambda}^*_+}p^2\sigma_p^2 + \sum_{p\in \Lambda^*_+} N^\kappa \widehat{V}(p/N^{1-\kappa})\sigma_p\gamma_p\nonumber\\ &+ \sum_{p\in P_L}N^\kappa \widehat{V}(p/N^{1-\kappa}) \sigma_p^2 +\frac{1}{2N}\sum_{\substack{p,r\in \Lambda^*_+\\r\neq p}} N^\kappa \widehat{V}(r/N^{1-\kappa})\sigma_p\sigma_{p-r}\gamma_p\gamma_{p-r} \nonumber\\ & - \frac 1 N \sum_{v \in P_L} \sigma^2_v \sum_{p \in P_L^c} N^\kappa \widehat V(p/N^{1-\kappa}) \eta_p . \end{align} $$

Then, under the assumptions of Theorem 1.1, for all $\kappa \in (1/2; 2/3)$ and $\varepsilon> 0$ with $3\kappa -2 + 4\varepsilon < 0$ and N sufficiently large, we have

(3.7) $$ \begin{align} \frac{\langle \xi_\nu, \mathcal{G}_N \xi_\nu \rangle}{\| \xi_\nu \|^2} \leq \; &C_{\mathcal{G}_N} + \frac{\langle \xi_\nu, (\mathcal{K}+\mathcal{V}_N^{(H)} +\mathcal{C}_N) \xi_\nu\rangle}{\| \xi_\nu \|^2}\nonumber \\ &+ C N^{5\kappa/2} \cdot \max \{ N^{-\varepsilon}, N^{9\kappa - 5 + 6\varepsilon} , N^{21 \kappa /4 - 3 + 3 \varepsilon } \}. \end{align} $$

Proposition 3.2. Under the assumptions of Theorem 1.1, we have

$$ \begin{align*} \frac{\langle \xi_\nu, (\mathcal{K}+\mathcal{V}^{(H)}_N +\mathcal{C}_{N}) \xi_\nu \rangle }{\|\xi_\nu\|^2} \leq \; & \frac1N\sum_{v\in P_L}\sigma_v^2\sum_{r\in P_H} N^\kappa \widehat V(r/N^{1-\kappa})(\eta_r+\eta_{r+v}) \\ &+ C N^{5\kappa/2} \cdot \max \{ N^{-\varepsilon} ,\, N^{12\kappa-7 +5\varepsilon}\} \end{align*} $$

for all $\kappa \in (1/2; 2/3)$ and all $\varepsilon> 0$ so small that $3\kappa -2 + 4\varepsilon < 0$ .

Let us now use the statements of Proposition 3.1 and Proposition 3.2 to obtain an upper bound for the energy of the trial state $\Psi _N$ and prove Theorem 2.5. From Proposition 3.1 and Proposition 3.2 we find

(3.8) $$ \begin{align} E_N^\Psi \leq \; & \frac{N^{1+\kappa}}{2}\,\widehat{V}(0)+ \sum_{p \in \Lambda^*_+} p^2 \sigma_p^2 + \sum_{p \in \Lambda^*_+} N^\kappa \widehat V(p/N^{1-\kappa}) \gamma_p \sigma_p \nonumber \\ & + \sum_{v \in P_L} N^\kappa \widehat V(v/N^{1-\kappa}) \sigma^2_v + \frac 1 {2N} \sum_{\substack{p,q \in \Lambda^*_+\nonumber\\ p \neq q}} N^\kappa \widehat V( (p-q)/N^{1-\kappa}) \gamma_p \gamma_{q} \sigma_p \sigma_{q}\nonumber\\ & - \frac 1 N \sum_{v \in P_L} \sigma^2_v \sum_{p \in P_L^c} N^\kappa \widehat V(p/N^{1-\kappa}) \eta_p \nonumber\\ & + \frac1N\sum_{v\in P_L}\sigma_v^2\sum_{r\in P_H} N^\kappa \widehat V(r/N^{1-\kappa})(\eta_r+\eta_{r+v})\, + \mathcal{E} \end{align} $$

with

$$ \begin{align*} \mathcal{E} \leq C N^{5\kappa/2} \cdot \max \{ N^{-\varepsilon},\,N^{9\kappa-5 +6\varepsilon} ,\, N^{21\kappa/4-3+3\varepsilon}\} \end{align*} $$

for all $\kappa \in (1/2; 2/3)$ and all $\varepsilon> 0$ with $3\kappa -2 + 4\varepsilon < 0$ (in this range, $12\kappa -7+5\varepsilon < 9\kappa -5+3\varepsilon $ ). Since $|\sigma _p - \eta _p| \leq C |\eta _p|^3$ for all $p \in P_L^c$ , with (2.8) we find

(3.9) $$ \begin{align} \sum_{p \in \Lambda^*_+} p^2 \sigma_p^2 \leq \sum_{v \in P_L} v^2 \sigma_v^2 + \sum_{p \in P_L^c} p^2 \eta_p^2 + C N^{5\kappa/2 - 3\varepsilon} . \end{align} $$

Similarly, with $|\gamma _p \sigma _p - \eta _p| \leq C \eta _p^3$ for all $p \in P_L^c $ , the last term on the first line of (3.8) can be written as

(3.10) $$ \begin{align} \sum_{p \in \Lambda^*_+} N^\kappa &\widehat V(p/N^{1-\kappa})\gamma_p \sigma_p \nonumber \\ & \leq \sum_{v \in P_L} N^\kappa \widehat V(v/N^{1-\kappa}) \gamma_v \sigma_v + \sum_{p \in P_L^c} N^\kappa \widehat V(p/N^{1-\kappa}) \eta_p + C N^{5\kappa/2-3\varepsilon} . \end{align} $$

Next, we focus on the last term on the second line of (3.8). We define $\mathcal {E}_1$ through the identity

(3.11) $$ \begin{align} &\frac 1 {2N} \sum_{\substack{p,q \in \Lambda^*_+\\ p \neq q}} N^\kappa \widehat V((p-q)/N^{1-\kappa}) \gamma_p \gamma_{q} \sigma_p \sigma_{q}\nonumber \\ &= \frac 1 {N} \!\!\sum_{\substack{p \in P_L^c, \\ q \in P_L}} N^\kappa \widehat V((p-q)/N^{1-\kappa}) \eta_p \gamma_{q} \sigma_{q} + \frac 1 {2N}\!\! \sum_{\substack{p,q \in P_L^c\\ p \neq q}} \!N^\kappa \widehat V((p-q)/N^{1-\kappa}) \eta_p \eta_q \!+ \mathcal{E}_1. \end{align} $$

Using again $|\gamma _p \sigma _p - \eta _p | \leq C |\eta _p|^3$ for all $p \in P_L^c$ , the estimate

(3.12) $$ \begin{align} \mathop{\mathrm{sup}}\limits_{r\in \Lambda^*}\,\sum_{s \in \Lambda^*_+} N^\kappa |\widehat V((r-s)/N^{1-\kappa})| |\eta_s| \leq C N^{1+\kappa} \end{align} $$

and the bounds from Lemma 2.4, we conclude (using the condition $3\kappa -2 + 5\varepsilon < 0$ ) that

$$ \begin{align*} \mathcal{E}_1 &\leq C \left[ N^{\kappa} \| \eta_{L^c} \|_\infty \| \eta_{L^c} \|^2 + N^{\kappa-1} \| \eta_{L^c} \|_\infty \| \eta_{L^c} \|^2 \| \sigma_L \gamma_L \|_1 + N^{\kappa-1} \| \sigma_L \gamma_L \|_1^2 \right] \leq C N^{5\kappa/2-\varepsilon}. \end{align*} $$

To prove (3.12) we use (2.8) and we remark that

$$ \begin{align*} \sum_{s \in \Lambda^* : |s| \leq N^{1-\kappa}} N^\kappa |\widehat V((r-s)/N^{1-\kappa})| |\eta_s| \leq C N^{2\kappa} \sum_{s \in \Lambda^*_+: |s| \leq N^{1-\kappa}} |s|^{-2} \leq C N^{1+\kappa} \end{align*} $$

and that, rescaling variables (setting ${\tilde {r}} = r/ N^{1-\kappa }$ ) and using an integral approximation,

$$ \begin{align*} \sum_{s \in \Lambda^*_+ : |s|> N^{1-\kappa}} N^\kappa |\widehat V((r-s)/N^{1-\kappa})| |\eta_s| &\leq C N^{1+\kappa} \sum_{s \in \Lambda^* / N^{1-\kappa}: |s| > 1} N^{-3(1-\kappa)} \, |\widehat{V} ({\tilde{r}} - s)| |s|^{-2} \\ &\leq C N^{1+\kappa} \int_{|s| > 1} |\widehat{V} ({\tilde{r}} - s)| |s|^{-2} ds \leq C_q N^{1+\kappa} \| \widehat{V} \|_q \end{align*} $$

for any $q < 3$ . With the assumption $V \in L^{q'} (\mathbb {R}^3)$ , for some $q'> 3/2$ , (3.12) follows by the Hausdorff–Young inequality.

Finally, we remark that the terms in the last two lines of (3.8) can be combined, using (2.8) and the bound $\| \sigma _L \|^2 \leq C N^{3\kappa /2}$ from Lemma 2.4, into

(3.13) $$ \begin{align} - \frac 1 N \sum_{v \in P_L} \sigma^2_v \sum_{p \in P_L^c} N^\kappa \widehat V(p/N^{1-\kappa}) \eta_p& +\frac1N\sum_{v\in P_L}\sigma_v^2\sum_{r\in P_H} N^\kappa \widehat V(r/N^{1-\kappa})(\eta_r+\eta_{r+v})\nonumber\\ &\leq \frac 1 N \sum_{v \in P_L} \sigma^2_v \sum_{r \in P_L^c} N^\kappa \widehat V(r/N^{1-\kappa}) \eta_{r+v} + C N^{5\kappa/2 - \varepsilon}. \end{align} $$

Inserting (3.9), (3.10), (3.11) and (3.13) into (3.8), we obtain

(3.14) $$ \begin{align} E_N^\Psi \leq & \, \frac{N^{1+\kappa}}{2} \widehat{V} (0)+ \sum_{p \in P_L^c}\big[ p^2 \eta_p + N^\kappa \widehat V(p/N^{1-\kappa}) + \frac 1 {2N} \sum_{r \in P_L^c} N^\kappa \widehat V((r-p)/N^\kappa) \eta_r \big] \eta_p \nonumber\\ & + \sum_{v \in P_L} \Big[ v^2 \sigma_v^2 + \big(\sigma^2_v + \gamma_v \sigma_v \big)\Big( N^\kappa \widehat V(v/N^{1-\kappa}) +\frac 1 N \sum_{r \in P_L^c} N^\kappa \widehat V((r-v)/N^{1-\kappa})\eta_{r} \Big) \Big] \nonumber\\ & + C N^{5\kappa/2} \max \{ N^{-\varepsilon},\,N^{9\kappa-5 +6\varepsilon} ,\, N^{21\kappa/4-3+3\varepsilon}\} . \end{align} $$

Let us now consider the first square bracket on the right-hand side of (3.14). Using the scattering equation (2.9) we obtain

(3.15) $$ \begin{align} &\sum_{p \in P_L^c} \Big[ p^2 \eta_p + N^\kappa \widehat V(p/N^{1-\kappa}) + \frac 1 {2N} \sum_{r \in P_L^c} N^\kappa \widehat V( (r-p)/N^{1-\kappa}) \eta_r \Big] \eta_p \nonumber\\ &\qquad\qquad= \frac 1 2 \sum_{p \in P_L^c} N^\kappa \widehat V(p/N^{1-\kappa}) \eta_p - \frac 1 {2N} \sum_{\substack{p \in P_L^c\\ v \in P_L} } N^\kappa \widehat V((p-v)/N^{1-\kappa}) \eta_v \eta_p+ \mathcal{E}_2 \end{align} $$

with

$$ \begin{align*} \mathcal{E}_2 =\; N^{3-2\kappa} \lambda_{\ell} \sum_{p \in P_L^c} \big( \widehat \chi_\ell \ast \widehat f_{N,\ell}\big)_p \eta_p - \frac 1 {2N} \sum_{p \in P_L^c} N^\kappa \widehat V(p/N^{1-\kappa})\eta_p \eta_0 . \end{align*} $$

Using $N^{3-3\kappa } \lambda _{\ell } \leq C$ (Lemma 2.1), $\| \widehat \chi _\ell \ast \widehat f_{N,\ell }\| = \| \chi _\ell f_{N,\ell } \| \leq C$ and $\|\eta _{L^c}\|^2 \leq C N^{3\kappa /2 -\varepsilon }$ (Lemma 2.4) in the first term, (2.10) and (3.12) in the second term, we find $\mathcal {E}_2 \leq C N^{5\kappa /2-\varepsilon }$ (using that $3\kappa -2 + 4\varepsilon < 0$ and $\kappa> 1/2$ ).

As for the second square bracket on the right-hand side of (3.14), we write

(3.16) $$ \begin{align} \sum_{v \in P_L} \big(\sigma^2_v + \gamma_v \sigma_v \big) \Big( N^\kappa \widehat V(v/N^{1-\kappa}) & +\frac 1 N \sum_{r \in P_L^c} N^\kappa \widehat V((r-v)/N^{1-\kappa})\eta_{r} \Big) \nonumber\\ &= \sum_{v \in P_L} \big(\sigma^2_v + \gamma_v \sigma_v \big) N^\kappa \big( \widehat V(\cdot/N^{1-\kappa}) \ast \widehat f_{N,\ell} \big)_v + \mathcal{E}_3 \end{align} $$

with

$$ \begin{align*} \mathcal{E}_3 =\; -\frac 1 N \hskip -0.2cm \sum_{v,\,r\in P_L}\hskip -0.2cm N^\kappa \widehat V((r-v)/N^{1-\kappa}) \big(\sigma^2_v + \gamma_v \sigma_v \big)\eta_{r} - \frac {\eta_0} N \sum_{v \in P_L} N^\kappa \widehat V_{N}(v/N^{1-\kappa}) \big(\sigma^2_v + \gamma_v \sigma_v \big) . \end{align*} $$

With (2.8), Lemma 2.4, $|\eta _0| \leq C N^\kappa $ and the assumption $3\kappa -2 + 5\varepsilon < 0$ , we obtain $\mathcal {E}_3 \leq N^{5\kappa /2-\varepsilon }$ .

Inserting (3.15) and (3.16) in (3.14) and completing sums over p on the right-hand side of (3.15), we arrive at

(3.17) $$ \begin{align} E_N^\Psi \leq \; & \frac{N}{2} \big( N^\kappa \widehat V_{N}(\cdot/N^{1-\kappa}) \ast \widehat f_{N,\ell} \big)_0 \nonumber \\ &+ \sum_{v \in P_L} \Big[ v^2 \sigma_v^2 + \big(\sigma^2_v + \gamma_v \sigma_v \big) \big( N^\kappa \widehat V(\cdot/N^{1-\kappa}) \ast \widehat f_{N,\ell} \big)_v - \frac {1} 2 N^\kappa \big( \widehat V(\cdot/N^{1-\kappa}) \ast \widehat f_{N,\ell} \big)_v \eta_v \Big] \nonumber \\ &+ C N^{5\kappa/2} \max \{ N^{-\varepsilon},\,N^{9\kappa-5 +6\varepsilon} ,\, N^{21\kappa/4-3+3\varepsilon}\}. \end{align} $$

Let us now introduce the notation $\hat {g}_p = ( N^\kappa \widehat V(\cdot /N^{1-\kappa }) \ast \widehat f_{N,\ell })_p$ . Notice that

(3.18) $$ \begin{align} \big|\, \widehat g_0 - 8 \pi \mathfrak{a} N^\kappa \,\big| \leq C N^{2\kappa -1}, \qquad \big|\, \widehat g_p - \widehat g_0 \, \big| \leq C |p| N^{2\kappa -1} . \end{align} $$

With the expression (2.28), we obtain

$$ \begin{align*} \sum_{v \in P_L} \Big[ v^2 \sigma_v^2 + \big(\sigma^2_v + \gamma_v \sigma_v \big) \widehat g_v \Big] & =\frac 12 \sum_{v \in P_L} \left[ -v^2 - \widehat g_v + \frac{v^4 + v^2 (8\pi \frak{a} N^\kappa + \widehat g_v)}{\sqrt{v^4 + 16 \pi \frak{a} N^\kappa v^2}} \right] \\ & = \frac 12 \sum_{v \in P_L} \left[ \sqrt{v^4 + 16 \pi \frak{a} N^\kappa v^2} - v^2 - 8 \pi \mathfrak{a} N^\kappa \right] + \mathcal{E}_{4} \end{align*} $$

where, with (3.18) and $3\kappa -2 + 4\varepsilon < 0$ , we find

$$ \begin{align*} \mathcal{E}_{4} &= \frac 12 \sum_{v\in P_L} \left[ 8\pi \frak{a} N^\kappa - \widehat g_v + \frac{\widehat g_v - 8\pi \frak{a} N^\kappa}{\sqrt{1 + 16 \pi \frak{a} N^\kappa / v^2}} \right] \\ &\leq C \sum_{v\in P_L} | 8\pi \frak{a} N^\kappa - \widehat g_v | \;\frac{|\sqrt{1 + 16 \pi \frak{a} N^\kappa / v^2} - 1 |}{\sqrt{1 + 16 \pi \frak{a} N^\kappa /v^2}} \\ &\leq C N^{2\kappa-1} \Big[ \sum_{|v| < N^{\kappa/2}} (1+ |v|) + \sum_{v \in P_L : |v|> N^{\kappa/2}} \frac{N^{\kappa/2}}{|v|^2} \Big] \leq C N^{5\kappa/2-\varepsilon} . \end{align*} $$

Moreover, from (3.17) and with the scattering equation (2.9), we obtain

$$ \begin{align*} - \frac{N^\kappa}{2} \sum_{v \in P_L} \widehat{g}_v \eta_v \leq \sum_{v \in P_L} \frac{(8\pi \frak{a} N^\kappa )^2}{4v^2} + C N^{5\kappa/2-\varepsilon}. \end{align*} $$

Thus,

$$ \begin{align*} E^\Psi_N \leq \; & 4 \pi \mathfrak{a} N^{1+\kappa} + \frac 12 \sum_{v \in P_L} \Big[ \sqrt{v^4 + 16 \pi \mathfrak{a} N^\kappa v^2} - v^2 - 8 \pi \mathfrak{a} N^\kappa + \frac{(8 \pi \mathfrak{a} N^\kappa )^2}{2v^2} \Big] \\ &+ C N^{5\kappa/2} \max \{ N^{-\varepsilon},\,N^{9\kappa-5 +6\varepsilon} ,\, N^{21\kappa/4-3+3\varepsilon}\} . \end{align*} $$

With

$$ \begin{align*} \left| \sqrt{v^4 + 16 \pi \mathfrak{a} N^\kappa v^2} - v^2 - 8 \pi \mathfrak{a} N^\kappa + \frac{(8 \pi \mathfrak{a} N^\kappa )^2}{2v^2} \right| \leq C \frac{N^{2\kappa}}{|v|^4} \end{align*} $$

we can replace, up to an error of order $N^{5\kappa /2-\varepsilon }$ , the sum over $P_L$ with a sum over all $\Lambda ^*_+$ . With the rescaling $v \to N^{\kappa /2} v$ , we arrive at

(3.19) $$ \begin{align} E^\Psi_N \leq \; & 4 \pi \mathfrak{a} N^{1+\kappa} + \frac{N^\kappa}{2} \sum_{v \in 2\pi N^{-\kappa/2} \mathbb{Z}^3} \Big[ \sqrt{v^4 + 16 \pi \mathfrak{a} v^2} - v^2 - 8 \pi \mathfrak{a} + \frac{(8 \pi \mathfrak{a} )^2}{2v^2} \Big] \nonumber \\ &+ C N^{5\kappa/2} \max \{ N^{-\varepsilon},\,N^{9\kappa-5 +6\varepsilon} ,\, N^{21\kappa/4-3+3\varepsilon}\}. \end{align} $$

Recognising that (3.19) defines a Riemann sum and explicitly computing

$$ \begin{align*} \frac{1}{2 (2\pi)^3} \int dv \, \Big[ \sqrt{v^4 + 16 \pi \mathfrak{a} v^2} - v^2 - 8 \pi \mathfrak{a} + \frac{(8 \pi \mathfrak{a} )^2}{2v^2} \Big] = 4 \pi \frak{a} \cdot \frac{128}{15 \sqrt{\pi}} \frak{a}^{3/2} \end{align*} $$

we conclude that

$$ \begin{align*} E^\Psi_N \leq \; & 4 \pi \mathfrak{a} N^{1+\kappa} \cdot \left[ 1 + \frac{128}{15 \sqrt{\pi}} \, (\frak{a}^3 N^{3\kappa-2})^{1/2} \right] \\ &+ C N^{5\kappa/2} \max \{ N^{-\varepsilon},\,N^{9\kappa-5 +6\varepsilon} ,\, N^{21\kappa/4-3+3\varepsilon}\} . \end{align*} $$

To compare the Riemann sum in (3.19) with the integral, we first removed contributions arising from $|v| \leq N^{-\varepsilon }$ using that $|F(v)| \leq C / v^2$ , for small v, with the definition $F(v) = \sqrt {v^4 + 16 \pi \mathfrak {a} v^2} - v^2 - 8 \pi \mathfrak {a} + (8 \pi \mathfrak {a} )^2 / 2v^2$ . For $|v|> N^{-\varepsilon }$ , we use that $|\nabla F (v)| \leq C |v|^{-3} (1+v^2)^{-1}$ to compare the value of $F(q)$ with $F(v)$ , for all q in the cube of size $2\pi N^{-\kappa /2}$ centred at v.

4 Bogoliubov transformation

In this section, we show Proposition 3.1. From the definition (3.1) and from (3.3), we obtain (since $T_\nu $ does not act on the zero momentum mode and since $a_0 \xi _\nu = 0$ )

$$ \begin{align*} \frac{\langle \xi_\nu, \mathcal{G}_N \xi_\nu \rangle}{\| \xi_\nu \|^2} = \frac{N_0^2}{2} N^{\kappa - 1} \widehat{V} (0) + \sum_{j=2}^4 \frac{\langle \xi_\nu, \mathcal{G}_N^{(j)} \xi_\nu \rangle}{\| \xi_\nu \|^2} \end{align*} $$

with $\mathcal {G}_N^{(j)} = T_\nu ^* \mathcal {L}_N^{(j)} T_\nu $ , for $j=2,3,4$ .

We start from the contribution of $\mathcal {G}_N^{(2)}$ . We write $\mathcal {L}_N^{(2)}= \mathcal {K} + \mathcal {L}_N^{(2,V)}$ with

$$ \begin{align*} \mathcal{L}_N^{(2,V)} =\frac{N_0}{N}\sum_{p\in \Lambda^*} N^\kappa \big( \hat{V}(p/N^{1-\kappa}) + \hat V(0) \big) a_p^*a_p + \frac{N_0}{2N}\sum_{p\in \Lambda^*}N^\kappa \hat{V}(p/N^{1-\kappa})(a_p^*a_{-p}^*+\mbox{h.c.} ). \end{align*} $$

Using (2.14) we get

$$ \begin{align*} T^*_{\nu} \mathcal{K} T_\nu - \left[ \mathcal{K} + \sum_{p \in \Lambda_+^*} p^2 \sigma_p^2 \right] = 2 \sum_{p\in \Lambda^*_+} p^2 \sigma_p^2 a^*_p a_p + \sum_{p \in \Lambda^*_+} p^2 \big[\gamma_p\sigma_p ( a^*_p a^*_{-p}+\mbox{h.c.}) \big] := E_1 + E_2. \end{align*} $$

From (2.15), $\langle \xi _\nu , \text {E}_2 \xi _\nu \rangle = 0$ ( $\xi _\nu $ is a superposition of states with $3m$ particles, for $m \in \mathbb {N}$ ). To bound the expectation of $E_1$ on $\xi _\nu $ we notice that $\langle \xi _\nu ,a^*_pa_p\xi _\nu \rangle =0$ if $p\in \Lambda ^*_+\backslash (P_S\cup P_H)$ . Moreover, proceeding as in (2.31), we have

$$ \begin{align*} \mathop{\mathrm{sup}}\limits_{p \in P_S} ( p^2 \sigma_p^2) \leq \mathop{\mathrm{sup}}\limits_{p \in P_S: |p|\leq N^{\kappa/2}} N^{\kappa/2} |p| + \mathop{\mathrm{sup}}\limits_{p \in P_S : |p|> N^{\kappa/2}} \frac{N^{2\kappa}}{p^2} \leq C N^{\kappa} \end{align*} $$

while

$$ \begin{align*} \mathop{\mathrm{sup}}\limits_{p \in P_H} (p^2 \sigma_p^2) \leq \mathop{\mathrm{sup}}\limits_{|p| \geq N^{1-\kappa-\varepsilon}} \frac{N^{2\kappa}}{p^2} \leq C N^{-2+4\kappa+2\varepsilon} \leq C N^\kappa \end{align*} $$

because, by assumption, $3\kappa -2+2\varepsilon < 0$ . Hence,

(4.1) $$ \begin{align} |{\bigl\langle\xi_\nu, E_{1}\xi_\nu\bigr\rangle}|\leq C N^\kappa \, \| \mathcal{N}^{1/2}\xi_\nu \|^2 . \end{align} $$

We now consider the contribution from $ \mathcal {L}_N^{(2,V)}$ . Using again $\langle \xi _\nu , a_p^* a_{-p}^* \xi _\nu \rangle = 0$ for all $p \in \Lambda _+^*$ and $\langle \xi _\nu , a_p^* a_{p} \xi _\nu \rangle = 0$ for all $p \in \Lambda ^*_+\backslash (P_S \cup P_H)$ , a straightforward computation shows that

$$ \begin{align*} &\frac{\langle \xi_\nu, T^*_\nu \mathcal{L}_N^{(2,V)} T_\nu \xi_\nu \rangle}{\| \xi_\nu \|^2} \\ &\hspace{2cm} = \frac{N_0}{N} \sum_{p \in \Lambda^*_+} N^\kappa \widehat V(p/N^{1-\kappa}) \gamma_p \sigma_p + \frac{N_0}{N} \sum_{p \in \Lambda^*_+} N^\kappa \big(\widehat V(0) +\widehat V(p/N^{1-\kappa}) \big) \sigma^2_p \\ & \hspace{2.3cm} + \frac{N_0}{N} \sum_{p\in P_S \cup P_H} N^\kappa \big[\widehat V(p/N^{\kappa-1})(\gamma_p+\sigma_p)^2 +\widehat V(0) (\gamma^2_p + \sigma^2_p) \big] \frac{\langle \xi_\nu, a^*_p a_p \xi_\nu \rangle}{\| \xi_\nu \|^2} . \end{align*} $$

With the bounds $\|\gamma _S \|^2_\infty $ , $\|\sigma _S\|^2_\infty $ , $\|\sigma _H\|_\infty ^2$ , $\|\gamma _H\|_\infty ^2 \leq C N^{\varepsilon }$ from Lemma 2.4, with (4.1) and with the estimate $\langle \xi _\nu , \mathcal {N} \xi _\nu \rangle \leq C N^{9\kappa /2 -2+\varepsilon } \| \xi _\nu \|^2$ from Proposition 2.2, we conclude that

(4.2) $$ \begin{align} \frac{\langle \xi_\nu, \mathcal{G}^{(2)}_N \xi_\nu \rangle}{\| \xi_\nu \|^2} \leq \; &\frac{\langle \xi_\nu, \mathcal{K} \xi_\nu \rangle}{\| \xi_\nu \|^2} + \sum_{p \in \Lambda^*_+} p^2 \sigma_p^2 + \frac{N_0}{N} \sum_{p \in \Lambda^*_+} N^\kappa \widehat V(p/N^{1-\kappa}) \gamma_p \sigma_p \nonumber\\ &+ \frac{N_0}{N} \sum_{p \in \Lambda^*_+} N^\kappa \big(\widehat V(0) +\widehat V(p/N^{1-\kappa}) \big) \sigma^2_p + C N^{5\kappa/2-\varepsilon} \end{align} $$

using again the condition $3\kappa - 2 +4\varepsilon < 0$ .

Next, we study the contribution of $\mathcal {G}_N^{(3)} = T_\nu ^* \mathcal {L}^{(3)}_N T_\nu $ , with $\mathcal {L}^{(3)}_N$ as in (3.3). Recall the operator $\mathcal {C}_N$ , defined in (3.5). Taking into account the fact that $\xi _\nu $ is a superposition of vectors with $2m$ particles with momenta in $P_H$ and m particles with momenta in $P_S$ , for $m \in \mathbb {N}$ , we obtain that

$$ \begin{align*} \langle \xi_\nu, \mathcal{G}_N^{(3)} \xi_\nu \rangle = \langle \xi_\nu , \mathcal{C}_N \xi_\nu \rangle + \sum_{j=1}^3 \left[ \langle \xi_\nu, \text{F}_j \xi_\nu \rangle + \text{h.c.} \right] \end{align*} $$

with

$$ \begin{align*} \text{F}_1 &= \frac{\sqrt{N_0}}{N} \sum_{p,r \in P_H : p+r \in P_S} N^\kappa \widehat V(r/N^{1-\kappa}) \gamma_{p+r} \sigma_p \sigma_r \, a^*_{p+r} a^*_{-p} a^*_{-r} \\ \text{F}_2 &= \frac{\sqrt{N_0}}{N} \sum_{p \in P_H , r \in P_S : p+r \in P_H} N^\kappa \left[ \widehat V(r/N^{1-\kappa}) + \widehat{V} (p / N^{1-\kappa}) \right] \gamma_{p+r} \sigma_p \sigma_r \, a^*_{p+r} a^*_{-p} a^*_{-r} \\ \text{F}_3 &= \frac{\sqrt{N_0}}{N} \sum_{p \in P_H, r \in P_S : p+r \in P_H} N^\kappa \left[ \widehat V(r/N^{1-\kappa}) + \widehat V(p /N^{1-\kappa}) \right] \sigma_{p+r} \gamma_p \gamma_r \, a_{-p-r} a_{p} a_{r} . \end{align*} $$

Using $\|a^*_{-r} (\mathcal {N}+1)^{1/2}\xi _\nu \|\leq \|a_{-r} (\mathcal {N}+1)^{1/2}\xi _\nu \| + \|(\mathcal {N}+1)^{1/2}\xi _\nu \|$ , we can bound

$$ \begin{align*} \langle \xi_\nu , \text{F}_1 \xi_\nu \rangle \leq \; &C N^{\kappa -1/2} \| \gamma_S \|_\infty \sum_{p,r \in P_H} |\sigma_r| |\sigma_p | \, \| a_{p+r} a_{-p} (\mathcal{N} + 1)^{-1/2} \xi_\nu \| \\ &\hspace{4cm} \times \left[ \| a_{-r} (\mathcal{N} + 1)^{1/2} \xi_\nu \| + \| (\mathcal{N} + 1)^{1/2} \xi_{\nu} \| \right] \\ \leq \; &C N^{\kappa -1/2} \| \gamma_S \|_\infty \| \sigma_H \|_\infty \| \sigma_H \| \, \| (\mathcal{N} + 1)^{1/2} \xi_\nu \| \| (\mathcal{N}+ 1) \xi_\nu \| \\ &+ C N^{\kappa -1/2} \| \gamma_S \|_\infty \| \sigma_H \|^2 \, \| (\mathcal{N} + 1)^{1/2} \xi_\nu \|^2. \end{align*} $$

With Lemma 2.4 and Proposition 2.2, we obtain

$$ \begin{align*} \frac{\langle \xi_\nu , \text{F}_1 \xi_\nu \rangle}{\| \xi_\nu \|^2} \leq CN^{37\kappa /4 - 4 +5\varepsilon/2} + C N^{17\kappa/2 - 7/2 + 5\varepsilon/2} \leq C N^{5\kappa/2} \cdot N^{21\kappa /4 - 7/2 + 3\varepsilon/2} \end{align*} $$

from the assumption that $3\kappa -2 +4\varepsilon < 0$ . Similarly, we find

$$ \begin{align*} \langle \xi_\nu , \text{F}_2 \xi_\nu \rangle \leq \; & C N^{\kappa -1/2} \| \gamma_H \|_\infty \| \sigma_S \|_\infty \| \sigma_H \| \, \| (\mathcal{N}+ 1)^{1/2} \xi_\nu \| \| (\mathcal{N}+ 1) \xi_\nu \| \\ &+ C N^{\kappa -1/2} \| \gamma_H \|_\infty \| \sigma_S \| \| \sigma_H \| \, \| (\mathcal{N} + 1)^{1/2} \xi_\nu \|^2 \\ \leq \; &C \big[ N^{37\kappa/4-4+3\varepsilon} + N^{31\kappa/4-3+3\varepsilon/2} \big] \| \xi_\nu \|^2 \leq C N^{5\kappa/2} \cdot N^{21\kappa/4 - 3 + 3 \varepsilon /2} \| \xi_\nu \|^2 \end{align*} $$

and also

$$ \begin{align*} \langle \xi_\nu , \text{F}_3 \xi_\nu \rangle \leq \; & C N^{\kappa -1/2} \| \gamma_H \|_\infty \| \gamma_S \|_\infty \| \sigma_H \| \, \| (\mathcal{N}+ 1)^{1/2} \xi_\nu \| \| (\mathcal{N}+ 1) \xi_\nu \| \\ &+ C N^{\kappa -1/2} \| \gamma_H \|_\infty \| \gamma_S \| \| \sigma_H \| \, \| (\mathcal{N} + 1)^{1/2} \xi_\nu \|^2 \\ \leq \; &C N^{5\kappa/2} \cdot N^{21\kappa/4 - 3 + 3 \varepsilon} \| \xi_\nu \|^2 . \end{align*} $$

Summarising, we have

(4.3) $$ \begin{align} \frac{\langle \xi_\nu, \mathcal{G}_N^{(3)} \xi_\nu \rangle}{ \| \xi_\nu \|^2} \leq \frac{\langle \xi_\nu , \mathcal{C}_N \xi_\nu \rangle}{\| \xi_\nu \|^2} + C N^{5\kappa/2} \cdot N^{21\kappa/4 - 3 + 3 \varepsilon} . \end{align} $$

Finally, let us consider $\mathcal {G}_N^{(4)} = T_\nu ^* \mathcal {L}_N^{(4)} T_\nu $ . We decompose $\langle \xi _\nu , \mathcal {G}_N^{(4)} \xi _\nu \rangle = \sum _{j=1}^3 \langle \xi _\nu , \text {G}_j \xi _\nu \rangle $ with

$$ \begin{align*} \text{G}_1 &= \frac 1 {2N} \sum_{\substack{r \in \Lambda^*,\, p,q \in \Lambda^*_+ \\ -r \neq q,p}} N^\kappa \widehat V(r/N^{1-\kappa}) \gamma_p \gamma_q \gamma_{p+r} \gamma_{q+r} a^*_{p+r} a^*_q a_{p} a_{q+r} \\ \text{G}_2 & = \frac 1 {2N} \sum_{\substack{r \in \Lambda^*,\, p,q \in \Lambda^*_+ \\ r \neq q,-p}} N^\kappa \widehat V(r/N^{1-\kappa}) \big( \gamma_{p+r} \sigma_q a^*_{p+r} a_{-q} + \sigma_{p+r} \gamma_q a_{-p-r} a^*_q \big) \\[-0.5cm] & \hskip5.5cm\times \big( \gamma_{p} \sigma_{q+r} a_{p}a^*_{-q-r} + \sigma_{p} \gamma_{q+r} a^*_{-p} a_{q+r}\big)\\[0.2cm] \text{G}_3 & = \frac 1 {2N} \sum_{\substack{r \in \Lambda^*,\, p,q \in \Lambda^*_+ \\ r \neq q,-p}} N^\kappa \widehat V(r/N^{1-\kappa}) \sigma_p \sigma_q \sigma_{p+r} \sigma_{q+r} a_{p+r} a_{q} a^*_{p} a^*_{q+r}. \end{align*} $$

To estimate contributions from $\text {G}_3$ , we arrange terms in normal order. We find

$$ \begin{align*} \text{G}_3 = \; & \frac 1 {2N} \sum_{\substack{r \in \Lambda^*,\, p,q \in \Lambda^*_+ \\ -r \neq q,p}} N^\kappa \widehat V(r/N^{1-\kappa}) \sigma_p \sigma_q \sigma_{p+r} \sigma_{q+r} a^*_{p} a^*_{q+r} a_{p+r} a_{q} \\ &+ \frac 1 {2N} \sum_{\substack{r \in \Lambda^*,\, p \in \Lambda^*_+ \\ p \neq -r}} N^\kappa \widehat V(r/N^{1-\kappa}) \sigma_p^2 \sigma^2_{p+r} \big( a^*_p a_p + a^*_{p+r}a_{p+r}\big) \\ &+ \frac 1 {N} \sum_{\substack{p,q \in \Lambda^*_+ }} N^\kappa \widehat V(0) \sigma_p^2 \sigma^2_{q} a^*_p a_p \\ &+ \frac 1 {2N} \sum_{\substack{r \in \Lambda^*,\, p \in \Lambda^*_+ :\\ p\not = -r}} N^\kappa \widehat V(r/N^{1-\kappa}) \sigma^2_p \sigma^2_{p+r} + \frac 1 {2N} \sum_{\substack{p,q \in \Lambda^*_+}} N^\kappa \widehat V(0) \sigma^2_p \sigma^2_{q} . \end{align*} $$

Since $a_p\,\xi _\nu =0$ if $p\in \Lambda ^*_+\backslash (P_S\cup P_H)$ and $\|\sigma _H\|_\infty \leq \|\sigma _S\|_\infty $ , we find, by Cauchy–Schwarz,

(4.4) $$ \begin{align} \langle \xi_\nu, \text{G}_3 \xi_\nu \rangle &\leq C N^{\kappa -1} \left[ \| \sigma_S \|_\infty^2 \| \sigma \|^2 \| (\mathcal{N}+1) \xi_\nu \|^2 + \| \sigma \|^4 \| \xi_\nu \|^2 \right] \nonumber\\ &\leq C N^{5\kappa/2} \cdot \max \{ N^{-\varepsilon} , N^{9\kappa -5 + 3\varepsilon} \} \| \xi_\nu \|^2 \end{align} $$

using Proposition 2.2 and $3\kappa -2 + 4\varepsilon < 0$ . We proceed similarly for $\text {G}_2$ . Through normal ordering, we get

$$ \begin{align*} \text{G}_2 = \; &\frac 1 {N} \sum_{\substack{r \in \Lambda^*,\, p,q \in \Lambda^*_+ \\ -r \neq q,p}} N^\kappa \widehat V(r/N^{1-\kappa}) \gamma_p \gamma_{p+r}\sigma_q \sigma_{q+r} a^*_{p+r} a^*_{-q-r} a_{p} a_{-q} \\ &+ \frac 1 {N} \sum_{\substack{r \in \Lambda^*,\, p,q \in \Lambda^*_+ \\ -r \neq q,p}} N^\kappa \widehat V(r/N^{1-\kappa}) \gamma_{p+r} \gamma_{q+r}\sigma_p \sigma_{q} a^*_{p+r} a^*_{-p} a_{-q} a_{q+r} \\ &+ \frac 1 {N} \sum_{\substack{p,q \in \Lambda^*_+ }} N^\kappa \widehat V(0) \gamma_p^2 \sigma^2_q a^*_p a_p + \frac 1 {N} \sum_{\substack{r \in \Lambda^*,\,p \in \Lambda^*_+ }} N^\kappa \widehat V(r/N^{1-\kappa}) \gamma_p^2 \sigma^2_{p+r} a^*_p a_p \\ &+ \frac 2 {N} \sum_{\substack{r \in \Lambda^*,\,p \in \Lambda^*_+ }} N^\kappa \widehat V(r/N^{1-\kappa}) \gamma_p \sigma_p \gamma_{p+r} \sigma_{p+r} a^*_p a_p \\ &+ \frac 1 {2N} \sum_{\substack{p,r \in \Lambda^*_+}} N^\kappa \widehat V(r/N^{1-\kappa}) \gamma_p \gamma_{p+r} \sigma_p \sigma_{p+r} . \end{align*} $$

Keeping the last contribution intact and estimating the term on the fourth line distinguishing the two cases $(p+r) \in P_S$ and $(p+r) \in P_H$ , we arrive at

$$ \begin{align*} \langle \xi_\nu , \text{G}_2 \xi_\nu \rangle \leq \; &\frac 1 {2N} \sum_{\substack{p,r \in \Lambda^*_+}} N^\kappa \widehat V(r/N^{1-\kappa}) \gamma_p \gamma_{p+r} \sigma_p \sigma_{p+r} \| \xi_\nu \|^2 \\ &+ C N^{\kappa -1} \| \gamma_S \|_\infty^2 \| \sigma \|^2 \| (\mathcal{N}+1) \xi_\nu \|^2 \\ &+ C N^{\kappa -1} \| \gamma_{S \cup H} \|_\infty \| \sigma_{S \cup H} \|_\infty \\ & \hspace{2cm} \times \left[ \| \gamma_S \sigma_S \|_1 + \| \gamma_H \|_\infty \mathop{\mathrm{sup}}\limits_p \sum_{r \in \Lambda^*} \widehat{V} (r/ N^{1-\kappa}) |\eta_{p+r}| \right] \| \mathcal{N}^{1/2} \xi_\nu \|^2 . \end{align*} $$

With the bounds in Lemma 2.4 and in Proposition 2.2 and with (3.12), we conclude that

(4.5) $$ \begin{align} \langle \xi_\nu , \text{G}_2 \xi_\nu \rangle \leq \frac 1 {2N} \sum_{\substack{p,r \in \Lambda^*_+}} N^\kappa \widehat V(r/N^{1-\kappa}) \gamma_p \gamma_{p+r} \sigma_p \sigma_{p+r} \| \xi_\nu \|^2 + C N^{5\kappa /2} \cdot N^{9\kappa -5 + 3\varepsilon} \| \xi_\nu \|^2. \end{align} $$

Finally, we consider $\text {G}_1$ . Recalling that $a_p \xi _\nu = 0$ if $p \in \Lambda ^*_+ \backslash (P_S \cup P_H)$ and observing that $\langle \xi _\nu , a^*_{p+r}a^*_qa_pa_{q+r}\xi _\nu \rangle \neq 0$ only if the operator $a^*_{p+r}a^*_q a_p a_{q+r}$ preserves the number of particles in $P_S$ and in $P_H$ , we arrive at

$$ \begin{align*} \langle \xi_\nu, \text{G}_1 \xi_\nu \rangle \leq \; &\frac 1 {2N} \sum_{\substack{r \in \Lambda^*,\, p,q \in P_H: \\p+r,q+r\in P_H}} N^\kappa \widehat V(r/N^{1-\kappa}) \gamma_p \gamma_q \gamma_{p+r} \gamma_{q-r} \langle \xi_\nu , a^*_{p+r} a^*_q a_{p} a_{q+r} \xi_\nu \rangle \\ &+ C N^{\kappa -1 } \| \gamma_{S \cup H} \|_\infty^2 \| \gamma_S \|^2 \| (\mathcal{N} + 1) \xi \|^2. \end{align*} $$

With $|\gamma _p \gamma _q \gamma _{p+r} \gamma _{q+r} - 1| \leq C \| \eta _H \|_\infty ^2$ for all $p,q \in P_H$ , with $(p+r), (q+r) \in P_H$ and using the estimate (see the proof of (3.12))

$$ \begin{align*} \mathop{\mathrm{sup}}\limits_{p\in \Lambda^*} \sum_{r \in \Lambda^*_+ : r \not = p} \frac{N^\kappa |\widehat{V} (r/N^{1-\kappa})|}{|p-r|^2} \leq C N \end{align*} $$

we conclude that

$$ \begin{align*} \langle \xi_\nu, \text{G}_1 \xi_\nu \rangle \leq \langle \xi_\nu, \mathcal{V}_N^{(H)} \xi_\nu \rangle + C \| \eta_H \|_\infty^2 \| \mathcal{N}^{1/2} \mathcal{K}^{1/2} \xi_\nu \|^2 + C N^{\kappa -1 } \| \gamma_{S \cup H} \|_\infty^2 \| \gamma_S \|^2 \| \mathcal{N} \xi \|^2 \end{align*} $$

with $\mathcal {V}_N^{(H)}$ defined as in (3.4). With Lemma 2.4 and Proposition 2.2, we find (using the assumption $3\kappa - 2 + 4 \varepsilon < 0$ )

$$ \begin{align*} \langle \xi_\nu, \text{G}_1 \xi_\nu \rangle \leq \langle \xi_\nu, \mathcal{V}_N^{(H)} \xi_\nu \rangle + C N^{5\kappa/2} \cdot N^{9 \kappa - 5 + 6\varepsilon} \| \xi_\nu \|^2. \end{align*} $$

With (4.4) and (4.5), we have shown that

$$ \begin{align*} \frac{\langle \xi_\nu, \mathcal{G}_N^{(4)} \xi_\nu \rangle}{\| \xi_\nu \|^2} \leq \frac{\langle \xi_\nu, \mathcal{V}_N^{(H)} \xi_\nu \rangle}{\| \xi_\nu \|^2} + C N^{5\kappa/2} \cdot \max \{ N^{-\varepsilon}, N^{9\kappa - 5 + 6 \varepsilon} \}. \end{align*} $$

Combining the last bound with (4.2) and (4.3), we obtain

$$ \begin{align*} \frac{\langle \xi_\nu, \mathcal{G}_N \xi_\nu \rangle}{\| \xi_\nu \|^2} \leq \; &\widetilde{C}_N + \frac{\langle \xi_\nu, (\mathcal{K} +\mathcal{V}_N^{(H)} + \mathcal{C}_N) \xi_\nu \rangle}{\| \xi_\nu \|^2} \\ &+ C N^{5\kappa/2} \cdot \max \{ N^{-\varepsilon}, N^{9\kappa - 5 + 6\varepsilon} , N^{21 \kappa /4 - 3 + 3 \varepsilon } \} \end{align*} $$

where we defined

(4.6) $$ \begin{align} \widetilde{C}_{N} =\; & \frac{N_0^2 }{2N} N^\kappa \widehat{V}(0) +\sum_{p\in {\Lambda}^*_+}p^2\sigma_p^2 + \frac{N_0}{N}\sum_{p\in \Lambda^*_+} N^\kappa \big( \widehat{V}(p/N^{1-\kappa}) + \widehat{V}(0) \big)\sigma_p^2\nonumber\\ & + \frac{N_0}{N}\sum_{p\in \Lambda^*_+}N^\kappa \widehat{V}(p/N^{1-\kappa})\sigma_p\gamma_p +\frac{1}{2N}\sum_{\substack{p,r\in \Lambda^*_+\\r\neq p}}N^\kappa \widehat{V}(r/N^{1-\kappa})\sigma_p\sigma_{p+r}\gamma_p\gamma_{p+r}. \end{align} $$

Inserting $N_0 = N - \| \sigma _L \|^2$ and recalling from Lemma 2.4 that $\| \sigma _L \|^2 \leq C N^{3\kappa /2}$ and $\| \sigma _{L^c} \|^2 \leq C N^{3\kappa /2 - \varepsilon }$ , we obtain $\widetilde {C}_N = C_{\mathcal {G}_N} + \mathcal {O} (N^{5\kappa /2 - \varepsilon })$ , with $C_{\mathcal {G}_N}$ as defined in (3.6) (with the assumption $3\kappa - 2 + 4\varepsilon < 0$ ). To handle the first term on the second line of (4.6), we used that $|\sigma _p \gamma _p - \eta _p| \leq C \eta _p^3 \leq C N^{3\kappa }/|p|^6$ , for $p \in P_L^c$ . This completes the proof of Proposition 3.1.

5 Cubic conjugation

In this section we prove Proposition 2.2 and Proposition 3.2, as a consequence of the following lemma.

Lemma 5.1. Let $A_\nu $ be defined in (2.15) and $\mathcal {K}$ , $\mathcal {V}_N^{(H)}$ and $\mathcal {C}_N$ be defined in (3.4) and (3.5), respectively. Then, for $\xi _\nu =e^{A_\nu }\Omega $ ,

(5.1) $$ \begin{align} \frac{\langle \xi_\nu, \mathcal{K} \xi_\nu \rangle}{\| \xi_\nu \|^2} \leq\; & \frac 2 {N} \sum_{\substack{v \in P_S, r \in P_H:\\r+v \in P_H}} r^2\eta_r (\eta_r+\eta_{r+v})\sigma_v^2 + \mathcal{E} , \end{align} $$

(5.2) $$ \begin{align} \frac{\langle \xi_\nu, \mathcal{C}_N \xi_\nu \rangle}{\| \xi_\nu \|^2} \leq \; & \frac 2 {N} \sum_{\substack{v \in P_S, r \in P_H :\\ r+v \in P_H}} N^\kappa \widehat V(r/N^{1-\kappa})(\eta_r +\eta_{r+v}) \sigma_v^2 + \mathcal{E} , \end{align} $$

(5.3) $$ \begin{align} \frac{\langle \xi_\nu, \mathcal{V}_N^{(H)} \xi_\nu \rangle}{\| \xi_\nu \|^2} \leq \;& \frac 1 {N^2} \sum_{\substack{v \in P_S, r \in P_H: \\ r+v \in P_H}} \, \big( N^\kappa \widehat V(\cdot/N^{1-\kappa}) \ast \eta \big)_r (\eta_r+\eta_{r+v}) \sigma_v^2 + \mathcal{E} , \end{align} $$

with

$$ \begin{align*} \mathcal{E} \leq C N^{5\kappa/2} \cdot \max \{ N^{-\varepsilon},\, N^{12\kappa-7 +5\varepsilon} \} \end{align*} $$

for all $\kappa \in (1/2 ;2/3)$ , $\varepsilon>0$ so small that $3\kappa - 2 + 4 \varepsilon < 0$ and N large enough.

With Lemma 5.1, we can immediately show Proposition 3.2.

Proof Proof of Proposition 3.2.

From Lemma 5.1 we have

$$ \begin{align*} &\frac{\langle \xi_\nu, (\mathcal{K} + \mathcal{V}_N^{(H)} + \mathcal{C}_{N} ) \xi_\nu \rangle}{\| \xi_\nu \|^2} \\ & \leq \frac 2 N \sum_{v \in P_S} \sigma_v^2 \sum_{\substack{r \in P_H: \\ r+v \in P_H}}\Big[ r^2 \eta_r + N^\kappa \widehat V(r/N^{1-\kappa}) + \frac {N^\kappa} {2N} \big(\widehat V(\cdot/N^{1-\kappa}) \ast \eta \big)_r \Big] (\eta_r+\eta_{r+v}) + \mathcal{E} \end{align*} $$

with $\mathcal {E} \leq CN^{5\kappa /2} \cdot \max \{ N^{-\varepsilon }, N^{12\kappa -7 + 5\varepsilon } \}$ . With the scattering equation (2.9), we obtain

$$ \begin{align*} \frac{\langle \xi_\nu, (\mathcal{K} + \mathcal{V}_N^{(H)} + \mathcal{C}_{N} ) \xi_\nu \rangle}{\| \xi_\nu \|^2} \leq \frac1N\sum_{v\in P_S}\sigma_v^2\sum_{\substack{r \in P_H: \\r+v\in P_H}} N^\kappa \widehat V(r/N^{1-\kappa})(\eta_r+\eta_{r+v})+ \mathcal{E}' \end{align*} $$

with

$$ \begin{align*} \mathcal{E}' \leq \; \frac 1 N \sum_{v \in P_S} \sigma_v^2 \sum_{\substack{r \in P_H:\, r+v \in P_H}} N^{3-2\kappa} \lambda_{\ell} (\widehat \chi_\ell \ast \widehat f_{N,\ell})_r \eta_r + \mathcal{E} . \end{align*} $$

Using $|N^{3-3\kappa }\lambda _{\ell }|\leq C$ and $\|\widehat {\chi }_\ell \ast \widehat f_{N,\ell }\|\leq C$ , we conclude

$$ \begin{align*} &\frac{\langle \xi_\nu, (\mathcal{K} + \mathcal{V}_N^{(H)} + \mathcal{C}_{N} ) \xi_\nu \rangle}{\| \xi_\nu \|^2} \\ &\hspace{.3cm} \leq \frac1N\sum_{v\in P_S}\sigma_v^2\sum_{\substack{r \in P_H: \\r+v\in P_H}} N^\kappa \widehat V(r/N^{1-\kappa})(\eta_r+\eta_{r+v}) + C N^{5\kappa/2} \cdot \max \{ N^{-\varepsilon}, N^{12\kappa -7 + 5\varepsilon} \}. \end{align*} $$

Finally, with (3.12) and the expression (2.28) for $\sigma ^2_v$ , we can extend the sum over $v \in P_S$ to a sum over all $v \in P_L$ , without changing the size of the error. This completes the proof of Proposition 3.2.

We still have to show Proposition 2.2 and Lemma 5.1.

5.1 Expectation of the particle number and kinetic energy

In this section we prove (5.1) and Proposition 2.2. We start by computing the expectation $\langle \xi _\nu , \mathcal {K} \xi _\nu \rangle $ . We proceed as we did in (2.19)–(2.22) to compute $\| \xi _\nu \|^2$ . With $\mathcal {K} a_{r+v}^* a_{-r}^* a_{-v}^* = a_{r+v}^* a_{-r}^* a_{-v}^* (\mathcal {K} + (r+v)^2 + r^2 + v^2)$ we obtain

$$ \begin{align*} {\bigl\langle\xi_{\nu},\mathcal{K} \xi_{\nu}\bigr\rangle} = &\; \sum_{m\geq 1}\frac1{2^m (m-1)!}\frac1{N^m} \sum_{\substack{ v_1 \in P_S, r_1 \in P_H : \\ r_1 +v_1 \in P_H } } \cdots \sum_{\substack{ v_m \in P_S, r_m \in P_H : \\ r_m +v_m \in P_H } } \theta \big( \{ r_j, v_j \}_{j=1}^{m} \big) \\&\hspace{3cm} \times [r_m^2+v_m^2+(r_m+v_m)^2]\, \prod_{i=1}^m (\eta_{r_i} + \eta_{r_i + v_i})^2 \sigma_{v_i}^2 \, \end{align*} $$

with the cutoff $\theta $ introduced in (2.18). Since all terms are positive, we can find an upper bound for $\langle \xi _\nu , \mathcal {K} \xi _\nu \rangle $ by replacing $\theta (\{r_j,v_j\}_{j=1}^m)$ with $\theta (\{r_j,v_j\}_{j=1}^{m-1})$ , removing conditions involving momenta with index m. Recalling (2.22), we find

$$ \begin{align*} {\bigl\langle\xi_{\nu},\mathcal{K} \xi_{\nu}\bigr\rangle} &\leq \frac{1}{2N} \sum_{\substack{v\in P_S,r\in P_H:\\r+v\in P_H}}[r^2+v^2+(r+v)^2] (\eta_r + \eta_{r+v})^2 \sigma_v^2 \,\|\xi_\nu\|^2 \\ &\leq \frac{2}{N} \sum_{v \in P_S , r\in P_H} r^2 \eta_r(\eta_r + \eta_{r+v}) \sigma_v^2 \,\|\xi_\nu\|^2 + \mathcal{E} \end{align*} $$

with (using Lemma 2.4 and the assumption $3\kappa -2 +4\varepsilon < 0$ )

$$ \begin{align*} \frac{\mathcal{E}}{\|\xi_\nu\|^2} =& \frac2N \sum_{\substack{v\in P_S,r\in P_H : \\ r+v \in P_H}} \hspace{-.3cm} (v ^2+r\cdot v) \eta_r (\eta_r+\eta_{r+v})\sigma_v^2 \\&\leq \frac C N( \| \sigma_S \|_{H^1}^2 \| \eta_H \|^2 +\|\sigma_S\|\|\eta_H\|\|\sigma_S\|_{H^1}\|\eta_H\|_{H^1})\leq C N^{4\kappa -1+\varepsilon}\leq C N^{5\kappa/2-\varepsilon}. \end{align*} $$

This proves (5.1). In particular, (5.1) implies, together with Lemma 2.4, that

(5.4) $$ \begin{align} \frac{\langle \xi_\nu , \mathcal{K} \xi_\nu \rangle}{\| \xi_\nu \|^2} \leq C N^{-1} \| \eta_H \|_{H^1}^2 \| \sigma_S \|^2 \leq C N^{5\kappa /2}, \end{align} $$

which shows (2.24) with $j=1$ in Proposition 2.2.

Analogously, we find

$$ \begin{align*} \langle \xi_\nu, \mathcal{K} \mathcal{N} \xi_\nu \rangle & \leq \sum_{m\geq 1}\frac{3m}{2^m (m-1)!}\frac1{N^m} \sum_{\substack{ v_1 \in P_S, r_1 \in P_H : \\ r_1 +v_1 \in P_H } } \cdots \sum_{\substack{ v_m \in P_S, r_m \in P_H : \\ r_m +v_m \in P_H } } \theta \big( \{ r_j, v_j \}_{j=1}^{m} \big) \\&\hspace{3cm} \times [r_m^2+v_m^2+(r_m+v_m)^2]\, \prod_{i=1}^m (\eta_{r_i} + \eta_{r_i + v_i})^2 \sigma_{v_i}^2 . \end{align*} $$

Writing $m = 1+ (m-1)$ and bounding $\theta ( \{ r_j, v_j \}_{j=1}^{m} )$ by $\theta ( \{ r_j, v_j \}_{j=1}^{m-2} )$ , we obtain

$$ \begin{align*} \langle \xi_\nu, \mathcal{K} \mathcal{N} \xi_\nu \rangle \leq \; & 3 \langle \xi_\nu , \mathcal{K} \xi_\nu \rangle + \frac{3}{4N^2} \sum_{r,r' \in P_H, v,v' \in P_S} [ r^2 + v^2 + (r+v)^2] \\ &\hspace{5cm} \times (\eta_r + \eta_{r+v})^2 (\eta_{r'} + \eta_{r' + v'})^2 \sigma_v^2 \sigma_{v'}^2 \| \xi_\nu \|^2. \end{align*} $$

With (5.4) and with the bounds for $\| \eta _H \|_{H^1}^2, \| \eta _H \|^2, \| \sigma _S \|^2$ from Lemma 2.4, we find

(5.5) $$ \begin{align} \frac{\langle \xi_\nu, \mathcal{K} \mathcal{N} \xi_\nu \rangle}{\| \xi_\nu \|^2} \leq C N^{5\kappa /2} \cdot N^{9\kappa/2 - 2 + \varepsilon}, \end{align} $$

which shows (2.24) with $j=2$ .

To show (2.23) we observe that, by (2.15), the operator $A_\nu $ only creates particles with momenta in $P_S \cup P_H$ and for each particle with momentum in $P_S$ , it creates two particles with momenta in $P_H$ . Since $|p|> N^{1-\kappa -\varepsilon }$ for all $p \in P_H$ , we find, by (5.4),

$$ \begin{align*} \langle \xi_\nu, \mathcal{N} \xi_\nu \rangle &= \sum_{p \in P_S \cup P_H} \langle \xi_\nu, a^*_p a_p \xi_\nu \rangle = \frac{3}{2} \sum_{p \in P_H} \langle \xi_\nu , a^*_p a_p \xi_\nu \rangle \\ &\leq C N^{-2+2\kappa+2\varepsilon} \langle \xi_\nu, \mathcal{K} \xi_\nu \rangle \leq N^{9\kappa/2 - 2 + 2\varepsilon} \| \xi_\nu \|^2, \end{align*} $$

proving (2.23) for $j=1$ . Analogously, we find

$$ \begin{align*} \langle \xi_\nu, \mathcal{N}^2 \xi_\nu \rangle &= \; \sum_{p\in P_S\cup P_H} \langle \mathcal{N}^{1/2}\xi_\nu, a^*_p a_p \mathcal{N}^{1/2}\xi_\nu \rangle \\ &\leq \; \frac32 \sum_{p\in P_H} \langle \mathcal{N}^{1/2}\xi_\nu, a^*_p a_p \mathcal{N}^{1/2} \xi_\nu \rangle \leq C N^{-2+2\kappa+2\varepsilon} \langle \xi_\nu, \mathcal{K} \mathcal{N} \xi_\nu \rangle . \end{align*} $$

By (5.5), we obtain (2.23) with $j=2$ . This completes the proof of Proposition 2.2.

5.2 Expectation of the cubic term

The goal of this section is to show (5.2). From (3.5), we have (using the reality of $\eta _p, \gamma _p, \sigma _p$ )

$$ \begin{align*} &\langle\xi_\nu,\mathcal{C}_N\xi_\nu\rangle\\ &=2\frac{\sqrt{N_0}}{ N}\sum_{m\geq 1}\frac1{m!(m-1)!}\sum_{\substack{p,r \in P_H\\ p+r \in P_S}} N^\kappa \widehat V(r/N^{1-\kappa})\, \sigma_{p+r} \gamma_p\gamma_r \langle A_\nu^{m} \xi_\nu, \,a^*_{p+r} a^*_{-p} a^*_{-r} A_\nu^{m-1}\xi_\nu\rangle. \end{align*} $$

Proceeding as in the previous section, we get

$$ \begin{align*}\begin{aligned} & {\bigl\langle\xi_{\nu},\mathcal{C}_N\xi_{\nu}\bigr\rangle}\\ &= 2 \sqrt{\frac{N_0}{N}}\,\sum_{m\geq 1}\frac1{2^{m-1} (m-1)!}\frac1{N^m}\sum_{\substack{ v_1 \in P_S, r_1 \in P_H : \\ r_1 +v_1 \in P_H } } \cdots \sum_{\substack{ v_m \in P_S, r_m \in P_H : \\ r_m +v_m \in P_H } } \hskip -0.5cm \theta\big( \{r_j, v_j \}_{j=1}^{m} \big) \\ & \hskip 0.5cm \times N^\kappa \widehat V(r_m/N^{1-\kappa})\,\big(\eta_{r_m} +\eta_{r_m+v_m}\big)\gamma_{r_m}\gamma_{r_m+v_m} \sigma^2_{v_m}\, \prod_{i=1}^{m-1} (\eta_{r_i} + \eta_{r_i + v_i})^2 \sigma_{v_i}^2. \end{aligned}\\[-17pt]\end{align*} $$

To reconstruct the norm $\| \xi _\nu \|^2$ on the right-hand side, we need to free the momenta with index m. To this end, we recall the defintion (2.18) to write

(5.6) $$ \begin{align} \theta\big( \{r_j, v_j \}_{j=1}^{m} \big) = \theta\big( \{r_j, v_j \}_{j=1}^{m-1} \big)\, \theta_m\big( \{ r_j, v_j \}_{j=1}^{m} \big)\\[-17pt]\nonumber \end{align} $$

with

$$ \begin{align*} \theta_m\big( \{r_j, v_j \}_{j=1}^{m} \big)= \prod_{i,j=1}^{m-1}\prod_{\substack{p_i,p_j,p_m:\\p_\ell\in\{-r_\ell,r_\ell+v_\ell\}}}\delta_{p_i\neq -p_j+v_m}\delta_{-p_m+v_i\neq p_j} \\[-17pt]\end{align*} $$

collecting all conditions involving $\{r_m, v_m\}$ . Writing $\theta _m = 1 + [ \theta _m - 1]$ , we split $\langle \xi _\nu , \mathcal {C}_N \xi _\nu \rangle = I_{\mathcal {C}} + J_{\mathcal {C}}$ with (recall the expression (2.22) for $\| \xi _\nu \|^2$ )

$$ \begin{align*} I_{\mathcal{C}} = \; 2\,\sqrt{\frac{N_0}{N}}\,\sum_{\substack{ v \in P_S,\, r \in P_H: \\r+v \in P_H}} N^{\kappa-1}\widehat V(r/N^{1-\kappa})\,\big(\eta_{r}+\eta_{r+v}\big)\gamma_{r}\gamma_{r+v} \sigma^2_{v}\, \| \xi_\nu\|^2 \\[-17pt]\end{align*} $$

and

$$ \begin{align*}\begin{aligned} J_{\mathcal{C}} =\; & 2\sqrt{\frac{N_0}{N}}\sum_{m\geq 1}\frac1{2^{m-1} (m-1)!}\frac1{N^m} \sum_{\substack{ v_1 \in P_S, r_1 \in P_H : \\ r_1 +v_1 \in P_H } } \cdots \sum_{\substack{ v_m \in P_S, r_m \in P_H : \\ r_m +v_m \in P_H } } \theta\big( \{r_j, v_j \}_{j=1}^{m-1} \big) \\ & \times \Big[ \theta_m\big( \{r_j, v_j \}_{j=1}^{m} \big) -1 \Big] N^\kappa\widehat V(r_m/N^{1-\kappa})\,\big(\eta_{r_m} +\eta_{r_m+v_m}\big)\gamma_{r_m}\gamma_{r_m+v_m} \sigma^2_{v_m} \\ &\hspace{8cm} \times \prod_{i=1}^{m-1} (\eta_{r_i} + \eta_{r_i + v_i})^2 \sigma_{v_i}^2 . \end{aligned}\end{align*} $$

With $|\sqrt {N_0/N} - 1| \leq C \| \sigma _L \|^2/N$ and $|\gamma _r \gamma _{r+v} - 1| \leq C N^{2\kappa } / |r|^4$ for all $r \in P_H, v\in P_S$ , we obtain (using (3.12) and the assumption $3\kappa - 2 + 4\varepsilon < 0$ ) that

(5.7) $$ \begin{align} \frac{I_{\mathcal{C}}}{\| \xi_\nu \|^2} \leq \frac{2}{N} \sum_{\substack{ v \in P_S,\, r \in P_H: \\r+v \in P_H}} N^{\kappa}\widehat V(r/N^{1-\kappa})\,\big(\eta_{r}+\eta_{r+v}\big) \sigma^2_{v} + C N^{5\kappa /2 - \varepsilon}.\\[-17pt]\nonumber \end{align} $$

To complete the proof of (5.2), we focus now on the error term $J_{\mathcal {C}}$ . We observe that

(5.8) $$ \begin{align} | \theta_m\big( \{r_j, v_j \}_{j=1}^{m} \big) -1 | \leq& \sum_{j=1}^{m-1} \Big[ \delta_{v_j , v_m} + \sum_{ \substack{ p_m \in \{ -r_m, r_m+v_m \} \\p_j \in \{ -r_j, r_j+v_j\} }}\delta_{p_m, p_j} \Big] \nonumber\\ & + \sum_{\substack{j,k=1\\j\neq k}}^{m-1} \Big[ \sum_{ \substack{ p_j \in \{ -r_j, r_j+v_j \} \\ p_k \in \{ -r_k, r_k+v_k\} }} \delta_{v_m, p_j+p_k} + \sum_{\substack{ p_m \in \{ -r_m, r_m+v_m \} \\ p_j \in \{ -r_j, r_j+v_j \} }}\delta_{p_m, -p_j+v_k} \Big] . \end{align} $$

We bound $|J_{\mathcal {C}}| \leq \text {X}_1+ \text {X}_2$ , with $\text {X}_1$ denoting the contribution arising from the first term on the right-hand side of (5.8) (this term involves two indices, m and j) and $\text {X}_2$ indicating the contribution from the second term on the right-hand side of (5.8) (this term involves three indices, $m,j,k$ ). We can estimate

$$ \begin{align*} \text{X}_1 \leq \; &C \sum_{m\geq 2}\frac1{2^{m-2} (m-2)!}\frac1{N^m}\sum_{\substack{ v_1 \in P_S, r_1 \in P_H : \\ r_1 +v_1 \in P_H } } \cdots \sum_{\substack{ v_m \in P_S, r_m \in P_H : \\ r_m +v_m \in P_H } } \theta\big( \{r_j, v_j \}_{j=1}^{m-1} \big) \\ &\times N^\kappa |\widehat V(r_m/N^{1-\kappa})| \,\big| \eta_{r_m} +\eta_{r_m+v_m}\big| |\gamma_{r_m}| |\gamma_{r_m+v_m}| \sigma^2_{v_m} \prod_{i=1}^{m-1} (\eta_{r_i} + \eta_{r_i + v_i})^2 \sigma_{v_i}^2 \\ & \times \Big[ \delta_{v_m, v_{m-1}} +\sum_{\substack{p_{m-1},p_m:\\p_\ell\in \{-r_\ell,r_\ell+v_\ell\}}}\hspace{-0.3cm}\delta_{p_m, p_{m-1}}\Big] . \end{align*} $$

With $\theta \big ( \{r_j, v_j \}_{j=1}^{m-1} \big ) \leq \theta \big ( \{r_j, v_j \}_{j=1}^{m-2} \big )$ , we reconstruct $\| \xi _\nu \|^2$ . Since $\| \gamma _H \|_\infty \leq C$ , we end up with

$$ \begin{align*} \frac{\text{X}_1}{\| \xi_\nu \|^2} \leq\; & \frac{C}{N^2} \sum_{r , r' \in P_H, v,v' \in P_S} N^\kappa |\widehat{V} (r/N^{1-\kappa}) |\eta_r + \eta_{r+v}| |\eta_{r'} + \eta_{r'+v'}|^2 \sigma_v^2 \sigma_{v'}^2 \\ &\hspace{6cm} \times \Big[ \delta_{v,v'} + \sum_{\substack{p \in \{-r, r+v\} \\ p' \in \{ -r', r' +v' \}}} \delta_{p,p'} \Big] \\ \leq \; &C N^{2\kappa-2} \| \sigma_S \|_\infty^2 \| \sigma_S \|^2 \| \eta_H \|^2 \sum_{r \in P_H} \frac{|\widehat{V} (r/N^{1-\kappa})|}{r^2} + C N^{4\kappa -2} \| \sigma_S \|^4 \sum_{r \in P_H} |r|^{-6} \\ \leq \; &C N^{11\kappa/2-2+2\varepsilon} + C N^{10\kappa-5+3\varepsilon} \leq C N^{5\kappa/2-\varepsilon}, \end{align*} $$

where we used Lemma 2.4, (3.12), the assumption $3\kappa -2 + 4\varepsilon < 0$ and the remark that $|\eta _{r+v}| \leq CN^\kappa |r|^{-2}$ , for all $r \in P_H$ and $v \in P_S$ . We can proceed similarly to estimate $X_2$ . In the second term on the right-hand side of (5.8), we have to sum over $(m-1) (m-2)/2$ pairs of indices $j,k$ . With $\theta \big ( \{r_j, v_j \}_{j=1}^{m-1} \big ) \leq \theta \big ( \{r_j, v_j \}_{j=1}^{m-3} \big )$ and again with Lemma 2.4 and(3.12), we arrive at

$$ \begin{align*} \frac{\text{X}_2}{\| \xi_\nu \|^2} \leq\; & \frac{C}{N^3} \sum_{\substack{r , r' , r'' \in P_H, \\ v,v', v'' \in P_S}} N^\kappa |\widehat{V} (r/N^{1-\kappa}) |\eta_r + \eta_{r+v}| |\eta_{r'} + \eta_{r'+v'}|^2 |\eta_{r''} + \eta_{r''+v''}|^2 \sigma_v^2 \sigma_{v'}^2 \sigma_{v''}^2 \\ &\hspace{4.5cm} \times \Big[ \sum_{ \substack{p \in \{ -r , r+v \}, \\ p' \in \{ -r', r' + v'\}}} \delta_{p,-p'+v''} + \sum_{\substack{p' \in \{-r', r'+v'\}, \\ p'' \in \{-r'', r'' + v''\}}} \delta_{v,p'+p''} \Big] \\ \leq \; &C N^{4\kappa - 3} \| \sigma_S \|^6 \| \eta_H \|^2 \sum_{r \in P_H} |r|^{-6} + C N^{6\kappa -3} \| \sigma_S \|^6 \sum_{r \in P_H} \frac{|\widehat{V} (r/N^{1-\kappa})|}{r^2} \sum_{r' \in P_H} |r'|^{-8} \\ \leq \; &C N^{29\kappa/2 -7 + 5\varepsilon} \leq C N^{5\kappa/2} \cdot N^{12 \kappa -7 + 5\varepsilon} . \end{align*} $$

Thus, $|J_{\mathcal {C}}| / \| \xi _\nu \|^2 \leq N^{5\kappa /2} \cdot \max \{ N^{-\varepsilon } , N^{12\kappa -7 + 5\varepsilon } \}$ . With (5.7), this implies (5.2).

5.3 Expectation of the quartic term

In this section we show the bound (5.3) for the expectation of $\mathcal {V}_N^{(H)}$ . Pairing momenta in $P_S$ , as we did in (2.20) and in the previous subsections, we obtain

(5.9) $$ \begin{align} \begin{aligned} {\bigl\langle\xi_\nu,\mathcal{V}_N^{(H)} \xi_\nu\bigr\rangle}=&\frac1{2N}\sum_{m\geq 1}\frac1{m!}\frac1{N^m} \sum_{\substack{v_1 \in P_S,\, r_1, {\tilde{r}}_1 \in P_H:\\ r_1+v_1,\, {\tilde{r}}_1 + v_1 \in P_H}} \cdots\sum_{\substack{v_m \in P_S,\, r_m, {\tilde{r}}_m \in P_H:\\ r_m+v_m,\, {\tilde{r}}_m + v_m \in P_H}} \\ &\times \theta\big( \{r_j, v_j \}_{j=1}^{m} \big) \theta\big( \{{\tilde{r}}_j, v_j \}_{j=1}^{m} \big) \prod_{i=1}^m\eta_{r_i}\eta_{\tilde{r}_i}\sigma_{v_i}^2\hspace{-0.3cm}\sum_{\substack{r\in \Lambda^*, p,q \in P_H: \\p+r,\,q+r\in P_H}}\hspace{-0.2cm} N^\kappa \widehat V(r/N^{1-\kappa})\\ &\times{\bigl\langle\Omega,A_{r_1,v_1}\dots A_{r_m,v_m}\,a^*_{p+r}a^*_{q} a_p a_{q+r} A^*_{\tilde{r}_1,v_1}\dots A^*_{\tilde{r}_m,v_m}\Omega\bigr\rangle} \end{aligned}\end{align} $$

where we use the notation $A_{r_i, v_i} = a_{r_i+v_i} a_{-r_i}$ that was already introduced in (2.20). Next we observe that because of the cutoffs $\theta (\{r_j,v_j\}_{j=1}^m)$ and $\theta (\{{\tilde {r}}_j,v_j\}_{j=1}^m)$ , at most two indices $i,j \in \{ 1, \dots , m \}$ can be involved in contractions with the observable $a_{p+r}^* a_q^* a_p a_{q+r}$ . We distinguish two possible cases:

1. 1) There exists an index $i \in \{ 1,\dots ,m \}$ such that $a_p$ , $a_{q+r}$ are contracted with $A^*_{{\tilde {r}}_i,v_i}$ and $a^*_{q}$ , $a^*_{p+r}$ are contracted with $A_{r_i,v_i}$ .

2. 2) There are two indices $i \neq j \in \{ 1, \dots , m \}$ such that the operators $a_p$ and $a_{q+r}$ are contracted with $a^*_{{\tilde {p}}_i}$ and $a^*_{{\tilde {p}}_j}$ for some ${\tilde {p}}_\ell \in \{-{\tilde {r}}_\ell , {\tilde {r}}_\ell + v_\ell \}, \ell =i,j$ and the operators $a^*_q, a^*_{p+r}$ are contracted with $a_{p_i}, a_{p_j}$ , with $p_\ell \in \{-r_\ell ,r_\ell +v_\ell \}, \ell =i,j$ . Note that in this case the operators $a^*_{-{\tilde {p}}_i+v_i}, a^*_{-{\tilde {p}}_j+v_j}$ have to be contracted with $a_{-p_i+v_i}, a_{-p_j+v_j}.$

We denote by $\text {V}_1$ and $\text {V}_2$ the contributions to ${\bigl \langle \xi _\nu ,\mathcal {V}_N^{(H)} \xi _\nu \bigr \rangle }$ arising from the two cases described above. Let us first consider $\text {V}_1$ . There are m choices (all leading to the same contribution) for the index $i \in \{ 1, \dots , m \}$ labelling momenta to be contracted with the observable. Let us fix $i = m$ . Then we have $p={\tilde {p}}_m, q+r=-{\tilde {p}}_m+v_m$ with ${\tilde {p}}_m \in \{-{\tilde {r}}_m,{\tilde {r}}_m +v_m \}$ and $p+r=p_m, q=-p_m+v_m$ with $p_m\in \{-r_m,r_m+v_m\}$ . Note that the choice of p and $p+r$ also determines q and $q+r$ , since we always have $q = v_m - (p+r)$ . The presence of the cutoffs immediately implies that $A_{r_j,v_j}$ is fully contracted with $A^*_{{\tilde {r}}_j , v_j}$ , for all $j \not = m$ . We find

(5.10) $$ \begin{align} &\langle \xi_{\nu}, \text{V}_1 \xi_{\nu} \rangle \nonumber\\ &=\frac1{2N}\sum_{m\geq 1}\frac1{(m-1)!}\frac1{N^m} \;\sum_{\substack{v_1 \in P_S, r_1, {\tilde{r}}_1 \in P_H: \nonumber\\ r_1 + v_1 ,\, {\tilde{r}}_1 + v_1 \in P_H}} \cdots \sum_{\substack{v_{m} \in P_S, r_{m}, {\tilde{r}}_{m} \in P_H: \\ r_{m} + v_{m} ,\, {\tilde{r}}_{m} + v_{m} \in P_H}}\hskip -0.5cm\theta\big( \{r_j, v_j \}_{j=1}^{m} \big) \theta\big( \{{\tilde{r}}_j, v_j \}_{j=1}^{m} \big) \nonumber\\[6pt] &\, \times \hskip -0.1cm \prod_{j=1}^{m-1}\hskip -0.1cm\eta_{r_j}\eta_{{\tilde{r}}_j}(\delta_{r_j,{\tilde{r}}_j}+\delta_{-r_j,{\tilde{r}}_j+v_j})\sigma_{v_j}^2\, \eta_{r_m} \eta_{{\tilde{r}}_m}\sigma_{v_m}^2\hspace{-.5cm} \sum_{\substack{r\in \Lambda^*,\,p\in P_H:\\p-v_m,p+r\in P_H}} \hskip -0.7cm N^\kappa\widehat{V}(r/N^{1-\kappa}) \hskip -0.5cm \sum_{\substack{p_m\in \{-r_m,r_m+v_m\}\\{\tilde{p}}_m\in \{-{\tilde{r}}_m,{\tilde{r}}_m+v_m\}}}\hspace{-0.8cm}\delta_{p,p_m}\delta_{p+r,{\tilde{p}}_m} . \end{align} $$

Since here (in contrast to the previous subsections) the contraction does not fix ${\tilde {r}}_m$ to be either $r_m$ or $-(r_m + v_m)$ , we cannot erase the cutoff $\theta (\{ {\tilde {r}}_j , v_j \}_{j=1}^m)$ . With the decomposition (5.6), we can replace, on the right-hand side of (5.10),

$$ \begin{align*} \theta\big( \{r_j, v_j \}_{j=1}^{m} \big) \theta\big( \{{\tilde{r}}_j, v_j \}_{j=1}^{m} \big) = \theta \big( \{r_j, v_j \}_{j=1}^{m-1} \big) \theta_m \big( \{ r_j, v_j \}_{j=1}^{m} \big) \theta_m \big( \{{\tilde{r}}_j, v_j \}_{j=1}^{m} \big). \end{align*} $$

Writing

$$ \begin{align*} \theta_m \big( \{{\tilde{r}}_j, v_j \}_{j=1}^{m} \big) \theta_m \big( \{{\tilde{r}}_j, v_j \}_{j=1}^{m} \big) = 1 + \Big[ \theta_m \big( \{ r_j, v_j \}_{j=1}^{m} \big) \theta_m \big( \{{\tilde{r}}_j, v_j \}_{j=1}^{m} \big) - 1 \Big] \end{align*} $$

we split (as we did in the last subsection) $\langle \xi _\nu , \text {V}_1 \xi _\nu \rangle = I_{\mathcal {V}} + J_{\mathcal {V}}$ , with

(5.11) $$ \begin{align} I_{\mathcal{V}} = \frac 1 {N^2} \sum_{r \in \Lambda^*} \sum_{\substack{ v\in P_S, p \in P_H:\\ p+r,p-v,\, p+r-v \in P_H}} \hskip -0.5cm N^\kappa \widehat V(r/N^{1-\kappa}) \eta_{p} \big( \eta_{p+r} +\eta_{p+r-v} \big) \sigma_{v}^2 \, \| \xi_\nu \|^2 \end{align} $$

and

(5.12) $$ \begin{align} J_{\mathcal{V}} =&\frac1{2N}\sum_{m\geq 1}\frac1{2^{m-1} (m-1)!}\frac1{N^m}\sum_{\substack{v_1 \in P_S, r_1 \in P_H: \\ r_1 + v_1 \in P_H}} \cdots \sum_{\substack{v_{m-1} \in P_S, r_{m-1} \in P_H: \\ r_{m-1} + v_{m-1} \in P_H}} \sum_{\substack{v_{m} \in P_S , r_{m}, {\tilde{r}}_{m} \in P_H: \\ r_{m} + v_{m} ,\, {\tilde{r}}_{m} + v_{m} \in P_H}} \nonumber\\ & \times \theta\big( \{r_j, v_j \}_{j=1}^{m-1} \big) \big[ \theta_m(\{r_j, v_j\}_{j=1}^{m})\theta_m(\{ r^\sharp_j ,v_j\}_{j=1}^m)-1\big] \prod_{i=1}^{m-1} (\eta_{r_i} + \eta_{r_i + v_i})^2 \sigma_{v_i}^2 \, \nonumber\\ &\times \eta_{r_m} \eta_{{\tilde{r}}_m} \sigma_{v_m}^2 \sum_{\substack{r\in \Lambda^*}} N^\kappa \widehat V(r/N^{1-\kappa})\hspace{-0.5cm} \sum_{\substack{ p \in P_H :\, p+r,\\ p-v_m,\, p+r-v_m\in P_H}} \sum_{\substack{p_m\in \{-r_m,r_m+v_m\}\\{\tilde{p}}_m\in \{-{\tilde{r}}_m,{\tilde{r}}_m+v_m\}}}\hspace{-0.3cm}\delta_{p,p_m}\delta_{p+r,{\tilde{p}}_m} \end{align} $$

where $r^\sharp _j = r_j$ for $j=1,\dots , m-1$ and $r^\sharp _m = {\tilde {r}}_m$ in the argument of $\theta _m$ . Observing that, with Lemma 2.4 and (3.12),

$$ \begin{align*} \frac 1 {N^2} \sum_{r \in \Lambda^*} & \sum_{\substack{v \in P_S, p \in \Lambda^* : \\ p\in P_H^c \text{ or } p-v \in P_H^c}} N^\kappa |\widehat V(r/N^{1-\kappa})| |\eta_{p}| \big( |\eta_{p+r}| + |\eta_{p+r-v}| \big) \sigma_{v}^2 \\ &\leq C N^{-2+2\kappa} \| \sigma_S \|^2 \Big[ \sum_{|p| \leq N^{1-\kappa -\varepsilon}} |p|^{-2} \Big] \mathop{\mathrm{sup}}\limits_{p\in \Lambda^*} \sum_{r \in \Lambda^*} |\widehat V(r/N^{1-\kappa})| |\eta_{p+r}| \leq C N^{5\kappa/2 - \varepsilon}, \end{align*} $$

we conclude from (5.11) (switching $p+r \to p$ and $v \to -v$ ) that

(5.13) $$ \begin{align} \frac{I_{\mathcal{V}}}{\| \xi_\nu \|^2} \leq \frac 1 {N^2} \sum_{\substack{v \in P_S, p \in P_H: \\ p+v \in P_H}} \, \big( N^\kappa \widehat V(\cdot/N^{1-\kappa}) \ast \eta \big)_p (\eta_p +\eta_{p+v}) \sigma_v^2 + C N^{5\kappa/2-\varepsilon}. \end{align} $$

Let us now focus on the term $J_{\mathcal {V}}$ . With

(5.14) $$ \begin{align} \Big| \theta_m \big( &\{r_j, v_j \}_{j=1}^{m} \big) \theta_m\big( \{{\tilde{r}}_j, v_j \}_{j=1}^{m} \big) -1 \Big| \nonumber\\ \leq \; &\sum_{j=1}^{m-1} \delta_{v_m, v_j} + \sum_{j=1}^{m-1} \Big[ \sum_{\substack{p_j \in \{ - r_j, r_j + v_j \} \\ p_m \in \{ - r_m , r_m + v_m \}}} \delta_{p_m, p_j} + \sum_{\substack{p_j \in \{ - r_j, r_j + v_j \} \\ {\tilde{p}}_m \in \{ - {\tilde{r}}_m , {\tilde{r}}_m + v_m \}}} \delta_{{\tilde{p}}_m, p_j} \Big] \nonumber \\ &+ \sum_{\substack{j,k=1 \\ j \not = k}}^{m-1} \Big[ \sum_{\substack{p_j \in \{ - r_j, r_j + v_j \} \\ p_m \in \{ - r_m , r_m + v_m \}}} \delta_{p_m, -p_j + v_k} + \sum_{\substack{p_j \in \{ - r_j, r_j + v_j \} \\ {\tilde{p}}_m \in \{ - {\tilde{r}}_m , {\tilde{r}}_m + v_m \}}} \delta_{{\tilde{p}}_m, -p_j + v_k} \Big]\nonumber \\ &+ \sum_{\substack{j,k=1 \\ j \not = k}}^{m-1} \sum_{\substack{p_j \in \{ - r_j, r_j + v_j \} \\ p_k \in \{ - r_k , r_k + v_k \}}} \delta_{v_m , p_j + p_k} \end{align} $$

we can bound $|J_{\mathcal {V}}| \leq \text {W}_1 + \text {W}_2 + \text {W}_3 + \text {W}_4$ , with $W_\ell $ indicating the contribution to (5.12) arising from the $\ell $ th term on the right-hand side of (5.14).

The term $\text {W}_1$ contains the sum of $(m-1)$ identical contributions, corresponding to $j \in \{ 1, \dots , m-1 \}$ in the first term on the right-hand side of (5.14). Let us fix $j=m-1$ . Estimating $\theta \big ( \{r_j, v_j \}_{j=1}^{m-1} \big ) \leq \theta \big ( \{r_j, v_j \}_{j=1}^{m-2} \big )$ and reconstructing the expression (2.22) for $\| \xi _\nu \|^2$ , we can bound (the momenta $r',r'',\tilde {r}''$ correspond to $r_{m-1}, r_m, \tilde {r}_m$ )

$$ \begin{align*} \frac{\text{W}_1}{\| \xi_\nu \|^2} \leq C N^{-3} \sum_{r \in \Lambda^*} N^\kappa |\widehat{V} (r/N^{1-\kappa})| \hspace{-.3cm} \sum_{\substack{r', r'', {\tilde{r}}'' \in P_H \\ v' \in P_S}} \hspace{-.3cm} (\eta_{r'} + \eta_{r'+v'})^2 |\eta_{r''}| |\eta_{{\tilde{r}}''}| \sigma_{v'}^4 \hspace{-.4cm} \sum_{\substack{p'' \in \{ -r'', r''+v' \} \\ {\tilde{p}}'' \in \{ -{\tilde{r}}'', {\tilde{r}}'' +v' \}}} \hspace{-.5cm} \delta_{p''+r ,{\tilde{p}}''}. \end{align*} $$

With Lemma 2.4 and with the estimate

(5.15) $$ \begin{align} \mathop{\mathrm{sup}}\limits_{v \in P_S \cup \{ 0 \}} \frac{1}{N^2} \sum_{\substack{r \in \Lambda^*, q \in P_H : \\ q-r \in P_H}} N^\kappa |\widehat{V} (r/N^{1-\kappa})| |\eta_{q-v} | |\eta_{q-r}| \leq C N^{\kappa}, \end{align} $$

which can be shown similarly to (3.12) (using $V \in L^q (\mathbb {R}^3)$ , for some $q> 3/2$ ), we find

(5.16) $$ \begin{align} \frac{\text{W}_1}{\| \xi_\nu \|^2} \leq C N^{\kappa -1} \| \eta_H \|^2 \| \sigma_S \|_\infty^2 \| \sigma_S \|^2 \leq C N^{11\kappa/2 -2 + 2\varepsilon} \leq C N^{5\kappa/2-\varepsilon} \end{align} $$

since $3\kappa - 2 + 4\varepsilon < 0$ . Analogously, we bound, with (3.12) and Lemma 2.4,

(5.17) $$ \begin{align} \frac{\text{W}_2}{\| \xi_\nu \|^2} \leq \; &C N^{-3} \sum_{r \in \Lambda^*} N^\kappa |\widehat{V} (r/N^{1-\kappa})| \hspace{-.3cm} \sum_{\substack{r', r'', {\tilde{r}}'' \in P_H \\ v', v'' \in P_S}} \hspace{-.3cm} (\eta_{r'} + \eta_{r'+v'})^2 |\eta_{r''}| |\eta_{{\tilde{r}}''}| \sigma_{v'}^2 \sigma_{v''}^2 \hspace{-.3cm}\nonumber\\ &\times \sum_{\substack{p'' \in \{ -r'', r''+v'' \} \\ {\tilde{p}}'' \in \{ -{\tilde{r}}'', {\tilde{r}}'' +v'' \}}} \hspace{-.3cm} \delta_{p'' + r , {\tilde{p}}''} \Big[ \sum_{\substack{p' \in \{ -r', r'+v' \} \\ p'' \in \{ -r'', r'' +v'' \}}} \delta_{p', p''} + \sum_{\substack{p' \in \{ -r', r'+v' \} \\ {\tilde{p}}'' \in \{ -{\tilde{r}}'', {\tilde{r}}'' +v'' \}}} \delta_{p', {\tilde{p}}''} \Big] \nonumber\\ \leq \; &C N^{-3+5\kappa} \| \sigma_S \|^4 \Big[\sum_{r' \in P_H} |r'|^{-6}\Big] \Big[ \mathop{\mathrm{sup}}\limits_{r' \in \Lambda^*} \sum_{r \in \Lambda^* , r \not = -r'} \frac{|\widehat{V} (r/N^{1-\kappa})|}{|r+r'|^{2}} \Big] \leq C N^{5\kappa/2-\varepsilon}. \end{align} $$

As for $\text {W}_3$ , there are $(m-1)(m-2)$ possible choices of the indices $j,k$ in (5.14), all leading to the same contribution. We fix $j = m-1$ and $k =m-2$ . Estimating now $\theta \big ( \{r_j, v_j \}_{j=1}^{m-1} \big ) \leq \theta \big ( \{r_j, v_j \}_{j=1}^{m-3} \big )$ , we obtain, with (3.12),

(5.18) $$ \begin{align} \frac{\text{W}_3}{\| \xi_\nu \|^2} \leq \; &C N^{-4} \sum_{r \in \Lambda^*} N^\kappa |\widehat{V} (r/N^{1-\kappa})| \nonumber\\ &\times \sum_{\substack{r', r'', r''', {\tilde{r}}''' \in P_H \nonumber\\ v', v'', v''' \in P_S}} \hspace{-.3cm} (\eta_{r'} + \eta_{r'+v'})^2 (\eta_{r''} + \eta_{r''+v''})^2 |\eta_{r'''}| |\eta_{{\tilde{r}}'''}| \sigma_{v'}^2 \sigma_{v''}^2 \sigma_{v'''}^2 \hspace{-.5cm}\nonumber\\ &\times \sum_{\substack{p''' \in \{ -r''', r'''+ v''' \} \nonumber\\ {\tilde{p}}''' \in \{ -{\tilde{r}}''' , {\tilde{r}}''' + v''' \}}} \hspace{-.5cm} \delta_{p'''+r , {\tilde{p}}'''} \Big[ \sum_{\substack{p' \in \{ -r', r'+v' \}\nonumber \\ p''' \in \{ -r''', r''' +v''' \}}} \hspace{-.5cm} \delta_{p''', -p'+v''} + \sum_{\substack{p' \in \{ -r', r'+v' \}\nonumber \\ {\tilde{p}}''' \in \{ -{\tilde{r}}''', {\tilde{r}}''' +v''' \}}} \hspace{-.3cm} \delta_{{\tilde{p}}''' , - p' + v''} \Big] \\ \leq \; &C N^{-3+ 4\kappa} \| \sigma_S \|^6 \| \eta_H \|^2 \sum_{r' \in P_H} |r'|^{-6} \leq C N^{5\kappa/2} \cdot N^{12\kappa - 7 +4\varepsilon}. \end{align} $$

Analogously, with Lemma 2.4 and (5.15), we find

$$ \begin{align*} \frac{\text{W}_4}{\| \xi_\nu \|^2} \leq \; &C N^{-4} \sum_{r \in \Lambda^*} N^\kappa |\widehat{V} (r/N^{1-\kappa})| \\ &\times \sum_{\substack{r', r'', r''', {\tilde{r}}''' \in P_H \\ v', v'', v''' \in P_S}} \hspace{-.3cm} (\eta_{r'} + \eta_{r'+v'})^2 (\eta_{r''} + \eta_{r''+v''})^2 |\eta_{r'''}| |\eta_{{\tilde{r}}'''} | \sigma_{v'}^2 \sigma_{v''}^2 \sigma_{v'''}^2 \hspace{-.5cm}\\ &\times \sum_{\substack{p''' \in \{ -r''', r'''+ v''' \} \\ {\tilde{p}}''' \in \{ -{\tilde{r}}''' , {\tilde{r}}'''+ v''' \}}} \hspace{-.5cm} \delta_{p''' + r , {\tilde{p}}''' } \sum_{\substack{p' \in \{ -r', r'+v' \} \\ p'' \in \{ -r'', r'' +v'' \}}} \hspace{-.5cm} \delta_{v''', p' +p''} \\ \leq \; &C N^{-2+5\kappa}\| \sigma_S \|^6 \sum_{r' \in P_H} |r'|^{-8} \leq C N^{5\kappa/2} \cdot N^{12\kappa -7 + 5\varepsilon}. \end{align*} $$

Together with (5.16), (5.17), (5.18), we conclude that

(5.19) $$ \begin{align} | J_{\mathcal{V}} | \leq C N^{5\kappa/2} \cdot \max \{ N^{-\varepsilon} , N^{12\kappa - 7 + 5\varepsilon} \}. \end{align} $$

Finally, we consider the term $\text {V}_2$ , associated with the second case listed after (5.9). We fix $i =m$ and $j=m-1$ and we consider all possible contractions of $a_p$ with $a^*_{{\tilde {p}}_m}$ , of $a_{q+r}$ with $a^*_{{\tilde {p}}_{m-1}}$ and of $a_q^*, a_{p+r}^*$ with $a_{p_m}, a_{p_{m-1}}$ , where ${\tilde {p}}_\ell \in \{-{\tilde {r}}_\ell ,{\tilde {r}}_\ell +v_\ell \}$ and $p_\ell \in \{ -r_\ell , r_\ell + v_\ell \}$ , for $\ell =m,m-1$ . We obtain

$$ \begin{align*}\begin{aligned} &{\bigl\langle\xi_\nu,\text{V}_2\, \xi_\nu\bigr\rangle} \\ & = \frac{1}{2N} \sum_{m\geq 2}\frac1{(m-2)!}\frac1{N^m} \sum_{\substack{v_1 \in P_S, r_1, {\tilde{r}}_1 \in P_H: \\ r_1 + v_1 ,\, {\tilde{r}}_1 + v_1 \in P_H}} \cdots \sum_{\substack{v_{m} \in P_S, r_{m}, {\tilde{r}}_{m} \in P_H: \\ r_{m} + v_{m} ,\, {\tilde{r}}_{m} + v_{m} \in P_H}}\hskip -0.5cm\theta\big( \{r_j, v_j \}_{j=1}^{m} \big) \theta\big( \{{\tilde{r}}_j, v_j \}_{j=1}^{m} \big) \\ & \hskip 0.5cm \times \prod_{i=1}^{m-2}\eta_{r_i}\eta_{\tilde{r}_i} \big( \delta_{{\tilde{r}}_i,r_i}+\delta_{-{\tilde{r}}_i,r_i+v_i}\big)\sigma_{v_i}^2 \, \prod_{j=m,m-1} \eta_{r_j} \eta_{{\tilde{r}}_j} \sigma_{v_j}^2\\ &\hspace{0.5cm}\times \sum_{\substack{r \in \Lambda^*,\, p,q \in P_H: \\ p-r, q-r \in P_H }} N^\kappa \widehat V(r/N^{1-\kappa}) \sum_{\substack{{\tilde{p}}_\ell \in \{ -{\tilde{r}}_\ell, {\tilde{r}}_\ell+v_\ell\} \\ \ell=m-1,m } } \delta_{p, {\tilde{p}}_m} \delta_{q+r, {\tilde{p}}_{m-1}} \\ & \hspace{0.5cm} \times \hskip -0.5cm \sum_{\substack{ p_\ell \in \{ - r_\ell, r_\ell+v_\ell\} \\ \ell=m-1,m } } \big(\delta_{q, p_m} \delta_{p+r, p_{m-1}} + \delta_{q, p_{m-1}} \delta_{p+r, p_{m}} \big) \big( \delta_{{\tilde{p}}_m, p_m} + \delta_{-{\tilde{p}}_m+v_m, -p_{m-1}+v_{m-1}} \big). \end{aligned}\end{align*} $$

Estimating $\theta \big ( \{r_j, v_j \}_{j=1}^{m} \big ) \theta \big ( \{{\tilde {r}}_j, v_j \}_{j=1}^{m} \big ) \leq \theta \big ( \{r_j, v_j \}_{j=1}^{m-2} \big )$ and using Lemma 2.4 and the condition $3\kappa - 2 + 4\varepsilon < 0$ , we find

$$ \begin{align*}\begin{aligned} \frac{|{\bigl\langle\xi_\nu,\text{V}_2\, \xi_\nu\bigr\rangle}|}{\| \xi_\nu \|^2} \\ \leq \; & C N^{-3} \sum_{r \in \Lambda^*} N^\kappa \widehat{V} (r/N^{1-\kappa}) \sum_{\substack{r',{\tilde{r}}',r'', {\tilde{r}}'' \in P_H \\ v',v'' \in P_S}} |\eta_{r'}| |\eta_{{\tilde{r}}'}| |\eta_{r''}| |\eta_{{\tilde{r}}''}| \sigma_{v'}^2 \sigma_{v''}^2 \\ &\times \sum_{\substack{p' \in \{ -r' , r' + v' \} \\ p'' \in \{ -r'' , r'' + v'' \}}} \sum_{\substack{{\tilde{p}}' \in \{ -{\tilde{r}}' , {\tilde{r}}' + v' \} \\ {\tilde{p}}'' \in \{ -{\tilde{r}}'' , {\tilde{r}}'' + v'' \} }} (\delta_{{\tilde{p}}', p''+r} \delta_{{\tilde{p}}'' + r, p'} + \delta_{{\tilde{p}}', p'+r} \delta_{{\tilde{p}}'' + r, p''}) \\ &\hspace{6cm} \times (\delta_{{\tilde{p}}'', p''} + \delta_{-{\tilde{p}}'' + v'', -p'+v'}) \\ \leq \; &C N^{-3+\kappa} \| \eta_H \|^4 \| \sigma_S \|^4 \leq C N^{10\kappa - 5 + 2\varepsilon} \leq C N^{5\kappa/2 -\varepsilon}. \end{aligned} \end{align*} $$

With (5.13) and (5.19), we obtain (5.3).