# A Short Proof of Bose-Einstein Condensation in the Gross-Pitaevskii Regime
and Beyond

Christian Brennecke Institute for Applied Mathematics, University of Bonn,
Endenicher Allee 60, 53115 Bonn, Germany Morris Brooks Institute for
Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich,
Switzerland Cristina Caraci22footnotemark: 2 Jakob Oldenburg22footnotemark: 2

###### Abstract

We consider dilute Bose gases on the three dimensional unit torus that
interact through a pair potential with scattering length of order
$N^{\kappa-1}$, for some $\kappa>0$. For the range
$\kappa\in[0,\frac{1}{43})$, [1] proves complete BEC of low energy states into
the zero momentum mode based on a unitary renormalization through operator
exponentials that are quartic in creation and annihilation operators. In this
paper, we give a new and self-contained proof of BEC of the ground state for
$\kappa\in[0,\frac{1}{20})$ by combining some of the key ideas of [1] with the
novel diagonalization approach introduced recently in [16], which is based on
the Schur complement formula. In particular, our proof avoids the use of
operator exponentials and is significantly simpler than [1].

##  1 Introduction and Main Result

We consider $N$ interacting bosons in
$\Lambda:=\mathbb{T}^{3}=\mathbb{R}^{3}/\mathbb{Z}^{3}$ with Hamiltonian

$H_{N}=\sum_{i=1}^{N}-\Delta_{x_{i}}+\sum_{1\leq i<j\leq
N}N^{2-2\kappa}V(N^{1-\kappa}(x_{i}-x_{j})),$ (1)

acting in $L^{2}_{s}(\Lambda^{N})$, the Hilbert space consisting of functions
in $L^{2}(\Lambda^{N})$ that are invariant with respect to permutations of the
$N$ particles. We assume the interaction potential $V\in
L^{1}(\mathbb{R}^{3})$ to have compact support, to be radial and to be
pointwise non-negative.

Note that analyzing $H_{N}$ is equivalent to analyzing the Hamiltonian of $N$
bosons interacting through the unscaled potential $V$ in
$\mathbb{R}^{3}/L\mathbb{Z}^{3}$ for $L=N^{1-\kappa}$. In this sense, we
consider regimes of strongly diluted systems of bosons with number of
particles density $N^{3\kappa-2}\ll 1$ (as long as $\kappa<\frac{2}{3}$). The
case $\kappa=0$ corresponds to the Gross-Pitaevskii (GP) regime and the case
$\kappa=\frac{2}{3}$ corresponds to the usual thermodynamic limit (with number
of particles density equal to one).

In this paper, we are interested in understanding low energy properties of the
Bose gas in regimes that interpolate between the GP and thermodynamic limits.
Based on [20, 31], it is well-known that the ground state energy
$E_{N}:=\inf\text{spec}(H_{N})$ is equal to

$E_{N}=4\pi\mathfrak{a}N^{1+\kappa}+o(N^{1+\kappa}),$

where $\mathfrak{a}$ denotes the scattering length of the potential $V$ and
where $o(N^{1+\kappa})$ denotes an error of subleading order, that is
$\lim_{N\to\infty}o(N^{1+\kappa})/N^{1+\kappa}=0$. Recall that under our
assumptions the scattering length of $V$ is characterized by

$\begin{split}8\pi\mathfrak{a}&=\inf\bigg{\\{}\int_{\mathbb{R}^{3}}dx\,\Big{(}2|\nabla
f(x)|^{2}+V(x)|f(x)|^{2}\Big{)}:\lim_{|x|\to\infty}f(x)=1\bigg{\\}}.\end{split}$

A question closely related to the computation of the ground state energy is
whether the ground state exhibits Bose-Einstein condensation (BEC). If
$\psi_{N}$ denotes the ground state vector, this means that the largest
eigenvalue of the associated reduced one particle density matrix
$\gamma_{N}^{(1)}=\text{tr}_{2,\dots,N}|\psi_{N}\rangle\langle\psi_{N}|$
remains of size one in the limit $N\to\infty$:

$\liminf_{N\to\infty}\|\gamma_{N}^{(1)}\|_{\text{op}}>0.$

Proving BEC in the thermodynamic limit is a difficult open problem in
mathematical physics. For strongly diluted systems, on the other hand, there
has recently been great progress in proving that low energy states exhibit
BEC. The first proof of BEC has been obtained in [27] in the GP regime111It is
worth to point out that the arguments of [27], which build on energy bounds
from [31, 30], can in fact be used to prove BEC in the parameter range
$\kappa\in[0,\frac{1}{10})$; see also [29, Chapter 7]., implying that for
$\varphi_{0}:=1_{|\Lambda}\in L^{2}(\Lambda)$ one has that

$\lim_{N\to\infty}\langle\varphi_{0},\gamma_{N}^{(1)}\varphi_{0}\rangle=1.$
(2)

This result has later been extended to approximate ground states in [28, 34]
and the works [5, 8] have proved (2) with the optimal rate of convergence.
Since then, several generalizations and simplified proofs have been obtained
in [32, 1, 21, 25, 13, 17, 9, 33]. Notice that such results can be used to
derive the low energy excitation spectrum of $H_{N}$ in accordance with
Bogoliubov theory [10], see e.g. [7, 8, 35, 14, 26, 11, 18, 2, 16, 19].

In recent years, progress has also been made in regimes that interpolate
between the GP and thermodynamic limits. Based on unitary renormalizations
developed first in the dynamical context [4, 12] and in the context of the
derivation of the excitation spectrum in the GP regime [6, 7], the work [1]
proves BEC for approximate ground states in regimes
$\kappa\in[0,\frac{1}{43})$. A different method, that is based on box
localization arguments, has been introduced in [21] which proves BEC in the
larger parameter range $\kappa\in[0,\frac{2}{5}+\epsilon)$, for some
sufficiently small $\epsilon>0$. This result represents currently the best
available parameter range and it is closely tied to the computation of the
second order correction to the ground state energy, which turns out to be of
order $N^{5\kappa/2}$ [36, 23, 3, 24].

The methods introduced in [1] and [21] have both certain advantages. While
[21] obtains the currently best parameter range and applies to a large class
of potentials including hard-core interactions, it is based on box
localization arguments and therefore involves the change of boundary
conditions222To be more precise, the localization procedure of [21] replaces
the standard Laplacian in the periodic setting by a more involved localized
kinetic energy operator, see [21, Eq. (2.7)]. For a recent overview that
focuses on the key steps of the energy bounds in the simpler translation
invariant setting, see [22].. This makes the derivation of suitable lower
bounds more complicated, compared to the translation invariant setting, and
essentially restricts the method to obtaining lower bounds while upper bounds
require separate tools. The method of [1], on the other hand, does not require
localization and enables both upper and lower bounds at the same time.
However, it only applies to soft potentials satisfying some mild integrability
assumption. Moreover, controlling the error terms in the operator expansions
quickly becomes rather challenging and this is among the main reasons why the
method only works in a much more restricted parameter range.

In this paper, our goal is to revisit the strategy of [1]. However, instead of
renormalizing the system through unitary conjugations by quartic operator
exponentials, we proceed as in [16] whose renormalization is based on the
Schur complement formula applied to the two body problem and on lifting it in
a suitable sense to the $N$ body setting. As a consequence, our proof becomes
significantly simpler and shorter compared to the one in [1]. Although our
results are still only valid in a small parameter range compared to [21], our
arguments are elementary, self-contained and do neither require box
localization methods nor operator exponential expansions.

######  Theorem 1.

Let $H_{N}$ be defined as in (1) for $\kappa\in[0,\frac{1}{20})$ and denote by
$\gamma_{N}^{(1)}$ the one particle reduced density associated to its
normalized ground state vector $\psi_{N}$. Then

$\lim_{N\to\infty}\langle\varphi_{0},\gamma_{N}^{(1)}\varphi_{0}\rangle=1.$

Remarks:

  1. 1)

Theorem 1 applies to the ground state vector $\psi_{N}$ of $H_{N}$. With some
additional effort that involves the use of number of particles localization
arguments, we expect that our results could also be proved for approximate
ground states $\phi_{N}$ that satisfy
$\langle\phi_{N},H_{N}\phi_{N}\rangle\leq 4\pi\mathfrak{a}N^{1+\kappa}+o(N)$.
To keep our arguments as short and simple as possible, we omit the details and
focus on the ground state vector $\psi_{N}$.

  2. 2)

In our proof of Theorem 1, we assume the relatively mild a priori information
that the ground state energy $E_{N}$ is bounded from above by $E_{N}\leq
4\pi\mathfrak{a}N^{1+\kappa}+o(N)$, if $\kappa<\frac{1}{20}$. Based on ideas
similar to those presented below, this could be proved with little additional
effort in a self-contained way. Since this has already been explained in [16]
(which obtains a more precise upper bound on $E_{N}$ for all
$\kappa<\frac{2}{13}$ based on the evaluation of the energy of suitable trial
states, see [16, Theorem 3]), however, we refer the interested reader to [16]
for the details.

##  2 Proof of Theorem 1

In the following, let us denote by $a_{k}$ and $a^{*}_{k}$ the annihilation
and, respectively, creation operators associated with the plane waves
$x\mapsto\varphi_{k}(x):=e^{ikx}\in L^{2}(\Lambda)$ of momentum $k$, for
$k\in\Lambda^{*}:=2\pi\mathbb{Z}^{3}$. They satisfy the canonical commutation
relations $[a_{p},a_{q}^{*}]=\delta_{p,q}$ and
$[a_{p},a_{q}]=[a_{p}^{*},a_{q}^{*}]=0$, and they can be used to express
$H_{N}$ as

$H_{N}=\sum_{r\in\Lambda^{*}}|r|^{2}a^{*}_{r}a_{r}+\frac{N^{\kappa}}{2N}\sum_{p,q,r\in\Lambda^{*}}\widehat{V}(r/N^{1-\kappa})a^{*}_{p+r}a^{*}_{q-r}a_{p}a_{q},$

where $\widehat{V}(r)=\int_{\mathbb{R}^{3}}dx\,e^{-irx}V(x)$ denotes the
standard Fourier transform of $V$.

Now, denote by $V_{N}$ the two body operator that multiplies by
$N^{2-2\kappa}V(N^{1-\kappa}(x_{1}-x_{2}))$ in $L^{2}(\Lambda^{2})$ and define
for $\alpha\in[0,1-\kappa]$ the low momentum set

$\text{P}_{\text{L}}:=\big{\\{}p\in\Lambda^{*}:|p|\leq N^{\alpha}\big{\\}}.$
(3)

Denote, moreover, by $\Pi_{\text{L}}:L^{2}(\Lambda^{2})\to L^{2}(\Lambda^{2})$
the orthogonal projection onto

$\overline{\text{span}(\varphi_{k}\otimes\varphi_{l}:k,l\in\text{P}_{\text{L}})}$

and set $\Pi_{\text{H}}:=1-\Pi_{\text{L}}$. Then, as explained in detail in
[16], a straightforward application of the Schur complement formula implies
the many body lower bound

$\begin{split}H_{N}\geq\sum_{r\in\Lambda_{+}^{*}}|r|^{2}c^{*}_{r}c_{r}+\frac{N^{\kappa}}{2N}\\!\\!\\!\\!\sum_{\begin{subarray}{c}p,q,r\in\Lambda^{*}:\\\
p,q,p+r,q-r\in\text{P}_{\text{L}}\end{subarray}}\\!\\!\\!\\!\\!\langle\varphi_{p+r}\otimes\varphi_{q-r},V_{\text{ren}}\varphi_{p}\otimes\varphi_{q}\rangle
a^{*}_{p+r}a^{*}_{q-r}a_{p}a_{q}-R_{N},\end{split}$ (4)

where we set $\Lambda_{+}^{*}:=\Lambda^{*}\setminus\\{0\\}$ as well as

$\begin{split}c_{r}&:=a_{r}+\frac{N^{\kappa}}{N}\sum_{\begin{subarray}{c}(p,q)\in\text{P}_{\text{L}}^{2}\end{subarray}}\langle\varphi_{p+q-r}\otimes\varphi_{r},\eta\,\varphi_{p}\otimes\varphi_{q}\rangle
a^{*}_{p+q-r}a_{p}a_{q},\\\
\eta&:=N^{1-\kappa}\,\Pi_{\text{H}}\big{[}\Pi_{\text{H}}(-\Delta_{x_{1}}-\Delta_{x_{2}}+V_{N})\Pi_{\text{H}}\big{]}^{-1}\Pi_{\text{H}}V_{N}\Pi_{\text{L}},\\\
V_{\text{ren}}&:=N^{1-\kappa}\big{(}V_{N}-V_{N}\Pi_{\text{H}}\big{[}\Pi_{\text{H}}(-\Delta_{x_{1}}-\Delta_{x_{2}}+V_{N})\Pi_{\text{H}}\big{]}^{-1}\Pi_{\text{H}}V_{N}\big{)},\end{split}$
(5)

and where the three body error term $R_{N}$ is given by

$\begin{split}R_{N}&:=\frac{N^{2\kappa}}{N^{2}}\\!\\!\\!\sum_{\begin{subarray}{c}r,p,q,s,t\in\Lambda^{*}\end{subarray}}\\!\\!\\!|r|^{2}\langle\eta\,\varphi_{p}\otimes\varphi_{q},\varphi_{p+q-r}\otimes\varphi_{r}\rangle\langle\varphi_{s+t-r}\otimes\varphi_{r},\eta\,\varphi_{s}\otimes\varphi_{t}\rangle\\\
&\hskip 85.35826pt\times
a_{p}^{*}a^{*}_{q}a^{*}_{s+t-r}a_{p+q-r}a_{s}a_{t}.\end{split}$ (6)

Notice that we used that both $\eta$ and $V_{\text{ren}}$ preserve the total
momentum in $L^{2}(\Lambda^{2})$.

Let us briefly comment on the main ideas leading to (4). Viewing
$V_{N}=\Pi_{\text{L}}V_{N}\Pi_{\text{L}}+(\Pi_{\text{H}}V_{N}\Pi_{\text{L}}+\text{h.c.})+\Pi_{\text{H}}V_{N}\Pi_{\text{H}}$
and hence the Hamiltonian
$\mathcal{H}_{2}:=-\Delta_{x_{1}}-\Delta_{x_{2}}+V_{N}$ of the two body
problem as a block matrix, one can block-diagonalize the latter using the
Schur complement formula. This renormalizes the low-momentum interaction to
$N^{\kappa-1}\Pi_{\text{L}}V_{\text{ren}}\Pi_{\text{L}}$ while the large
momentum interaction $\Pi_{\text{H}}V_{N}\Pi_{\text{H}}$ is left untouched.
The (non-symmetric) map that block-diagonalizes $\mathcal{H}_{2}$ is of the
form $\mathcal{S}_{\eta}=1+N^{\kappa-1}\eta$ and, in order to obtain an
analogous renormalization of the many body interaction, it seems natural to
lift $\mathcal{S}_{\eta}$ to the unitary generalized Bogoliubov transformation

$\begin{split}&\mathcal{U}_{\eta}:=\exp(\mathcal{D}_{\eta}-\mathcal{D}_{\eta}^{*})\,\big{(}\approx
1+\mathcal{D}_{\eta}-\mathcal{D}_{\eta}^{*}\big{)},\text{ where }\\\
&\mathcal{D}_{\eta}:=\frac{N^{\kappa}}{2N}\sum_{\begin{subarray}{c}p,q,r\in\Lambda^{*}:(p,q)\in\text{P}_{\text{L}}^{2},\\\
(p-r,q+r)\in(\text{P}_{\text{L}}^{2})^{c}\end{subarray}}\langle\varphi_{p-r}\otimes\varphi_{q+r},\eta\,\varphi_{p}\otimes\varphi_{q}\rangle
a^{*}_{p-r}a^{*}_{q+r}a_{p}a_{q}.\end{split}$

On a conceptual level, this approach corresponds to the one pursued in [1] (in
particular, the role of $\eta$ defined in (5) is similar to that of
$\eta_{\text{H}}$ defined in [1] through the zero energy scattering equation).
Compared to that, a key idea of [16] is to expand $H_{N}$ directly around
powers of suitably modified creation and annihilation operators, including
e.g.
$c_{r}=a_{r}+[a_{r},\mathcal{D}_{\eta}]\,(\approx\mathcal{U}_{\eta}^{*}a_{r}\,\mathcal{U}_{\eta}$).
This leads to the low momentum renormalization of the many body interaction in
a simple way and avoids the use of operator exponential expansions. Notice
that this approach is reminiscent of previously introduced ideas in [15, 23].
Finally, let us stress that, although the bound (4) is all we need in view of
Theorem 1, [16] derives in fact exact algebraic identities. Similarly as in
[1], what is dropped in (4) is the non-renormalized high momentum part of the
potential energy.

Proceeding as in [16, Lemma 1], let us record the useful upper bounds

$\begin{split}\big{|}\langle\varphi_{k_{1}}\otimes\varphi_{k_{2}},V_{\text{ren}}\varphi_{k_{3}}\otimes\varphi_{k_{4}}\rangle\big{|}&\leq
C,\\\
\big{|}\langle\varphi_{k_{1}}\otimes\varphi_{k_{2}},V_{\text{ren}}\varphi_{k_{3}}\otimes\varphi_{k_{4}}\rangle-8\pi\mathfrak{a}\big{|}\\!&\leq
CN^{\kappa-1}\Big{(}N^{\alpha}+\sum_{i=1}^{4}N^{-\alpha}|k_{i}|^{2}\Big{)},\end{split}$
(7)

for all $k_{1},k_{2},k_{3},k_{4}\in\Lambda^{*}$ satisfying
$k_{1}+k_{2}=k_{3}+k_{4}$ and
$\langle\varphi_{k_{1}}\otimes\varphi_{k_{2}},V_{\text{ren}}\varphi_{k_{3}}\otimes\varphi_{k_{4}}\rangle=0$
in case $k_{1}+k_{2}\neq k_{3}+k_{4}$. The bounds (7) imply in particular that

$\big{|}\langle\varphi_{k_{1}}\otimes\varphi_{k_{2}},\eta\,\varphi_{k_{3}}\otimes\varphi_{k_{4}}\rangle\big{|}\leq\frac{C\,\delta_{k_{1}+k_{2},k_{3}+k_{4}}}{|k_{1}|^{2}+|k_{2}|^{2}}\textbf{1}_{(\text{P}_{\text{L}}^{2})^{c}}((k_{1},k_{2}))\textbf{1}_{\text{P}_{\text{L}}^{2}}((k_{3},k_{4})).$
(8)

For completeness, we prove (7) and (8) in Appendix A, following [16, Appendix
A].

Based on (4), (7) and (8), the proof of Theorem 1 follows by carefully
estimating the three terms on the r.h.s. in (4) and by combining these
estimates with some mild a priori information on the ground state energy.
Before summarizing the key steps, let us introduce the following additional
notation: for every $\zeta\geq 0$, we set

$\mathcal{N}_{>\zeta}:=\sum_{r\in\Lambda^{*}:|r|>\zeta}a^{*}_{r}a_{r}$

and similarly, we define $\mathcal{N}_{\geq\zeta},\mathcal{N}_{<\zeta}$ and
$\mathcal{N}_{\leq\zeta}$. Moreover, we set $\mathcal{N}:=\mathcal{N}_{\geq
0}\ (\equiv N)$, $\mathcal{N}_{+}:=\mathcal{N}_{>0}$ and
$\mathcal{K}:=\sum_{r\in\Lambda^{*}_{+}}|r|^{2}a^{*}_{r}a_{r}$. It is an
elementary observation that

$1-\langle\varphi_{0},\gamma_{N}^{(1)}\varphi_{0}\rangle=N^{-1}\langle\psi_{N},\mathcal{N}_{+}\psi_{N}\rangle.$

The key tool we use below to control the error terms is to estimate the number
of particles with large momenta in terms of the energy: a standard Markov
bound implies

$\mathcal{N}_{>N^{\beta}}\leq N^{-2\beta}\mathcal{K}\leq N^{-2\beta}H_{N}.$
(9)

In particular $\langle\psi_{N},\mathcal{N}_{>N^{\beta}}\psi_{N}\rangle\leq
CN^{1+\kappa-2\beta}\ll N$ as soon as $2\beta>\kappa$ if $\psi_{N}$ denotes an
approximate ground state vector. In Lemma 5, we slightly generalize the bound
(9) to products of the kinetic energy with number of particles operators for
large momenta.

######  Lemma 2.

Suppose $\delta\in(\frac{\kappa}{2},\alpha)$, then we have that

$\sum_{r\in\Lambda_{+}^{*}}|r|^{2}c^{*}_{r}c_{r}\geq
4\pi^{2}\big{(}\mathcal{N}_{<N^{\delta}}-a^{*}_{0}a_{0}\big{)}+\mathcal{E}_{\delta}$

for a self-adjoint operator $\mathcal{E}_{\delta}$ which satisfies for some
$C>0$ and $N$ large enough that

$\pm\mathcal{E}_{\delta}\leq
CN^{\kappa+\frac{\delta}{2}-\frac{3\alpha}{2}-1}(\mathcal{K}+N)\mathcal{N}_{>N^{\alpha}/3}.$

###### Proof.

Using positivity and the definition of the operators $c_{r}$ in (5), we get

$\begin{split}&\sum_{r\in\Lambda_{+}^{*}}|r|^{2}c^{*}_{r}c_{r}-4\pi^{2}\big{(}\mathcal{N}_{<N^{\delta}}-a^{*}_{0}a_{0}\big{)}\\\
&\;\geq\frac{4\pi^{2}N^{\kappa}}{N}\\!\\!\\!\\!\\!\\!\\!\sum_{p,q,r\in\Lambda^{*}:0<|r|<N^{\delta}}\\!\\!\\!\langle\varphi_{p+q-r}\otimes\varphi_{r},\eta\,\varphi_{p}\otimes\varphi_{q}\rangle
a^{*}_{r}a^{*}_{p+q-r}a_{p}a_{q}+\text{h.c.}\ \end{split}$

With the bound (8) and Cauchy-Schwarz, we then obtain for $\xi\in
L^{2}_{s}(\Lambda^{N})$ that

$\begin{split}&\bigg{|}N^{\kappa-1}\sum_{p,q,r\in\Lambda^{*}:0<|r|<N^{\delta}}\langle\varphi_{p+q-r}\otimes\varphi_{r},\eta\,\varphi_{p}\otimes\varphi_{q}\rangle\langle\xi
a^{*}_{r}a^{*}_{p+q-r}a_{p}a_{q}\xi\rangle\bigg{|}\\\ &\;\leq
CN^{\kappa-2\alpha-1}\\!\\!\\!\sum_{\begin{subarray}{c}(p,q,r)\in
P_{L}^{3}:\\\
0<|r|<N^{\delta},\,|p|>N^{\alpha}/3,\,p+q-r\in\text{P}_{\text{L}}^{c}\end{subarray}}\frac{|r|}{|q|+1}\|a_{r}a_{p+q-r}\xi\|\frac{|q|+1}{|r|}\|a_{p}a_{q}\xi\|\\\
&\;\leq
CN^{\kappa+\frac{\delta}{2}-\frac{3\alpha}{2}-1}\langle\xi,(\mathcal{K}+N)\mathcal{N}_{>N^{\alpha}/3}\xi\rangle.\end{split}$

Notice that due to the constraint $p+q-r\in\text{P}_{\text{L}}^{c}$ and the
condition $|r|<N^{\delta}$ for $\delta<\alpha$, at least one of the momenta
$p$ and $q$ has to be larger than $N^{\alpha}/3$ for large $N$. ∎

######  Lemma 3.

There exists a constant $C>0$ such that

$\begin{split}&\frac{N^{\kappa}}{2N}\\!\\!\\!\sum_{\begin{subarray}{c}p,q,r\in\Lambda^{*}:\\\
p,q,p+r,q-r\in\text{P}_{\text{L}}\end{subarray}}\\!\\!\\!\\!\\!\langle\varphi_{p+r}\otimes\varphi_{q-r},V_{\emph{ren}}\varphi_{p}\otimes\varphi_{q}\rangle
a^{*}_{p+r}a^{*}_{q-r}a_{p}a_{q}\\\ &\;\geq 4\pi
N^{1+\kappa}-CN^{\kappa}\mathcal{N}_{>N^{\alpha}}-CN^{\kappa+3\alpha}-CN^{2\kappa+2\alpha-1}(\mathcal{K}+N)\end{split}$
(10)

###### Proof.

We use (7) together with the fact that $|p|,|q|,|r|\leq 2N^{\alpha}$ to
replace $V_{\text{ren}}$ as follows: for every $\xi\in
L^{2}_{s}(\Lambda^{N})$, we have that

$\begin{split}&\frac{N^{\kappa}}{2N}\\!\sum_{\begin{subarray}{c}p,q,r\in\Lambda^{*}:\\\
p,q,p+r,q-r\in
P_{L}\end{subarray}}\\!\big{|}\langle\varphi_{p+r}\otimes\varphi_{q-r},V_{\text{ren}}\varphi_{p}\otimes\varphi_{q}\rangle-8\pi\mathfrak{a}\big{|}|\langle\xi,a^{*}_{p+r}a^{*}_{q-r}a_{p}a_{q}\xi\rangle|\\\
&\leq CN^{2\kappa+\alpha-2}\\!\sum_{\begin{subarray}{c}p,q,r\in\Lambda^{*}:\\\
p,q,p+r,q-r\in
P_{L}\end{subarray}}\\!\frac{|p+r|+1}{|p|+1}\|a_{p+r}a_{q-r}\xi\|\frac{|p|+1}{|p+r|+1}\|a_{p}a_{q}\xi\|\\\
&\leq CN^{2\kappa+2\alpha-1}\langle\xi,(\mathcal{K}+N)\xi\rangle.\end{split}$

The lemma now follows by combining this estimate with

$\begin{split}&\frac{4\pi\mathfrak{a}N^{\kappa}}{N}\\!\\!\\!\sum_{\begin{subarray}{c}p,q,r\in\Lambda^{*}:\\\
p,q,p+r,q-r\in\text{P}_{\text{L}}\end{subarray}}\\!\\!\\!\\!\\!a^{*}_{p+r}a^{*}_{q-r}a_{p}a_{q}\\\
&\;=\frac{4\pi\mathfrak{a}N^{\kappa}}{N}\sum_{r\in\Lambda^{*}}\bigg{(}\sum_{\begin{subarray}{c}q\in\text{P}_{\text{L}}:\\\
q+r\in\text{P}_{\text{L}}\end{subarray}}a^{*}_{q}a_{q+r}\bigg{)}^{*}\bigg{(}\sum_{\begin{subarray}{c}q\in\text{P}_{\text{L}}:\\\
q+r\in\text{P}_{\text{L}}\end{subarray}}a^{*}_{q}a_{q+r}\bigg{)}-\frac{4\pi\mathfrak{a}N^{\kappa}}{N}\\!\\!\\!\sum_{\begin{subarray}{c}p,r\in\Lambda^{*}:\\\
p,p+r\in\text{P}_{\text{L}}\end{subarray}}\\!\\!\\!\\!\\!a^{*}_{p+r}a_{p+r}\\\
&\;\geq\frac{4\pi\mathfrak{a}N^{\kappa}}{N}\big{(}N-\mathcal{N}_{>N^{\alpha}}\big{)}^{2}-\frac{4\pi\mathfrak{a}N^{\kappa}}{N}\\!\\!\\!\sum_{\begin{subarray}{c}p,r\in\Lambda^{*}:\\\
p,p+r\in\text{P}_{\text{L}}\end{subarray}}\\!\\!\\!\\!\\!a^{*}_{p+r}a_{p+r}\\\
&\;\geq 4\pi
N^{1+\kappa}-CN^{\kappa}\mathcal{N}_{>N^{\alpha}}-CN^{\kappa+3\alpha}.\qed\end{split}$

######  Lemma 4.

Let $R_{N}$ be as in (6) and let $0\leq\beta<\alpha$. Then, there exists $C>0$
such that for $N$ large enough, we have that

$\begin{split}\pm R_{N}&\leq
CN^{2\kappa-2\alpha-2}\Big{(}N^{4\alpha}(\mathcal{N}_{>N^{\beta}}+N^{3\alpha})+N^{\frac{5}{2}\alpha+\frac{3}{2}\beta+\frac{1}{2}}(\mathcal{N}_{>N^{\beta}}+N^{3\alpha})^{\frac{1}{2}}+N^{\frac{3}{2}\alpha+\frac{5}{2}\beta+1}\Big{)}\\\
&\hskip
58.32823pt\times\big{(}\mathcal{K}+N+N^{5\beta}\big{)}\big{(}\mathcal{N}_{>N^{\alpha}/3}+1\big{)}.\end{split}$

###### Proof.

Given $\xi\in L^{2}_{s}(\Lambda^{N})$, we apply the bound (8) to get

$\begin{split}|\langle\xi,R_{N}\xi\rangle|&\leq\frac{CN^{2\kappa}}{N^{2}}\\!\\!\\!\sum_{\begin{subarray}{c}r\in\Lambda^{*}_{+},p,q,s,t\in\Lambda^{*}:\\\
(p+q-r,r),(s+t-r,r)\in(\text{P}_{\text{L}}^{2})^{c},\\\
(p,q),(s,t)\in\text{P}_{\text{L}}^{2}\end{subarray}}\\!\\!\\!\frac{|r|^{2}|\langle\xi,a_{p}^{*}a_{q}^{*}a_{s+t-r}^{*}a_{p+q-r}a_{s}a_{t}\xi\rangle|}{(|p+q-r|^{2}+|r|^{2})(|s+t-r|^{2}+|r|^{2})}\\\
&\leq
CN^{2\kappa-2\alpha-2}\\!\\!\\!\sum_{\begin{subarray}{c}r\in\Lambda^{*}_{+},p,q,s,t\in\Lambda^{*}:\\\
(p+q-r,r),(s+t-r,r)\in(\text{P}_{\text{L}}^{2})^{c},\\\
(p,q),(s,t)\in\text{P}_{\text{L}}^{2}\end{subarray}}\\!\\!\\!|\langle\xi,a_{p}^{*}a_{q}^{*}a_{s+t-r}^{*}a_{p+q-r}a_{s}a_{t}\xi\rangle|.\end{split}$

In order to control the sum on the right hand side, we split it according to
two types of restrictions: first, consider another scale $N^{\beta}$, for
$\beta<\alpha$, and consider the cases in which the momenta
$p,q,s,t\in\text{P}_{\text{L}}^{4}$ are smaller or greater than $N^{\beta}$.
We consider the cases

$\begin{split}&(1)\;\;|p|,|q|,|s|,|t|\leq N^{\beta},\\\
&(3)\;\;|p|,|q|>N^{\beta}\text{ and }|s|,|t|\leq N^{\beta},\\\
&(5)\;\;|p|,|q|,|s|>N^{\beta}\text{ and }|t|\leq N^{\beta},\end{split}\hskip
28.45274pt\begin{split}&(2)\;\;|p|>N^{\beta}\text{ and }|q|,|s|,|t|\leq
N^{\beta},\\\ &(4)\;\;|p|,|s|>N^{\beta}\text{ and }|q|,|t|\leq N^{\beta},\\\
&(6)\;\;|p|,|q|,|s|,|t|>N^{\beta}.\end{split}$ (11)

Furthermore, the conditions
$(p+q-r,r),(s+t-r,r)\in(\text{P}_{\text{L}}^{2})^{c}$ imply that at least one
of $p,q,p+q-r$ and one of $s,t,s+t-r$ is greater than $N^{\alpha}/3$: we
consider the cases

$\begin{split}&(a)\;\;|p+q-r|,|s+t-r|>N^{\alpha}/3,\\\
&(c)\;\;|p+q-r|,|s|>N^{\alpha}/3,\end{split}\hskip
28.45274pt\begin{split}&(b)\;\;|p|,|s+t-r|>N^{\alpha}/3,\\\
&(d)\;\;|p|,|s|>N^{\alpha}/3.\end{split}$ (12)

Now, using symmetries among and within the pairs
$(p,q)\in\text{P}_{\text{L}}^{2}$ and $(s,t)\in\text{P}_{\text{L}}^{2}$, one
readily sees that for N large enough, such that $N^{\beta}<N^{\alpha}/3$, we
have that

$\begin{split}&\sum_{\begin{subarray}{c}r\in\Lambda^{*}_{+},p,q,s,t\in\Lambda^{*}:\\\
(p+q-r,r),(s+t-r,r)\in(\text{P}_{\text{L}}^{2})^{c},\\\
(p,q),(s,t)\in\text{P}_{\text{L}}^{2}\end{subarray}}\\!\\!\\!|\langle\xi,a_{p}^{*}a_{q}^{*}a_{s+t-r}^{*}a_{p+q-r}a_{s}a_{t}\xi\rangle|\\\
&\;\leq
C\bigg{(}\sum_{j=1}^{6}\Sigma_{ja}(\xi)+\sum_{j=2}^{6}\Sigma_{jb}(\xi)+\Sigma_{5c}(\xi)+\sum_{j=4}^{6}\Sigma_{jd}(\xi)\bigg{)},\end{split}$

where $\Sigma_{j\alpha}$, for $j\in\\{1,\ldots,6\\}$ and
$\alpha\in\\{a,b,c,d\\}$, refers to the contribution

$\Sigma_{j\alpha}(\cdot):=\sum_{\begin{subarray}{c}r\in\Lambda^{*}_{+},p,q,s,t\in\Lambda^{*}:p,q,s,t,\\\
p+q-r,s+t-r\text{ satisfy }j)\text{ and
}\alpha)\end{subarray}}\\!\\!\\!|\langle\cdot,a_{p}^{*}a_{q}^{*}a_{s+t-r}^{*}a_{p+q-r}a_{s}a_{t}\cdot\rangle|\geq
0.$

Here, the restriction labels $j\in\\{1,\ldots,6\\}$ and
$\alpha\in\\{a,b,c,d\\}$ refer to (11) and (12), respectively. Applying basic
Cauchy-Schwarz estimates as in Lemmas 2 and 3, we find

$\begin{split}\Sigma_{1a}&\leq
N^{4\beta+1}(\mathcal{K}+N)(\mathcal{N}_{>N^{\alpha}/3}+1),\\\
\Sigma_{2a},\Sigma_{3a}&\leq
N^{2\alpha+2\beta+\frac{1}{2}}(\mathcal{K}+N)(\mathcal{N}_{>N^{\alpha}/3}+1)(\mathcal{N}_{>N^{\beta}}+1)^{\frac{1}{2}},\\\
\Sigma_{2b}&\leq
N^{\frac{3}{2}\alpha+\frac{5}{2}\beta+1}(\mathcal{K}+N)(\mathcal{N}_{>N^{\alpha}/3}+1),\\\
\Sigma_{3b}&\leq\big{(}N^{\frac{3}{2}\alpha+\frac{5}{2}\beta+\frac{1}{2}}(\mathcal{N}_{>N^{\beta}}+N^{3\alpha})^{\frac{1}{2}}+N^{3\alpha+\beta}(\mathcal{N}_{>N^{\beta}}+N^{3\alpha})\big{)}\\\
&\hskip
14.22636pt\times(\mathcal{K}+N+N^{5\beta})(\mathcal{N}_{>N^{\alpha}/3}+1),\\\
\Sigma_{4a},\Sigma_{5a},\Sigma_{6a}&\leq
N^{4\alpha}(\mathcal{K}+N)(\mathcal{N}_{>N^{\alpha}/3}+1)(\mathcal{N}_{>N^{\beta}}+1),\\\
\Sigma_{4b},\Sigma_{5b},\Sigma_{6b}&\leq\big{(}N^{\frac{5}{2}\alpha+\frac{3}{2}\beta+\frac{1}{2}}(\mathcal{N}_{>N^{\beta}}+1)^{\frac{1}{2}}+N^{4\alpha}(\mathcal{N}_{>N^{\beta}}+1)\big{)}(\mathcal{K}+N)(\mathcal{N}_{>N^{\alpha}/3}+1),\\\
\Sigma_{5c}&\leq\big{(}N^{2\alpha+2\beta+\frac{1}{2}}(\mathcal{N}_{>N^{\beta}}+1)^{\frac{1}{2}}+N^{\frac{7}{2}\alpha+\frac{1}{2}\beta}(\mathcal{N}_{>N^{\beta}}+1)\big{)}(\mathcal{K}+N)(\mathcal{N}_{>N^{\alpha}/3}+1),\\\
\Sigma_{4d},\Sigma_{5d},\Sigma_{6d}&\leq\big{(}N^{\frac{5}{2}\alpha+\frac{3}{2}\beta+\frac{1}{2}}(\mathcal{N}_{>N^{\beta}}+1)^{\frac{1}{2}}+N^{4\alpha}(\mathcal{N}_{>N^{\beta}}+1)+N^{\alpha+3\beta+1}\big{)}\\\
&\hskip
14.22636pt\times(\mathcal{K}+N)(\mathcal{N}_{>N^{\alpha}/3}+1).\end{split}$

Here, an inequality of the form $\Sigma_{j\alpha}\leq\mathcal{L}$ for a non-
negative self-adjoint operator $\mathcal{L}$ refers to the statement that
$\Sigma_{j\alpha}(\xi)\leq\langle\xi,\mathcal{L}\,\xi\rangle$, for all $\xi\in
L^{2}_{s}(\Lambda^{N})$. Note one subtlety: for $\Sigma_{3b}$ all momenta
appearing in creation operators are high and in order to efficiently use the
kinetic energy we bound this contribution not in the straightforward way, but
rather by

$\begin{split}\Sigma_{3b}&\leq\\!\\!\\!\sum_{\begin{subarray}{c}r\in\Lambda^{*}_{+},p,q,s,t\in\Lambda^{*}:p,q,s,t,\\\
p+q-r,s+t-r\text{ satisfy }3)\text{ and
}b)\end{subarray}}\\!\\!\\!\frac{|s|+1}{|t|+1}\|(\mathcal{N}_{s}+1)^{\frac{1}{2}}a_{p}a_{s+t-r}\cdot\|\frac{|t|+1}{|s|+1}\|(\mathcal{N}_{q}+1)^{\frac{1}{2}}a_{p+q-r}a_{t}\cdot\|\\\
&\leq\big{(}N^{\frac{3}{2}\alpha+\frac{5}{2}\beta+\frac{1}{2}}(\mathcal{N}_{>N^{\beta}}+N^{3\alpha})^{\frac{1}{2}}\\!+\\!N^{3\alpha+\beta}(\mathcal{N}_{>N^{\beta}}+N^{3\alpha})\big{)}(\mathcal{K}+\\!N\\!+\\!N^{5\beta})(\mathcal{N}_{>N^{\alpha}/3}\\!+\\!1),\end{split}$

where we set $\mathcal{N}_{s}:=a_{s}^{*}a_{s}$ and where, to obtain the second
line, we have also distinguished the cases in which $|p+q-r|$ is larger or
smaller than $N^{\beta}$.

Collecting the above estimates and multiplying by a factor
$N^{2\kappa-2\alpha-2}$, we arrive at

$\begin{split}&N^{2\kappa-2\alpha-2}\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}r\in\Lambda^{*}_{+},p,q,s,t\in\Lambda^{*}:\\\
(p+q-r,r),(s+t-r,r)\in(\text{P}_{\text{L}}^{2})^{c},\\\
(p,q),(s,t)\in\text{P}_{\text{L}}^{2}\end{subarray}}\\!\\!\\!|\langle\xi,a_{p}^{*}a_{q}^{*}a_{s+t-r}^{*}a_{p+q-r}a_{s}a_{t}\xi\rangle|\\\
&\leq
CN^{2\kappa-2\alpha-2}\langle\xi,\big{(}N^{4\alpha}(\mathcal{N}_{>N^{\beta}}+N^{3\alpha})+N^{\frac{5}{2}\alpha+\frac{3}{2}\beta+\frac{1}{2}}(\mathcal{N}_{>N^{\beta}}+N^{3\alpha})^{\frac{1}{2}}+N^{\frac{3}{2}\alpha+\frac{5}{2}\beta+1}\big{)}\\\
&\hskip
28.45274pt\times(\mathcal{K}+N+N^{5\beta})(\mathcal{N}_{>N^{\alpha}/3}+1)\xi\rangle.\qed\end{split}$

Before concluding Theorem 1, the last ingredient that we need is some mild a
priori information on the energy of the ground state vector $\psi_{N}$, as
remarked around Eq. (9).

######  Lemma 5.

Let $\psi_{N}$ denote the normalized ground state vector of $H_{N}$, defined
in (1), and let $\beta\geq 0$. Then $\psi_{N}$ satisfies the a priori bounds

$\begin{split}\langle\psi_{N},\mathcal{N}_{>N^{\beta}}\psi_{N}\rangle&\leq
CN^{1+\kappa-2\beta},\\\
N^{-1}\langle\psi_{N},\mathcal{K}\mathcal{N}_{>N^{\beta}}\psi_{N}\rangle&\leq
CN^{1+2\kappa-2\beta}+CN^{\frac{3}{2}\kappa+\frac{1}{2}\beta},\\\
N^{-2}\langle\psi_{N},\mathcal{K}\mathcal{N}^{2}_{>N^{\beta}}\psi_{N}\rangle&\leq
CN^{1+3\kappa-4\beta}+CN^{\beta+2\kappa}+CN^{\frac{5}{2}\kappa-\frac{3}{2}\beta}.\end{split}$

###### Proof.

The first bound is a direct consequence of (9) and the fact that $E_{N}\leq
CN^{1+\kappa}$. For the bound on $\mathcal{K}\mathcal{N}_{\geq N^{\beta}}$, we
use a commutator argument as in [6, 7, 1]. We bound

$\begin{split}N^{-1}\langle\psi_{N},\mathcal{K}\mathcal{N}_{>N^{\beta}}\psi_{N}\rangle&\leq
N^{-1}\langle\psi_{N},\mathcal{K}\psi_{N}\rangle^{\frac{1}{2}}\langle\psi_{N},\mathcal{N}_{>N^{\beta}}\mathcal{K}\mathcal{N}_{>N^{\beta}}\psi_{N}\rangle^{\frac{1}{2}}\\\
&\leq
CN^{\frac{\kappa}{2}-\frac{1}{2}}\langle\psi_{N},\mathcal{N}_{>N^{\beta}}\mathcal{K}\mathcal{N}_{>N^{\beta}}\psi_{N}\rangle^{\frac{1}{2}}\end{split}$

and then

$\begin{split}\frac{1}{N}\langle\psi_{N},\mathcal{N}_{>N^{\beta}}\mathcal{K}\mathcal{N}_{>N^{\beta}}\psi_{N}\rangle&\leq\frac{1}{N}\langle\psi_{N},\mathcal{N}_{>N^{\beta}}H_{N}\mathcal{N}_{>N^{\beta}}\psi_{N}\rangle\\\
&=\frac{E_{N}}{N}\langle\psi_{N},\mathcal{N}^{2}_{>N^{\beta}}\psi_{N}\rangle+\frac{1}{N}\langle\psi_{N},\mathcal{N}_{>N^{\beta}}\big{[}H_{N},\mathcal{N}_{>N^{\beta}}\big{]}\psi_{N}\rangle\\\
&\leq
CN^{\kappa-2\beta}\langle\psi_{N},\mathcal{K}\mathcal{N}_{>N^{\beta}}\psi_{N}\rangle+\frac{1}{N}\langle\psi_{N},\mathcal{N}_{>N^{\beta}}\big{[}H_{N},\mathcal{N}_{>N^{\beta}}\big{]}\psi_{N}\rangle.\end{split}$

To estimate the commutator contribution on the r.h.s. in the previous
equation, we write

$H_{N}-\mathcal{K}=\frac{1}{2}\int_{\Lambda^{2}}dxdy\,N^{2-2\kappa}V(N^{1-\kappa}(x-y))\check{a}_{x}^{*}\check{a}_{y}^{*}\check{a}_{x}\check{a}_{y}=:\mathcal{V}_{N},$

where $\check{a}_{x}:=\sum_{p\in\Lambda^{*}}e^{ipx}a_{p}$ denotes the usual
operator valued distribution annihilating a particle at $x\in\Lambda$, and we
note $[\mathcal{K},\mathcal{N}_{>N^{\beta}}]=0$ as well as
$[\mathcal{V}_{N},\mathcal{N}_{>N^{\beta}}]=[\mathcal{N}_{\leq
N^{\beta}},\mathcal{V}_{N}]$ with

$\begin{split}\big{[}\mathcal{N}_{\leq
N^{\beta}},\mathcal{V}_{N}\big{]}=\sum_{p\in\Lambda^{*}:|p|\leq
N^{\beta}}\int_{\Lambda^{2}}dxdy\,N^{2-2\kappa}V(N^{1-\kappa}(x-y))e^{ipx}\check{a}_{p}^{*}\check{a}_{y}^{*}\check{a}_{x}\check{a}_{y}+\text{h.c.}\
\end{split}$

Now, basic Cauchy-Schwarz estimates imply that

$\begin{split}&N^{-1}\big{|}\big{\langle}\psi_{N},\mathcal{N}_{>N^{\beta}}\big{[}\mathcal{N}_{\leq
N^{\beta}},\mathcal{V}_{N}\big{]}\psi_{N}\big{\rangle}\big{|}\\\ &\;\leq
CN^{-1}\sum_{p\in\Lambda^{*}:|p|\leq
N^{\beta}}\int_{\Lambda^{2}}dxdy\,N^{2-2\kappa}V(N^{1-\kappa}(x-y))\|a_{p}\check{a}_{y}\,\mathcal{N}_{>N^{\beta}}\psi_{N}\|\|\check{a}_{x}\check{a}_{y}\psi_{N}\|\\\
&\hskip 11.38092pt+CN^{-1}\sum_{p\in\Lambda^{*}:|p|\leq
N^{\beta}}\int_{\Lambda^{2}}dxdy\,N^{2-2\kappa}V(N^{1-\kappa}(x-y))\|a_{p}\check{a}_{y}\psi_{N}\|\|\check{a}_{x}\check{a}_{y}\,\mathcal{N}_{>N^{\beta}}\psi_{N}\|\\\
&\leq
CN^{\frac{\beta}{2}+\frac{\kappa}{2}-1}\big{(}\|(\mathcal{K}+a^{*}_{0}a_{0})^{\frac{1}{2}}\mathcal{N}_{>N^{\beta}}\psi_{N}\|\|\mathcal{V}_{N}^{1/2}\psi_{N}\|+\|(\mathcal{K}+a^{*}_{0}a_{0})^{\frac{1}{2}}\psi_{N}\|\|\mathcal{V}_{N}^{1/2}\mathcal{N}_{>N^{\beta}}\psi_{N}\|\big{)}\\\
&\leq
CN^{\frac{\beta}{2}+\kappa-\frac{1}{2}}\|H_{N}^{\frac{1}{2}}\mathcal{N}_{>N^{\beta}}\psi_{N}\|+CN^{\frac{\beta}{2}+\kappa}\|\mathcal{N}_{>N^{\beta}}\psi_{N}\|.\end{split}$

Using a simple contraction argument, this implies that

$\frac{1}{N}\langle\psi_{N},\mathcal{N}_{>N^{\beta}}H_{N}\mathcal{N}_{>N^{\beta}}\psi_{N}\rangle\leq
CN^{\kappa-2\beta}\langle\psi_{N},\mathcal{K}\mathcal{N}_{>N^{\beta}}\psi_{N}\rangle+CN^{\beta+2\kappa}$

and, as a consequence, that

$\begin{split}N^{-1}\langle\psi_{N},\mathcal{K}\mathcal{N}_{>N^{\beta}}\psi_{N}\rangle&\leq
CN^{1+2\kappa-2\beta}+CN^{\frac{3}{2}\kappa+\frac{1}{2}\beta},\\\
N^{-2}\langle\psi_{N},\mathcal{N}_{>N^{\beta}}\mathcal{K}\mathcal{N}_{>N^{\beta}}\psi_{N}\rangle&\leq
CN^{1+3\kappa-4\beta}+CN^{\beta+2\kappa}+CN^{\frac{5}{2}\kappa-\frac{3}{2}\beta}.\qed\end{split}$

We are now ready to prove our main result.

###### Proof of Theorem 1.

Let $\psi_{N}$ denote the normalized ground state vector of $H_{N}$, given
some parameter $\kappa\in[0,\frac{1}{20})$. Let $\text{P}_{\text{L}}$ be
defined as in (3) and choose

$\alpha:=(1+\epsilon)\frac{41}{10}\kappa$

for some sufficiently small $\epsilon>0$; in particular
$\alpha\in[0,1-\kappa]$. Now, by (4), we have that

$\begin{split}\langle\psi_{N},H_{N}\psi_{N}\rangle&\geq\sum_{r\in\Lambda_{+}^{*}}|r|^{2}\langle\psi_{N},c^{*}_{r}c_{r}\psi_{N}\rangle-\langle\psi_{N},R_{N}\psi_{N}\rangle\\\
&\hskip
11.38092pt+\frac{N^{\kappa}}{2N}\\!\\!\\!\sum_{\begin{subarray}{c}p,q,r\in\Lambda^{*}:\\\
p,q,p+r,q-r\in\text{P}_{\text{L}}\end{subarray}}\\!\\!\\!\\!\\!\langle\varphi_{p+r}\otimes\varphi_{q-r},V_{\text{ren}}\varphi_{p}\otimes\varphi_{q}\rangle\langle\psi_{N},a^{*}_{p+r}a^{*}_{q-r}a_{p}a_{q}\psi_{N}\rangle\end{split}$

and our goal is to estimate the terms on the right hand side. We start with
the kinetic energy term. Combining the bounds from Lemmas 2 and 5, we find
that

$\begin{split}&\sum_{r\in\Lambda_{+}^{*}}|r|^{2}\langle\psi_{N},c^{*}_{r}c_{r}\psi_{N}\rangle-4\pi^{2}\langle\psi_{N},\mathcal{N}_{+}\psi_{N}\rangle\\\
&\;\geq-
CN^{1+\kappa-2\delta}-CN^{1+3\kappa+\frac{1}{2}\delta-\frac{7}{2}\alpha}-CN^{\frac{5}{2}\kappa+\frac{1}{2}\delta-\alpha}=o(N),\end{split}$

where we used (9), the choice $\frac{\kappa}{2}<\delta<\alpha$ and the
identity
$\mathcal{N}_{<N^{\delta}}-a^{*}_{0}a_{0}=\mathcal{N}_{+}-\mathcal{N}_{\geq
N^{\delta}}$.

Proceeding similarly for the remaining error terms, we obtain from Lemma 3
that

$\begin{split}&\frac{N^{\kappa}}{2N}\\!\\!\\!\sum_{\begin{subarray}{c}p,q,r\in\Lambda^{*}:\\\
p,q,p+r,q-r\in\text{P}_{\text{L}}\end{subarray}}\\!\\!\\!\\!\\!\langle\varphi_{p+r}\otimes\varphi_{q-r},V_{\text{ren}}\varphi_{p}\otimes\varphi_{q}\rangle\langle\psi_{N},a^{*}_{p+r}a^{*}_{q-r}a_{p}a_{q}\psi_{N}\rangle\\\
&\;\geq
4\pi\mathfrak{a}N^{1+\kappa}-CN^{\kappa+3\alpha}=4\pi\mathfrak{a}N^{1+\kappa}+o(N)\end{split}$

and from Lemma 4, assuming $\beta=(1+\epsilon)\frac{5}{2}\kappa$ for
sufficiently small $\epsilon>0$, that

$\begin{split}|\langle\psi_{N},R_{N}\psi_{N}\rangle|&\leq
CN^{1+5\kappa-2\beta}+CN^{\frac{1}{2}+\frac{9}{2}\kappa+\frac{5}{2}\alpha-2\beta}+CN^{1+\frac{9}{2}\kappa+\frac{1}{2}\beta-\frac{3}{2}\alpha}\\\
&\hskip
14.22636pt+CN^{\frac{1}{2}+4\kappa+\alpha+\frac{1}{2}\beta}+CN^{1+4\kappa+\frac{5}{2}\beta-\frac{5}{2}\alpha}\\\
&=o(N)+CN^{\frac{1}{2}+\frac{5}{2}\alpha-\frac{1}{2}\kappa+O(\epsilon)}+CN^{\frac{1}{2}+\frac{21}{4}\kappa+\alpha+O(\epsilon)}=o(N).\end{split}$

Combining this with the ground state energy upper bound $E_{N}\leq
4\pi\mathfrak{a}N^{1+\kappa}+o(N)$, as pointed out in the second remark after
Theorem 1, we get

$\begin{split}4\pi\mathfrak{a}N^{1+\kappa}+o(N)&\geq\langle\psi_{N},H_{N}\psi_{N}\rangle\geq
4\pi\mathfrak{a}N^{1+\kappa}+4\pi^{2}\langle\psi_{N},\mathcal{N}_{+}\psi_{N}\rangle+o(N)\end{split}$

and thus conclude that

$\lim_{N\to\infty}N^{-1}\langle\psi_{N},\mathcal{N}_{+}\psi_{N}\rangle=\lim_{N\to\infty}\big{(}1-\langle\varphi_{0},\gamma_{N}^{(1)}\varphi_{0}\rangle\big{)}=0.\qed$

Acknowledgements. C. B. acknowledges support by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy – GZ 2047/1, Projekt-ID 390685813. M.B., C.C and J.O.
acknowledge partial support from the Swiss National Science Foundation through
the Grant “Dynamical and energetic properties of Bose- Einstein condensates”,
from the European Research Council through the ERC-AdG CLaQS, grant agreement
n. 834782, and from the NCCR SwissMAP. C.C. acknowledges the GNFM Gruppo
Nazionale per la Fisica Matematica - INDAM.

##  Appendix A Proof of the Bounds (7) and (8)

###### Proof of the Bounds (7) and (8).

Throughout this appendix, we assume $p,q,s,t\in\Lambda^{*}$ and we abbreviate
$\langle\mathcal{L}\rangle_{pq,st}:=\langle\varphi_{p}\otimes\varphi_{q},\mathcal{L}\,\varphi_{s}\otimes\varphi_{t}\rangle$
for every operator $\mathcal{L}$ on $L^{2}(\Lambda^{2})$. Let us begin with a
few elementary observations: it is clear that the operator $N^{1-\kappa}V_{N}$
preserves the total momentum and that

$\langle
N^{1-\kappa}V_{N}\rangle_{pq,st}\leq\delta_{p+q,s+t}\int_{\Lambda^{2}}dx_{1}dx_{2}\,N^{3-3\kappa}V(N^{1-\kappa}(x_{1}-x_{2}))\leq\delta_{p+q,s+t}\|V\|_{1}.$

Combining this with the fact that $-\Delta_{x_{1}}-\Delta_{x_{2}}+V_{N}$ and
hence its pseudo-inverse

$\mathcal{R}:=\Pi_{\text{H}}\big{[}\Pi_{\text{H}}(-\Delta_{x_{1}}-\Delta_{x_{2}}+V_{N})\Pi_{\text{H}}\big{]}^{-1}\Pi_{\text{H}}$

from $\Pi_{\text{H}}L^{2}(\Lambda^{2})$ to $\Pi_{\text{H}}L^{2}(\Lambda^{2})$
also preserve the total momentum, we get that

$\begin{split}\langle V_{\text{ren}}\rangle_{pq,st}&\leq
C\delta_{pq,st}\sup_{p,q\in\Lambda^{*}}\Big{(}1+N^{1-\kappa}\big{\langle}V_{N}^{1/2}\Pi_{\text{H}}V_{N}^{1/2}\mathcal{R}V_{N}^{1/2}\Pi_{\text{H}}V_{N}^{1/2}\big{\rangle}_{pq,pq}\\\
&\hskip
85.35826pt+N^{1-\kappa}\big{\langle}V_{N}^{1/2}\Pi_{\text{L}}V_{N}^{1/2}\mathcal{R}V_{N}^{1/2}\Pi_{\text{L}}V_{N}^{1/2}\big{\rangle}_{pq,pq}\Big{)}.\end{split}$

To control the right hand side, we make use of the operator inequalities

$0\leq\Pi_{\text{H}}V_{N}^{1/2}\mathcal{R}V_{N}^{1/2}\Pi_{\text{H}}\leq
1\;\text{ and }\;\mathcal{R}\leq N^{-2\alpha}\Pi_{\text{H}}\leq N^{-2\alpha}.$

This implies on the one hand that

$N^{1-\kappa}\big{\langle}V_{N}^{1/2}\Pi_{\text{H}}V_{N}^{1/2}\mathcal{R}V_{N}^{1/2}\Pi_{\text{H}}V_{N}^{1/2}\big{\rangle}_{pq,pq}\leq\langle
N^{1-\kappa}V_{N}\rangle_{pq,pq}=\|V\|_{1}$

and on the other hand that

$N^{1-\kappa}\big{\langle}V_{N}^{1/2}\Pi_{\text{L}}V_{N}^{1/2}\mathcal{R}V_{N}^{1/2}\Pi_{\text{L}}V_{N}^{1/2}\big{\rangle}_{pq,pq}\leq
N^{-2\alpha}\|V\|_{1}\|\Pi_{\text{L}}V_{N}^{1/2}\varphi_{p}\otimes\varphi_{q}\|^{2}_{\infty}.$

Looking at the Fourier expansion

$\Pi_{\text{L}}V_{N}^{1/2}\varphi_{p}\otimes\varphi_{q}=\sum_{s\in\text{P}_{\text{L}}:p+q-s\in\text{P}_{\text{L}}}N^{2\kappa-2}\widehat{V^{1/2}}((s-p)/N^{1-\kappa})\varphi_{s}\otimes\varphi_{p+q-s},$

the assumptions that $V\in L^{1}(\mathbb{R}^{3})$ having compact support and
that $\alpha\leq 1-\kappa$ imply that
$\|\Pi_{\text{L}}V_{N}^{1/2}\varphi_{p}\otimes\varphi_{q}\|_{\infty}\leq
CN^{\alpha}$ so that altogether $|\langle V_{\text{ren}}\rangle_{pq,st}|\leq
C\delta_{pq,st}$.

Next, let us switch to the second bound in (7). We first show that

$|\langle V_{\text{ren}}\rangle_{00,00}-8\pi\mathfrak{a}|\leq
CN^{\alpha+\kappa-1}.$ (13)

Up to minor modifications, this bound follows as in [16, Appendix A], so let
us focus on the key steps. Denote by $f$ the zero energy scattering solution
in $\mathbb{R}^{3}$ such that

$(-2\Delta+V)f=0$

with $\lim_{|x|\to\infty}f(x)=1$. It is well known (see e.g. [29, Appendix C])
that $0\leq f\leq 1$, that $f$ is radial and that for $x\in\mathbb{R}^{3}$
outside the support of $V$, we have that $f(x)=1-\mathfrak{a}/|x|.$ Moreover,
a basic integration by parts shows that

$8\pi\mathfrak{a}=\int_{\mathbb{R}^{3}}dx\,V(x)f(x)=\int_{\Lambda}dx\,N^{3-3\kappa}(Vf)(N^{1-\kappa}x).$

Let us denote $w:=1-f$ which is easily seen to satisfy the bounds

$w(x)\leq\frac{C}{|x|},\hskip
14.22636pt|\widehat{w}(p)|\leq\frac{C}{1+|p|^{2}}$

for some constant $C>0$ (e.g. based on the identity $w=(-2\Delta)^{-1}Vf$).
Moreover, pick a smooth bump function $\chi\in C^{\infty}_{c}(B_{1/2}(0))$
such that $\chi(x)=1$ if $|x|\leq\frac{1}{4}$ and define

$(x_{1},x_{2})\mapsto\phi_{N}(x_{1}-x_{2}):=\chi(x_{1}-x_{2})w(N^{1-\kappa}(x_{1}-x_{2}))\in
L^{2}(\Lambda^{2}).$

By slight abuse of notation, we identify $\phi_{N}$ with the associated
multiplication operator in $L^{2}(\Lambda^{2})$. As explained in [16], we then
have the identity

$\mathcal{R}V_{N}\varphi_{0}\otimes\varphi_{0}=\phi_{N}\varphi_{0}\otimes\varphi_{0}+\mathcal{R}\zeta_{N}\varphi_{0}\otimes\varphi_{0}-(1-\mathcal{R}V_{N})\Pi_{\text{L}}\phi_{N}\varphi_{0}\otimes\varphi_{0},$

where $\zeta_{N}(x_{1}-x_{2}):=N^{\kappa-1}\zeta(x_{1}-x_{2})$ for

$x\mapsto\zeta(x):=2\mathfrak{a}\frac{(\Delta\chi)(x)}{|x|}-4\mathfrak{a}\frac{(\nabla\chi)(x)\cdot(x)}{|x|^{3}}\in
C^{\infty}_{0}\big{(}B_{1/2}(0)\cap\overline{B}^{c}_{1/4}(0)\big{)}.$

Using that $8\pi\mathfrak{a}=N^{1-\kappa}\langle
V_{N},(1-\phi_{N})\rangle_{00,00}$, this yields

$\langle
V_{\text{ren}}\rangle_{00,00}=8\pi\mathfrak{a}+N^{1-\kappa}\langle\mathcal{R}V_{N},\zeta_{N}\rangle_{00,00}+N^{1-\kappa}\langle
V_{N},(1-\mathcal{R}V_{N})\Pi_{\text{L}}\phi_{N}\rangle_{00,00}.$

Now observe that for $|p|>N^{\alpha}$, we have that

$\begin{split}\langle\mathcal{R}V_{N}\rangle_{-pp,00}&=\frac{-\langle
V_{N}\mathcal{R}V_{N}\rangle_{-pp,00}}{2|p|^{2}}+\frac{\langle
V_{N}-\Pi_{\text{L}}(1-V_{N}\mathcal{R})V_{N}\rangle_{-pp,00}}{2|p|^{2}}\\\
&=\frac{\langle
V_{N}(1-\mathcal{R}V_{N})\rangle_{-pp,00}}{2|p|^{2}}\end{split}$ (14)

and otherwise $\langle\mathcal{R}V_{N}\rangle_{-pp,00}=0$ (by definition of
$\mathcal{R}$) s.t.
$N^{1-\kappa}\langle\mathcal{R}V_{N},\zeta_{N}\rangle_{00,00}\leq
CN^{\kappa-1}$. Similarly, $|\widehat{\phi}_{N}(p)|\leq
CN^{\kappa-1}(1+|p|^{2})^{-1}$ and Cauchy-Schwarz imply that

$\begin{split}&|N^{1-\kappa}\langle
V_{N},(1-\mathcal{R}V_{N})\Pi_{\text{L}}\phi_{N}\rangle_{00,00}|\\\ &\;\leq
C\|V\|_{1}\big{(}\|\Pi_{\text{L}}\phi_{N}\|_{\infty}+N^{-\alpha}\|\Pi_{\text{L}}V_{N}^{1/2}\Pi_{\text{L}}\phi_{N}\|_{\infty}\big{)}\leq
CN^{\alpha+\kappa-1}.\end{split}$

Combining the previous estimates yields (13). In fact, using
$N^{1-\kappa}\langle
V_{N}(1-\phi_{N})\rangle_{p+r\,q-r,pq}=\widehat{Vf}(r/N^{1-\kappa})$ we can
also compute $\langle V_{\text{ren}}\rangle_{00,00}$ to higher precision and
obtain that

$\begin{split}\langle
V_{\text{ren}}\rangle_{00,00}=&8\pi\mathfrak{a}+\frac{N^{\kappa-1}}{2}\sum_{\begin{subarray}{c}0\neq
s\in\text{P}_{\text{L}}\end{subarray}}\frac{(8\pi\mathfrak{a})^{2}}{|s|^{2}}+O(N^{\kappa-1})+O(N^{2\kappa+2\alpha-2}).\end{split}$

We omit the details as the second term is irrelevant for our range of
$\kappa$, it only becomes relevant if one wants to consider the Lee-Huang-Yang
order.

To get (7), we combine (13) with two further steps. On the one hand, we have
that

$|\langle V_{\text{ren}}\rangle_{pq,st}-\langle
V_{\text{ren}}\rangle_{(p+q)0,(s+t)0}|\leq
CN^{\kappa-1}\big{(}|p|+|q|+|s|+|t|\big{)}$ (15)

whenever $p+q=s+t$. This bound follows very similarly as the first bound in
(7): since $V_{N}$ is a multiplication operator, (15) clearly holds if we
replace $V_{\text{ren}}$ by $N^{1-\kappa}V_{N}$. Hence, it is enough to prove
(15) for $V_{\text{ren}}$ replaced by $N^{1-\kappa}V_{N}\mathcal{R}V_{N}$. In
this case, we write

$\begin{split}&N^{1-\kappa}\langle
V_{N}\mathcal{R}V_{N}\rangle_{pq,st}-N^{1-\kappa}\langle
V_{N}\mathcal{R}V_{N}\rangle_{(p+q)0,(s+t)0}\\\
&\;=N^{1-\kappa}\langle\varphi_{p}\otimes\varphi_{q},V_{N}\mathcal{R}V_{N}(\varphi_{s}\otimes\varphi_{t}-\varphi_{s+t}\otimes\varphi_{0})\rangle\\\
&\hskip
11.38092pt+N^{1-\kappa}\langle(\varphi_{p}\otimes\varphi_{q}-\varphi_{p+q}\otimes\varphi_{0}),V_{N}\mathcal{R}V_{N}\varphi_{s+t}\otimes\varphi_{0}\rangle.\end{split}$

Now, given any pair $k,l\in\Lambda^{*}$, a direct computation shows that

$\begin{split}N^{1-\kappa}\|V_{N}^{1/2}(\varphi_{k}\otimes\varphi_{l}-\varphi_{k+l}\otimes\varphi_{0})\|^{2}&=2\widehat{V}(0)-2\widehat{V}(l/N^{1-\kappa})\leq\frac{C|l|^{2}}{N^{2-2\kappa}}.\end{split}$

Note that the last step follows from a second order Taylor expansion and the
fact that
$(\nabla_{p}\widehat{V}(./N^{1-\kappa})(0)=N^{2-2\kappa}\int_{\mathbb{R}^{3}}dx\,(-ix)V(x)=0$,
$V$ being radial. Similarly, we get

$\begin{split}&N^{-2\alpha}\|\Pi_{\text{L}}V_{N}^{1/2}(\varphi_{k}\otimes\varphi_{l}-\varphi_{k+l}\otimes\varphi_{0})\|^{2}_{\infty}\\\
&=N^{-2\alpha}\Big{\|}\sum_{\begin{subarray}{c}s\in\text{P}_{\text{L}}:\\\
k+l-s\in\text{P}_{\text{L}}\end{subarray}}\\!\\!\\!\\!\\!\\!N^{2\kappa-2}\big{(}\widehat{V^{1/2}}((s-k)/N^{1-\kappa})-\widehat{V^{1/2}}((s-k-l)/N^{1-\kappa})\big{)}\varphi_{s}\otimes\varphi_{k+l-s}\Big{\|}_{\infty}^{2}\\\
&\leq C|l|^{2}/N^{2-2\kappa}.\end{split}$

Proceeding now as in the proof of the first bound in (7), we obtain (15).

Combining (15) with (13), the second bound in (7) thus follows if we prove
that

$|\langle V_{N}\mathcal{R}V_{N}\rangle_{p0,p0}-\langle
V_{N}\mathcal{R}V_{N}\rangle_{00,00}|\leq CN^{2\kappa-\alpha-2}|p|^{2}$ (16)

for every $p\in\Lambda^{*}_{+}$. This can be proved similarly as detailed in
[16, Appendix A]: define

$-\Delta^{(p)}:=(-i\nabla_{x_{1}}+p)^{2}-\Delta_{x_{2}},\hskip
14.22636pt\mathcal{R}^{(p)}:=\Pi_{\text{H}}^{+}\big{[}\Pi_{\text{H}}^{+}(-\Delta^{(p)}+V_{N})\Pi_{\text{H}}^{+}\big{]}^{-1}\Pi_{\text{H}}^{+},$

where the orthogonal projection $\Pi_{\text{H}}^{+}$ maps onto

$\overline{\text{span}\\{\varphi_{k}\otimes\varphi_{l}:(k,l)\in(\text{P}_{\text{L}}^{2})^{c}\text{
and }l\neq 0\\}}.$

Notice that this ensures $\langle\xi,-\Delta^{(p)}\xi\rangle\geq 4\pi^{2}$ for
every $\xi\in\Pi_{\text{H}}^{+}L^{2}(\Lambda^{2})$, by construction of the
projection $\Pi_{\text{H}}^{+}$. In particular, $\mathcal{R}^{(p)}$ is well
defined. Now, based on the observation

$\langle V_{N}\mathcal{R}V_{N}\rangle_{p0,p0}=\langle
V_{N}e^{-ipx_{1}}\mathcal{R}e^{ipx_{1}}V_{N}\rangle_{00,00}$

and the fact that
$-\Delta^{(p)}=e^{-ipx_{1}}(-\Delta_{x_{1}}-\Delta_{x_{2}})e^{ipx_{1}}$ for
$p\in\Lambda^{*}$, it follows that

$\begin{split}&\langle
V_{N}e^{-ipx_{1}}\mathcal{R}e^{ipx_{1}}V_{N}\rangle_{00,00}-\langle
V_{N}\mathcal{R}^{(p)}V_{N}\rangle_{00,00}\\\ &=-\langle
V_{N}\mathcal{R}^{(p)}e^{-ipx_{1}}\Pi_{\text{L}}(1-V_{N}\mathcal{R})V_{N}\rangle_{00,p0}+\langle
V_{N}(1-\mathcal{R}^{(p)}V_{N})\Pi_{\text{L}}^{+}e^{-ipx_{1}}\mathcal{R}V_{N}\rangle_{00,p0}.\end{split}$

Here, we set $\Pi_{\text{L}}^{+}:=1-\Pi_{\text{H}}^{+}$. Since
$V_{N}\mathcal{R}^{(p)}$ preserves the total momentum and projects onto a
subset of $(\text{P}_{\text{L}}^{2})^{c}$, we have that

$\begin{split}&\langle
V_{N}\mathcal{R}^{(p)}e^{-ipx_{1}}\Pi_{\text{L}}(1-V_{N}\mathcal{R})V_{N}\rangle_{00,p0}\\\
&=\sum_{q\in\text{P}_{\text{L}}^{c}\cap\Lambda^{*}_{+};\,s,t\in\text{P}_{\text{L}}}\langle
V_{N}\mathcal{R}^{(p)}\rangle_{00,-qq}\langle\varphi_{-q}\otimes\varphi_{q},\varphi_{s-p}\otimes\varphi_{t}\rangle\langle(1-V_{N}\mathcal{R})V_{N}\rangle_{st,p0}=0.\end{split}$

On the other hand, using that $\mathcal{R}=\Pi_{\text{H}}\mathcal{R}$ so that

$\begin{split}\langle\mathcal{R}V_{N}\rangle_{(p-q)q,p0}&=\frac{\langle
V_{N}-V_{N}\mathcal{R}V_{N}\rangle_{(p-q)q,p0}}{|p-q|^{2}+|q|^{2}}-\frac{\langle\Pi_{\text{L}}(1-V_{N}\mathcal{R})V_{N}\rangle_{(p-q)q,p0}}{|p-q|^{2}+|q|^{2}}\\\
&=\frac{\langle
V_{N}-V_{N}\mathcal{R}V_{N}\rangle_{(p-q)q,p0}}{|p-q|^{2}+|q|^{2}}\end{split}$

if $(p-q,q)\in(\text{P}_{\text{L}}^{2})^{c}$ and
$\langle\mathcal{R}V_{N}\rangle_{(p-q)q,p0}=0$ otherwise, we obtain that

$\begin{split}\big{\|}\Pi_{\text{L}}^{+}e^{-ipx_{1}}\mathcal{R}V_{N}e^{ipx_{1}}\big{\|}_{\infty}&\leq\sum_{s\in\text{P}_{\text{L}}}|\langle\mathcal{R}V_{N}\rangle_{(p-s)s,p0}|\leq
CN^{\alpha+\kappa-1},\\\
\big{\|}\Pi_{\text{L}}^{+}V_{N}^{1/2}\Pi_{\text{L}}^{+}e^{-ipx_{1}}\mathcal{R}V_{N}e^{ipx_{1}}\big{\|}_{\infty}&\leq\sum_{s,t\in\text{P}_{\text{L}}}\frac{C}{N^{3-3\kappa}(1+|t|^{2})}\leq
CN^{2\alpha+\kappa-1}.\end{split}$

Hence, arguing similarly as in the previous steps, we find that

$\begin{split}&|\langle
V_{N}(1-\mathcal{R}^{(p)}V_{N})\Pi_{\text{L}}^{+}e^{-ipx_{1}}\mathcal{R}V_{N}\rangle_{00,p0}|\\\
&\leq\frac{CN^{\kappa}}{N}\big{(}\big{\|}\Pi_{\text{L}}^{+}e^{-ipx_{1}}\mathcal{R}V_{N}e^{ipx_{1}}\big{\|}_{\infty}+N^{-\alpha}\big{\|}\Pi_{\text{L}}^{+}V_{N}^{1/2}\Pi_{\text{L}}^{+}e^{-ipx_{1}}\mathcal{R}V_{N}e^{ipx_{1}}\big{\|}_{\infty}\big{)}\leq
CN^{\alpha+2\kappa-2}.\end{split}$

Notice here that we used additionally the operator inequalities
$-\Delta^{(p)}\geq N^{2\alpha}\Pi_{\text{H}}^{+}$ and, as a consequence,
$\mathcal{R}^{(p)}\leq N^{-2\alpha}\Pi_{\text{H}}^{+}\leq N^{-2\alpha}$ in the
image

$\Pi_{\text{H}}^{+}\big{(}\textbf{1}_{\text{P}=0}\,L^{2}(\Lambda^{2})\big{)}=\Pi_{\text{H}}^{+}\,\overline{\text{span}\\{\varphi_{s}\otimes\varphi_{-s}:s\in\Lambda^{*}\\}}=\overline{\text{span}\\{\varphi_{s}\otimes\varphi_{-s}:|s|>N^{\alpha}\\}}$

of the space of zero total momentum
$\text{P}:=-i\nabla_{x_{1}}-i\nabla_{x_{2}}$ under $\Pi_{\text{H}}^{+}$, and
that both

$V_{N}^{1/2}\Pi_{\text{L}}^{+}V_{N}^{1/2}\Pi_{\text{L}}^{+}e^{-ipx_{1}}\mathcal{R}V_{N}e^{ipx_{1}}\in
L^{2}(\Lambda^{2})\;\;\text{ and
}\;\;V_{N}^{1/2}\Pi_{\text{L}}^{+}V_{N}^{1/2}\in L^{2}(\Lambda^{2})$

have zero total momentum.

Collecting the previous bounds, proving (16) reduces to proving that

$|\langle V_{N}\mathcal{R}^{(p)}V_{N}\rangle_{00,00}-\langle
V_{N}\mathcal{R}V_{N}\rangle_{00,00}|\leq CN^{2\kappa-\alpha-2}|p|^{2}.$

To show this, we use that

$|\langle\mathcal{R}^{(sp)}V_{N}\rangle_{-qq,00}|=\frac{|\langle
V_{N}-V_{N}\mathcal{R}^{(sp)}V_{N}\rangle_{-qq,00}|}{|sp-q|^{2}+|q|^{2}}\leq\frac{CN^{\kappa-1}}{|sp-q|^{2}+|q|^{2}}$

for all $s\in[0,1]$ and $|q|>N^{\alpha}$ (otherwise
$\langle\mathcal{R}^{(sp)}V_{N}\rangle_{-qq,00}=0$). Together with

$\langle V_{N}\mathcal{R}^{(p)}V_{N}\rangle_{00,00}=\langle
V_{N}\mathcal{R}^{(-p)}V_{N}\rangle_{00,00}$

and a second order Taylor expansion, we find that

$|\langle V_{N}\mathcal{R}^{(p)}V_{N}\rangle_{00,00}-\langle
V_{N}\mathcal{R}V_{N}\rangle_{00,00}|\leq
CN^{2\kappa-2}|p|^{2}\int_{|q|>N^{\alpha}}dq\,|q|^{-4}\leq
N^{2\kappa-\alpha-2}|p|^{2},$

which implies (16) and thus (7).

Finally, (8) is a direct consequence of the identity
$(-\Delta_{x_{1}}-\Delta_{x_{2}})\eta=\Pi_{\text{H}}V_{\text{ren}}\Pi_{\text{L}}$
and the bound (7) implying that
$(|p|^{2}+|q|^{2})|\langle\eta\rangle_{pq,st}|\leq
C\delta_{pq,st}\textbf{1}_{(\text{P}_{\text{L}}^{2})^{c}}((p,q))\textbf{1}_{\text{P}_{\text{L}}^{2}}((s,t))$.
∎

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