# Bogoliubov Theory for Trapped Bosons
in the Gross-Pitaevskii Regime

Christian Brennecke1, Benjamin Schlein2, Severin Schraven3111Corresponding
author<EMAIL_ADDRESS>

Institute for Applied Mathematics, University of Bonn,
Endenicher Allee 60, 53115 Bonn, Germany1

Institute of Mathematics, University of Zurich,
Winterthurerstrasse 190, 8057 Zurich, Switzerland2,3

###### Abstract

We consider systems of $N$ bosons in $\mathbb{R}^{3}$, trapped by an external
potential. The interaction is repulsive and has a scattering length of the
order $N^{-1}$ (Gross-Pitaevskii regime). We determine the ground state energy
and the low-energy excitation spectrum up to errors that vanish in the limit
$N\to\infty$.

##  1 Introduction and Main Results

We consider trapped bosons described by the Hamilton operator

$H_{N}=\sum_{j=1}^{N}\left[-\Delta_{x_{j}}+V_{\text{ext}}(x_{j})\right]+\sum_{1\leq
i<j\leq N}N^{2}V(N(x_{i}-x_{j})),$ (1.1)

acting on a dense subspace of $L^{2}_{s}(\mathbb{R}^{3N})$, the subspace of
$L^{2}(\mathbb{R}^{3N})$ consisting of functions that are symmetric with
respect to permutations of the $N$ particles. Here, $V_{\text{ext}}\in
L^{\infty}_{loc}(\mathbb{R}^{3})$ is a trapping potential with the property
that $V_{\text{ext}}(x)\to\infty$ for $|x|\to\infty$ (later, we will formulate
more precise assumptions on $V_{\text{ext}}$). As for the interaction $V$, we
assume that $V\in L^{3}(\mathbb{R}^{3})$ and that it is pointwise non-
negative, spherically symmetric and compactly supported.

The scattering length $\mathfrak{a}_{0}$ of $V$ is defined through the zero-
energy scattering equation

$\left[-\Delta+\frac{1}{2}V(x)\right]f(x)=0$ (1.2)

with the boundary condition $f(x)\to 1$, as $|x|\to\infty$. For $|x|$ large
enough, we have

$f(x)=1-\frac{\mathfrak{a}_{0}}{|x|}$

for a constant $\mathfrak{a}_{0}$ which is determined by the potential $V$.
This constant is called the scattering length of $V$. It is straightforward to
show that

$\displaystyle 8\pi\mathfrak{a}_{0}=\int_{\mathbb{R}^{3}}V(x)f(x)dx$

and, by scaling, (1.2) implies that

$\left[-\Delta+\frac{1}{2}N^{2}V(Nx)\right]f(Nx)=0.$

This means that the scattering length of the interaction $N^{2}V(N.)$
appearing in (1.1) is given by $\mathfrak{a}_{0}/N$, which characterises the
Gross-Pitaeavskii regime.

From [20, 19, 22] it is well-known that the ground state energy $E_{N}$ of the
Hamilton operator (1.1) satisfies

$\lim_{N\to\infty}\frac{E_{N}}{N}=\inf_{\psi\in
H^{1}(\mathbb{R}^{3}):\|\psi\|_{2}=1}\mathcal{E}_{GP}(\psi),$ (1.3)

where $\mathcal{E}_{GP}$ denotes the Gross-Pitaevskii functional, defined by

$\mathcal{E}_{GP}(\psi)=\int_{\mathbb{R}^{3}}\left(|\nabla\psi(x)|^{2}+V_{\text{ext}}(x)|\psi(x)|^{2}+4\pi\mathfrak{a}_{0}|\psi(x)|^{4}\right)dx.$
(1.4)

We remark that $\mathcal{E}_{GP}$ admits a unique normalized, strictly
positive minimizer which we denote for the rest of this paper by
$\varphi_{0}\in L^{2}(\mathbb{R}^{3})$. It satisfies the Euler-Lagrange
equation

$\displaystyle-\Delta\varphi_{0}+V_{\text{ext}}\varphi_{0}+8\pi\mathfrak{a}_{0}|\varphi_{0}|^{2}\varphi_{0}=\varepsilon_{GP}\varphi_{0}$

with the Lagrange multiplier
$\varepsilon_{GP}=\mathcal{E}_{GP}(\varphi_{0})+4\pi\mathfrak{a}_{0}\|\varphi_{0}\|_{4}^{4}$.
In the Appendix, more precisely in Lemma A.1 and Lemma A.2, we establish
regularity and decay properties of the minimizer $\varphi_{0}$, which will be
useful for our analysis.

From [18, 19, 22], it is also known that every approximate ground state of
(1.1), ie. every sequence of normalized wave functions $\psi_{N}\in
L^{2}_{s}(\mathbb{R}^{3N})$ satisfying

$\lim_{N\to\infty}\frac{1}{N}\langle\psi_{N},H_{N}\psi_{N}\rangle=\mathcal{E}_{\text{GP}}(\varphi_{0})\,,$

exhibits complete Bose-Einstein condensation in $\varphi_{0}$. In other words,
denoting by
$\gamma_{N}^{(1)}=\operatorname{tr}_{2,\dots,N}|\psi_{N}\rangle\langle\psi_{N}|$
the one–particle reduced density associated with a sequence of approximate
ground states $\psi_{N}$, we have

$\lim_{N\to\infty}\langle\varphi_{0},\gamma_{N}^{(1)}\varphi_{0}\rangle=1.$
(1.5)

For translation invariant systems (particles trapped in $\Lambda=[0;1]^{3}$
with periodic boundary conditions), (1.3) and (1.5) have been proved in [3, 6,
1, 16] to hold with the optimal rate of convergence. This result was
generalized in [21] (extending the approach of [9, 14]) to trapped systems
described by (1.1), under the assumption of a sufficiently small scattering
length $\mathfrak{a}_{0}$. In [11], we recently established (1.3) and (1.5)
with optimal rate of convergence and with no assumption on the size of
$\mathfrak{a}_{0}$. These bounds (obtained through a generalization of the
methods of [3, 6]) lead to the a-priori estimates stated below in Theorem 2.6
and Proposition 2.9 and represent a crucial ingredient for our analysis, aimed
at understanding the low-energy spectrum of (1.1), in the limit $N\to\infty$.

To state our main theorem, we introduce the notation

$H_{\text{GP}}=-\Delta+V_{\text{ext}}+8\pi\mathfrak{a}_{0}\varphi_{0}^{2}-\varepsilon_{GP},\hskip
14.22636ptE=\big{(}H_{\text{GP}}^{1/2}(H_{\text{GP}}+16\pi\mathfrak{a}_{0}\varphi_{0}^{2})H_{\text{GP}}^{1/2}\big{)}^{1/2}.$
(1.6)

The operators $H_{\text{GP}}$ and $E$ have discrete spectrum and in both
cases, $\varphi_{0}$ is the unique, positive and normalized ground state
vector with $H_{\text{GP}}\varphi_{0}=E\varphi_{0}=0$.

Let us make our assumptions on the interaction potential $V\in
L^{3}(\mathbb{R}^{3})$ and the trapping potential
$V_{\text{ext}}:\mathbb{R}^{3}\to\mathbb{R}$ more precise. Our standing
assumptions throughout the rest of this paper are as follows

$\begin{split}(1)\;&V\in L^{3}(\mathbb{R}^{3}),V(x)\geq 0\text{ for
}a.e.\;x\in\mathbb{R}^{3},V(x)=V(y)\text{ if }|x|=|y|,\\\ &V\text{ compactly
supported},\\\ (2)\;&V_{\text{ext}}\in
C^{1}\big{(}\mathbb{R}^{3};[0;\infty)\big{)},V_{\text{ext}}(x)\to\infty\text{
as }|x|\to\infty,\\\
&\exists\;C>0\;\forall\;x,y\in\mathbb{R}^{3}:V_{\text{ext}}(x+y)\leq
C(V_{\text{ext}}(x)+C)(V_{\text{ext}}(y)+C),\\\ &\nabla V_{\text{ext}}\text{
has at most exponential growth as }|x|\to\infty,\\\
(3)\;&(H_{\text{GP}}+1)^{-3/4-\varepsilon}e^{-\alpha|x|}\text{ is a Hilbert-
Schmidt operator, for any $\varepsilon>0$}\\\ &\text{and some
$\alpha>0$}.\end{split}$ (1.7)

It is worth mentioning that the assumption (2) implies that $V_{\text{ext}}$
grows at most exponentially fast (see [11, Appendix A] for a proof). Combining
this with the fact that $\varphi_{0}$ decays faster than any exponential (see
(A.1)) implies that we can control any $L^{p}$-norm of
$V_{\text{ext}}\varphi_{0}$.

Our assumptions are technical and we have not attempted to optimize them. The
assumption that $V_{\text{ext}}\geq 0$ is for computational convenience only
and can always be arranged by an energy shift (notice that we do not need to
assume $V_{\text{ext}}\geq 0$ in [11]).

######  Theorem 1.1.

Let $V$ and $V_{\text{ext}}$ be as in (1.7). Then, there exists $\rho>0$ such
that, in the limit $N\to\infty$, the ground state energy $E_{N}$ of $H_{N}$,
defined in (1.1), is given by

$\begin{split}E_{N}&=N\mathcal{E}_{GP}(\varphi_{0})-4\pi\mathfrak{a}_{0}\|\varphi_{0}\|_{4}^{4}+E_{\text{Bog}}+\mathcal{O}\big{(}N^{-\rho}\big{)},\end{split}$
(1.8)

where $E_{\text{Bog}}$ a the finite constant (independent of $N$) given by

$\begin{split}E_{\text{Bog}}=\frac{1}{2}\lim_{\delta\to
0}&\left\\{\operatorname{tr}_{\perp\varphi_{0}}\Big{[}\Big{(}H_{\text{GP}}^{1/2}(H_{\text{GP}}+16\pi\mathfrak{a}_{0}1\\!\\!1_{\delta}\varphi_{0}^{2})H_{\text{GP}}^{1/2}\Big{)}^{1/2}-H_{\text{GP}}-8\pi\mathfrak{a}_{0}1\\!\\!1_{\delta}\varphi_{0}^{2}\Big{]}\right.\\\
&\hskip
156.49014pt\left.+\frac{(8\pi\mathfrak{a}_{0})^{2}}{2}\operatorname{tr}\Big{[}\varphi_{0}^{2}1\\!\\!1_{\delta}\frac{1}{-\Delta}1\\!\\!1_{\delta}\varphi_{0}^{2}\Big{]}\right\\}\end{split}$
(1.9)

with $1\\!\\!1_{\delta}$ denoting the approximation of the identity operator
with integral kernel
$1\\!\\!1_{\delta}(x;y)=(2\pi\delta)^{-3/2}e^{-(x-y)^{2}/2\delta^{2}}$ (but
any reasonable choice of the approximating sequence would do). Here,
$\operatorname{tr}_{\bot\varphi_{0}}$ denotes the trace over the orthogonal
complement $\\{\varphi_{0}\\}^{\bot}$.

Moreover, the spectrum of $H_{N}-E_{N}$ below a threshold $\zeta>0$ (assuming
$\zeta\leq CN^{-\rho/5-\varepsilon}$ for some $\varepsilon>0$) consists of
eigenvalues of the form

$\sum_{i=1}^{\infty}n_{i}e_{i}+\mathcal{O}\big{(}N^{-\rho}(1+\zeta^{5}+N^{-5\rho}\zeta^{30})\big{)},$
(1.10)

where the $(e_{j})_{j\in\mathbb{N}}$ denote the eigenvalues of the operator
$E$, defined in (1.6), and where $n_{i}\in\mathbb{N}$ with $n_{i}\neq 0$ for
finitely many $i\in\mathbb{N}$.

Remark: In (1.9), we cannot take directly the limit $\delta\to 0$, replacing
$1\\!\\!1_{\delta}$ with the identity, because the resulting operator is not
of trace class and the trace would not be well-defined. Nevertheless, we will
show in Section 3.3 that the limit defining $E_{\text{Bog}}$ exists and that
it is explicitly given by

$\begin{split}E_{\emph{Bog}}&=\frac{\kappa^{2}}{4}(8\pi\mathfrak{a}_{0})^{2}\operatorname{tr}\left[\varphi_{0}^{2}(-\Delta+\kappa^{2})^{-1}(-\Delta)^{-1}\varphi_{0}^{2}\right]\\\
&\hskip
14.22636pt+\frac{1}{4}(8\pi\mathfrak{a}_{0})^{2}\left\|\varphi_{0}(-\Delta+\kappa^{2})^{-1}[\varphi_{0}^{2},-\Delta](-\Delta+\kappa^{2})^{-1/2}\right\|_{\text{HS}}^{2}\\\
&\hskip
14.22636pt+\frac{1}{4}(8\pi\mathfrak{a}_{0})^{2}\left\|\varphi_{0}(-\Delta+\kappa^{2})^{-1}\nabla\varphi_{0}\right\|_{\text{HS}}^{2}\\\
&\hskip
14.22636pt+\frac{(8\pi\mathfrak{a}_{0})^{2}}{4}\operatorname{tr}\,\Big{[}\varphi_{0}^{2}\Big{(}\frac{1}{-\Delta+\kappa^{2}}-\frac{1}{H_{\text{GP}}+\kappa^{2}}\Big{)}\varphi_{0}^{2}\Big{]}\\\
&\hskip
14.22636pt+\frac{(8\pi\mathfrak{a}_{0})^{2}}{4\kappa^{2}}\|\varphi_{0}^{3}\|^{2}+\frac{(8\pi\mathfrak{a}_{0})^{2}}{4}\|(H_{\text{GP}}+\kappa^{2})^{-1/2}Q\varphi_{0}^{3}\|^{2}\\\
&\hskip
14.22636pt-\frac{(8\pi\mathfrak{a}_{0})^{2}\kappa^{2}}{4}\operatorname{tr}_{\perp\varphi_{0}}\Big{[}\varphi_{0}^{2}Q\frac{1}{H_{\text{GP}}(H_{\text{GP}}+\kappa^{2})}Q\varphi_{0}^{2}\Big{]}\\\
&\hskip
14.22636pt-\frac{(8\pi\mathfrak{a}_{0})^{2}}{\pi}\int_{0}^{\infty}ds\,\sqrt{s}\,\operatorname{tr}_{\perp\varphi_{0}}\frac{H_{\text{GP}}^{1/2}}{s+H_{\text{GP}}^{2}}\Big{[}\Big{[}\varphi_{0}^{2},\frac{H_{\text{GP}}}{s+H_{\text{GP}}^{2}}\Big{]},\varphi_{0}^{2}\Big{]}\frac{H_{\text{GP}}^{1/2}}{s+H_{\text{GP}}^{2}}\\\
&\hskip
14.22636pt-\frac{4(8\pi\mathfrak{a}_{0})^{3}}{\pi}\int_{0}^{\infty}ds\,\sqrt{s}\operatorname{tr}_{\perp\varphi_{0}}\Big{[}\frac{1}{s+H_{\text{GP}}^{2}}H_{\text{GP}}^{1/2}\varphi_{0}^{2}H_{\text{GP}}^{1/2}\Big{]}^{3}\\\
&\hskip
170.71652pt\times\frac{1}{s+H_{\text{GP}}^{1/2}(H_{\text{GP}}+16\pi\mathfrak{a}_{0}\varphi_{0}^{2})H_{\text{GP}}^{1/2}}\end{split}$
(1.11)

for any $\kappa>0$ large enough. In Sect. 3.3 we also prove that all
contributions on the r.h.s. of (1.11) are finite.

Theorem 1.1 extends to the Gross-Pitaevskii regime earlier results obtained
for systems of bosons in the mean-field limit. The low-energy spectrum of
Hamilton operators describing mean-field bosons has been established in the
translation invariant case in [25, 12] and for systems trapped by external
fields in [15, 17]. More precise expansions in powers of $1/N$ for the ground
state energy and low energy excitations of mean field Hamiltonians has been
recently obtained in [24, 8].

For translation invariant systems consisting of $N$ bosons moving in the box
$\Lambda=[0;1]^{3}$ with periodic boundary conditions, Theorem 1.1 has been
recently shown in [5], using a rigorous version of Bogoliubov theory, see [7].
While the strategy that we are going to use to show Theorem 1.1 is similar to
the one developed in [5], the lack of translation invariance makes the proof
more complicated.

Let us briefly explain the main steps. First of all, in Section 2, we are
going to factor out the Bose-Einstein condensate and restrict our attention on
its orthogonal excitations. We do so through a unitary transformation $U_{N}$,
first introduced in [17], mapping the Hilbert space
$L^{2}_{s}(\mathbb{R}^{3N})$ onto the truncated Fock space $\mathcal{F}^{\leq
N}_{\perp\varphi_{0}}$ constructed on the orthogonal complement of the
condensate wave function $\varphi_{0}$. This procedure, which can be seen as a
rigorous version of Bogoliubov c-number substitution, leads to the excitation
Hamiltonian $\mathcal{L}_{N}=U_{N}H_{N}U_{N}^{*}$. In second quantized form,
the operator $\mathcal{L}_{N}$ can be written as the sum of a constant and of
terms that are linear, quadratic, cubic and quartic in (modified) creation and
annihilation operators. In contrast with the translation invariant case, where
$\mathcal{L}_{N}$ could be easily expressed in momentum space, here we have to
use operator valued distributions, localized in position space.

It turns out that (similarly to the translation invariant case) cubic and
quartic terms in the excitation Hamiltonian $\mathcal{L}_{N}$ still contain
important contributions to the ground state energy and to the energy of low-
lying excited eigenvalues. To extract these contributions, we will conjugate
$\mathcal{L}_{N}$ with a (generalized) Bogoliubov transformation having the
form $e^{B}$, with $B$ quadratic in (modified) creation and annihilation
operators.

Through the unitary operator $e^{B}$, we aim at removing short-scale
correlations among particles; to reach this goal, we choose the integral
kernel $\eta$ defining $B$ through the ground state solution of the Neumann
problem associated with the interaction potential $N^{2}V(N.)$, in the ball of
radius $\ell$, for a fixed $0<\ell<1$.

Conjugation with $e^{B}$ leads to a new excitation Hamiltonian
$\mathcal{G}_{N}=e^{-B}\mathcal{L}_{N}e^{B}$, with renormalized quadratic off-
diagonal terms. The properties of $\mathcal{G}_{N}$ are stated in Prop. 2.5,
which can be shown similarly as in the translation invariant setting. For
completeness, we include a sketch of the proof, but we defer it to Section 4.

Looking at the form of $\mathcal{G}_{N}$ in Prop. 2.5, we will observe that,
while conjugation with $e^{B}$ renormalizes quadratic terms, it does not
change cubic terms. Similarly as in the translation invariant case, these
terms cannot be neglected; they still contribute to the low-energy spectrum of
the Hamilton operator. To extract the important contributions from the cubic
terms, we conjugate $\mathcal{G}_{N}$ with another unitary operator, this time
given by the exponential $e^{A}$ of a cubic expression in (modified) creation
and annihilation operators. This leads to a second renormalized excitation
Hamiltonian $\mathcal{J}_{N}=e^{-A}\mathcal{G}_{N}e^{A}$, whose properties are
stated in Prop 2.8. As already observed in [26] (in a slightly different
context) and then in [5], it is important that scattering events described by
$A$ involve two particles with high momenta and one with small momentum (and
one particle from the condensate). For technical reasons, we need to restrict
to small momenta using a Gaussian, rather than a sharp cutoff (this guarantees
that the cutoff decays sufficiently fast, also in position space). For this
reason, the proof of Prop. 2.8, given in Section 5, is a bit more complicated
than the proof of the corresponding result in the translation invariant
setting (where sharp cutoffs could be used).

To control error terms produced by conjugation with $e^{B}$ and $e^{A}$, we
will make use of strong a-priori bounds stated in Theorem 2.6 and Proposition
2.9; these important estimates follow from the proof of complete Bose-Einstein
condensation obtained in [11].

It follows from Prop. 2.8 that, up to negligible contributions, the
renormalized excitation Hamiltonian $\mathcal{J}_{N}$ is quadratic in creation
and annihiliation operators (in fact, $\mathcal{J}_{N}$ still contains a
positive quartic term, which however can be neglected when proving lower
bounds and which only produces small errors when evaluated on trial states
that are appropriate to show upper bounds on the low-energy eigenvalues). In
Section 3, we proceed therefore with the diagonalization of the quadratic part
of $\mathcal{J}_{N}$. This is the main novelty of our analysis; the lack of
translation invariant makes it much more involved than for particles on the
torus. This is already clear by looking at the operators $H_{\text{GP}}$ and
$E$, defined in (1.6), which determine the ground state energy and the excited
eigenvalues of the Hamiltonian. In the translation invariant case,
$\varphi_{0}\equiv 1$ and, in momentum space, $H_{\text{GP}}=p^{2}$ and
$E=(|p|^{4}+16\pi\mathfrak{a}_{0}p^{2})^{1/2}$. For systems trapped by
external potentials, on the other hand, $\varphi_{0}$ is not a constant and it
does not commute with $H_{\text{GP}}$.

Finally, let us remark that while writing up the last details of our
manuscript, a result similar to (1.10) was posted in [23]. While the strategy
used in [23] is similar to ours (and to the one previously developed in [5],
in the translation invariant setting), the details of its implementation are
different. Let us explain the main differences.

  * •

After removing the condensate through conjugation with the unitary map
$U_{N}$, the authors of [23] extend the resulting excitation Hamiltonian,
defined on the truncated Fock space with at most $N$ particles, to the full
Fock space. This step, which can be justified with a localization argument in
the number of particles operator, allows them to work with standard creation
and annihilation operators $a^{*},a$, rather than with the modified fields
$b^{*},b$ defined on $\mathcal{F}_{\perp\varphi_{0}}^{\leq N}$ (see (2.4)
below), and simplifies a bit the analysis (notice however that the action of
generalized Bogoliubov transformation, defined in terms of modified creation
and annihilation operators, $b^{*},b$ can be controlled quite well with bounds
like those collected in Lemma 2.4, which are, by now, standard tools; for this
reason, working with $b^{*},b$ and with generalized Bogoliubov transformations
is, at least conceptually, not much more difficult than working with
$a^{*},a$).

  * •

Another important difference between our work and [23] is the choice of the
integral kernel $\eta$ defining the (standard or generalized) Bogoliubov
transformation and the cubic transformation renormalizing the excitation
Hamiltonian. In our work, $\eta$ is chosen through the ground state solution
of the Neumann problem, defined on a ball of radius $\ell$, for a fixed
$0<\ell<1$, independent of $N$. In [23], on the other hand, $\eta$ is defined
through the solution of the zero-energy scattering equation (1.2), cutting off
correlations at distance $1/N\ll\ell\ll 1$. While the choice of $\eta$ in
terms of the solution of the Neumann problem has the advantage that the cutoff
at $|x|=\ell$ does not contribute directly to the kinetic energy (in the sense
that the radial derivative of $\eta$ vanishes at $|x|=\ell$), the definition
in terms of the zero-energy scattering equation allows for slightly more
general interaction potentials (the result of [23] only requires $V\in
L^{1}(\mathbb{R}^{3})$; in Theorem 1.1, on the other hand, we assume $V\in
L^{3}(\mathbb{R}^{3})$, because this condition is needed to derive important
properties of the solution of the Neumann problem). Also the different choice
of $\ell$ produces important differences between the two approaches. We choose
$\ell$ of order one, independent of $N$. This means that after quadratic and
cubic renormalizations, the resulting excitation Hamiltonian only requires
diagonalization on momenta of order one. In [23], the authors choose
$\ell=N^{-\alpha}$ for some $0<\alpha<1$. The advantage of this choice is that
the integral kernel $\eta$ has small Hilbert-Schmidt norm; as a consequence,
in a commutator expansion, terms of higher order in $\eta$ are negligible. The
price to pay for this choice of $\ell$ is that, in the diagonalization, we
need to act also on large momenta (the diagonalization, in this case, needs to
renormalize also all momenta scales larger that one and smaller than
$\ell^{-1}$). At the end, choosing $\ell$ small, as done in [23], seems to
make the analysis slightly simpler.

  * •

While we and the authors of [23] both apply the same general strategy to
diagonalize the renormalized excitation Hamiltonian, in our paper we make an
additional effort to identify all non-vanishing contributions to the ground
state energy $E_{N}$. In particular, our expression (1.8), with the Bogoliubov
energy given by (1.9) or by (1.11), shows that the ground state energy, in the
limit $N\to\infty$, only depends on the interaction potential through its
scattering length $\mathfrak{a}_{0}$ (notice that also the minimizer
$\varphi_{0}$ of the Gross-Pitaevskii energy functional only depends on $V$
through its scattering length). In [23], on the other hand, the ground state
energy is left as a complicated expression depending on $N$ and the limit is
not further investigated (there is also no proof of the fact that
$E_{N}-N\mathcal{E}_{\text{GP}}(\varphi_{0})$ approaches a limit, as
$N\to\infty$). In fact, it should be noted that our assumptions that $\nabla
V_{\text{ext}}$ grows at most exponentially and that
$(H_{\text{GP}}+1)^{-3/4-\varepsilon}e^{-\alpha|x|}$ is a Hilbert-Schmidt
operator, for any $\varepsilon>0$ and some $\alpha>0$, are only needed to
study the behaviour of the ground state energy $E_{N}$, in the limit
$N\to\infty$, and to establish the validity of (1.8), with $E_{\text{Bog}}$ as
in (1.11).

Acknowledgements. BS acknowledges financial support from the NCCR SwissMAP and
from the ERC through the ERC-AdG CLaQS (grant agreement No. 834782). BS and SS
also acknowledge financial support from the Swiss National Science Foundation
through the Grant “Dynamical and energetic properties of Bose-Einstein
condensates”.

##  2 Excitation Hamiltonians

To prove Theorem 1.1 it is convenient to factor out the Bose-Einstein
condensate and to focus instead on its orthogonal excitations. Recalling that
$\varphi_{0}\in L^{2}(\mathbb{R}^{3})$ denotes the unique normalized minimizer
of the Gross-Pitaevskii functional (1.4), we consider the unitary map
$U_{N}:L^{2}_{s}(\mathbb{R}^{3N})\to\mathcal{F}_{\perp\varphi_{0}}^{\leq N}$
mapping $L^{2}_{s}(\mathbb{R}^{3N})$ into the truncated Fock space

$\mathcal{F}_{\perp\varphi_{0}}^{\leq
N}=\bigoplus_{n=0}^{N}L^{2}_{\perp\varphi_{0}}(\mathbb{R}^{3})^{\otimes_{s}n}$

constructed on the orthogonal complement
$L^{2}_{\perp\varphi_{0}}(\mathbb{R}^{3})$ of $\varphi_{0}$. The map $U_{N}$,
first introduced in [17], is defined by
$U_{N}\psi_{N}=\\{\alpha_{0},\alpha_{1},\dots,\alpha_{N}\\}$, if

$\psi_{N}=\alpha_{0}\varphi_{0}^{\otimes
N}+\alpha_{1}\otimes_{s}\varphi_{0}^{\otimes(N-1)}+\dots+\alpha_{N}$

with $\alpha_{j}\in L^{2}_{\perp\varphi_{0}}(\mathbb{R}^{3})^{\otimes j}$, for
all $j=0,1,\dots,N$ (here $\otimes_{s}$ denotes the symmetric tensor product).
The action of $U_{N}$ on creation and annihilation operators is given by

$\begin{split}U_{N}a^{*}(\varphi_{0})a(\varphi_{0})U_{N}^{*}&=N-\mathcal{N},\\\
U_{N}a^{*}(f)a(\varphi_{0})U_{N}^{*}&=a^{*}(f)\sqrt{N-\mathcal{N}},\\\
U_{N}a^{*}(\varphi_{0})a(g)U_{N}^{*}&=\sqrt{N-\mathcal{N}}a(g),\\\
U_{N}a^{*}(f)a(g)U_{N}^{*}&=a^{*}(f)a(g)\end{split}$ (2.1)

for all $f,g\in L^{2}_{\perp\varphi_{0}}(\mathbb{R}^{3})$, where $\mathcal{N}$
denotes the number of particles operator in
$\mathcal{F}_{\bot\varphi_{0}}^{\leq N}$.

With $U_{N}$, we define the excitation Hamiltonian
$\mathcal{L}_{N}=U_{N}H_{N}U_{N}^{*}$, acting on a dense subspace of
$\mathcal{F}_{\bot\varphi_{0}}^{\leq N}$. Using the relations (2.1),
$\mathcal{L}_{N}$ can be written as

$\mathcal{L}_{N}=\mathcal{L}^{(0)}_{N}+\mathcal{L}^{(1)}_{N}+\mathcal{L}^{(2)}_{N}+\mathcal{L}^{(3)}_{N}+\mathcal{L}^{(4)}_{N},$
(2.2)

where, in the sense of forms in $\mathcal{F}_{\bot\varphi_{0}}^{\leq N}$, we
have that

$\begin{split}\mathcal{L}_{N}^{(0)}&=\big{\langle}\varphi_{0},\big{[}-\Delta+V_{\text{ext}}+\frac{1}{2}\big{(}N^{3}V(N\cdot)*|\varphi_{0}|^{2}\big{)}\big{]}\varphi_{0}\big{\rangle}(N-\mathcal{N})\\\
&\qquad-\frac{1}{2}\big{\langle}\varphi_{0},\big{(}N^{3}V(N\cdot)*|\varphi_{0}|^{2}\big{)}\varphi_{0}\big{\rangle}(\mathcal{N}+1)(1-\mathcal{N}/N),\\\
\mathcal{L}_{N}^{(1)}&=\sqrt{N}b\left(\left(N^{3}V(N\cdot)*|\varphi_{0}|^{2}-8\pi\mathfrak{a}_{0}|\varphi_{0}|^{2}\right)\varphi_{0}\right)\\\
&\hskip
14.22636pt-\frac{\mathcal{N}+1}{\sqrt{N}}b\left(\left(N^{3}V(N\cdot)*|\varphi_{0}|^{2}\right)\varphi_{0}\right)+\text{h.c.},\\\
\mathcal{L}_{N}^{(2)}&=\int dx\;a_{x}^{*}(-\Delta_{x})a_{x}+\int
dx\;V_{\text{ext}}(x)a_{x}^{*}a_{x}\\\ &\hskip 14.22636pt+\int
dxdy\;N^{3}V(N(x-y))\varphi_{0}^{2}(y)\Big{(}b_{x}^{*}b_{x}-\frac{1}{N}a_{x}^{*}a_{x}\Big{)}\\\
&\hskip 14.22636pt+\int
dxdy\;N^{3}V(N(x-y))\varphi_{0}(x)\varphi_{0}(y)\Big{(}b_{x}^{*}b_{y}-\frac{1}{N}a_{x}^{*}a_{y}\Big{)}\\\
&\hskip 14.22636pt+\frac{1}{2}\int
dxdy\;N^{3}V(N(x-y))\varphi_{0}(y)\varphi_{0}(x)\Big{(}b_{x}^{*}b_{y}^{*}+b_{x}b_{y}\Big{)},\\\
\mathcal{L}_{N}^{(3)}&=\int
N^{5/2}V(N(x-y))\varphi_{0}(y)\big{(}b_{x}^{*}a^{*}_{y}a_{x}+a^{*}_{x}a_{y}b_{x}\big{)}dxdy,\\\
\mathcal{L}_{N}^{(4)}&=\frac{1}{2}\int
dxdy\;N^{2}V(N(x-y))a_{x}^{*}a_{y}^{*}a_{y}a_{x}.\end{split}$ (2.3)

For $x\in\mathbb{R}^{3}$, we introduced here operator-valued distributions

$b_{x}=\sqrt{\frac{N-\mathcal{N}}{N}}\,a_{x},\qquad
b^{*}_{x}=a^{*}_{x}\,\sqrt{\frac{N-\mathcal{N}}{N}}$ (2.4)

with $a_{x}^{*},a_{x}$ denoting standard creation and annihilation operators.
From the canonical commutation relations $[a_{x},a_{y}^{*}]=\delta(x-y)$,
$[a_{x},b_{y}]=[a_{x}*,a_{y}^{*}]=0$, we obtain

$[b_{x},b_{y}^{*}]=\left(1-\frac{\mathcal{N}}{N}\right)\delta(x-y)-\frac{1}{N}a_{y}^{*}a_{x},\hskip
28.45274pt[b_{x},b_{y}]=[b_{x}^{*},b_{y}^{*}]=0$ (2.5)

and

$\displaystyle[b_{x},a_{y}^{*}a_{z}]$
$\displaystyle=\delta(x-y)b_{z},\qquad[b_{x}^{*},a_{y}^{*}a_{z}]=-\delta(x-z)b_{y}^{*}.$

For $f\in L^{2}_{\perp\varphi_{0}}(\mathbb{R}^{3})$, it is also useful to
define the operators

$b(f)=\int\overline{f(x)}\,b_{x}\,dx=\sqrt{\frac{N-\mathcal{N}}{N}}a(f),\qquad
b(f)=\int f(x)\,b^{*}_{x}\,dx=a^{*}(f)\sqrt{\frac{N-\mathcal{N}}{N}}.$

Modified creation and annihilation operators can be controlled as the standard
creation and annihilation operators, ie.

$\begin{split}\|b(f)\xi\|\leq\|f\|\|\mathcal{N}^{1/2}\xi\|,\qquad\|b^{*}(f)\xi\|\leq\|f\|\|(\mathcal{N}+1)^{1/2}\xi\|.\end{split}$

where we recall that $\mathcal{N}$ denotes the number of particles operator on
$\mathcal{F}_{\perp\varphi_{0}}^{\leq N}$.

To determine the spectrum of $\mathcal{L}_{N}$ (and thus the spectrum of
(1.1)), a naive application of Bogoliubov method would suggest to ignore all
contributions except the constant and quadratic terms and to diagonalize the
resulting Fock space Hamiltonian. This, however, does not yield the correct
spectrum. In fact, it does not even produce the right leading order
contribution to the ground state energy. The cubic and quartic terms in (2.2)
contain relevant contributions to the energy. The point is that unitary
conjugation with the map $U_{N}$ (which leads to the excitation Hamiltonian
$\mathcal{L}_{N}$) removes products of the condensate wave function, but it
leaves correlations in the excitation vector
$U_{N}\psi_{N}=\\{\alpha_{0},\dots,\alpha_{N}\\}$. To extract the important
contributions to the energy from cubic and quartic terms in (2.3), we need
therefore to factor out some of the correlations among the particles. To this
end, we follow the strategy introduced in [6] and we conjugate
$\mathcal{L}_{N}$ with a generalized Bogoliubov transformation.

To define the kernel of the Bogoliubov transformation, we consider the Neumann
ground state $f_{\ell}$ that solves

$\left[-\Delta+\frac{1}{2}V\right]f_{\ell}=\lambda_{\ell}f_{\ell}$ (2.6)

on the ball $|x|\leq N\ell$, for some fixed $0<\ell<1$. As explained below, we
will choose the parameter $\ell>0$ sufficiently small, but independent of $N$.
To simplify notation, we do not indicate the $N$-dependence in $f_{\ell}$ and
$\lambda_{\ell}$. By radial symmetry of $f_{\ell}$, we can normalize it so
that $f_{\ell}(x)=1$ if $|x|=N\ell$. By scaling, $f_{\ell}(N.)$ solves

$\left[-\Delta+\frac{N^{2}}{2}V(Nx)\right]f_{\ell}(Nx)=N^{2}\lambda_{\ell}f_{\ell}(Nx)$

on the ball where $|x|\leq\ell$. We then extend $f_{\ell}(N.)$ to
$\mathbb{R}^{3}$, by setting $f_{N,\ell}(x)=f_{\ell}(Nx)$ if $|x|\leq\ell$ and
$f_{N,\ell}(x)=1$ for $x\in\mathbb{R}^{3}$ with $|x|>\ell$. Thus, $f_{N,\ell}$
solves

$\left(-\Delta+\frac{N^{2}}{2}V(N.)\right)f_{N,\ell}=N^{2}\lambda_{\ell}f_{N,\ell}\chi_{\ell},$
(2.7)

where $\chi_{\ell}$ denotes the characteristic function of the ball
$B_{\ell}(0)\subset\mathbb{R}^{3}$. Finally, we let $w_{\ell}=1-f_{\ell}$.
Notice that $w_{\ell}(N.)$ is support in $B_{\ell}(0)$. Defining the Fourier
transform of $w_{\ell}$ through

$\widehat{w}_{\ell}(p)=\int dx\;w_{\ell}(x)e^{-2\pi ipx},$

we obtain

$\int dx\;w_{\ell}(Nx)e^{-2\pi ipx}=\frac{1}{N^{3}}\widehat{w}_{\ell}(p/N).$

From (2.7), we find

$\displaystyle\begin{split}-4\pi^{2}p^{2}\widehat{w}_{\ell}(p/N)+\frac{N^{2}}{2}(\widehat{V}(./N)\ast\widehat{f}_{N,\ell})(p)=N^{5}\lambda_{\ell}(\widehat{\chi}_{\ell}\ast\widehat{f}_{N,\ell})(p).\end{split}$

The next lemma collects important properties of $f_{\ell},w_{\ell}$ and the
Neumann eigenvalue $\lambda_{\ell}$. Its proof is based on [13, Lemma A.1],
[6, Lemma 4.1] and can be found in [5, Appendix B] and [11, Lemma 3.2].

######  Lemma 2.1.

Let $V\in L^{3}(\mathbb{R}^{3})$ be non-negative, compactly supported and
spherically symmetric. Fix $\ell>0$ and let $f_{\ell}$ denote the solution of
(2.6).

  1. i)

We have that

$\displaystyle\lambda_{\ell}=\frac{3\mathfrak{a}_{0}}{(\ell
N)^{3}}\left(1+\frac{9}{5}\frac{\mathfrak{a}_{0}}{\ell
N}+\mathcal{O}\big{(}\mathfrak{a}_{0}^{2}/(\ell N)^{2}\big{)}\right).$

  2. ii)

We have $0\leq f_{\ell},w_{\ell}\leq 1$ and there exists a constant $C>0$ such
that

$\left|\int_{\mathbb{R}^{3}}V(x)f_{\ell}(x)dx-8\pi\mathfrak{a}_{0}\Big{(}1+\frac{3}{2}\frac{\mathfrak{a}_{0}}{\ell
N}\Big{)}\right|\leq\frac{C\mathfrak{a}_{0}^{3}}{(\ell N)^{2}}$ (2.8)

for all $\ell\in(0;1)$, $N\in\mathbb{N}$.

  3. iii)

There exists a constant $C>0$ such that

$w_{\ell}(x)\leq\frac{C}{|x|+1}\quad\text{ and }\quad|\nabla
w_{\ell}(x)|\leq\frac{C}{|x|^{2}+1}$ (2.9)

for all $x\in\mathbb{R}^{3}$, $\ell\in(0;1)$ and $N\in\mathbb{N}$ large
enough. Moreover,

$\displaystyle\Big{|}\frac{1}{(N\ell)^{2}}\int_{\mathbb{R}^{3}}w_{\ell}(x)dx-\frac{2}{5}\pi\mathfrak{a}_{0}\Big{|}\leq\frac{C\mathfrak{a}_{0}^{2}}{N\ell}$

for all $\ell\in(0;1)$ and $N\in\mathbb{N}$ large enough.

  4. iv)

There exists a constant $C>0$ such that

$|\widehat{w}_{\ell}(p)|\leq\frac{C}{|p|^{2}}$ (2.10)

for all $p\in\mathbb{R}^{3}$, $\ell\in(0;1)$ and $N\in\mathbb{N}$ large
enough.

We define the kernel $k\in L_{s}^{2}(\mathbb{R}^{3}\times\mathbb{R}^{3})$
through

$k(x;y)=-Nw_{\ell}(N(x-y))\varphi_{0}(x)\varphi_{0}(y).$ (2.11)

With $Q=1-|\varphi_{0}\rangle\langle\varphi_{0}|$ denoting the orthogonal
projection onto the orthogonal complement of $\varphi_{0}$, we set

$\eta(x;y)=\big{[}\big{(}Q\otimes Q\big{)}k\big{]}(x;y).$ (2.12)

The following lemma (whose proof can be found in [2, Lemma 3.3], [10, Lemma
4.3] and in [11, Lemma 2.2]) summarizes important properties of the kernels
$k,\eta$ and $\mu=\eta-k$.

######  Lemma 2.2.

Assume (1.7) and let $\ell\in(0;1)$. Then, there exists $C>0$, uniform in $N$,
$\ell\in(0;1)$ and $x\in\mathbb{R}^{3}$, such that for $i=1,2$ it holds true
that

$\begin{split}\|\eta\|&\leq C\ell^{1/2},\hskip 14.22636pt\|k\|\leq
C\ell^{1/2},\hskip 14.22636pt\|\mu\|\leq C\ell^{1/2},\\\ \|\eta_{x}\|&\leq
C|\varphi_{0}(x)|,\hskip 14.22636pt\|k_{x}\|\leq C|\varphi_{0}(x)|,\hskip
14.22636pt\|\mu_{x}\|\leq C|\varphi_{0}(x)|,\\\ \|\nabla_{i}\eta\|&\leq
C\sqrt{N},\hskip 14.22636pt\,\|\nabla_{i}k\|\leq C\sqrt{N},\hskip
14.22636pt\|\nabla_{i}\mu\|\leq C,\hskip 14.22636pt\|\Delta_{i}\mu\|\leq
C.\end{split}$ (2.13)

Moreover, for $x,y\in\mathbb{R}^{3}$ and $\alpha_{1},\alpha_{2}\in\\{0,1,2\\}$
we have the pointwise bounds

$\begin{split}&|\eta(x;y)|,|k(x;y)|\leq
C\frac{\varphi_{0}(x)\varphi_{0}(y)}{|x-y|+N^{-1}},\,|\mu(x;y)|\leq
C\varphi_{0}(x)\varphi_{0}(y),\\\
&|\nabla_{1}\eta(x;y)|,|\nabla_{1}k(x;y)|\leq
C\frac{\varphi_{0}(x)\varphi_{0}(y)}{|x-y|^{2}+N^{-2}}+C\frac{|\nabla\varphi_{0}(x)|\varphi_{0}(y)}{|x-y|+N^{-1}},\\\
&|\nabla_{1}^{\alpha_{1}}\nabla_{2}^{\alpha_{2}}\mu(x;y)|\leq
C\sum_{j_{1}=0}^{\alpha_{1}}|\nabla^{j_{1}}\varphi_{0}(x)|\sum_{j_{2}=0}^{\alpha_{2}}|\nabla^{j_{2}}\varphi_{0}(y)|\end{split}$
(2.14)

(and similarly for $\nabla_{2}\eta$, $\nabla_{2}k$). Finally, identifying the
kernel $\eta$ with the corresponding Hilbert-Schmidt operator and denoting by
$\eta^{(n)}$ the kernel of its $n$-th power, we find, for every $n\geq 2$ and
$x,y\in\mathbb{R}^{3}$ that $\|\nabla_{i}\eta^{(n)}\|\leq C^{n}$,
$\|\Delta_{i}\eta^{(n)}\|\leq C^{n}$ and that

$\begin{split}&|\eta^{(n)}(x;y)|\leq
C\|\eta\|^{n-2}|\varphi_{0}(x)||\varphi_{0}(y)|,\\\
&\|\nabla_{x}\eta^{(n)}_{x}\|\leq C(|\nabla\varphi_{0}(x)|+\varphi_{0}(x)),\\\
&\|\Delta_{x}\eta^{(n)}_{x}\|\leq
C(|\Delta\varphi_{0}(x)|+|\nabla\varphi_{0}(x)|+\varphi_{0}(x)).\end{split}$
(2.15)

We introduce now the antisymmetric operator

$B=\frac{1}{2}\int
dxdy\,\big{[}\eta(x;y)b_{x}^{*}b_{y}^{*}-\text{h.c.}\big{]}$ (2.16)

and we consider the generalized Bogoliubov transformation $e^{B}$. Since
$\eta\in(Q\otimes Q)L^{2}(\mathbb{R}^{3}\times\mathbb{R}^{3})$, the unitary
operator $e^{B}$ maps the Hilbert space $\mathcal{F}^{\leq
N}_{\perp\varphi_{0}}$ back into itself. Another important property of $e^{B}$
is the fact that it preserves, approximately, the number of particles (ie. the
number of excitations of the Bose-Einstein condensate). The following lemma
was proved in [10, Lemma 3.1].

######  Lemma 2.3.

Let $\eta\in L_{s}^{2}(\mathbb{R}^{3}\times\mathbb{R}^{3})$. Let $B$ be the
antisymmetric operator defined in (2.16). For every $n\in\mathbb{N}$ there
exists a constant $C>0$ (depending only on $n$ and on $\|\eta\|$) such that

$e^{-B}(\mathcal{N}+1)^{n}e^{B}\leq C(\mathcal{N}+1)^{n}$

as an operator inequality on $\mathcal{F}_{\perp\varphi_{0}}^{\leq N}$.

We will also need more precise information on the action of the $e^{B}$.
Keeping in mind the fact that, if we replaced in (2.16) the
$b^{*},b$-operators with standard creation and annihilation operators, $e^{B}$
would be a Bogoliubov transformation with explicit action, we decompose, for
an arbitrary $f\in L^{2}_{\perp\varphi_{0}}(\mathbb{R}^{3})$,

$e^{-B}b(f)e^{B}=b(\cosh_{\eta}(f))+b^{*}(\sinh_{\eta}(\overline{f}))+d_{\eta}(f),$
(2.17)

where

$\begin{split}\cosh_{\eta}=\sum_{n=0}^{\infty}\frac{\eta^{(2n)}}{(2n)!},\hskip
14.22636pt\sinh_{\eta}=\sum_{n=0}^{\infty}\frac{\eta^{(2n+1)}}{(2n+1)!}.\end{split}$
(2.18)

We also introduce the operator-valued distributions
$d_{\eta,x},d^{*}_{\eta,x}$ defined through

$e^{-B}b_{x}e^{B}=b(\cosh_{\eta,x})+b^{*}(\sinh_{\eta,x})+d_{\eta,x}.$ (2.19)

We will use the short-hand notations

$\gamma=\cosh_{\eta},\qquad\sigma=\sinh_{\eta},\qquad\text{p}_{\eta}=\gamma-1,\qquad\text{r}_{\eta}=\sigma-\eta.$
(2.20)

It is a simple consequence of Lemma 2.2 that, under the same assumptions as in
Lemma 2.2, there is a constant $C>0$ such that, for all
$\alpha_{1},\alpha_{2}\in\\{0,1,2\\}$ with $\alpha_{1}+\alpha_{2}\leq 2$, we
have

$\|\nabla_{1}^{\alpha_{1}}\nabla_{2}^{\alpha_{2}}\emph{p}_{\eta}\|,\|\nabla_{1}^{\alpha_{1}}\nabla_{2}^{\alpha_{2}}\emph{r}_{\eta}\|\leq
C.$

Moreover, we have the bounds $|\emph{p}(x;y)|,|\emph{r}(x;y)|\leq
C\varphi_{0}(x)\varphi_{0}(y)$ for all $x,y\in\mathbb{R}^{3}$.

The idea that on states with few excitations $e^{B}$ acts almost as a
Bogoliubov transformation is confirmed by the next lemma, which bounds the
norm of the operators $d_{\eta}(f)$, in terms of (appropriate powers of) the
number of particles operator. This lemma, whose proof is an adaptation of the
translation-invariant case [6, Lemma 3.4], requires $\|\eta\|$ to be
sufficiently small. From Lemma 2.2, this can be achieved requiring that
$\ell>0$ is small enough.

######  Lemma 2.4.

Let $n\in\mathbb{Z}$, $f\in L^{2}(\mathbb{R}^{3})$. Let $\eta\in
L^{2}(\mathbb{R}^{3}\times\mathbb{R}^{3})$ be as defined in (2.16), with
$\ell>0$ small enough. Let $d_{\eta}(f)$ be as in (2.17) (and $d_{\eta,x}$ as
defined in (2.19)). Then there exists $C>0$ such that

$\displaystyle\begin{split}\|(\mathcal{N}+1)^{n/2}d_{\eta}(f)\xi\|&\leq\frac{C}{N}\|f\|\|(\mathcal{N}+1)^{(n+3)/2}\xi\|,\\\
\|(\mathcal{N}+1)^{n/2}d_{\eta}(f)^{*}\xi\|&\leq\frac{C}{N}\|f\|\|(\mathcal{N}+1)^{(n+3)/2}\xi\|\end{split}$

and such that, for all $x\in\mathbb{R}^{3}$, we have

$\begin{split}\|(\mathcal{N}+1)^{n/2}d_{\eta,x}\xi\|&\leq\frac{C}{N}\Big{[}\|a_{x}(\mathcal{N}+1)^{(n+2)/2}\xi\|+\varphi_{0}(x)\|(\mathcal{N}+1)^{(n+3)/2}\xi\|\Big{]}.\end{split}$
(2.21)

Furthermore, if we set
$\overline{d}_{\eta,x}=d_{\eta,x}+(\mathcal{N}/N)b^{*}(\eta_{x})$, we find

$\begin{split}&\|(\mathcal{N}+1)^{n/2}a_{y}\overline{d}_{\eta,x}\xi\|\\\
&\leq\frac{C}{N}\Big{[}\varphi_{0}(x)\varphi_{0}(y)\|(\mathcal{N}+1)^{(n+2)/2}\xi\|+|\eta(x;y)|\|(\mathcal{N}+1)^{(n+2)/2}\xi\|\\\
&\hskip
42.67912pt+\varphi_{0}(y)\|a_{x}(\mathcal{N}+1)^{(n+1)/2}\xi\|+\varphi_{0}(x)\|a_{y}(\mathcal{N}+1)^{(n+3)/2}\xi\|\\\
&\hskip
42.67912pt+\|a_{x}a_{y}(\mathcal{N}+1)^{(n+2)/2}\xi\|\Big{]}\end{split}$
(2.22)

and, finally, we have that

$\begin{split}&\|(\mathcal{N}+1)^{n/2}d_{\eta,x}d_{\eta,y}\xi\|\\\
&\leq\frac{C}{N^{2}}\Big{[}\varphi_{0}(x)\varphi_{0}(y)\|(\mathcal{N}+1)^{(n+6)/2}\xi\|+|\eta(x;y)|\|(\mathcal{N}+1)^{(n+4)/2}\xi\|\\\
&\hskip
34.14322pt+\varphi_{0}(y)\|a_{x}(\mathcal{N}+1)^{(n+5)/2)}\xi\|+\varphi_{0}(x)\|a_{y}(\mathcal{N}+1)^{(n+5)/2}\xi\|\\\
&\hskip
34.14322pt+\|a_{x}a_{y}(\mathcal{N}+1)^{(n+4)/2}\xi\|\Big{]}.\end{split}$
(2.23)

Using the generalized Bogoliubov transformation $e^{B}$, we define the
renormalized excitation Hamiltonian

$\mathcal{G}_{N}=e^{-B}\mathcal{L}_{N}e^{B}.$ (2.24)

In the next proposition, we collect some important properties of
$\mathcal{G}_{N}$. Here and in the rest of the paper, we are going to use the
notation

$\begin{split}\mathcal{K}&=\int dx\;a^{*}_{x}(-\Delta_{x})a_{x},\hskip
99.58464pt\mathcal{V}_{\text{ext}}=\int
dx\,V_{\text{ext}}(x)a^{*}_{x}a_{x},\\\ \mathcal{V}_{N}&=\frac{1}{2}\int
dxdy\;N^{2}V(N(x-y))a^{*}_{x}a^{*}_{y}a_{y}a_{x},\hskip
14.22636pt\mathcal{H}_{N}=\mathcal{K}+\mathcal{V}_{\text{ext}}+\mathcal{V}_{N}.\end{split}$

######  Proposition 2.5.

Assume (1.7), let $\eta$ be defined as in (2.12) and let $E_{N}$ denote the
ground state energy of $\mathcal{G}_{N}$. Then the following holds true:

  * a)

We have that $|E_{N}-N\mathcal{E}_{GP}(\varphi_{0})|\leq C$ and

$\mathcal{G}_{N}-E_{N}=\mathcal{H}_{N}+\Delta_{N},$ (2.25)

where the error term $\Delta_{N}$ is such that for every $\delta>0$, there
exists $C>0$ with

$\pm\Delta_{N}\leq\delta\mathcal{H}_{N}+C(\mathcal{N}+1).$ (2.26)

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

$\pm\text{ad}^{\,(k)}_{\,i\mathcal{N}}(\mathcal{G}_{N})=\pm\text{ad}^{\,(k)}_{\,i\mathcal{N}}(\Delta_{N})=\pm\big{[}i\mathcal{N},\dots\big{[}i\mathcal{N},\Delta_{N}\big{]}\dots\big{]}\leq
C(\mathcal{H}_{N}+1).$ (2.27)

  * b)

Let $\sigma$ and $\gamma$ be defined as in (2.18) and let
$\kappa_{\mathcal{G}_{N}}$ denote the constant

$\begin{split}\kappa_{\mathcal{G}_{N}}&=N\big{\langle}\varphi_{0},(-\Delta+V_{\text{ext}}+\widehat{V}(0)\varphi_{0}^{2}/2)\varphi_{0}\big{\rangle}-4\pi\mathfrak{a}_{0}\|\varphi_{0}\|_{4}^{4}\\\
&\hskip
14.22636pt+\operatorname{tr}\big{(}\sigma(-\Delta+V_{\text{ext}}-\varepsilon_{GP})\sigma\big{)}\\\
&\hskip
14.22636pt+\operatorname{tr}\big{(}\gamma\big{[}N^{3}V(N(x-y))\varphi_{0}(x)\varphi_{0}(y)\big{]}\sigma)\\\
&\hskip
14.22636pt+\operatorname{tr}\big{(}\sigma\big{[}N^{3}V(N.)\ast\varphi_{0}^{2}+N^{3}V(N(x-y))\varphi_{0}(x)\varphi_{0}(y)\big{]}\sigma\big{)}\\\
&\hskip 14.22636pt+\frac{1}{2}\int
dxdy\,N^{2}V(N(x-y))|\langle\sigma_{x},\gamma_{y}\rangle|^{2}.\end{split}$
(2.28)

Here, $(N^{3}V(N.)\ast\varphi_{0}^{2})$ acts as multiplication operator and we
identify kernels like $N^{3}V(N(x-y))\varphi_{0}(x)\varphi_{0}(y)$ with their
associated Hilbert Schmidt operators. Let

$\begin{split}\Phi&=\gamma\big{(}-\Delta+V_{\text{ext}}-\varepsilon_{GP}\big{)}\gamma+\sigma\big{(}-\Delta+V_{\text{ext}}-\varepsilon_{GP}\big{)}\sigma\\\
&\hskip
11.38092pt+\gamma\big{(}N^{3}V(N.)\ast\varphi_{0}^{2}+N^{3}V(N(x-y))\varphi_{0}(x)\varphi_{0}(y)\big{)}\gamma\\\
&\hskip
11.38092pt+\sigma\big{(}N^{3}V(N.)\ast\varphi_{0}^{2}+N^{3}V(N(x-y))\varphi_{0}(x)\varphi_{0}(y)\big{)}\sigma\\\
&\hskip
11.38092pt+\Big{(}\gamma\big{(}N^{3}(Vf_{\ell})(N(x-y))\varphi_{0}(x)\varphi_{0}(y)\big{)}\sigma+\emph{h.c.}\Big{)},\end{split}$
(2.29)

and

$\begin{split}\Gamma&=\gamma\big{(}N^{3}(Vf_{\ell})(N(x-y))\varphi_{0}(x)\varphi_{0}(y)\big{)}\gamma\\\
&\hskip
14.22636pt+\sigma\\!\big{(}N^{3}(Vf_{\ell})(N(x-y))\varphi_{0}(x)\varphi_{0}(y)\big{)}\\!\sigma\\\
&\hskip
14.22636pt+\big{[}\sigma\big{(}-\Delta+V_{\text{ext}}-\varepsilon_{GP}\big{)}\gamma+\emph{h.c.}\big{]}\\\
&\hskip
14.22636pt+\big{[}\sigma\big{(}N^{3}V(N.)\ast\varphi_{0}^{2}+N^{3}V(N(x-y))\varphi_{0}(x)\varphi_{0}(y)\big{)}\gamma+\emph{h.c.}\big{]}.\\\
\end{split}$ (2.30)

With $\Phi$ and $\Gamma$, we then define the quadratic Fock space Hamiltonian

$\begin{split}\mathcal{Q}_{\mathcal{G}_{N}}=\int
dxdy\,\Phi(x;y)b^{*}_{x}b_{y}+\frac{1}{2}\int
dxdy\,\Gamma(x;y)\big{(}b^{*}_{x}b^{*}_{y}+b_{x}b_{y}\big{)},\end{split}$
(2.31)

and we denote by $\mathcal{C}_{\mathcal{G}_{N}}$ the cubic operator

$\mathcal{C}_{\mathcal{G}_{N}}=\int
dxdy\,N^{5/2}V(N(x-y))\varphi_{0}(y)b^{*}_{x}b^{*}_{y}\big{(}b(\gamma_{x})+b^{*}(\sigma_{x})\big{)}+\emph{h.c.}$
(2.32)

Then, we have that

$\begin{split}\mathcal{G}_{N}&=\kappa_{\mathcal{G}_{N}}+\mathcal{Q}_{\mathcal{G}_{N}}+\mathcal{C}_{\mathcal{G}_{N}}+\mathcal{V}_{N}+\mathcal{E}_{\mathcal{G}_{N}}\end{split}$
(2.33)

for a self-adjoint operator $\mathcal{E}_{\mathcal{G}_{N}}$ that satisfies in
$\mathcal{F}_{\bot\varphi_{0}}^{\leq N}$ the operator bound

$\pm\mathcal{E}_{\mathcal{G}_{N}}\leq
CN^{-1/2}(\mathcal{H}_{N}+\mathcal{N}^{2}+1)(\mathcal{N}+1).$ (2.34)

The proof of Prop. 2.5 is similar to the proof of [5, Prop. 3.2]. For
completeness, we sketch it in Sect. 4.

Combining Prop. 2.5 with the lower bound

$\mathcal{G}_{N}-E_{N}\geq c\mathcal{N}-C$ (2.35)

from [11, Eq. (3.1)] (and from Lemma 2.3 in the present paper), we obtain
strong a-priori estimates on the number and on the energy of excitations of
the Bose-Einstein condensate in low-energy states of (1.1). These bounds are
crucial for the rest of out analysis.

######  Theorem 2.6.

Assume (1.7), let $\eta$ be defined as in (2.12) and let $E_{N}$ denote the
ground state energy of $H_{N}$, defined in (1.1). Let $\psi_{N}\in
L^{2}_{s}(\mathbb{R}^{3N})$ with $\|\psi_{N}\|=1$ belong to the spectral
subspace of $H_{N}$ with energies below $E_{N}+\zeta$, for some $\zeta>0$,
i.e.

$\psi_{N}={\bf 1}_{(-\infty;E_{N}+\zeta]}(H_{N})\psi_{N}.$

Let $\xi_{N}=e^{-B}U_{N}\psi_{N}\in\mathcal{F}_{\bot\varphi_{0}}^{\leq N}$ be
the renormalized excitation vector associated with $\psi_{N}$. Then, for every
$j\in\mathbb{N}$ there exists a constant $C>0$ such that

$\displaystyle\langle\xi_{N},(\mathcal{H}_{N}+1)(\mathcal{N}+1)^{j}\xi_{N}\rangle\leq
C(1+\zeta^{j+1}).$

###### Proof.

The proof carries over from [5, Prop. 4.1], once we show that

$\mathcal{G}_{N}-E_{N}\geq c\,\mathcal{H}_{N}-C$ (2.36)

for two positive constants $c,C>0$, independent of $N$. Indeed, this enables
us to use the induction argument from [5, Prop. 4.1] combined with the bounds
(2.27). The bound (2.36) can be shown interpolating (2.35) with the estimate

$\mathcal{G}_{N}-E_{N}\geq\frac{1}{2}\mathcal{H}_{N}-C(\mathcal{N}+1)$

which follows from (2.25) and (2.26). ∎

Although Prop. 2.5 and Theorem 2.6 provide strong control on low-energy states
of $\mathcal{G}_{N}$, this is not enough to deduce Theorem 1.1. The reason is
that the cubic operator $\mathcal{C}_{\mathcal{G}_{N}}$ and the quartic
operator $\mathcal{V}_{N}$ appearing on the r.h.s. of (2.32) still contain
energy contributions of order one. To extract them, we follow again the
strategy introduced in [6] and we conjugate $\mathcal{G}_{N}$ with an
additional unitary transformation, this time generated by an operator cubic in
the modified creation and annihilation fields.

In order to define our cubic transformation, we need to introduce some
notation. For an exponent $\varepsilon>0$ to be specified later on, we use
$\chi_{H}(p)=\chi(|p|>N^{\varepsilon})$ to restrict to high momenta
$|p|>N^{\varepsilon}$. We consider the kernel

$\widetilde{k}_{H}(x;y)=\left[-Nw_{\ell}(N.)*\check{\chi}_{H}\right](x-y)\varphi_{0}(y).$
(2.37)

On the other hand, to restrict to small momenta, we use a Gaussian cutoff. For
$0<\tau<\varepsilon$, we set $g_{L}(p)=e^{-p^{2}/N^{2\tau}}$. The advantage of
$g_{L}$, with respect to a sharp cutoff, is that it decays fast also in
position space. In particular, from the assumptions (1.7) on $V_{\text{ext}}$,
we find

$|V_{\text{ext}}*\check{g}_{L}|\leq C(V_{\text{ext}}+1).$ (2.38)

Recalling the definition of $\sigma,\gamma$ in (2.20), we introduce the
notation

$\sigma_{L}=\sigma*_{2}\check{g}_{L},\qquad\gamma_{L}=\gamma*_{2}\check{g}_{L}$
(2.39)

where $*_{2}$ is the convolution in the second variable. In particular, for
every $x\in\mathbb{R}^{3}$, we have $\sigma_{L,x}=\sigma_{x}*\check{g}_{L}$
and $\gamma_{L,x}=\gamma_{x}*\check{g}_{L}$.

In the next lemma, we collect useful bounds for various kernels.

######  Lemma 2.7.

We assume that $0<3\tau\leq\varepsilon$. Let $\widetilde{k}_{H},\sigma_{L}$
and $\gamma_{L}$ as in (2.37) and (2.39). Then we have

$\begin{split}&\|\widetilde{k}_{H}\|\leq
CN^{-\varepsilon/2},\quad\sup_{x}\|\widetilde{k}_{H}(x,.)\|\leq
CN^{-\varepsilon/2},\quad\|\widetilde{k}_{H}(.,y)\|\leq
CN^{-\varepsilon/2}\varphi_{0}(y)\end{split}$ (2.40)

for all $y\in\mathbb{R}^{3}$. Setting
$\widetilde{k}(x;y)=-Nw_{\ell}(N(x-y))\varphi_{0}(y)$ (no cutoff), we have

$\|\nabla_{x}(\widetilde{k}-\widetilde{k}_{H})(x,.)\|\leq
CN^{\varepsilon/2},\qquad|(\widetilde{k}-\widetilde{k}_{H})(x;y)|\leq
CN^{\varepsilon}\varphi_{0}(y).$ (2.41)

Moreover, we find

$\begin{split}\|\sigma_{L,x}\|,\|\big{[}(\gamma-1)*_{2}\check{g}_{L}\big{]}_{x}\|\leq
C\varphi_{0}(x),\end{split}$ (2.42)

for all $x\in\mathbb{R}^{3}$, which implies
$\|\sigma_{L}\|,\|(\gamma-1)*_{2}\check{g}_{L}\|,\|\gamma_{L}\|_{\text{op}}\leq
C$, and also

$\displaystyle\begin{split}\|V_{\text{ext}}\big{[}(\gamma-1)*_{2}\check{g}_{L}\big{]}\|,\|V_{\text{ext}}\sigma_{L}\|\leq
C.\end{split}$

Finally, we have

$\|\nabla_{x}\sigma_{L,x}\|\leq
CN^{\tau/2}\big{[}\varphi_{0}(x)+|\nabla\varphi_{0}(x)|\big{]},\quad\|\nabla_{x}\big{[}(\gamma-1)_{x}*_{2}\check{g}_{L}\big{]}\|\leq
C\big{[}\varphi_{0}(x)+|\nabla\varphi_{0}(x)|\big{]}.$ (2.43)

###### Proof.

Using (2.10) we compute

$\displaystyle\|Nw_{\ell}(N.)*\check{\chi}_{H}\|^{2}\leq C\int_{|p|\geq
N^{\varepsilon}}|p|^{-4}dp=CN^{-\varepsilon}.$

This implies all the bounds in (2.40). To show (2.41), we observe that
$(\widetilde{k}-\widetilde{k}_{H})(x;y)=\big{[}-Nw_{\ell}(N.)*\check{\chi}_{H^{c}}\big{]}(x-y)\varphi_{0}(y)$,
with $\chi_{H_{c}}$ the indicator function of the set
$\\{p\in\mathbb{R}^{3}:|p|\leq N^{\varepsilon}\\}$ and that, from (2.10),

$\big{\|}\nabla\big{[}-Nw_{\ell}(N.)*\check{\chi}_{H^{c}}\big{]}\big{\|}^{2}_{2}=\int_{|p|\leq
N^{\varepsilon}}dp\left[\frac{|p|}{N^{2}}|\hat{w}_{\ell}(p/N)|\right]^{2}\leq
CN^{\varepsilon}$

and

$\|-Nw_{\ell}(N.)*\check{\chi}_{H^{c}}\|_{\infty}\leq C\int_{|p|\leq
N^{\varepsilon}}\frac{1}{N^{2}}\hat{w}_{\ell}(p/N)dp\leq CN^{\varepsilon}.$

To prove (2.42), we estimate, with (2.13), (2.15), Parseval’s identity and
$|g_{L}|\leq 1$,

$\|\sigma_{L,x}\|=\|\widehat{\sigma_{x}}g_{L}\|\leq\|\sigma_{x}\|\leq\sum_{n\geq
0}\frac{1}{(2n+1)!}\|\eta^{(2n+1)}_{x}\|\leq C\varphi_{0}(x).$

The bound for $(\gamma-1)*_{2}\check{g}_{L}$ can be proven similarly. Finally,
we show (2.43). With (2.15), it is easy to check that
$\|\nabla_{x}(\sigma_{L}-k*_{2}\check{g}_{L})_{x}\|\leq
C(\varphi_{0}(x)+|\nabla\varphi_{0}(x)|)$, with $k$ as defined in (2.11). We
have

$\begin{split}\nabla_{x}(k*_{2}\check{g}_{L})(x;y)=&-\int N^{2}\nabla
w_{\ell}(N(x-z-y))\varphi_{0}(x)\varphi_{0}(y+z)\check{g}_{L}(z)dz\\\ &-\int
Nw_{\ell}(N(x-z-y))\nabla\varphi_{0}(x)\varphi_{0}(y+z)\check{g}_{L}(z)dz.\end{split}$
(2.44)

The $L^{2}$-norm squared (in the variable $y$, for fixed $x$) of the first
term can be bounded, with (2.9), by

$\begin{split}\int
dy&dz_{1}dz_{2}\frac{1}{|x-z_{1}-y|^{2}|x-z_{2}-y|^{2}}\varphi_{0}(x)^{2}\varphi_{0}(y+z_{1})\varphi_{0}(y+z_{2})\check{g}_{L}(z_{1})\check{g}_{L}(z_{2})\\\
\leq\;&C\int
dz_{1}dz_{2}\,\frac{1}{|z_{1}-z_{2}|}\check{g}_{L}(z_{1})\check{g}_{L}(z_{2})\leq
CN^{\tau}\int
dw_{1}dw_{2}\frac{1}{|w_{1}-w_{2}|}e^{-w_{1}^{2}}e^{-w_{2}^{2}}\leq
CN^{\tau}\end{split}$

where we switched to new variables $w_{i}=z_{i}N^{\tau}$, for $i=1,2$. The
second term on the r.h.s. of (2.44) can be controlled analogously. ∎

We introduce the antisymmetric operator

$A=\frac{1}{\sqrt{N}}\int
dxdy\,\widetilde{k}_{H}(x;y)\widetilde{b}_{x}^{*}\widetilde{b}_{y}^{*}\left[\widetilde{b}(\gamma_{L,x})+\widetilde{b}^{*}(\sigma_{L,x})\right]-\text{h.c.}$
(2.45)

In order to preserve the space $\mathcal{F}_{\perp\varphi_{0}}^{\leq N}$ and
stay orthogonal to $\varphi_{0}$, we used here the operators

$\widetilde{b}_{x}=b(Q_{x})=\int dz\,Q(x;z)b_{z}$

with $Q=1-|\varphi_{0}\rangle\langle\varphi_{0}|$. Using $A$, we define the
cubically renormalized excitation Hamiltonian
$\mathcal{J}_{N}=e^{-A}\mathcal{G}_{N}e^{A}$.

The next proposition summarises important properties of the operator
$\mathcal{J}_{N}$.

######  Proposition 2.8.

Assume (1.7) and let $0<6\tau\leq\varepsilon\leq\frac{1}{2}$. Then we have

$\mathcal{J}_{N}=\kappa_{\mathcal{J}_{N}}+\mathcal{Q}_{\mathcal{J}_{N}}+\mathcal{V}_{N}+\mathcal{E}_{\mathcal{J}_{N}},$
(2.46)

where

$\begin{split}\kappa_{\mathcal{J}_{N}}&=\kappa_{\mathcal{G}_{N}}-\emph{tr}(\sigma
N^{3}(Vw_{\ell})(N.)*\varphi_{0}^{2}\sigma)-\emph{tr}(\sigma
N^{3}(Vw_{\ell})(N(x-y))\varphi_{0}(x)\varphi_{0}(y)\sigma)\end{split}$ (2.47)

with $\kappa_{\mathcal{G}_{N}}$ defined in (2.28) and where the quadratic
operator $\mathcal{Q}_{\mathcal{J}_{N}}$ is given by

$\begin{split}\mathcal{Q}_{\mathcal{J}_{N}}=\int
dxdy\,\widetilde{\Phi}(x;y)b^{*}_{x}b_{y}+\frac{1}{2}\int
dxdy\,\widetilde{\Gamma}(x;y)\big{(}b^{*}_{x}b^{*}_{y}+b_{x}b_{y}\big{)}\end{split}$
(2.48)

for

$\begin{split}\widetilde{\Phi}&=\gamma\big{(}-\Delta+V_{\text{ext}}-\varepsilon_{GP}\big{)}\gamma+\sigma\big{(}-\Delta+V_{\text{ext}}-\varepsilon_{GP}\big{)}\sigma\\\
&\hskip
11.38092pt+\gamma\big{(}8\pi\mathfrak{a}_{0}\varphi_{0}^{2}+N^{3}(Vf_{\ell})(N(x-y))\varphi_{0}(x)\varphi_{0}(y)\big{)}\gamma\\\
&\hskip
11.38092pt+\sigma\big{(}8\pi\mathfrak{a}_{0}\varphi_{0}^{2}+N^{3}(Vf_{\ell})(N(x-y))\varphi_{0}(x)\varphi_{0}(y)\big{)}\sigma\\\
&\hskip
11.38092pt+\Big{(}\gamma\big{(}N^{3}(Vf_{\ell})(N(x-y))\varphi_{0}(x)\varphi_{0}(y)\big{)}\sigma+\emph{h.c.}\Big{)},\end{split}$
(2.49)

and

$\begin{split}\widetilde{\Gamma}&=\gamma\big{(}N^{3}(Vf_{\ell})(N(x-y))\varphi_{0}(x)\varphi_{0}(y)\big{)}\gamma\\\
&\hskip
14.22636pt+\sigma\\!\big{(}N^{3}(Vf_{\ell})(N(x-y))\varphi_{0}(x)\varphi_{0}(y)\big{)}\\!\sigma\\\
&\hskip
14.22636pt+\big{[}\sigma\big{(}-\Delta+V_{\text{ext}}-\varepsilon_{GP}\big{)}\gamma+\emph{h.c.}\big{]}\\\
&\hskip
14.22636pt+\big{[}\sigma\big{(}8\pi\mathfrak{a}_{0}\varphi_{0}^{2}+N^{3}(Vf_{\ell})(N(x-y))\varphi_{0}(x)\varphi_{0}(y)\big{)}\gamma+\emph{h.c.}\big{]}.\\\
\end{split}$ (2.50)

Moreover, the self-adjoint operator $\mathcal{E}_{\mathcal{J}_{N}}$ is bounded
by

$\pm\mathcal{E}_{\mathcal{J}_{N}}\leq\frac{C}{N^{\min\\{\frac{\tau}{2},\frac{1}{2}-\varepsilon\\}}}(\mathcal{H}_{N}+1)(\mathcal{N}+1)^{3}.$
(2.51)

The proof of Prop. 2.8, which is similar to the proof of [5, Prop. 3.3], is
sketched it in Sect. 4.

In order to compute the spectrum of $\mathcal{J}_{N}$, we need to have good
control on the number of excitations of wave functions in low-energy spectral
subspaces of $\mathcal{J}_{N}$. This is summarized in the following
proposition whose proof is deferred to Section 5.

######  Proposition 2.9.

Assume (1.7), let $\eta$ be defined as in (2.12), $A$ as defined in (2.45) and
let $E_{N}$ denote the ground state energy of $H_{N}$. Let $\psi_{N}\in
L^{2}_{s}(\mathbb{R}^{3N})$ with $\|\psi_{N}\|=1$ belong to the spectral
subspace of $H_{N}$ with energies below $E_{N}+\zeta$, for some $\zeta>0$,
i.e.

$\psi_{N}={\bf 1}_{(-\infty;E_{N}+\zeta]}(H_{N})\psi_{N}.$

Let $\xi_{N}=e^{-A}e^{-B}U_{N}\psi_{N}\in\mathcal{F}_{\bot\varphi_{0}}^{\leq
N}$ be the renormalized excitation vector associated with $\psi_{N}$. Then,
for any $j\in\mathbb{N}$ there exists a constant $C>0$ such that

$\langle\xi_{N},(\mathcal{N}+1)^{j}(\mathcal{H}_{N}+1)\xi_{N}\rangle\leq
C(1+\zeta^{j+3}).$

##  3 Bounds on $\sigma(H_{N})$ and Proof of Theorem 1.1

In this section, we conclude the proof of Theorem 1.1 and determine the ground
state energy $E_{N}$ of $H_{N}$ as well as the spectrum of $H_{N}-E_{N}$ below
a threshold $\zeta>0$, up to errors that vanish in the limit $N\to\infty$. Our
proof is based on Proposition 2.8, recalling that
$\mathcal{J}_{N}=e^{-A}e^{-B}U_{N}H_{N}U_{N}^{*}e^{B}e^{A}$ is unitarily
equivalent to $H_{N}$. For the rest of this section, we assume that the
parameter $\ell\in(0;1)$ introduced in (2.6) is sufficiently small (by Lemma
2.2, this guarantees that $\|\eta\|$ is small enough). We also assume that the
parameters introduced around (2.37) are fixed as $\varepsilon=6/13$ and
$\tau=\varepsilon/6$, so that we can apply the results of Proposition 2.5 and
Proposition 2.8. From (2.51), we have

$\pm\mathcal{E}_{\mathcal{J}_{N}}\leq
CN^{-\rho}(\mathcal{H}_{N}+1)(\mathcal{N}+1)^{3},$ (3.1)

with $\rho=1/26$.

To compute the spectrum of $H_{N}$, we proceed in two main steps, proving
first lower and then upper bounds. To show the lower bound, we start from
(2.46), we drop the non-negative potential energy $\mathcal{V}_{N}$ in (2.46)
and then we diagonalize the remaining quadratic Fock space Hamiltonian. This
is discussed in the Section 3.1. For the upper bound, we need additionally to
control the expectation of the potential energy on low energy states; this is
discussed in Section 3.2. Combining the two results, we obtain asymptotically
matching lower and upper bounds on the min-max values of $\mathcal{J}_{N}$. To
conclude Theorem 1.1, it then only remains to verify Eq. (1.8) which
determines the ground state energy $E_{N}$ up to errors that vanish as
$N\to\infty$. This is the content of Section 3.3.

Before we start with the lower bound, it is convenient to switch to the full
excitation Fock space $\mathcal{F}_{\bot\varphi_{0}}$, replacing the modified
creation and annihilation operators in (2.48) by the standard ones. This will
enable us to diagonalize the resulting quadratic Fock space Hamiltonian
exactly. To this end, let us denote by

$\widetilde{\mathcal{Q}}_{\mathcal{J}_{N}}=\int
dxdy\,\widetilde{\Phi}(x,y)a_{x}^{*}a_{y}+\frac{1}{2}\int dxdy\
\widetilde{\Gamma}(x,y)(a_{x}^{*}a_{y}^{*}+\text{h.c.})$ (3.2)

the quadratic Fock space Hamiltonian that is obtained from
$\mathcal{Q}_{\mathcal{J}_{N}}$ replacing $b,b^{*}$ operators by the $a,a^{*}$
fields. We claim that

$\mathcal{Q}_{\mathcal{J}_{N}}=\widetilde{\mathcal{Q}}_{\mathcal{J}_{N}}+\widetilde{\mathcal{E}}_{N}$
(3.3)

for an error $\widetilde{\mathcal{E}}_{N}$ that is bounded, in the sense of
forms in $\mathcal{F}_{\bot\varphi_{0}}^{\leq N}$, by

$\pm\widetilde{\mathcal{E}}_{N}\leq
CN^{-1/2}(\mathcal{K}+\mathcal{V}_{\text{ext}}+\mathcal{N}+1)(\mathcal{N}+1).$
(3.4)

To prove the bound (3.4), consider first the non-diagonal contribution to
$\mathcal{Q}_{\mathcal{J}_{N}}$. From the definition (2.4) and using that
$|\sqrt{1-x/N}\sqrt{1-x/N-1/N}-1|\leq Cx/N$ for $0\leq x\leq N$, we bound

$\pm\bigg{(}\int dxdy\
\widetilde{\Gamma}(x,y)(b_{x}^{*}b_{y}^{*}-a_{x}^{*}a_{y}^{*}+\text{h.c.})\bigg{)}\leq
CN^{-1}\|\widetilde{\Gamma}\|_{\text{HS}}(\mathcal{N}+1)^{2}.$ (3.5)

In particular, this error is small if we show that

$\|\widetilde{\Gamma}\|_{\text{HS}}\leq C$ (3.6)

for some constant $C>0$, independent of $N\in\mathbb{N}$. Here, we identify
the kernel $\widetilde{\Gamma}$ with the corresponding operator on
$L^{2}_{\bot\varphi_{0}}(\mathbb{R}^{3})$ and $\|\cdot\|_{\text{HS}}$ denotes
the Hilbert-Schmidt norm. To control $\|\widetilde{\Gamma}\|_{\text{HS}}$, we
consider the different contributions to $\widetilde{\Gamma}$ in (2.50). We
first note that the operator with kernel
$N^{3}(Vf_{\ell})(N(x-y))\varphi_{0}(x)\varphi_{0}(y)$ is bounded, uniformly
in $N$. Analogously, $\|8\pi\mathfrak{a}_{0}\varphi_{0}^{2}\|_{op}\leq C$.
From Lemma 2.2, we have $\|\sigma\|_{\text{HS}}<\infty$, and also $\|\sigma
V_{\text{ext}}\|_{\text{HS}},\|\sigma\Delta\gamma-k\Delta\|_{\text{HS}}<\infty$.
Thus, to show (3.6), we only need to control the norm of the operator on
$L^{2}_{\perp\varphi_{0}}(\mathbb{R}^{3})$ with kernel

$\begin{split}N^{3}(Vf_{\ell})(N(x-y))&\varphi_{0}(x)\varphi_{0}(y)+(\Delta_{x}+\Delta_{y})\big{(}Nw_{\ell}(N(x-y))\varphi_{0}(x)\varphi_{0}(y)\big{)}\\\
=\;&N^{3}\left[2\Delta
w_{\ell}(N(x-y))+(Vf_{\ell})(N(x-y))\right]\varphi_{0}(x)\varphi_{0}(y)\\\
&+Nw_{\ell}(N(x-y))\big{[}\Delta\varphi_{0}(x)\varphi_{0}(y)+\varphi_{0}(x)\Delta\varphi_{0}(y)\big{]}\\\
&+2N^{2}\nabla
w_{\ell}(N(x-y))\cdot\big{[}\nabla\varphi_{0}(x)\varphi_{0}(y)-\varphi_{0}(x)\nabla\varphi_{0}(y)\big{]}.\end{split}$
(3.7)

With the scattering equation (2.6) (and using the bounds in Lemma 2.1), we can
show that the Hilbert-Schmidt norm of the operator associated with the kernel
on the first line is bounded, uniformly in $N$. The operator associated with
the kernel on the second line is also Hilbert-Schmidt, with norm bounded
uniformly in $N$. As for the kernel on the last line, we can write

$\begin{split}\nabla_{i}\varphi_{0}(x)&\varphi_{0}(y)-\varphi_{0}(x)\nabla_{i}\varphi_{0}(y)\\\
&=\int_{0}^{1}ds\,\frac{d}{ds}\,\nabla_{i}\varphi_{0}(sx+(1-s)y)\varphi_{0}((1-s)x+sy)\\\
&=\sum_{j=1}^{3}(x_{j}-y_{j})\int_{0}^{1}ds\,\Big{[}\big{(}\nabla_{i}\nabla_{j}\varphi_{0}\big{)}(sx+(1-s)y)\varphi_{0}((1-s)x+sy)\\\
&\hskip
142.26378pt+\big{(}\nabla_{i}\varphi_{0}\big{)}(sx+(1-s)y)\big{(}\nabla_{j}\varphi_{0}\big{)}((1-s)x+sy)\Big{]}.\end{split}$

With the factor $(x-y)$ multiplying $N^{2}(\nabla w_{\ell})(N(x-y))$, and with
the estimates (2.9), (A.2), we conclude that also the last term on the right
hand side of (3.7) has a bounded Hilbert-Schmidt norm. This concludes the
proof of (3.6) and shows that the error (3.5) is small. Similarly, we can also
bound the diagonal part of $\mathcal{Q}_{\mathcal{J}_{N}}$. We find

$\pm\bigg{(}\int dxdy\
\widetilde{\Phi}(x,y)(b_{x}^{*}b_{y}-a_{x}^{*}a_{y})\bigg{)}\leq
CN^{-1}(\mathcal{K}+\mathcal{V}_{\text{ext}}+\mathcal{N}+1)(\mathcal{N}+1).$

which completes the proof of (3.4).

###  3.1 Lower Bound on $\sigma(H_{N})$

In this section, we compare the min-max values of $\mathcal{J}_{N}$ with those
of the quadratic operator $\widetilde{\mathcal{Q}}_{\mathcal{J}_{N}}$, defined
through (3.2) as a quadratic form in the full excitation Fock space
$\mathcal{F}_{\bot\varphi_{0}}$. To obtain a lower bound on the spectrum of
$\mathcal{J}_{N}$, suppose that $\lambda_{n}(\mathcal{J}_{N})-E_{N}\leq\zeta$.
Then the min-max principle implies with (2.46), (3.1), (3.3), (3.4) and
Proposition 2.9

$\begin{split}\lambda_{n}(\mathcal{J}_{N})&=\kappa_{\mathcal{J}_{N}}+\,\min_{\begin{subarray}{c}Y\subset
P_{\zeta}(\mathcal{F}_{\bot\varphi_{0}}^{\leq N}),\\\
\dim(Y)=n\end{subarray}}\max_{\begin{subarray}{c}\xi\in Y:\\\
\|\xi\|=1\end{subarray}}\langle\xi,(\widetilde{\mathcal{Q}}_{\mathcal{J}_{N}}+\mathcal{V}_{N}+\mathcal{E}_{\mathcal{J}_{N}}+\widetilde{\mathcal{E}}_{N})\xi\rangle\\\
&\geq\kappa_{\mathcal{J}_{N}}+\min_{\begin{subarray}{c}Y\subset
P_{\zeta}(\mathcal{F}_{\bot\varphi_{0}}^{\leq N}),\\\
\dim(Y)=n\end{subarray}}\max_{\begin{subarray}{c}\xi\in Y:\\\
\|\xi\|=1\end{subarray}}\langle\xi,\widetilde{\mathcal{Q}}_{\mathcal{J}_{N}}\xi\rangle-
CN^{-\rho}(1+\zeta^{6})\\\
&\geq\kappa_{\mathcal{J}_{N}}+\min_{\begin{subarray}{c}X\subset\mathcal{F}_{\bot\varphi_{0}},\\\
\dim(Y)=n\end{subarray}}\max_{\begin{subarray}{c}\xi\in Y:\\\
\|\xi\|=1\end{subarray}}\langle\xi,\widetilde{\mathcal{Q}}_{\mathcal{J}_{N}}\xi\rangle-
CN^{-\rho}(1+\zeta^{6})\\\
&=\kappa_{\mathcal{J}_{N}}+\lambda_{n}(\widetilde{\mathcal{Q}}_{\mathcal{J}_{N}})-CN^{-\rho}(1+\zeta^{6}).\end{split}$
(3.8)

Here, we have abbreviated by $P_{\zeta}$ the spectral projection of
$\mathcal{J}_{N}-E_{N}$ associated with the interval $(-\infty;\zeta]$. Up to
the constant $\kappa_{\mathcal{J}_{N}}$ (and an error that vanishes as
$N\to\infty$), we thus obtain a lower bound on the min-max values of
$\mathcal{J}_{N}$ through those of
$\widetilde{\mathcal{Q}}_{\mathcal{J}_{N}}$.

Let us now diagonalize the quadratic operator
$\widetilde{\mathcal{Q}}_{\mathcal{J}_{N}}$. We adapt here the arguments used
in [15] for mean-field bosons. We recall the definitions of $H_{\text{GP}}$
and $E$ in (1.6) and of $\widetilde{\Phi}$, $\widetilde{\Gamma}$ in (2.49),
(2.50). We identify these operators with operators mapping
$L^{2}_{\bot\varphi_{0}}(\mathbb{R}^{3})$ back into itself and, by slight
abuse of notation, we still write
$H_{\text{GP}},E,\widetilde{\Phi},\widetilde{\Gamma}$ instead of
$H_{\text{GP}}Q$, $EQ,Q\widetilde{\Phi}Q,Q\widetilde{\Gamma}Q$ (recall that
$Q=1-|\varphi_{0}\rangle\langle\varphi_{0}|$). Notice in particular that both
$H_{\text{GP}}$ and $E$ are invertible in
$L^{2}_{\bot\varphi_{0}}(\mathbb{R}^{3})$. In addition to these operators, we
also define, on $L_{\perp\varphi_{0}}^{2}(\mathbb{R}^{3})$,

$\begin{split}\widetilde{D}&=\widetilde{\Phi}-\widetilde{\Gamma}=e^{-\eta}H_{\text{GP}}\,e^{-\eta},\,\,\,\,\widetilde{E}=\big{(}\widetilde{D}^{1/2}(\widetilde{D}+2\widetilde{\Gamma})\widetilde{D}^{1/2}\big{)}^{1/2}\,\,\,\,\text{
with}\\\
\widetilde{D}+2\widetilde{\Gamma}&=\widetilde{\Phi}+\widetilde{\Gamma}=e^{\eta}\big{(}H_{\text{GP}}+2N^{3}(Vf_{\ell})(N(x-y))\varphi_{0}(x)\varphi_{0}(y)\big{)}e^{\eta}\end{split}$
(3.9)

as well as

$\begin{split}A={\widetilde{D}}^{1/2}\widetilde{E}^{-1/2},\hskip
14.22636ptB=\big{(}A^{-1}\big{)}^{*}=\widetilde{D}^{-1/2}\widetilde{E}^{1/2},\hskip
14.22636pt\alpha=\log\big{(}|A^{*}|\big{)}.\end{split}$ (3.10)

Using the polar decomposition, we also denote by $W$ the partial isometry on
$L_{\perp\varphi_{0}}^{2}(\mathbb{R}^{3})$ that is defined through

$A=W|A|=|A^{*}|W,\hskip 14.22636ptB=|B^{*}|W.$ (3.11)

The next lemma collects basic properties of the operators introduced above.

######  Lemma 3.1.

Let $\widetilde{E},\widetilde{D}$ be defined in
$L_{\perp\varphi_{0}}^{2}(\mathbb{R}^{3})$ by (3.9). Let $\ell\in(0;1)$ be
small enough and let $N$ be large enough. Then:

  1. a)

There exists $c>0$ such that $\widetilde{E},\widetilde{D}\geq c>0$. In
particular, the operators $A,B,\alpha$ and $W$, defined in (3.10), (3.11) are
well-defined in $L^{2}_{\bot\varphi_{0}}(\mathbb{R}^{3})$. Moreover, $W$ is
unitary.

  2. b)

There exist $c,C>0$ such that $c\widetilde{D}^{2}\leq\widetilde{E}^{2}\leq
C\widetilde{D}^{2}$. In particular, $c^{1/2}\widetilde{D}\leq\widetilde{E}\leq
C^{1/2}\widetilde{D}$.

  3. c)

There exists $C>0$ such that $H_{\text{GP}}-C\leq\widetilde{D}\leq
H_{\text{GP}}+C$.

  4. d)

We have $\|A-1\|_{\text{HS}},\|B-1\|_{\text{HS}}\leq C$ for some $C>0$.

  5. e)

For every $\beta<1$, there exists $C>0$ such that
$\|\widetilde{D}^{\beta/2}\alpha\widetilde{D}^{\beta/2}\|_{\text{HS}}\leq C$.
Moreover, there exists $C>0$ such that
$\|\widetilde{D}^{1/2}\alpha\|_{\text{HS}}\leq C$ (from parts a), b),c), these
estimates remain true if we replace factors of $\widetilde{D}$ by the
identity, by factors of $\widetilde{E}$ or by factors of $H_{\text{GP}}$).

###### Proof.

We start with the proof of a). On $L^{2}_{\perp\varphi_{0}}(\mathbb{R}^{3})$,
we have $H_{\text{GP}}\geq c>0$.Thus

$\langle\psi,\widetilde{D}\psi\rangle=\langle
e^{-\eta}\psi,H_{\text{GP}}e^{-\eta}\psi\rangle\geq c\|e^{-\eta}\psi\|^{2}\geq
c(1-C\|\eta\|)\|\psi\|^{2}$

for all $\psi\in L^{2}_{\bot\varphi_{0}}(\mathbb{R}^{3})$. By Lemma 2.2,
$\|\eta\|\leq C\ell^{1/2}$. Thus $D\geq c>0$ follows choosing $\ell>0$ small
enough. To prove the claim for $\widetilde{E}$, we compare first the operator
with kernel $N^{3}V(N(x-y))\varphi_{0}(x)\varphi_{0}(y)$ with
$8\pi\mathfrak{a}_{0}\varphi_{0}^{2}$. To this end, we observe that, for an
arbitrary $\phi\in L^{2}_{\perp\varphi_{0}}(\mathbb{R}^{3})$,

$\begin{split}&\left|\int dxdy\
\overline{\phi}(x)\phi(y)\varphi_{0}(x)\varphi_{0}(y)N^{3}(Vf_{\ell})(N(x-y))-8\pi\mathfrak{a}_{0}\int
dx\ |\phi(x)|^{2}\varphi_{0}(x)^{2}\right|\\\ &\leq
CN^{-1}\|\phi\|^{2}+\left|\int dxdy\
(Vf_{\ell})(y)\overline{\phi(x)}\varphi_{0}(x)\big{(}\phi(x+y/N)\varphi_{0}(x+y/N)-\phi(x)\varphi_{0}(x)\big{)}\right|\\\
&\leq CN^{-1}\|\phi\|^{2}+CN^{-1}\|(-\Delta+1)^{1/2}\phi\|^{2}\leq
CN^{-1}\|H_{\text{GP}}^{1/2}\phi\|^{2}\end{split}$ (3.12)

and thus

$\begin{split}\langle\psi,(\widetilde{D}+2\widetilde{\Gamma})\psi\rangle&\geq\langle
e^{\eta}\psi,\big{(}H_{\text{GP}}(1-C/N)+16\pi\mathfrak{a}_{0}\varphi_{0}^{2}\big{)}e^{\eta}\psi\rangle\\\
&\geq\langle e^{\eta}\psi,H_{\text{GP}}(1-C/N)e^{\eta}\psi\rangle\geq
c(1-C\|\eta\|)\|\psi\|^{2}\end{split}$

for $N$ large enough. This shows that $\widetilde{E}\geq c\widetilde{D}\geq
c>0$. An immediate consequence is that $A,B,\alpha$ and $W$ as in (3.10),
(3.11) are well-defined. Moreover, $W$ is a unitary map, because
$\text{ker}(W)=\text{ker}(A)=\\{0\\}$ (recall that
$\widetilde{E},\widetilde{D}\geq c>0$ in
$L^{2}_{\bot\varphi_{0}}(\mathbb{R}^{3})$).

Let us switch to part b). We have

$\widetilde{E}^{2}=\widetilde{D}^{1/2}(\widetilde{D}+2\widetilde{\Gamma})\widetilde{D}^{1/2}=\widetilde{D}^{2}+2\widetilde{D}^{1/2}\widetilde{\Gamma}\widetilde{D}^{1/2}.$

With (3.6) (and with part a)), we conclude that

$\widetilde{E}^{2}\leq\widetilde{D}^{2}+C\widetilde{D}\leq C\widetilde{D}^{2}$

and also that $\widetilde{E}^{2}\geq\widetilde{D}^{2}-C\widetilde{D}$, which
implies (from the proof of a), we have $\widetilde{E}\geq c\widetilde{D}$)

$\widetilde{D}^{2}\leq\widetilde{E}^{2}+C\widetilde{D}\leq\widetilde{E}^{2}+C\widetilde{E}\leq
C\widetilde{E}^{2}.$

As for part c), we observe that

$\widetilde{D}=e^{\eta}H_{\text{GP}}e^{\eta}=H_{\text{GP}}+(e^{\eta}-1)H_{\text{GP}}+H_{\text{GP}}(e^{\eta}-1)+(e^{\eta}-1)H_{\text{GP}}(e^{\eta}-1)$

and that, from Lemma 2.2,
$(e^{\eta}-1)H_{\text{GP}}+H_{\text{GP}}(e^{\eta}-1)$ and
$(e^{\eta}-1)H_{\text{GP}}(e^{\eta}-1)$ are bounded operators (the kernel of
$k(-\Delta)+(-\Delta)k$ has the form
$(\Delta_{x}+\Delta_{y})(Nw_{\ell}(N(x-y))\varphi_{0}(x)\varphi_{0}(y)$ and
can be handled as in (3.7), since the term
$N^{3}(Vf_{\ell})(N(x-y))\varphi_{0}(x)\varphi_{0}(y)$ corresponds to a
bounded operator).

Let us now prove part d). We focus on the operator $A-1$. Using functional
calculus (as in [15, Lemma 3]) we write

$A-1=(\widetilde{D}^{1/2}-\widetilde{E}^{1/2})\widetilde{E}^{-1/2}=2\sqrt{2}\pi\int_{0}^{\infty}dt\,t^{1/4}\frac{\widetilde{D}^{1/2}}{t+\widetilde{D}^{2}}\widetilde{\Gamma}\widetilde{D}^{1/2}\frac{\widetilde{E}^{-1/2}}{t+\widetilde{E}^{2}}.$

With (3.6) and with the bounds
$\|\widetilde{D}^{1/2}/(t+\widetilde{D}^{2})\|_{\text{op}}\leq C(t+1)^{-3/4}$,
$\|(t+\widetilde{E}^{2})^{-1}\|_{\text{op}}\leq C(t+1)^{-1}$,
$\|\widetilde{D}^{1/2}\widetilde{E}^{-1/2}\|_{\text{op}}\leq C$ (by part b),
we obtain

$\|A-1\|_{\text{HS}}\leq
C\int_{0}^{\infty}dt\,t^{1/4}\Big{\|}\frac{\widetilde{D}^{1/2}}{t+\widetilde{D}^{2}}\Big{\|}_{\text{op}}\big{\|}\widetilde{\Gamma}\big{\|}_{\text{HS}}\|\widetilde{D}^{1/2}\widetilde{E}^{-1/2}\|_{\text{op}}\Big{\|}\frac{1}{t+\widetilde{E}^{2}}\Big{\|}\leq
C\,.$

Finally, let us prove part e). From part b), we have $0<c\leq AA^{*}\leq C$.
For $x\in[c;C]$, we can bound

$x-1-C(x-1)^{2}\leq\log x\leq x-1+C(1-x)^{2}.$

Thus

$\|\widetilde{D}^{\beta/2}\alpha\widetilde{D}^{\beta/2}\|_{\text{HS}}\leq
C\|\widetilde{D}^{\beta/2}(1-AA^{*})\widetilde{D}^{\beta/2}\|_{\text{HS}}+C\|\widetilde{D}^{\beta/2}(1-AA^{*})^{2}\widetilde{D}^{\beta/2}\|_{\text{HS}}.$
(3.13)

With functional calculus, we find

$\begin{split}\widetilde{D}^{\beta/2}(1-AA^{*})\widetilde{D}^{\beta/2}&=\widetilde{D}^{(\beta+1)/2}(\widetilde{D}^{-1}-\widetilde{E}^{-1})\widetilde{D}^{(\beta+1)/2}\\\
&=C\int_{0}^{\infty}\frac{dt}{t^{1/2}}\frac{\widetilde{D}^{1+\beta/2}}{t+\widetilde{D}^{2}}\widetilde{\Gamma}D^{1/2}\frac{1}{t+\widetilde{E}^{2}}\widetilde{D}^{(\beta+1)/2}.\end{split}$

Thus, with (3.6),
$\|\widetilde{D}^{1+\beta/2}/(t+\widetilde{D}^{2})\|_{\text{op}},\|\widetilde{E}^{1+\beta/2}/(t+\widetilde{E}^{2})\|_{\text{op}}\leq
C(t+1)^{-1/2+\beta/4}$ and
$\|\widetilde{D}^{1/2}\widetilde{E}^{-1/2}\|_{\text{op}},\|\widetilde{E}^{-(\beta+1)/2}\widetilde{D}^{(\beta+1)/2}\|_{\text{op}}\leq
C$ (by part b)), we find

$\|\widetilde{D}^{\beta/2}(1-AA^{*})\widetilde{D}^{\beta/2}\|_{\text{HS}}\leq
C\int_{0}^{\infty}dt\,t^{-1/2}(1+t)^{-1+\beta/2}\leq C$

for any $\beta<1$. Since
$\|\widetilde{D}^{\beta/2}(1-AA^{*})^{2}\widetilde{D}^{\beta/2}\|_{\text{HS}}\leq
C\|\widetilde{D}^{\beta/2}(1-AA^{*})\widetilde{D}^{\beta/2}\|^{2}_{\text{HS}}\leq
C$, it follows from (3.13) that
$\|\widetilde{D}^{\beta/2}\alpha\widetilde{D}^{\beta/2}\|_{\text{HS}}\leq C$,
for all $\beta<1$. The bound for $\|\widetilde{D}^{1/2}\alpha\|_{\text{HS}}$
can be proven similarly. ∎

Using the operators defined above, we can now construct a suitable Bogoliubov
transformation that diagonalizes $\widetilde{\mathcal{Q}}_{\mathcal{J}_{N}}$.
To this end, we denote by $(\varphi_{j})_{j\in\mathbb{N}}$ the eigenbasis of
$\widetilde{E}$, we set $a^{\sharp}_{j}=a^{\sharp}(\varphi_{j})$ for
$\sharp\in\\{\cdot,*\\}$,
$\alpha_{ij}=\langle\varphi_{i},\alpha\,\varphi_{j}\rangle$ and we define

$\mathcal{W}=\Gamma(W),\hskip
14.22636ptX=\frac{1}{2}\sum_{i,j=1}^{\infty}\alpha_{ij}a^{*}_{i}a^{*}_{j}-\text{h.c.},\hskip
14.22636pt\mathcal{U}=e^{X}\mathcal{W}.$ (3.14)

Note in particular that
$\mathcal{U}:\mathcal{F}_{\bot\varphi_{0}}\to\mathcal{F}_{\bot\varphi_{0}}$
and recall that $\mathcal{U}$ acts as

$\begin{split}&\mathcal{U}^{*}a^{*}(f)\mathcal{U}\\\
&=\frac{1}{2}a^{*}(W^{*}|A^{*}|f)+\frac{1}{2}a^{*}(W^{*}|A^{*}|^{-1}f)+\frac{1}{2}a(W^{*}|A^{*}|\bar{f})-\frac{1}{2}a^{*}(W^{*}|A^{*}|^{-1}\bar{f})\\\
&=\frac{1}{2}a^{*}(\widetilde{E}^{-1/2}\widetilde{D}^{1/2}f)+\frac{1}{2}a^{*}(\widetilde{E}^{1/2}\widetilde{D}^{-1/2}f)+\frac{1}{2}a(\widetilde{E}^{-1/2}\widetilde{D}^{1/2}\bar{f})-\frac{1}{2}a(\widetilde{E}^{1/2}\widetilde{D}^{-1/2}\bar{f})\end{split}$

for $f\in L^{2}_{\bot\varphi_{0}}(\mathbb{R}^{3})$, by the definitions in
(3.10). From [15], we recall that $\mathcal{U}$ is constructed so that
$\mathcal{U}^{*}\widetilde{\mathcal{Q}}_{\mathcal{J}_{N}}\mathcal{U}$ is
diagonal. Indeed, representing $\widetilde{\mathcal{Q}}_{\mathcal{J}_{N}}$ in
$\mathcal{F}_{\bot\varphi_{0}}$ as

$\widetilde{\mathcal{Q}}_{\mathcal{J}_{N}}=\sum_{i,j=1}^{\infty}\langle\varphi_{i},\widetilde{\Phi}\,\varphi_{j}\rangle\,a^{*}_{i}a_{j}+\frac{1}{2}\sum_{i,j=1}^{\infty}\langle\varphi_{i},\widetilde{\Gamma}\,\varphi_{j}\rangle\big{(}a^{*}_{i}a^{*}_{j}+\text{h.c.}\big{)},$

a lengthy, but straightforward calculation verifies that

$\begin{split}\mathcal{U}^{*}\widetilde{\mathcal{Q}}_{\mathcal{J}_{N}}\mathcal{U}&=\frac{1}{2}\operatorname{tr}_{\bot\varphi_{0}}\left(\frac{1}{2}(\widetilde{D}^{1/2}\widetilde{E}\widetilde{D}^{-1/2}+\widetilde{D}^{-1/2}\widetilde{E}\widetilde{D}^{1/2})-\widetilde{D}-\widetilde{\Gamma}\right)\\\
&\hskip
14.22636pt+\sum_{i,j=1}^{\infty}\frac{1}{2}\big{(}\widetilde{\Phi}_{ij}+\widetilde{\Gamma}_{ij}\big{)}a^{*}(\widetilde{E}^{-1/2}\widetilde{D}^{1/2}\varphi_{i})a(\widetilde{E}^{-1/2}\widetilde{D}^{1/2}\varphi_{j})\\\
&\hskip
14.22636pt+\sum_{i,j=1}^{\infty}\frac{1}{2}\big{(}\widetilde{\Phi}_{ij}-\widetilde{\Gamma}_{ij}\big{)}a^{*}(\widetilde{E}^{1/2}\widetilde{D}^{-1/2}\varphi_{i})a(\widetilde{E}^{1/2}\widetilde{D}^{-1/2}\varphi_{j})\\\
&=\frac{1}{2}\operatorname{tr}_{\bot\varphi_{0}}\big{(}\frac{1}{2}(\widetilde{D}^{1/2}\widetilde{E}\widetilde{D}^{-1/2}+\widetilde{D}^{-1/2}\widetilde{E}\widetilde{D}^{1/2})-\widetilde{D}-\widetilde{\Gamma}\big{)}+d\Gamma(\widetilde{E}),\end{split}$
(3.15)

where $\operatorname{tr}_{\bot\varphi_{0}}$ denotes the trace in
$L^{2}_{\bot\varphi_{0}}(\mathbb{R}^{3})$. As a consequence of (3.8), we
obtain that if $\lambda_{n}(\mathcal{J}_{N})-E_{N}\leq\zeta$, then

$\begin{split}\lambda_{n}(\mathcal{J}_{N})\geq\kappa_{\mathcal{J}_{N}}&+\frac{1}{2}\operatorname{tr}_{\bot\varphi_{0}}\left(\frac{1}{2}\left(\widetilde{D}^{1/2}\widetilde{E}\widetilde{D}^{-1/2}+\widetilde{D}^{-1/2}\widetilde{E}\widetilde{D}^{1/2}\right)-\widetilde{D}-\widetilde{\Gamma}\right)\\\
&+\lambda_{n}\big{(}d\Gamma(\widetilde{E})\big{)}-CN^{-\rho}(1+\zeta^{6}).\end{split}$
(3.16)

To obtain a lower bound for the eigenvalues of $\mathcal{J}_{N}$ (and hence of
$H_{N}$) matching the statement of Theorem 1.1, we still have to compare the
operator $\widetilde{E}$, defined in (3.9), with the operator $E$, defined in
(1.6). This is done in the next lemma.

######  Lemma 3.2.

On $L^{2}_{\bot\varphi_{0}}(\mathbb{R}^{3})$ we have that

$\begin{split}&(1-C/N)\big{(}\widetilde{D}^{1/2}e^{\eta}(H_{\text{GP}}+16\pi\mathfrak{a}_{0}\varphi_{0}^{2})e^{\eta}\widetilde{D}^{1/2}\big{)}^{1/2}\leq\widetilde{E}\\\
&\hskip
170.71652pt\leq(1+C/N)\big{(}\widetilde{D}^{1/2}e^{\eta}(H_{\text{GP}}+16\pi\mathfrak{a}_{0}\varphi_{0}^{2})e^{\eta}\widetilde{D}^{1/2}\big{)}^{1/2}\end{split}$

for a constant $C>0$, independent of $N$. Moreover, the operator $E$ defined
in (1.6) is unitarily equivalent to
$\big{(}\widetilde{D}^{1/2}e^{\eta}(H_{\text{GP}}+16\pi\mathfrak{a}_{0}\varphi_{0}^{2})e^{\eta}\widetilde{D}^{1/2}\big{)}^{1/2}$.
We conclude that

$(1-C/N)\lambda_{n}(E)\leq\lambda_{n}(\widetilde{E})\leq(1+C/N)\lambda_{n}(E).$

###### Proof.

Using the bound (3.12), we see that

$(1-C/N)e^{\eta}(H_{\text{GP}}+16\pi\mathfrak{a}_{0}\varphi_{0}^{2})e^{\eta}\leq(\widetilde{D}+2\widetilde{\Gamma})\leq(1+C/N)e^{\eta}(H_{\text{GP}}+16\pi\mathfrak{a}_{0}\varphi_{0}^{2})e^{\eta},$

which immediately implies the first claim. The statement about the unitary
equivalence of
$\big{(}\widetilde{D}^{1/2}e^{\eta}(H_{\text{GP}}+16\pi\mathfrak{a}_{0}\varphi_{0}^{2})e^{\eta}\widetilde{D}^{1/2}\big{)}^{1/2}$
and $E$ follows if we write

$\widetilde{D}^{1/2}=|H_{\text{GP}}^{1/2}e^{-\eta}|=U_{0}^{*}H_{\text{GP}}^{1/2}e^{-\eta}=e^{-\eta}H_{\text{GP}}^{1/2}U_{0}$

for some partial isometry $U_{0}$, by polar decomposition. Since $e^{-\eta}$
is bijective and $H_{\text{GP}}^{1/2}$ is injective, we see that
$\text{ker}\big{(}H_{\text{GP}}^{1/2}e^{-\eta}\big{)}=\\{0\\}$ and hence
$U_{0}$ is a unitary map. This implies that
$\big{(}\widetilde{D}^{1/2}e^{\eta}(H_{\text{GP}}+16\pi\mathfrak{a}_{0}\varphi_{0}^{2})e^{\eta}\widetilde{D}^{1/2}\big{)}^{1/2}=U_{0}^{*}EU_{0}$.
∎

With Lemma 3.2 and Eq. (3.16), we conclude for
$\lambda_{n}(\mathcal{J}_{N})-E_{N}\leq\zeta$ that

$\begin{split}\lambda_{n}(\mathcal{J}_{N})&\geq\kappa_{\mathcal{J}_{N}}+\frac{1}{2}\operatorname{tr}_{\bot\varphi_{0}}\left(\frac{1}{2}\left(\widetilde{D}^{1/2}\widetilde{E}\widetilde{D}^{-1/2}+\widetilde{D}^{-1/2}\widetilde{E}\widetilde{D}^{1/2}\right)-\widetilde{D}-\widetilde{\Gamma}\right)\\\
&\hskip
14.22636pt+\lambda_{n}\big{(}d\Gamma(E)\big{)}-CN^{-\rho}(1+\zeta^{6})\\\
&\hskip 14.22636pt-
CN^{-1}\Big{(}-\frac{1}{2}\operatorname{tr}_{\bot\varphi_{0}}\left[\frac{1}{2}\left(\widetilde{D}^{1/2}\widetilde{E}\widetilde{D}^{-1/2}+\widetilde{D}^{-1/2}\widetilde{E}\widetilde{D}^{1/2}\right)-\widetilde{D}-\widetilde{\Gamma}\right]\\\
&\hskip
71.13188pt+\lambda_{n}(\mathcal{J}_{N})-\kappa_{\mathcal{J}_{N}}\Big{)}.\end{split}$
(3.17)

In the next two sections, we will prove that the ground state energy of
$\mathcal{J}_{N}$ is equal to
$E_{N}=\kappa_{\mathcal{J}_{N}}+\frac{1}{2}\operatorname{tr}_{\bot\varphi_{0}}\left(\frac{1}{2}(\widetilde{D}^{1/2}\widetilde{E}\widetilde{D}^{-1/2}+\widetilde{D}^{-1/2}\widetilde{E}\widetilde{D}^{1/2})-\widetilde{D}-\widetilde{\Gamma}\right)+\mathcal{O}\big{(}N^{-\rho}(1+\zeta^{6})\big{)}$.
This will imply that the last term on the right hand side in (3.17) is a
negligible error of order $\mathcal{O}(N^{-1}\zeta)$. The bound (3.17) then
also shows that the spectrum of $\mathcal{J}_{N}$ above $E_{N}$ is bounded
from below by that of $d\Gamma(E)$, up to errors that vanish in the limit
$N\to\infty$. Note that the eigenvalues of $d\Gamma(E)$ are explicitly given
by finite sums of the form

$\sum_{j=1}^{\infty}n_{j}e_{j},$

where the $(e_{j})_{j\in\mathbb{N}}$ denote the eigenvalues of the operator
$E$ in $L^{2}_{\bot\varphi_{0}}(\mathbb{R}^{3})$, defined in (1.6), and
$n_{j}\in\mathbb{N}$ are non-zero for finitely many $j\in\mathbb{N}$.

###  3.2 Upper Bound on $\sigma(H_{N})$

In this section, we show that the spectrum of $d\Gamma(E)$ provides an upper
bound on the spectrum of $\mathcal{J}_{N}$ above its ground state energy. To
show this, we construct suitable trial states that approximately diagonalize
$\widetilde{\mathcal{Q}}_{\mathcal{J}_{N}}$, defined in (3.2), and we apply
Lemma 3.2. Since $\widetilde{\mathcal{Q}}_{\mathcal{J}_{N}}$ is defined in the
full excitation space $\mathcal{F}_{\bot\varphi_{0}}$, we can not directly
choose its eigenvectors as trial states, but have to truncate them first to
$\mathcal{F}_{\bot\varphi_{0}}^{\leq N}$. Furthermore, in contrast to the
lower bound, we also have to evaluate the potential energy $\mathcal{V}_{N}$
of the trial states and show that it is negligible in the limit $N\to\infty$.

We start the analysis with some preliminary results on the conjugation of
$\mathcal{N}$, $\mathcal{K}+\mathcal{V}_{\text{ext}}$ and $\mathcal{V}_{N}$ by
the unitary map $\mathcal{U}$, defined in (3.14).

######  Lemma 3.3.

For every $j\in\mathbb{N}$, there exists a constant $C>0$ such that in
$\mathcal{F}_{\perp\varphi_{0}}$ we have

$\mathcal{U}^{*}(\mathcal{N}+1)^{j}\mathcal{U}\leq C(\mathcal{N}+1)^{j}.$
(3.18)

Moreover, with $\widetilde{E}$ as defined in (3.9), we have that

$\mathcal{U}^{*}d\Gamma(\widetilde{E})(\mathcal{N}+1)^{j}\mathcal{U}\leq
C(d\Gamma(\widetilde{E})+\mathcal{N}+1)(\mathcal{N}+1)^{j}$ (3.19)

and consequently that

$\mathcal{U}^{*}(\mathcal{K}+\mathcal{V}_{\text{ext}})(\mathcal{N}+1)^{j}\mathcal{U}\leq
C(d\Gamma(\widetilde{E})+\mathcal{N}+1)(\mathcal{N}+1)^{j}.$ (3.20)

###### Proof.

Proceeding as in [10, Lemma 3.1] and using that $\|\alpha\|_{\text{HS}}\leq
C$, we readily find (recalling the notation introduced in (3.14)), that

$\sup_{s\in[-1;1]}\,e^{-sX}(\mathcal{N}+1)^{j}e^{sX}\leq
C(\mathcal{N}+1)^{j}.$

The bound (3.18) then follows from $\mathcal{N}=d\Gamma(1)$ so that
$\mathcal{W}^{*}(\mathcal{N}+1)^{j}\mathcal{W}=(\mathcal{N}+1)^{j}$.

Next, we note that, by (3.19) and Lemma 3.1 (part c)), we have

$\mathcal{U}^{*}(\mathcal{K}+\mathcal{V}_{\text{ext}})(\mathcal{N}+1)^{j}\mathcal{U}\leq
C\,(d\Gamma(\widetilde{E})+1)(\mathcal{N}+1)^{j}.$

Hence, (3.20) is a consequence of (3.19). To prove the latter, we expand for
$t\in[-1;1]$

$e^{-tX}d\Gamma(\widetilde{E})(\mathcal{N}+1)^{j}e^{tX}=d\Gamma(\widetilde{E})(\mathcal{N}+1)^{j}+\int_{0}^{t}ds\,e^{-sX}[d\Gamma(\widetilde{E})(\mathcal{N}+1)^{j},X]e^{sX}.$

Denoting (as we did around (3.14)) by $(\varphi_{k})_{k\in\mathbb{N}}$ the
eigenbasis of $\widetilde{E}$ and setting
$a^{\sharp}_{k}=a^{\sharp}(\varphi_{k})$, we find

$[d\Gamma(\widetilde{E})(\mathcal{N}+1)^{j},X]=\bigg{(}\sum_{k,l=1}^{\infty}\lambda_{k}(\widetilde{E})\alpha_{kl}a^{*}_{k}a^{*}_{l}+\text{h.c.}\bigg{)}(\mathcal{N}+1)^{j}+d\Gamma(\widetilde{E})[(\mathcal{N}+1)^{j},X].$

An application of Lemma 3.1 (part e)) and Cauchy-Schwarz shows that

$\begin{split}\sum_{k,l=1}^{\infty}\lambda_{k}(\widetilde{E})\big{|}\alpha_{kl}\langle\xi,a^{*}_{k}a^{*}_{l}(\mathcal{N}+1)^{j}\big{)}\xi\rangle\big{|}&\leq\langle\xi,\big{(}d\Gamma(\widetilde{E})+\|\widetilde{E}^{1/2}\alpha\|_{\text{HS}}^{2}\,(\mathcal{N}+1)\big{)}(\mathcal{N}+1)^{j}\xi\rangle\\\
&\leq\langle\xi,d\Gamma(\widetilde{E})(\mathcal{N}+1)^{j}\xi\rangle+C\langle\xi,(\mathcal{N}+1)^{j+1}\xi\rangle\end{split}$

for all $\xi\in\mathcal{F}_{\bot\varphi_{0}}$. Moreover, observing that

$[(\mathcal{N}+1)^{j},X]=\sum_{k,l=1}^{\infty}\alpha_{kl}a^{*}_{k}a^{*}_{l}\,j\,(\mathcal{N}+\Theta(\mathcal{N})+1)^{j-1}+\text{h.c.}$

for some $\Theta:\mathbb{N}\to(0;1)$ by the mean value theorem, we conclude
similarly that

$|\langle\xi,d\Gamma(\widetilde{E})[(\mathcal{N}+1)^{j},X]\xi\rangle|\leq
C\langle\xi,d\Gamma(\widetilde{E})(\mathcal{N}+1)^{j}\xi\rangle+C\langle\xi,(\mathcal{N}+1)^{j+1}\xi\rangle.$

Applying Gronwall to the map $t\mapsto
e^{-tX}d\Gamma(\widetilde{E})(\mathcal{N}+1)^{j}e^{tX}$ and using (3.18), we
find

$\begin{split}\sup_{s\in[-1,1]}e^{-sX}d\Gamma(\widetilde{E})(\mathcal{N}+1)^{j}e^{sX}&\leq
C(d\Gamma(\widetilde{E})+\mathcal{N}+1)(\mathcal{N}+1)^{j}.\end{split}$ (3.21)

To conclude the bound (3.19), we then notice that

$\mathcal{W}^{*}d\Gamma(\widetilde{E})(\mathcal{N}+1)^{j}\mathcal{W}=d\Gamma(W^{*}\widetilde{E}\,W)(\mathcal{N}+1)^{j}$
(3.22)

and that $W^{*}\widetilde{E}W\leq C\widetilde{E}$. From (3.11) and using Lemma
3.1, part b), this last bound follows if we can show that

$\Big{\|}\widetilde{E}^{-1/2}\frac{1}{\sqrt{\widetilde{E}^{-1/2}\widetilde{D}\widetilde{E}^{-1/2}}}\widetilde{E}^{1/2}\Big{\|}_{\text{op}}\leq
C$ (3.23)

uniformly in $N$. To prove (3.23), we observe first that

$\widetilde{E}^{-1/2}(\widetilde{E}-\widetilde{D})=C\int_{0}^{\infty}dt\,\sqrt{t}\,\widetilde{E}^{-1/2}\widetilde{D}^{1/2}\frac{1}{t+\widetilde{D}^{2}}\widetilde{\Gamma}\widetilde{D}^{1/2}\frac{1}{t+\widetilde{E}^{2}}$

which implies that
$\|\widetilde{E}^{-1/2}(\widetilde{E}-\widetilde{D})\|_{\text{op}}\leq C$.
Then we write

$\widetilde{E}^{-1/2}\frac{1}{\sqrt{\widetilde{E}^{-1/2}\widetilde{D}\widetilde{E}^{-1/2}}}\widetilde{E}^{1/2}-1=C\int_{0}^{\infty}\frac{dt}{t^{1/2}}\frac{1}{t+1}\,\widetilde{E}^{-1/2}\frac{1}{t+\widetilde{E}^{-1/2}\widetilde{D}\widetilde{E}^{-1/2}}\widetilde{E}^{-1/2}(\widetilde{E}-\widetilde{D})$

to conclude that

$\Big{\|}\widetilde{E}^{-1/2}\frac{1}{\sqrt{\widetilde{E}^{-1/2}\widetilde{D}\widetilde{E}^{-1/2}}}\widetilde{E}^{1/2}-1\Big{\|}_{\text{op}}\leq
C$

and therefore to obtain (3.23). ∎

As mentioned at the beginning of this section, below we have to evaluate the
potential energy of suitable trial states. To this end, we make use of the
following lemma that controls the conjugation of $\mathcal{V}_{N}$ by the
unitary map $\mathcal{U}$, defined in (3.14).

######  Lemma 3.4.

There exists $C>0$ such that in $\mathcal{F}_{\bot\varphi_{0}}$ we have

$\mathcal{U}^{*}\,\mathcal{V}_{N}\,\mathcal{U}\leq
CN^{-1/2}(d\Gamma(\widetilde{E})+1)^{2}.$

###### Proof.

Recall from (3.14) that $\mathcal{U}=e^{X}\mathcal{W}$. For $t\in[-1;1]$, we
have

$\begin{split}e^{-tX}\mathcal{V}_{N}e^{tX}=&\,\mathcal{V}_{N}+\int_{0}^{t}ds\,\int
dxdy\,N^{2}V(N(x-y))\alpha(x;y)\big{(}e^{-sX}a_{x}a_{y}e^{sX}+\text{h.c.}\big{)}\\\
&+\int_{0}^{t}ds\,\int
dxdy\,N^{2}V(N(x-y))\big{(}e^{-sX}a^{*}_{y}a(\alpha_{x})a_{x}a_{y}e^{sX}+\text{h.c.}\big{)}.\end{split}$
(3.24)

With the operator inequality

$v(x-y)\leq C\|v\|_{1}(-\Delta_{x}+1)^{\beta}(-\Delta_{y}+1)^{\beta}$ (3.25)

for any $\beta>3/4$, we find

$\begin{split}\Big{|}\int
dxdy\,N^{2}&V(N(x-y))\alpha(x;y)\langle\psi,a_{x}a_{y}\psi\rangle\Big{|}\\\
\leq\;&C\|\psi\|\|\mathcal{V}^{1/2}_{N}\psi\|\left[\int
dxdyN^{2}V(N(x-y))|\alpha(x;y)|^{2}\right]^{1/2}\\\
\leq\;&CN^{-1/2}\|(1-\Delta)^{\beta/2}\alpha(1-\Delta)^{\beta/2}\|_{\text{HS}}\|\psi\|\|\mathcal{V}^{1/2}_{N}\psi\|.\end{split}$

Using Lemma 3.1, part c) and then part e), to estimate
$\|(1-\Delta)^{\beta/2}\alpha(1-\Delta)^{\beta/2}\|_{\text{HS}}\leq\|\widetilde{D}^{\beta/2}\alpha\widetilde{D}^{\beta/2}\|_{\text{HS}}\leq
C$, we obtain

$\Big{|}\int
dxdy\,N^{2}V(N(x-y))\alpha(x;y)\langle\psi,a_{x}a_{y}\psi\rangle\Big{|}\leq
C\langle\psi,\mathcal{V}_{N}\psi\rangle+CN^{-1}\|\psi\|^{2}.$ (3.26)

Similarly, we can estimate

$\begin{split}\Big{|}\int
dxdy\,N^{2}&V(N(x-y))\langle\psi,a^{*}_{y}a(\alpha_{x})a_{x}a_{y}\psi\rangle\Big{|}\\\
\leq\;&C\|\mathcal{V}^{1/2}_{N}\psi\|\left[\int
dxdyN^{2}V(N(x-y))\|a^{*}(\alpha_{x})a_{y}\psi\|^{2}\right]^{1/2}\\\
\leq\;&CN^{-1/2}\|(1-\Delta)^{1/2}\alpha\|_{\text{HS}}\|\mathcal{V}^{1/2}_{N}\psi\|\|(\mathcal{K}+\mathcal{N})^{1/2}(\mathcal{N}+1)^{1/2}\psi\|\\\
\leq\;&C\langle\psi,\mathcal{V}_{N}\psi\rangle+CN^{-1}\langle\psi,(\mathcal{K}+\mathcal{N})(\mathcal{N}+1)\psi\rangle.\end{split}$
(3.27)

Inserting (3.26), (3.27) in (3.24), using Lemma 3.1 to replace $\mathcal{K}$
with $d\Gamma(\widetilde{E})$ and then applying (3.21), we arrive at

$\begin{split}\langle\psi,e^{-tX}\mathcal{V}_{N}e^{tX}\psi\rangle\leq\;&C\int_{0}^{t}ds\langle
e^{-sX}\psi,\mathcal{V}_{N}e^{sX}\psi\rangle+\langle\psi,\mathcal{V}_{N}\psi\rangle\\\
&+CN^{-1/2}\langle\psi,(d\Gamma(\widetilde{E})+1)(\mathcal{N}+1)\psi\rangle.\end{split}$

With Gronwall, we obtain

$\langle\psi,e^{-X}\mathcal{V}_{N}e^{X}\psi\rangle\leq
C\langle\psi,\mathcal{V}_{N}\psi\rangle+CN^{-1/2}\langle\psi,(d\Gamma(\widetilde{E})+1)(\mathcal{N}+1)\psi\rangle.$

Since $\mathcal{U}=e^{X}\mathcal{W}$, we find, using (3.22),

$\begin{split}\langle\psi,\mathcal{U}^{*}\mathcal{V}_{N}\mathcal{U}\psi\rangle&\leq
C\langle\psi,\mathcal{W}^{*}\mathcal{V}_{N}\mathcal{W}\psi\rangle+CN^{-1/2}\langle\psi,\mathcal{W}^{*}(d\Gamma(\widetilde{E})+1)(\mathcal{N}+1)\mathcal{W}\psi\rangle\\\
&\leq
C\langle\psi,\mathcal{W}^{*}\mathcal{V}_{N}\mathcal{W}\psi\rangle+CN^{-1/2}\langle\psi,(d\Gamma(\widetilde{E})+1)(\mathcal{N}+1)\psi\rangle.\end{split}$
(3.28)

To estimate the expectation of $\mathcal{W}^{*}\mathcal{V}_{N}\mathcal{W}$, we
observe that, on the $n$-particle sector, using (3.25) and Lemma 3.1,
$\mathcal{W}=\Gamma(W)$ and the bound $W^{*}\widetilde{E}W\leq C\widetilde{E}$
established after (3.22),

$\begin{split}\langle\psi^{(n)},\mathcal{W}^{*}\mathcal{V}_{N}\mathcal{W}\psi^{(n)}\rangle&=\sum_{i<j}^{n}\langle\mathcal{W}\psi^{(n)},N^{2}V(N(x_{i}-x_{j}))\mathcal{W}\psi^{(n)}\rangle\\\
&\leq
CN^{-1}\sum_{i<j}^{n}\langle\mathcal{W}\psi^{(n)},(1-\Delta_{x_{i}})(1-\Delta_{x_{j}})\mathcal{W}\psi^{(n)}\rangle\\\
&\leq
CN^{-1}\sum_{i<j}^{n}\langle\psi^{(n)},(W^{*}\widetilde{E}W)_{i}(W^{*}\widetilde{E}W)_{j}\psi^{(n)}\rangle\\\
&\leq
CN^{-1}\sum_{i<j}^{n}\langle\psi^{(n)},\widetilde{E}_{i}\widetilde{E}_{j}\psi^{(n)}\rangle\leq
CN^{-1}\langle\psi^{(n)}d\Gamma(\widetilde{E})^{2}\psi^{(n)}\rangle.\end{split}$

From (3.28), we obtain

$\langle\psi,\mathcal{U}^{*}\mathcal{V}_{N}\mathcal{U}\psi\rangle\leq
CN^{-1/2}\langle\psi,(d\Gamma(\widetilde{E})+1)^{2}\psi\rangle.$

∎

Equipped with Lemmas 3.3 and 3.4, let us now bound the spectrum
$\sigma(H_{N})$ from above. Like for the lower bound, we make use of the min-
max principle and we derive upper bounds on the eigenvalues of
$\mathcal{J}_{N}$. We first control the ground state energy $E_{N}$ and
afterwards the spectrum of $\mathcal{J}_{N}$ above $E_{N}$.

Recalling the definition of the unitary map
$\mathcal{U}:\mathcal{F}_{\bot\varphi_{0}}\to\mathcal{F}_{\bot\varphi_{0}}$ in
(3.14), the ground state energy of $\mathcal{J}_{N}$ is bounded from above by

$E_{N}=\lambda_{0}(\mathcal{J}_{N})\leq\frac{\langle\chi_{\leq
N}\mathcal{U}\Omega,\mathcal{J}_{N}\chi_{\leq
N}\mathcal{U}\Omega\rangle}{\langle\chi_{\leq N}\mathcal{U}\Omega,\chi_{\leq
N}\mathcal{U}\Omega\rangle}.$

Here, $\Omega=(1,0,0,\dots)\in\mathcal{F}_{\bot\varphi_{0}}$ denotes the
vacuum and we abbreviate $\chi_{\leq N}=1(\mathcal{N}\leq N)$. Similarly, we
write in the following $\chi_{>N}=1(\mathcal{N}>N)$ so that $1=\chi_{\leq
N}+\chi_{>N}$ in $\mathcal{F}_{\bot\varphi_{0}}$. By Markov’s inequality and
(3.18), we note that

$\|\chi_{\leq
N}\mathcal{U}\Omega\|^{2}=1-\|\chi_{>N}\mathcal{U}\Omega\|^{2}\geq
1-N^{-2}\|\mathcal{N}\mathcal{U}\Omega\|^{2}\geq 1-CN^{-2}.$

Using (2.46), (3.1), (3.3), (3.4), the fact that $\mathcal{J}_{N}\geq 0$ and
that $\mathcal{K},\mathcal{V}_{\text{ext}},\mathcal{V}_{N}\geq 0$ commute with
$\mathcal{N}$, we obtain

$\begin{split}E_{N}&\leq(1-C/N^{2})^{-1}\big{(}\kappa_{\mathcal{J}_{N}}+\langle\chi_{\leq
N}\mathcal{U}\Omega,\widetilde{\mathcal{Q}}_{\mathcal{J}_{N}}\chi_{\leq
N}\mathcal{U}\Omega\rangle\big{)}+C\,\langle\mathcal{U}\Omega,\mathcal{V}_{N}\mathcal{U}\Omega\rangle\\\
&\hskip
14.22636pt+CN^{-\rho}\langle\mathcal{U}\Omega,(\mathcal{H}_{N}+1)(\mathcal{N}+1)^{3}\mathcal{U}\Omega\rangle.\end{split}$

Applying Lemmas 3.3 and 3.4, this implies

$\begin{split}E_{N}&\leq(1-C/N^{2})^{-1}\big{(}\kappa_{\mathcal{J}_{N}}+\langle\chi_{\leq
N}\mathcal{U}\Omega,\widetilde{\mathcal{Q}}_{\mathcal{J}_{N}}\chi_{\leq
N}\mathcal{U}\Omega\rangle\big{)}+CN^{-\rho}.\end{split}$

To evaluate this further, we use that in the sense of forms in
$\mathcal{F}_{\bot\varphi_{0}}$, we have

$\pm\big{(}\chi_{\leq N}\widetilde{\mathcal{Q}}_{\mathcal{J}_{N}}\chi_{\leq
N}-\widetilde{\mathcal{Q}}_{\mathcal{J}_{N}}\big{)}\leq
CN^{-1}(\mathcal{K}+\mathcal{V}_{\text{ext}}+\mathcal{N}+1)(\mathcal{N}+1)$
(3.29)

which is a straightforward consequence of the bound (3.6) and Markov’s
inequality. Combining this with the identity (3.15), we find that

$\begin{split}E_{N}&\leq(1-C/N^{2})^{-1}\left(\kappa_{\mathcal{J}_{N}}+\frac{1}{2}\operatorname{tr}_{\bot\varphi_{0}}\left(\frac{1}{2}(\widetilde{D}^{1/2}\widetilde{E}\widetilde{D}^{-1/2}+\widetilde{D}^{-1/2}\widetilde{E}\widetilde{D}^{1/2})-\widetilde{D}-\widetilde{\Gamma}\right)\right)+CN^{-\rho}\\\
&\leq\kappa_{\mathcal{J}_{N}}+\frac{1}{2}\operatorname{tr}_{\bot\varphi_{0}}\left(\frac{1}{2}(\widetilde{D}^{1/2}\widetilde{E}\widetilde{D}^{-1/2}+\widetilde{D}^{-1/2}\widetilde{E}\widetilde{D}^{1/2})-\widetilde{D}-\widetilde{\Gamma}\right)+CN^{-\rho}.\end{split}$

Note that in the last step we used
$\kappa_{\mathcal{J}_{N}}+\frac{1}{2}\operatorname{tr}_{\bot\varphi_{0}}\left(\frac{1}{2}\widetilde{D}^{1/2}\widetilde{E}\widetilde{D}^{-1/2}+\frac{1}{2}\widetilde{D}^{-1/2}\widetilde{E}\widetilde{D}^{1/2}-\widetilde{D}-\widetilde{\Gamma}\right)=\mathcal{O}(N)$,
which is a consequence of the first inequality in the previous bound, the
lower bound (3.16) and the fact that $E_{N}=\mathcal{O}(N)$. Thus, in summary,
we obtain together with (3.16) that

$E_{N}=\kappa_{\mathcal{J}_{N}}+\frac{1}{2}\operatorname{tr}_{\bot\varphi_{0}}\left(\frac{1}{2}\left(\widetilde{D}^{1/2}\widetilde{E}\widetilde{D}^{-1/2}+\widetilde{D}^{-1/2}\widetilde{E}\widetilde{D}^{1/2}\right)-\widetilde{D}-\widetilde{\Gamma}\right)+\mathcal{O}(N^{-\rho}).$
(3.30)

To continue with the higher min-max values of $\mathcal{J}_{N}$, we assume
that $\lambda_{n}(\mathcal{J}_{N})$ is such that
$\lambda_{n}(\mathcal{J}_{N})-E_{N}\leq\zeta$. By (3.16) and the identity
(3.30), we also have

$\lambda_{n}\big{(}d\Gamma(\widetilde{E})\big{)}\leq\zeta+CN^{-\rho}(1+\zeta^{6}).$
(3.31)

Now, let us recall that

$\lambda_{n}\big{(}d\Gamma(\widetilde{E})\big{)}=\sum_{j=1}^{\infty}l_{j}^{(n)}\widetilde{e}_{j},$

where the $(\widetilde{e}_{j})_{j\in\mathbb{N}}$ denote the eigenvalues of
$\widetilde{E}$ and the $l^{(n)}_{j}\in\mathbb{N}$ are non-zero for finitely
many $j\in\mathbb{N}$. As we did in (3.14), let us also denote by
$(\varphi_{j})_{j\in\mathbb{N}}$ an orthonormal eigenbasis of $\widetilde{E}$
such that $\widetilde{E}\varphi_{j}=\widetilde{e}_{j}\varphi_{j}$, for
$j\in\mathbb{N}$. Then, we choose $n$ orthonormal eigenvectors
$(\xi_{k})_{k=1}^{n}$ of the form

$\xi_{k}=C_{k}\prod_{j=1}^{\infty}\big{(}a^{*}(\varphi_{j})\big{)}^{l^{(k)}_{j}}\Omega\in\mathcal{F}_{\bot\varphi_{0}},$

with $C_{k}$ ensuring that $\|\xi_{k}\|=1$ for each $k=1,\dots,n$ and such
that

$d\Gamma(\widetilde{E})\xi_{k}=\lambda_{k}\big{(}d\Gamma(\widetilde{E})\big{)}\xi_{k}\hskip
14.22636pt\text{for }k=1,\dots,n.$

Given the vectors $(\xi_{k})_{k=1}^{n}$, we define the linear space
$Y^{(n)}\subset\mathcal{F}_{\bot\varphi_{0}}^{\leq N}$ by

$Y^{(n)}=\chi_{\leq N}\,\mathcal{U}\,\text{span}(\xi_{1},\dots,\xi_{n}),$

which we use below in the min-max principle to get an upper bound on
$\lambda_{n}(\mathcal{J}_{N})$. Before evaluating the upper bound, note that
$\chi_{\leq N}\mathcal{U}\,:\,\text{span}\big{(}\xi_{1},\dots,\xi_{n})\to
Y^{(n)}$ is invertible, because we have for a normalized
$\xi\in\mathcal{U}\,\text{span}(\xi_{1},\dots,\xi_{n})$ and $m\in\mathbb{N}$
large enough (but fixed) that

$\|\chi_{\leq
N}\mathcal{U}\,\xi\|^{2}\geq\|\xi\|^{2}-\|\chi_{>N}\mathcal{U}\,\xi\|^{2}\geq
1-\big{(}\zeta+CN^{-\rho}(1+\zeta^{6})\big{)}^{m}N^{-m}>\frac{1}{2}$ (3.32)

for $N$ large enough. Indeed, the second bound follows from Markov’s
inequality, the assumption $\zeta\leq CN^{-\rho/5-\varepsilon}$ for some
$\varepsilon>0$, the bound (3.18), the fact that $\mathcal{N}$ commutes with
$d\Gamma(\widetilde{E})$, the operator bound $\mathcal{N}\leq
Cd\Gamma(\widetilde{E})$ in $\mathcal{F}_{\bot\varphi_{0}}$ and because
$\lambda_{k}\big{(}d\Gamma(\widetilde{E})\big{)}\leq\zeta+CN^{-\rho}(1+\zeta^{6})$
for each $k=1,\dots,n$ by (3.31). Thus, by (3.32), $Y^{(n)}$ is in particular
$n$-dimensional.

Now, let us control $\lambda_{n}(\mathcal{J}_{N})$ through the min-max
principle. We obtain with (2.46), (3.1), (3.3), (3.4), (3.29), (3.32) as well
as Lemmas 3.3 and 3.4 that for $m\in\mathbb{N}$ large (but fixed) and $N$
large enough

$\begin{split}\lambda_{n}(\mathcal{J}_{N})&\leq\sup_{\xi\in
Y^{(n)},\|\xi\|=1}\langle\xi,\mathcal{J}_{N}\xi\rangle=\sup_{\xi\in\,\text{span}(\xi_{1},\dots,\xi_{n}),\|\xi\|=1}\frac{\langle\chi_{\leq
N}\mathcal{U}\xi,\mathcal{J}_{N}\chi_{\leq
N}\mathcal{U}\xi\rangle}{\langle\chi_{\leq N}\mathcal{U}\xi,\chi_{\leq
N}\mathcal{U}\xi\rangle}\\\
&\leq\big{(}1-(N^{-1}\zeta+CN^{-1-\rho}(1+\zeta^{6}))^{m}\big{)}^{-1}\sup_{\xi\in\,\text{span}(\xi_{1},\dots,\xi_{n}),\|\xi\|=1}\big{(}\kappa_{\mathcal{J}_{N}}+\langle\mathcal{U}\xi,\widetilde{\mathcal{Q}}_{\mathcal{J}_{N}}\mathcal{U}\xi\rangle\big{)}\\\
&\hskip
14.22636pt+CN^{-\rho}\sup_{\xi\in\,\text{span}(\xi_{1},\dots,\xi_{n}),\|\xi\|=1}\langle\xi,(d\Gamma(\widetilde{E})+\mathcal{N}+1)(\mathcal{N}+1)^{4}\xi\rangle\\\
&\hskip
14.22636pt+CN^{-1/2}\sup_{\xi\in\,\text{span}(\xi_{1},\dots,\xi_{n}),\|\xi\|=1}\langle\xi,(d\Gamma(\widetilde{E})+1)^{2}\xi\rangle.\end{split}$
(3.33)

Note that we used here the bounds $d\Gamma(-\Delta+V_{\text{ext}})\leq
Cd\Gamma(\widetilde{E})$ and $\mathcal{V}_{N}\leq\mathcal{K}\mathcal{N}$. To
bound the right hand side in (3.33) further, we use that $\mathcal{N}$
commutes with $d\Gamma(\widetilde{E})$, $\mathcal{N}\leq
Cd\Gamma(\widetilde{E})$ in $\mathcal{F}_{\bot\varphi_{0}}$, the a priori
estimate (3.31) as well as the identities (3.15) and (3.30) so that

$\begin{split}\lambda_{n}(\mathcal{J}_{N})&\leq
E_{N}+\lambda_{n}\big{(}d\Gamma(\widetilde{E})\big{)}+CN^{-\rho}\big{(}1+\zeta^{5}+N^{-5\rho}\zeta^{30}\big{)}.\\\
\end{split}$

Finally, applying Lemma 3.2 and once again (3.31), we arrive at

$\begin{split}\lambda_{n}(\mathcal{J}_{N})&\leq
E_{N}+\lambda_{n}\big{(}d\Gamma(E)\big{)}+CN^{-\rho}\big{(}1+\zeta^{5}+N^{-5\rho}\zeta^{30}\big{)}.\end{split}$
(3.34)

###  3.3 Proof of Theorem 1.1

In this section, we conclude the proof of Theorem 1.1, based on the upper and
lower bounds on the spectrum $\sigma(H_{N})$ from Sections 3.1 and 3.2.

###### Proof of Theorem 1.1..

Recalling (3.30), we have shown that

$E_{N}=\kappa_{\mathcal{J}_{N}}+\frac{1}{2}\operatorname{tr}_{\bot\varphi_{0}}\left(\frac{1}{2}\left(\widetilde{D}^{1/2}\widetilde{E}\widetilde{D}^{-1/2}+\widetilde{D}^{-1/2}\widetilde{E}\widetilde{D}^{1/2}\right)-\widetilde{D}-\widetilde{\Gamma}\right)+\mathcal{O}(N^{-\rho})$
(3.35)

with $\kappa_{\mathcal{J}_{N}},\,\widetilde{\Gamma}$ from Proposition 2.8 and
with $\widetilde{D},\widetilde{E}$ defined in (3.9). Moreover, the bounds
(3.17), (3.34) show that if $\lambda_{n}(H_{N})-E_{N}\leq\zeta$, then we have
for sufficiently large $N$ that

$\lambda_{n}(H_{N})-E_{N}=\lambda_{n}\big{(}d\Gamma(E)\big{)}+\mathcal{O}\big{(}N^{-\rho}(1+\zeta^{5}+N^{-5\rho}\zeta^{30})\big{)},$

with $E$ defined in (1.6). The last identity proves the validity of (1.10) in
Theorem 1.1.

To finish the proof of the theorem, it now remains to verify the identity
(1.8). This follows from evaluating the constant
$\kappa_{\mathcal{J}_{N}}+\operatorname{tr}_{\bot\varphi_{0}}(\frac{1}{2}\widetilde{D}^{1/2}\widetilde{E}\widetilde{D}^{-1/2}+\frac{1}{2}\widetilde{D}^{-1/2}\widetilde{E}\widetilde{D}^{1/2}-\widetilde{D}-\widetilde{\Gamma})/2$
on the r.h.s. of (3.35), up to errors that vanish in the limit $N\to\infty$.
To this end, it will be convenient to abbreviate by
$\widetilde{K}_{N}:L^{2}(\mathbb{R}^{3})\to L^{2}(\mathbb{R}^{3})$ the
operator with kernel

$\widetilde{K}_{N}(x;y)=N^{3}(Vf_{\ell})(N(x-y))\varphi_{0}(x)\varphi_{0}(y)$

and to denote by $K_{N}:L^{2}_{\bot\varphi_{0}}(\mathbb{R}^{3})\to
L^{2}_{\bot\varphi_{0}}(\mathbb{R}^{3})$ the operator

$K_{N}=Q\,\widetilde{K}_{N}\,Q,$

where we recall that $Q=1-|\varphi_{0}\rangle\langle\varphi_{0}|$. Now,
inserting the definitions of $\kappa_{\mathcal{J}_{N}}$, $\widetilde{D}$ and
$\widetilde{\Gamma}$ into (3.35), we use cyclicity of the trace and
$1=f_{\ell}+w_{\ell}$, $\cosh^{2}=1+\sinh^{2}$ to write

$\begin{split}E_{N}&=N\int
dx\,\Big{(}|\nabla\varphi_{0}(x)|^{2}+V_{\text{ext}}(x)\varphi_{0}^{2}(x)+\frac{1}{2}\big{(}N^{3}(Vf_{\ell})(N.)\ast\varphi_{0}^{2}\big{)}(x)\varphi_{0}^{2}(x)\Big{)}\\\
&\hskip 14.22636pt+\frac{N}{2}\int
dxdy\,N^{3}(Vw_{\ell})(N(x-y))\varphi_{0}^{2}(y)\varphi_{0}^{2}(x)-4\pi\mathfrak{a}_{0}\int
dx\,\varphi_{0}^{4}(x)\\\ &\hskip 14.22636pt+\frac{1}{2}\int
dxdy\,N^{2}V(N(x-y))|\langle\sigma_{x},\gamma_{y}\rangle|^{2}\\\ &\hskip
14.22636pt+\operatorname{tr}\big{(}\gamma\big{[}N^{3}(Vw_{\ell})(N(x-y))\varphi_{0}(x)\varphi_{0}(y)\big{]}\sigma\big{)}\\\
&\hskip
14.22636pt+\frac{1}{2}\operatorname{tr}_{\bot\varphi_{0}}\left(\frac{1}{2}\left(\widetilde{D}^{1/2}\widetilde{E}\widetilde{D}^{-1/2}+\widetilde{D}^{-1/2}\widetilde{E}\widetilde{D}^{1/2}\right)-\widetilde{D}-\widetilde{\Gamma}\right)+\mathcal{O}(N^{-\rho}).\end{split}$

Applying Lemma 2.2, we also find that

$\begin{split}&\frac{1}{2}\int
dxdy\,N^{2}V(N(x-y))|\langle\sigma_{x},\gamma_{y}\rangle|^{2}\\\ &\hskip
42.67912pt=-\frac{N}{2}\int
dxdy\,N^{3}(Vw^{2}_{\ell})(N(x-y))\varphi_{0}^{2}(y)\varphi_{0}^{2}(x)\\\
&\hskip
56.9055pt-\operatorname{tr}\big{(}\gamma\big{[}N^{3}(Vw_{\ell})(N(x-y))\varphi_{0}(x)\varphi_{0}(y)\big{]}\sigma\big{)}+\mathcal{O}(N^{-1}),\end{split}$

which implies (recalling the definition (2.11) of the kernel $k$) that

$\begin{split}E_{N}&=N\int
dx\,\Big{(}|\nabla\varphi_{0}(x)|^{2}+V_{\text{ext}}(x)\varphi_{0}^{2}(x)+\frac{1}{2}\big{(}N^{3}(Vf_{\ell})(N.)\ast\varphi_{0}^{2}\big{)}(x)\varphi_{0}^{2}(x)\Big{)}\\\
&\hskip
14.22636pt-4\pi\mathfrak{a}_{0}\|\varphi_{0}\|_{4}^{4}+\frac{1}{2}\operatorname{tr}_{\bot\varphi_{0}}\left(\frac{1}{2}\left(\widetilde{D}^{1/2}\widetilde{E}\widetilde{D}^{-1/2}+\widetilde{D}^{-1/2}\widetilde{E}\widetilde{D}^{1/2}\right)-H_{\text{GP}}-K_{N}\right)\\\
&\hskip
14.22636pt-\frac{1}{2}\operatorname{tr}\big{(}\widetilde{K}_{N}k\big{)}+\mathcal{O}(N^{-\rho}).\end{split}$
(3.36)

To evaluate the right hand side in (3.36) we use Lemma 2.1 _ii)_ , switch to
Fourier space and apply Plancherel’s theorem to find

$\begin{split}&\int
dx\,\big{(}N^{3}(Vf_{\ell})(N.)\ast\varphi_{0}^{2}\big{)}(x)\varphi_{0}^{2}(x)=\int
dp\,\widehat{(Vf_{\ell})}(p/N)\big{|}\widehat{(\varphi_{0}^{2})}(p)\big{|}^{2}\\\
&=\widehat{(Vf_{\ell})}(0)\int dx\,\varphi_{0}^{4}(x)-N^{-1}\int
dx\,V(x)f_{\ell}(x)\,x\cdot\int
dy\,\varphi_{0}^{2}(y)\nabla(\varphi_{0}^{2})(y)+\mathcal{O}(N^{-2})\\\
&=\big{(}8\pi\mathfrak{a}_{0}+12\pi\mathfrak{a}^{2}_{0}\,\ell^{-1}N^{-1}\big{)}\|\varphi_{0}\|_{4}^{4}+\mathcal{O}(N^{-2}).\end{split}$

Note here that $\int dx\,V(x)f_{\ell}(x)x=0$ by radial symmetry of $V$ and
$f_{\ell}$.

Next, using the scattering equation (2.6), we write for $x\in\mathbb{R}^{3}$

$-Nw_{\ell}(Nx)=\int\frac{dy}{4\pi}\,\frac{1}{|x-y|}\Big{(}-\frac{1}{2}N^{3}(Vf_{\ell})(Ny)+N^{3}\lambda_{\ell}f_{\ell}(Ny)\chi_{\ell}(y)\Big{)}$

and obtain with similar arguments as above that

$\begin{split}-\frac{1}{2}\operatorname{tr}\big{(}\widetilde{K}_{N}k\big{)}=&\,\frac{1}{16\pi^{2}}\int
dpdq\,\big{|}\widehat{(\varphi_{0}^{2})}(p)\big{|}^{2}\widehat{(Vf_{\ell})}((p-q)/N)\widehat{(Vf_{\ell})}(q/N)|q|^{-2}\\\
&-\frac{N^{3}\lambda_{\ell}}{8\pi^{2}}\int
dpdq\,\big{|}\widehat{(\varphi_{0}^{2})}(p)\big{|}^{2}\widehat{(Vf_{\ell})}((p-q)/N)\widehat{\chi}_{\ell}(q)|q|^{-2}\\\
&+\frac{\lambda_{\ell}}{8\pi^{2}}\int
dpdq\,\big{|}\widehat{(\varphi_{0}^{2})}(p)\big{|}^{2}\widehat{(Vf_{\ell})}((p-q)/N)\widehat{w_{\ell}}(q/N)|q|^{-2}\\\
=&\,\frac{1}{16\pi^{2}}\int
dpdq\,\big{|}\widehat{(\varphi_{0}^{2})}(p)\big{|}^{2}\widehat{(Vf_{\ell})}((p-q)/N)\widehat{(Vf_{\ell})}(q/N)|q|^{-2}\\\
&-\frac{N^{3}\lambda_{\ell}}{8\pi^{2}}\int
dpdq\,\big{|}\widehat{(\varphi_{0}^{2})}(p)\big{|}^{2}\widehat{(Vf_{\ell})}((p-q)/N)\widehat{\chi}_{\ell}(q)|q|^{-2}+\mathcal{O}(N^{-1}),\end{split}$

where, in the last step, we used Lemma 2.1 _i)_ , the bound (2.10) and the
inequality $\|\widehat{w_{\ell}}\|_{\infty}\leq\|w_{\ell}\|_{1}\leq
CN^{2}\ell^{2}$. Since $|\widehat{\chi}_{\ell}(q)|\leq C|q|^{-2}$, Lemma 2.1
_i)_ , _ii)_ and a first order Taylor expansion also imply

$\begin{split}&\frac{N^{3}\lambda_{\ell}}{8\pi^{2}}\int
dpdq\,\big{|}\widehat{(\varphi_{0}^{2})}(p)\big{|}^{2}\widehat{(Vf_{\ell})}((p-q)/N)\widehat{\chi}_{\ell}(q)|q|^{-2}\\\
&=\frac{N^{3}\lambda_{\ell}}{8\pi^{2}}\int_{|q|\leq
N}dpdq\,\big{|}\widehat{(\varphi_{0}^{2})}(p)\big{|}^{2}\widehat{(Vf_{\ell})}((p-q)/N)\widehat{\chi}_{\ell}(q)|q|^{-2}\\\
&\hskip
14.22636pt+\frac{N^{3}\lambda_{\ell}}{8\pi^{2}}\int_{|q|>N}dpdq\,\big{|}\widehat{(\varphi_{0}^{2})}(p)\big{|}^{2}\widehat{(Vf_{\ell})}((p-q)/N)\widehat{\chi}_{\ell}(q)|q|^{-2}\\\
&=3\mathfrak{a}_{0}^{2}\,\pi^{-2}\ell^{-3}\\!\\!\int
dpdq\,\big{|}\widehat{(\varphi_{0}^{2})}(p)\big{|}^{2}\widehat{\chi}_{\ell}(q)|q|^{-2}+\mathcal{O}(\log
N/N)=6\pi\mathfrak{a}_{0}^{2}\,\ell^{-1}\|\varphi_{0}\|_{4}^{4}+\mathcal{O}(\log
N/N)\end{split}$

so that

$\begin{split}E_{N}&=N\mathcal{E}_{GP}(\varphi_{0})-4\pi\mathfrak{a}_{0}\|\varphi_{0}\|_{4}^{4}\\\
&\hskip 14.22636pt+\frac{1}{4}\int
dxdy\,N^{3}(Vf_{\ell})(N(x-y))\bigg{[}\frac{1}{4\pi|.|}\ast
N^{3}(Vf_{\ell})(N.)\bigg{]}(x-y)\varphi_{0}^{2}(x)\varphi_{0}^{2}(y)\\\
&\hskip
14.22636pt+\frac{1}{2}\operatorname{tr}_{\bot\varphi_{0}}\left(\frac{1}{2}\left(\widetilde{D}^{1/2}\widetilde{E}\widetilde{D}^{-1/2}+\widetilde{D}^{-1/2}\widetilde{E}\widetilde{D}^{1/2}\right)-H_{\text{GP}}-K_{N}\right)+\mathcal{O}(N^{-\rho}).\end{split}$
(3.37)

It remains to combine the last two lines in (3.37). We can rewrite

$\displaystyle\begin{split}&\frac{1}{4}\int
dxdy\,N^{3}(Vf_{\ell})(N(x-y))\bigg{[}\frac{1}{4\pi|.|}\ast
N^{3}(Vf_{\ell})(N.)\bigg{]}(x-y)\varphi_{0}^{2}(x)\varphi_{0}^{2}(y)\\\
&=\frac{1}{4}\operatorname{tr}\left[\widetilde{K}_{N}\varphi_{0}\big{[}(-\Delta)^{-1}\widehat{(Vf_{\ell})}(./N)\big{]}\varphi_{0}\right],\end{split}$

where $\varphi_{0}$ acts as multiplication operator in position space and
where $\widehat{(Vf_{\ell})}(./N)$ acts as Fourier multiplier. Proceeding
similarly as in [21, Eq. (64)], we first rewrite

$\begin{split}&\operatorname{tr}\left[\widetilde{K}_{N}\varphi_{0}\big{[}(-\Delta)^{-1}\widehat{(Vf_{\ell})}(./N)\big{]}\varphi_{0}\right]\\\
&=\operatorname{tr}\left[\frac{1}{2}\widetilde{K}_{N}\varphi_{0}\big{[}(-\Delta)^{-1}\widehat{(Vf_{\ell})}(./N)\big{]}\varphi_{0}+\frac{1}{2}\varphi_{0}\big{[}\widehat{(Vf_{\ell})}(./N)(-\Delta)^{-1}\big{]}\varphi_{0}\widetilde{K}_{N}\right]\\\
&=\operatorname{tr}\left[\frac{1}{2}\widetilde{K}_{N}\varphi_{0}\big{[}(-\Delta+\kappa^{2})^{-1}\widehat{(Vf_{\ell})}(./N)\big{]}\varphi_{0}+\frac{1}{2}\varphi_{0}\big{[}\widehat{(Vf_{\ell})}(./N)(-\Delta+\kappa^{2})^{-1}\big{]}\varphi_{0}\widetilde{K}_{N}\right]\\\
&\hskip
11.38092pt+\frac{\kappa^{2}}{2}\operatorname{tr}\left[\widetilde{K}_{N}\varphi_{0}\big{[}(-\Delta+\kappa^{2})^{-1}(-\Delta)^{-1}\widehat{(Vf_{\ell})}(./N)\big{]}\varphi_{0}+\text{h.c.}\right]\\\
\end{split}$

for a parameter $\kappa>0$ that later will be chosen large enough. Plugging in
$\widetilde{K}_{N}=\varphi_{0}\widehat{(Vf_{\ell})}(./N)\varphi_{0}$, we have

$\begin{split}&\frac{1}{2}\varphi_{0}\widehat{(Vf_{\ell})}(./N)\left(\varphi_{0}^{2}\frac{1}{-\Delta+\kappa^{2}}+\frac{1}{-\Delta+\kappa^{2}}\varphi_{0}^{2}\right)\widehat{(Vf_{\ell})}(./N)\varphi_{0}\\\
&=\varphi_{0}\widehat{(Vf_{\ell})}(./N)\varphi_{0}\left(\frac{1}{-\Delta+\kappa^{2}}\right)\varphi_{0}\widehat{(Vf_{\ell})}(./N)\varphi_{0}\\\
&\hskip
14.22636pt+\varphi_{0}\widehat{(Vf_{\ell})}(./N)\left(\frac{1}{-\Delta+\kappa^{2}}[\varphi_{0},-\Delta]\frac{1}{-\Delta+\kappa^{2}}[\varphi_{0},-\Delta]\frac{1}{-\Delta+\kappa^{2}}\right)\widehat{(Vf_{\ell})}(./N)\varphi_{0}\\\
&\hskip
14.22636pt+\varphi_{0}\widehat{(Vf_{\ell})}(./N)\left(\frac{1}{-\Delta+\kappa^{2}}|\nabla\varphi_{0}|^{2}\frac{1}{-\Delta+\kappa^{2}}\right)\widehat{(Vf_{\ell})}(./N)\varphi_{0}.\end{split}$

Inserting an additional projection
$Q=1-|\varphi_{0}\rangle\langle\varphi_{0}|$, we arrive at

$\begin{split}&\frac{1}{4}\operatorname{tr}\left[\widetilde{K}_{N}\varphi_{0}\big{[}(-\Delta)^{-1}\widehat{(Vf_{\ell})}(./N)\big{]}\varphi_{0}\right]=\frac{1}{4}\operatorname{tr}_{\bot\varphi_{0}}\left[K_{N}\frac{1}{H_{\text{GP}}}K_{N}\right]+\sum_{i=1}^{6}X_{i},\end{split}$
(3.38)

where we defined

$\begin{split}\text{X}_{1}&=\frac{\kappa^{2}}{8}\operatorname{tr}\left[\varphi_{0}\widehat{(Vf_{\ell})}(./N)\varphi_{0}^{2}(-\Delta+\kappa^{2})^{-1}(-\Delta)^{-1}\widehat{(Vf_{\ell})}(./N)\varphi_{0}+\text{h.c.}\right],\\\
\text{X}_{2}&=\frac{1}{4}\operatorname{tr}\left[\varphi_{0}\widehat{(Vf_{\ell})}(./N)\left(\frac{1}{-\Delta+\kappa^{2}}[\varphi_{0},-\Delta]\right)^{2}\frac{1}{-\Delta+\kappa^{2}}\widehat{(Vf_{\ell})}(./N)\varphi_{0}\right],\\\
\text{X}_{3}&=\frac{1}{4}\operatorname{tr}\left[\varphi_{0}\widehat{(Vf_{\ell})}(./N)\left(\frac{1}{-\Delta+\kappa^{2}}|\nabla\varphi_{0}|^{2}\frac{1}{-\Delta+\kappa^{2}}\right)\widehat{(Vf_{\ell})}(./N)\varphi_{0}\right],\\\
\text{X}_{4}&=\frac{1}{4}\operatorname{tr}\left[\varphi_{0}\widehat{(Vf_{\ell})}(./N)\varphi_{0}\left(\frac{1}{-\Delta+\kappa^{2}}-\frac{1}{H_{\text{GP}}+\kappa^{2}}\right)\varphi_{0}\widehat{(Vf_{\ell})}(./N)\varphi_{0}\right],\\\
\text{X}_{5}&=\frac{1}{4\kappa^{2}}\|\varphi_{0}\widehat{(Vf_{\ell})}(./N)\varphi_{0}^{2}\|^{2}+\frac{1}{4}\|(H_{\text{GP}}+\kappa^{2})^{-1/2}Q\varphi_{0}\widehat{(Vf_{\ell})}(./N)\varphi_{0}^{2}\|^{2},\\\
\text{X}_{6}&=-\frac{\kappa^{2}}{4}\operatorname{tr}\left[K_{N}\frac{1}{H_{\text{GP}}(H_{\text{GP}}+\kappa^{2})}K_{N}\right].\end{split}$
(3.39)

Observe here, under the assumption $V\in L^{2}(\mathbb{R}^{3})$, we have
$0\leq\operatorname{tr}_{\bot\varphi_{0}}(K_{N}H_{\text{GP}}^{-1}K_{N})=\|K_{N}H_{\text{GP}}^{-1/2}Q\|_{\text{HS}}^{2}\leq
CN<\infty$. Thus, the operator $K_{N}H_{\text{GP}}^{-1}K_{N}$ is of trace
class and the first term on the r.h.s. of (3.38) is well-defined.

Next, we show that the terms $\text{X}_{1}$ to $\text{X}_{6}$, defined in
(3.39), are $\mathcal{O}(1)$ and that, in the limit $N\to\infty$, we can
replace in each of them the Fourier multiplier $\widehat{(Vf_{\ell})}(./N)$ by
$8\pi\mathfrak{a}_{0}$. We start with $\text{X}_{1}$. Since
$\widehat{(Vf_{\ell})}(./N)$ and $\varphi_{0}$ have bounded operator norm, we
easily find (with Lemma A.1)

$|\text{X}_{1}|\leq
C\big{\|}\varphi_{0}(-\Delta+\kappa^{2})^{-1/2}(-\Delta)^{-1/2}\big{\|}_{\text{HS}}^{2}\leq
C.$

Similarly, we can bound, for any $\alpha<1$,

$\begin{split}\Big{|}\text{X}_{1}-\frac{\kappa^{2}}{4}&(8\pi\mathfrak{a}_{0})^{2}\operatorname{tr}\left[\varphi_{0}^{2}(-\Delta+\kappa^{2})^{-1}(-\Delta)^{-1}\varphi_{0}^{2}\right]\Big{|}\\\
\leq\;&C\|[\widehat{(Vf_{\ell})}(./N)-8\pi\mathfrak{a}_{0}]\varphi_{0}^{2}(-\Delta)^{-1/2}(-\Delta+\kappa^{2})^{-3/4+\varepsilon}\|_{\text{HS}}\\\
&+C\|(-\Delta)^{-1/2}(-\Delta+\kappa^{2})^{-3/4+\varepsilon}[\widehat{(Vf_{\ell})}(./N)-8\pi\mathfrak{a}_{0}]\varphi_{0}^{2}\|_{\text{HS}}\leq
CN^{-\alpha}\end{split}$ (3.40)

using that $|\widehat{(Vf_{\ell})}(p/N)-8\pi\mathfrak{a}_{0}|\leq
C(|p|/N)^{\alpha}$ (from the compact support of $V$),
$\|\Delta\varphi_{0}^{2}\|\leq C$ (from Lemma A.2) and where we have chosen
$\varepsilon>0$ small enough depending on $\alpha$.

For the contributions $\text{X}_{2},\text{X}_{3}$, we use that
$\widehat{(Vf_{\ell})}(./N)$ is real (by radial symmetry). We find

$\begin{split}\text{X}_{2}&=\frac{1}{4}\left\|\varphi_{0}\widehat{(Vf_{\ell})}(./N)(-\Delta+\kappa^{2})^{-1}[\varphi_{0},-\Delta](-\Delta+\kappa^{2})^{-1/2}\right\|_{\text{HS}}^{2},\\\
\text{X}_{3}&=\frac{1}{4}\operatorname{tr}\left\|\varphi_{0}\widehat{(Vf_{\ell})}(./N)(-\Delta+\kappa^{2})^{-1}\nabla\varphi_{0}\right\|_{\text{HS}}^{2}.\end{split}$

Proceeding as above, it is then straightforward to see that
$|\text{X}_{2}|,|\text{X}_{3}|\leq C$ and, using the bounds
$\|\nabla\varphi_{0}\|_{op}\leq C$,
$\|[\varphi_{0},-\Delta](-\Delta+\kappa^{2})^{-1/2}\|_{op}\leq C$, moreover,
that for any $\alpha<1/2$,

$\begin{split}&\Big{|}\text{X}_{2}-\frac{1}{4}(8\pi\mathfrak{a}_{0})^{2}\left\|\varphi_{0}(-\Delta+\kappa^{2})^{-1}[\varphi_{0},-\Delta](-\Delta+\kappa^{2})^{-1/2}\right\|_{\text{HS}}^{2}\Big{|}\leq
CN^{-\alpha},\\\
&\Big{|}\text{X}_{3}-\frac{1}{4}(8\pi\mathfrak{a}_{0})^{2}\left\|\varphi_{0}(-\Delta+\kappa^{2})^{-1}\nabla\varphi_{0}\right\|_{\text{HS}}^{2}\Big{|}\leq
CN^{-\alpha}.\end{split}$

To bound $\text{X}_{4}$, we write

$\begin{split}\text{X}_{4}&=\frac{1}{4}\operatorname{tr}\left[\varphi_{0}\widehat{(Vf_{\ell})}(./N)\varphi_{0}\frac{1}{-\Delta+\kappa^{2}}\big{(}V_{\text{ext}}+8\pi\mathfrak{a}_{0}\varphi_{0}^{2}-\varepsilon_{\text{GP}}\big{)}\frac{1}{H_{\text{GP}}+\kappa^{2}}\varphi_{0}\widehat{(Vf_{\ell})}(./N)\varphi_{0}\right].\\\
\end{split}$

Thus

$|\text{X}_{4}|\leq
C\big{\|}\varphi_{0}(-\Delta+\kappa^{2})^{-1}(V_{\text{ext}}+8\pi\mathfrak{a}\varphi_{0}^{2}-\varepsilon_{\text{GP}})\big{\|}_{\text{HS}}\big{\|}(H_{\text{GP}}+\kappa^{2})^{-1}\varphi_{0}\big{\|}_{\text{HS}}.$

Since $V_{\text{ext}}$ grows at most exponentially by (1.7) and, by (A.1),
$\varphi_{0}$ decays faster than any exponential, choosing $\kappa>0$ large
enough we obtain that

$\begin{split}\big{\|}\varphi_{0}(-\Delta+\kappa^{2})^{-1}&(V_{\text{ext}}+8\pi\mathfrak{a}\varphi_{0}^{2}-\varepsilon_{\text{GP}})\big{\|}_{\text{HS}}^{2}\\\
&\leq
C\int\varphi_{0}^{2}(x)\frac{e^{-2\kappa|x-y|}}{|x-y|^{2}}\big{[}V_{\text{ext}}(y)+8\pi\mathfrak{a}_{0}\varphi_{0}^{2}(y)-\varepsilon_{\text{GP}}\big{]}^{2}dxdy\leq
C<\infty.\end{split}$

From (1.7), we also have
$\|(H_{\text{GP}}+\kappa^{2})^{-1}\varphi_{0}\|_{\text{HS}}\leq C$ and thus
$|\text{X}_{4}|\leq C$. Proceeding similarly as in (3.40) (and writing
$\varphi_{0}(-\Delta+\kappa^{2})^{-1}=(-\Delta+\kappa^{2})^{-1}\varphi_{0}+(-\Delta+\kappa^{2})^{-1}[-\Delta,\varphi_{0}](-\Delta+\kappa^{2})^{-1}$),
we conclude that

$\Big{|}\text{X}_{4}-\frac{(8\pi\mathfrak{a}_{0})^{2}}{4}\operatorname{tr}\left[\varphi_{0}^{2}\Big{(}\frac{1}{-\Delta+\kappa^{2}}-\frac{1}{H_{\text{GP}}+\kappa^{2}}\Big{)}\varphi_{0}^{2}\right]\Big{|}\leq
CN^{-\alpha}$

for any $\alpha<1$. With (1.7), we can also show that
$\text{X}_{5},\text{X}_{6}$ are bounded and that

$\begin{split}\Big{|}\text{X}_{5}-\frac{(8\pi\mathfrak{a}_{0})^{2}}{4\kappa^{2}}\|\varphi_{0}^{3}\|-\frac{(8\pi\mathfrak{a}_{0})^{2}}{4}\|(H_{\text{GP}}+\kappa^{2})^{-1/2}Q\varphi_{0}^{3}\|^{2}\Big{|}&\leq
CN^{-1}.\\\
\Big{|}\text{X}_{6}+\frac{\kappa^{2}(8\pi\mathfrak{a}_{0})^{2}}{4}\operatorname{tr}_{\perp\varphi_{0}}\big{[}\varphi_{0}^{2}Q\frac{1}{H_{\text{GP}}(H_{\text{GP}}+\kappa^{2})}Q\varphi_{0}^{2}\Big{]}\Big{|}&\leq
CN^{-\alpha}.\end{split}$

for every $0<\alpha<1$. In fact, proceeding similarly as in (3.40) and using
that $\|\widehat{Vf_{\ell}}(p/N)-\widehat{Vf_{\ell}}(0)|\leq
C(|p|/N)^{\alpha}$ and that
$\||\nabla|^{\alpha}H^{-\alpha/2}_{\text{GP}}\|_{\text{op}}\leq C$, we can
estimate

$\begin{split}\Big{|}\text{X}_{6}+&\frac{\kappa^{2}(8\pi\mathfrak{a}_{0})^{2}}{4}\operatorname{tr}_{\perp\varphi_{0}}\big{[}\varphi_{0}^{2}Q\frac{1}{H_{\text{GP}}(H_{\text{GP}}+\kappa^{2})}Q\varphi_{0}^{2}\Big{]}\Big{|}\\\
&\leq\frac{C}{N^{\alpha}}\big{\|}H_{\text{GP}}^{\alpha/2}\varphi_{0}H_{\text{GP}}^{-1}(H_{\text{GP}}+\kappa^{2})^{-3/4+\varepsilon}\big{\|}_{\text{HS}}\big{\|}H_{\text{GP}}^{-1/2}(H_{\text{GP}}+\kappa^{2})^{-1/4-\varepsilon}K_{N}\big{\|}_{\text{HS}}.\end{split}$

The second norm on the r.h.s. is bounded, uniformly in $N$, by (1.7) (and by
the exponential decay of $\varphi_{0}$, shown in (A.1)).To bound the first
Hilbert-Schmidt norm, we write

$\begin{split}H_{\text{GP}}^{\alpha/2}\varphi_{0}&\frac{1}{H_{\text{GP}}^{1/2}(H_{\text{GP}}+\kappa^{2})^{3/4-\varepsilon}}\\\
&=\varphi_{0}\frac{H_{\text{GP}}^{\alpha/2}}{H_{\text{GP}}^{1/2}(H_{\text{GP}}+\kappa^{2})^{3/4-\varepsilon}}\\\
&+C\int_{0}^{\infty}ds\
s^{\alpha/2}\frac{1}{s+H_{\text{GP}}}[H_{\text{GP}},\varphi_{0}]\frac{1}{s+H_{\text{GP}}}\frac{1}{H_{\text{GP}}^{1/2}(H_{\text{GP}}+\kappa^{2})^{3/4-\varepsilon}}\end{split}$

for an appropriate constant $C>0$ and with
$[H_{\text{GP}},\varphi_{0}]=-2\nabla\cdot\nabla\varphi_{0}-\Delta\varphi_{0}$.
Estimating $\|(s+H_{\text{GP}})^{-1}\nabla\|_{\text{op}}\leq C(1+s)^{1/2}$,
$\|(s+H_{\text{GP}})^{-1}\|_{\text{op}}\leq C(1+s)^{-1}$ and again
$\|e^{-\alpha|x|}H_{\text{GP}}^{-1/2}(H_{\text{GP}}+\kappa^{2})^{-1/4-\varepsilon}\|_{\text{HS}}\leq
C$, we obtain the desired bound.

Let us now consider the trace on the last line of (3.37). From the proof of
Lemma 3.2, recall that
$\widetilde{D}^{1/2}=U_{0}^{*}H_{\text{GP}}^{1/2}e^{-\eta}$ for some unitary
$U_{0}:L^{2}_{\bot\varphi_{0}}(\mathbb{R}^{3})\rightarrow
L^{2}_{\bot\varphi_{0}}(\mathbb{R}^{3})$. Thus (recall that by the proof of
Lemma 3.1 _a)_ we have
$H_{\text{GP}}^{1/2}(H_{\text{GP}}+2K_{N})H_{\text{GP}}^{1/2}\geq 0$)

$\displaystyle\frac{1}{2}\operatorname{tr}_{\bot\varphi_{0}}\left(\frac{1}{2}\left(\widetilde{D}^{1/2}\widetilde{E}\widetilde{D}^{-1/2}+\widetilde{D}^{-1/2}\widetilde{E}\widetilde{D}^{1/2}\right)-H_{\text{GP}}-K_{N}\right)$
$\displaystyle=\frac{1}{2}\operatorname{tr}_{\bot\varphi_{0}}\Big{[}\frac{1}{2}\left(e^{-\eta}H_{\text{GP}}^{1/2}\big{(}H_{\text{GP}}^{1/2}(H_{\text{GP}}+2K_{N})H_{\text{GP}}^{1/2}\big{)}^{1/2}H_{\text{GP}}^{-1/2}e^{\eta}+\text{h.c.}\right)-H_{\text{GP}}-K_{N}\Big{]}.$

Next we want to remove the factors of $e^{\pm\eta}$. If all operators were
trace-class, we could use cyclicity of the trace. To overcome this issue, we
rewrite it as

$\begin{split}&\operatorname{tr}_{\bot\varphi_{0}}\left(\left[e^{-\eta}H_{\text{GP}}^{1/2}\big{(}H_{\text{GP}}^{1/2}(H_{\text{GP}}+2K_{N})H_{\text{GP}}^{1/2}\big{)}^{1/2}H_{\text{GP}}^{-1/2}e^{\eta}-H_{\text{GP}}-K_{N}\right]+\text{h.c.}\right)\\\
&=\operatorname{tr}_{\bot\varphi_{0}}\Big{(}\Big{[}H_{\text{GP}}^{1/2}\left(\big{(}H_{\text{GP}}^{1/2}(H_{\text{GP}}+2K_{N})H_{\text{GP}}^{1/2}\big{)}^{1/2}-H_{\text{GP}}\right)H_{\text{GP}}^{-1/2}-K_{N}\\\
&\hskip
48.36958pt+\left(e^{-\eta}-1\right)H_{\text{GP}}^{1/2}\left(\big{(}H_{\text{GP}}^{1/2}(H_{\text{GP}}+2K_{N})H_{\text{GP}}^{1/2}\big{)}^{1/2}-H_{\text{GP}}\right)H_{\text{GP}}^{-1/2}\left(e^{\eta}-1\right)\\\
&\hskip
48.36958pt+\left(e^{-\eta}-1\right)H_{\text{GP}}^{1/2}\left[\big{(}H_{\text{GP}}^{1/2}(H_{\text{GP}}+2K_{N})H_{\text{GP}}^{1/2}\big{)}^{1/2}-H_{\text{GP}}\right]H_{\text{GP}}^{-1/2}\\\
&\hskip
48.36958pt+H_{\text{GP}}^{1/2}\left[\big{(}H_{\text{GP}}^{1/2}(H_{\text{GP}}+2K_{N})H_{\text{GP}}^{1/2}\big{)}^{1/2}-H_{\text{GP}}\right]H_{\text{GP}}^{-1/2}\left(e^{-\eta}-1\right)\\\
&\hskip
48.36958pt+\left(e^{-\eta}-1+\eta\right)H_{\text{GP}}\left(e^{\eta}-1-\eta\right)+\left(1-\eta\right)H_{\text{GP}}\left(e^{-\eta}-1-\eta\right)\\\
&\hskip 48.36958pt+\left(e^{-\eta}-1+\eta\right)H_{\text{GP}}(1+\eta)-\eta
H_{\text{GP}}\eta\Big{]}+\text{h.c.}\Big{)}.\end{split}$ (3.41)

Now we want to move all factors involving $\eta$ to the right. For this we
will use that for all $\varepsilon>0$ we have

$\begin{split}&\|QH_{\text{GP}}^{1/2-\varepsilon}\left[\big{(}H_{\text{GP}}^{1/2}(H_{\text{GP}}+2K_{N})H_{\text{GP}}^{1/2}\big{)}^{1/2}-H_{\text{GP}}\right]H_{\text{GP}}^{-1/2}Q\|_{\text{HS}}<\infty,\\\
&\|QH_{\text{GP}}^{1/2}\left[\big{(}H_{\text{GP}}^{1/2}(H_{\text{GP}}+2K_{N})H_{\text{GP}}^{1/2}\big{)}^{1/2}-H_{\text{GP}}\right]H_{\text{GP}}^{-1/2-\varepsilon}Q\|_{\text{HS}}<\infty.\end{split}$
(3.42)

Before proving (3.42), we show how we apply it to (3.41). For the term on the
second line of the r.h.s of (3.41) we can use (3.42), the boundedness of
$e^{-\eta}-1$ and
$\|QH_{\text{GP}}^{\varepsilon}(e^{\eta}-1)\|_{\text{HS}}\leq
e^{\|\eta\|_{\text{op}}}\|H_{\text{GP}}^{\varepsilon-1}Q\|_{\text{op}}\|H_{\text{GP}}\eta\|_{\text{HS}}<\infty$
for $0\leq\varepsilon\leq 1$ to get

$\displaystyle\operatorname{tr}_{\bot\varphi_{0}}\left(\left(e^{-\eta}-1\right)H_{\text{GP}}^{1/2}\left(\big{(}H_{\text{GP}}^{1/2}(H_{\text{GP}}+2K_{N})H_{\text{GP}}^{1/2}\big{)}^{1/2}-H_{\text{GP}}\right)H_{\text{GP}}^{-1/2-\varepsilon}H_{\text{GP}}^{\varepsilon}\left(e^{\eta}-1\right)\right)$
$\displaystyle=\operatorname{tr}_{\bot\varphi_{0}}\left(H_{\text{GP}}^{1/2}\left(\big{(}H_{\text{GP}}^{1/2}(H_{\text{GP}}+2K_{N})H_{\text{GP}}^{1/2}\big{)}^{1/2}-H_{\text{GP}}\right)H_{\text{GP}}^{-1/2}\left(e^{\eta}-1\right)\left(e^{-\eta}-1\right)\right).$

Using additionally
$\|H_{\text{GP}}^{\varepsilon}(e^{-\eta}-1)H_{\text{GP}}^{\varepsilon}\|_{\text{HS}}\leq\|H_{\text{GP}}^{\varepsilon}\eta
H_{\text{GP}}^{\varepsilon}\|_{\text{HS}}+e^{\|\eta\|_{\text{op}}}\|H_{\text{GP}}\eta\|_{\text{HS}}^{2}<\infty$
for $0\leq\varepsilon<\frac{1}{2}$ we obtain

$\displaystyle\operatorname{tr}_{\bot\varphi_{0}}\left(\left(e^{-\eta}-1\right)H_{\text{GP}}^{1/2}\left[\big{(}H_{\text{GP}}^{1/2}(H_{\text{GP}}+2K_{N})H_{\text{GP}}^{1/2}\big{)}^{1/2}-H_{\text{GP}}\right]H_{\text{GP}}^{-1/2}\right)$
$\displaystyle=\operatorname{tr}_{\bot\varphi_{0}}\left(H_{\text{GP}}^{-\varepsilon}\left(H_{\text{GP}}^{1/2}\left[\big{(}H_{\text{GP}}^{1/2}(H_{\text{GP}}+2K_{N})H_{\text{GP}}^{1/2}\big{)}^{1/2}-H_{\text{GP}}\right]H_{\text{GP}}^{-1/2-\varepsilon}\right)H_{\text{GP}}^{\varepsilon}(e^{-\eta}-1)H_{\text{GP}}^{\varepsilon}\right)$
$\displaystyle=\operatorname{tr}_{\bot\varphi_{0}}\left(H_{\text{GP}}^{1/2}\left[\big{(}H_{\text{GP}}^{1/2}(H_{\text{GP}}+2K_{N})H_{\text{GP}}^{1/2}\big{)}^{1/2}-H_{\text{GP}}\right]H_{\text{GP}}^{-1/2}\left(e^{-\eta}-1\right)\right).$

Now we write $e^{-\eta}-1+\eta=g_{\eta}\eta^{2}$, where $g_{\eta}$ is a
bounded operator that commutes with $\eta$. Using that $\eta$ and $\eta
H_{\text{GP}}$ are both Hilbert-Schmidt, we obtain

$\displaystyle\operatorname{tr}_{\bot\varphi_{0}}\left(\left(e^{-\eta}-1+\eta\right)H_{\text{GP}}(1+\eta)\right)$
$\displaystyle=\operatorname{tr}_{\bot\varphi_{0}}\left(\eta H_{\text{GP}}\eta
g_{\eta}(1+\eta)\right)$
$\displaystyle=\operatorname{tr}_{\bot\varphi_{0}}\left(H_{\text{GP}}(1+\eta)\left(e^{-\eta}-1+\eta\right)\right).$

The remaining terms can be dealt with in a similar fashion. Thus, once we have
shown (3.42) we obtain

$\displaystyle\operatorname{tr}_{\bot\varphi_{0}}$
$\displaystyle\left(\left[e^{-\eta}H_{\text{GP}}^{1/2}\left(\big{(}H_{\text{GP}}^{1/2}(H_{\text{GP}}+2K_{N})H_{\text{GP}}^{1/2}\big{)}^{1/2}-H_{\text{GP}}\right)H_{\text{GP}}^{-1/2}e^{\eta}-K_{N}\right]+\text{h.c.}\right)$
$\displaystyle=\operatorname{tr}_{\bot\varphi_{0}}\left(\left[H_{\text{GP}}^{1/2}\left(\big{(}H_{\text{GP}}^{1/2}(H_{\text{GP}}+2K_{N})H_{\text{GP}}^{1/2}\big{)}^{1/2}-H_{\text{GP}}\right)H_{\text{GP}}^{-1/2}-K_{N}\right]+\text{h.c.}\right).$

Let us now prove (3.42). With the integral representation

$x^{1/2}=x\cdot
x^{-1/2}=\frac{1}{\pi}\int_{0}^{\infty}\frac{ds}{\sqrt{s}}\frac{x}{s+x}=\lim_{\Lambda\to\infty}\frac{1}{\pi}\int_{0}^{\Lambda}\frac{ds}{\sqrt{s}}-\frac{1}{\pi}\int_{0}^{\Lambda}ds\,\sqrt{s}\frac{1}{s+x}$

we can write

$\begin{split}\big{(}H_{\text{GP}}^{1/2}(H_{\text{GP}}&+2K_{N})H_{\text{GP}}^{1/2}\big{)}^{1/2}-H_{\text{GP}}\\\
&=-\frac{1}{\pi}\int_{0}^{\infty}ds\,\sqrt{s}\,\left[\frac{1}{s+H^{2}_{\text{GP}}+2H^{1/2}_{\text{GP}}K_{N}H^{1/2}_{\text{GP}}}-\frac{1}{s+H^{2}_{\text{GP}}}\right]\\\
&=\frac{2}{\pi}\int_{0}^{\infty}ds\,\sqrt{s}\,\frac{1}{s+H^{2}_{\text{GP}}}H_{\text{GP}}^{1/2}K_{N}H_{\text{GP}}^{1/2}\frac{1}{s+H^{2}_{\text{GP}}+2H^{1/2}_{\text{GP}}K_{N}H^{1/2}_{\text{GP}}}.\end{split}$
(3.43)

Using the resolvent identiy, we obtain

$\displaystyle
QH_{\text{GP}}^{1/2}\left[\big{(}H_{\text{GP}}^{1/2}(H_{\text{GP}}+2K_{N})H_{\text{GP}}^{1/2}\big{)}^{1/2}-H_{\text{GP}}\right]H_{\text{GP}}^{-1/2-\varepsilon}Q$
$\displaystyle=\frac{2}{\pi}\int_{0}^{\infty}ds\,\sqrt{s}\,Q\frac{1}{s+H^{2}_{\text{GP}}}H_{\text{GP}}K_{N}\frac{1}{s+H^{2}_{\text{GP}}}H_{\text{GP}}^{-\varepsilon}Q$
$\displaystyle\hskip
14.22636pt+\frac{4}{\pi}\int_{0}^{\infty}ds\,\sqrt{s}\,Q\frac{1}{s+H^{2}_{\text{GP}}}H_{\text{GP}}K_{N}H_{\text{GP}}^{1/2}\frac{1}{s+H_{\text{GP}}^{2}+2H_{\text{GP}}^{1/2}K_{N}H_{\text{GP}}^{1/2}}H_{\text{GP}}^{1/2}K_{N}\frac{H_{\text{GP}}^{-\varepsilon}}{s+H_{\text{GP}}^{2}}Q.$

Using that $\|Q[K_{N},H_{N}]H_{\text{GP}}^{-1/2}Q\|_{\text{HS}}<\infty$ we
obtain the second inequality in (3.42). Similarly one obtains the other
inequality in (3.42). Therefore, we can use cyclicity and get

$\displaystyle\frac{1}{2}\operatorname{tr}_{\bot\varphi_{0}}\left(\frac{1}{2}\left(\widetilde{D}^{1/2}\widetilde{E}\widetilde{D}^{-1/2}+\widetilde{D}^{-1/2}\widetilde{E}\widetilde{D}^{1/2}\right)-H_{\text{GP}}-K_{N}\right)$
$\displaystyle=\frac{1}{2}\operatorname{tr}_{\bot\varphi_{0}}\Big{[}\frac{1}{2}\left(H_{\text{GP}}^{1/2}\big{(}H_{\text{GP}}^{1/2}(H_{\text{GP}}+2K_{N})H_{\text{GP}}^{1/2}\big{)}^{1/2}H_{\text{GP}}^{-1/2}+\text{h.c.}\right)-H_{\text{GP}}-K_{N}\Big{]}$
$\displaystyle=\frac{1}{4}\operatorname{tr}_{\bot\varphi_{0}}\Big{[}\left(H_{\text{GP}}^{1/2}\left[\big{(}H_{\text{GP}}^{1/2}(H_{\text{GP}}+2K_{N})H_{\text{GP}}^{1/2}\big{)}^{1/2}-H_{\text{GP}}\right]H_{\text{GP}}^{-1/2}-K_{N}\right)+\text{h.c.}\Big{]}.$

Applying the resolvent identity twice to (3.42), we obtain

$\begin{split}H_{\text{GP}}^{1/2}&[\big{(}H_{\text{GP}}^{1/2}(H_{\text{GP}}+2K_{N})H_{\text{GP}}^{1/2}\big{)}^{1/2}-H_{\text{GP}}]H_{\text{GP}}^{-1/2}-K_{N}\\\
=\;&\frac{2}{\pi}\int_{0}^{\infty}ds\,\sqrt{s}\,H_{\text{GP}}\frac{1}{s+H^{2}_{\text{GP}}}K_{N}\frac{1}{s+H^{2}_{\text{GP}}}-K_{N}\\\
&-\frac{4}{\pi}\int_{0}^{\infty}ds\,\sqrt{s}\,\frac{1}{s+H^{2}_{\text{GP}}}H_{\text{GP}}K_{N}H_{\text{GP}}^{1/2}\frac{1}{s+H^{2}_{\text{GP}}}H_{\text{GP}}^{1/2}K_{N}\frac{1}{s+H^{2}_{\text{GP}}}\\\
&+\frac{8}{\pi}\int_{0}^{\infty}ds\,\sqrt{s}\,H_{\text{GP}}^{1/2}\Big{[}\frac{1}{s+H^{2}_{\text{GP}}}H_{\text{GP}}^{1/2}K_{N}H_{\text{GP}}^{1/2}\Big{]}^{3}\frac{1}{s+H^{2}_{\text{GP}}+2H^{1/2}_{\text{GP}}K_{N}H^{1/2}_{\text{GP}}}H_{\text{GP}}^{-1/2}.\end{split}$
(3.44)

Since

$\frac{2}{\pi}\int_{0}^{\infty}ds\,\sqrt{s}\,\frac{H_{\text{GP}}}{(s+H_{\text{GP}}^{2})^{2}}=1$

the trace of the first line on the r.h.s. of (3.44) vanishes, by cyclicity. To
apply cyclicity of the trace, we need here to be a bit careful, since $K_{N}$
is only guaranteed to be trace class, if $\widehat{(Vf_{\ell})}(./N)$ is
integrable, a property which does not follow from our assumptions (1.7) on the
interaction $V$. In general, we can justify cyclicity remarking that, for
self-adjoint operators $A,B$, we have the identity

$2ABA-A^{2}B-BA^{2}=-\big{[}A,\big{[}A,B\big{]}\big{]}.$

Thus

$\begin{split}&\left(\frac{2}{\pi}\int_{0}^{\infty}ds\,\sqrt{s}\,H_{\text{GP}}\frac{1}{s+H_{\text{GP}}^{2}}K_{N}\frac{1}{s+H_{\text{GP}}^{2}}-K_{N}\right)+\text{h.c.}\\\
&\hskip
14.22636pt=\left(\frac{1}{2}\left(H_{\text{GP}}K_{N}H_{\text{GP}}^{-1}-K_{N}\right)-\frac{1}{\pi}\int_{0}^{\infty}ds\,\sqrt{s}\,H_{\text{GP}}\Big{[}\frac{1}{s+H_{\text{GP}}^{2}},\Big{[}\frac{1}{s+H_{\text{GP}}^{2}},K_{N}\Big{]}\Big{]}\right)+\text{h.c.}\end{split}$

A tedious but straightforward analysis shows that the r.h.s. of the last
equation is trace class and that its trace norm depends continuously on
$Vf_{\ell}$ in the $L^{2}(\mathbb{R}^{3})$ topology. For this argument we use
that $(\partial_{i}\partial_{j}\varphi_{0})$ has exponential decay which can
be proved similarly like (A.1) and (A.2). With a simple approximation
argument, and cyclicity of the trace for the second and third term in (3.44),
we can therefore conclude that

$\begin{split}\frac{1}{4}\operatorname{tr}_{\bot\varphi_{0}}&\Big{[}\left(H_{\text{GP}}^{1/2}\left[\big{(}H_{\text{GP}}^{1/2}(H_{\text{GP}}+2K_{N})H_{\text{GP}}^{1/2}\big{)}^{1/2}-H_{\text{GP}}\right]H_{\text{GP}}^{-1/2}-K_{N}\right)+\text{h.c.}\Big{]}\\\
=&-\frac{2}{\pi}\int_{0}^{\infty}ds\,\sqrt{s}\,\operatorname{tr}_{\perp\varphi_{0}}\frac{1}{s+H^{2}_{\text{GP}}}H_{\text{GP}}^{1/2}K_{N}H_{\text{GP}}^{1/2}\frac{1}{s+H^{2}_{\text{GP}}}H_{\text{GP}}^{1/2}K_{N}H_{\text{GP}}^{1/2}\frac{1}{s+H^{2}_{\text{GP}}}\\\
&+\frac{4}{\pi}\int_{0}^{\infty}ds\,\sqrt{s}\,\operatorname{tr}_{\perp\varphi_{0}}\Big{[}\frac{1}{s+H^{2}_{\text{GP}}}H_{\text{GP}}^{1/2}K_{N}H_{\text{GP}}^{1/2}\Big{]}^{3}\frac{1}{s+H^{2}_{\text{GP}}+2H^{1/2}_{\text{GP}}K_{N}H^{1/2}_{\text{GP}}}.\end{split}$
(3.45)

To handle the first term on the r.h.s. we use cyclicity of the trace (from
(1.7), it follows that
$\|K_{N}H_{\text{GP}}^{-3/4-\varepsilon}Q\|_{\text{HS}}\leq C$ for any
$\varepsilon>0$, and therefore the operator under consideration is of trace
class, justifying cyclicity) and the fact that

$\frac{2}{\pi}\int_{0}^{\infty}ds\sqrt{s}\,\frac{1}{(s+H_{\text{GP}}^{2})^{3}}=\frac{1}{4H_{\text{GP}}^{3}}$

to write

$\begin{split}-\frac{2}{\pi}&\int_{0}^{\infty}ds\sqrt{s}\operatorname{tr}_{\perp\varphi_{0}}\frac{1}{s+H^{2}_{\text{GP}}}H_{\text{GP}}^{1/2}K_{N}H_{\text{GP}}^{1/2}\frac{1}{s+H^{2}_{\text{GP}}}H_{\text{GP}}^{1/2}K_{N}H_{\text{GP}}^{1/2}\frac{1}{s+H^{2}_{\text{GP}}}\\\
=\;&-\frac{1}{4}\operatorname{tr}_{\perp\varphi_{0}}K_{N}\frac{1}{H_{\text{GP}}}K_{N}-\frac{1}{\pi}\int_{0}^{\infty}ds\sqrt{s}\operatorname{tr}_{\perp\varphi_{0}}\frac{H_{\text{GP}}^{1/2}}{s+H_{\text{GP}}^{2}}\Big{[}\Big{[}K_{N},\frac{H_{\text{GP}}}{s+H_{\text{GP}}^{2}}\Big{]},K_{N}\Big{]}\frac{H_{\text{GP}}^{1/2}}{s+H_{\text{GP}}^{2}}.\end{split}$
(3.46)

The first term on the r.h.s. cancels precisely with the main contribution on
the r.h.s. of (3.38). Expanding the commutators in the second term, we
generate several contributions. All these contributions are bounded, uniformly
in $N$. As an example, we estimate the term

$\begin{split}\Big{|}\frac{1}{\pi}\int_{0}^{\infty}&ds\sqrt{s}\operatorname{tr}_{\perp\varphi_{0}}\frac{H_{\text{GP}}^{1/2}}{s+H_{\text{GP}}^{2}}\Big{[}K_{N},H_{\text{GP}}\Big{]}\frac{1}{s+H_{\text{GP}}^{2}}\Big{[}K_{N},H_{\text{GP}}\Big{]}\frac{H_{\text{GP}}^{3/2}}{(s+H_{\text{GP}}^{2})^{2}}\Big{|}\\\
&\leq
C\int_{0}^{\infty}ds\,\sqrt{s}\,\Big{\|}\frac{H_{\text{GP}}^{1/2}}{s+H_{\text{GP}}^{2}}e^{-\alpha|x|}\Big{\|}_{\text{HS}}\Big{\|}e^{\alpha|x|}[K_{N},H_{\text{GP}}]\frac{1}{(H_{\text{GP}}+1)^{1/2}}\Big{\|}_{\text{op}}\\\
&\hskip
28.45274pt\times\Big{\|}\frac{H_{\text{GP}}+1}{s+H_{\text{GP}}^{2}}\Big{\|}_{\text{op}}\Big{\|}\frac{1}{(H_{\text{GP}}+1)^{1/2}}[K_{N},H_{\text{GP}}]e^{\alpha|x|}\Big{\|}_{\text{op}}\Big{\|}e^{-\alpha|x|}\frac{H^{3/2}_{\text{GP}}}{(s+H_{\text{GP}}^{2})^{2}}\Big{\|}_{\text{HS}}\\\
&\leq C\int_{0}^{\infty}ds\,\sqrt{s}\,\frac{1}{(1+s)^{7/4-2\varepsilon}}\leq
C\end{split}$ (3.47)

choosing $\varepsilon>0$ small enough. Here we used that, for any $\alpha>0$,
$\|e^{\alpha|x|}[K_{N},H_{\text{GP}}](H_{\text{GP}}+1)^{-1/2}\|_{\text{op}}\leq
C$ (notice that the Laplacian commutes with the multiplier