# Bose gases in the Gross-Pitaevskii limit:
a survey of some rigorous results111Dedicated to Elliott Lieb on the occasion
of his ninetieth birthday, with admiration.

Benjamin Schlein222Institute of Mathematics, University of Zurich,
Winterthurerstrasse 190, 8057 Zurich.

###### Abstract

We review some mathematical work on the Bose gas in the Gross-Pitaevskii
regime. We start with the classical results by Lieb, Seiringer and Yngvason on
the ground state energy and by Lieb and Seiringer on the existence of Bose-
Einstein condensation. Afterwards, we discuss some more recent progress, based
on a rigorous version of Bogoliubov theory.

##  1 Trapped Bose gases: energy estimate and condensation

Bose-Einstein condensation for ideal gases. The phenomenon of Bose-Einstein
condensation has been first predicted by Einstein [23] in 1924, based on
previous work by Bose [13] on the foundations of quantum statistical
mechanics. Einstein considered the ideal Bose gas in a box
$\Lambda_{L}=[0;L]^{3}$, with periodic boundary conditions. In the grand-
canonical ensemble, the gas is described on the bosonic Fock space
$\mathcal{F}=\bigoplus_{n\geq 0}L^{2}(\Lambda_{L})^{\otimes_{s}n}$, by the
non-interacting Hamilton operator

$H=\sum_{p\in\Lambda_{L}^{*}}p^{2}a_{p}^{*}a_{p}$ (1.1)

where the sum runs over all momenta in
$\Lambda_{L}^{*}=(2\pi/L)\mathbb{Z}^{3}$ and where $a_{p}^{*},a_{p}$ denote
the usual creation and annihilation operators on $\mathcal{F}$, satisfying
canonical commutation relations. The partition function of the gas at inverse
temperature $\beta>0$ and chemical potential $\mu$ is given by

$Z(\beta,\mu)={\rm tr}_{\mathcal{F}}\;e^{-\beta(H-\mu\mathcal{N})}$

where $\mathcal{N}$ denotes the number of particles operator. Writing

$H-\mu\mathcal{N}=\sum_{p\in\Lambda^{*}_{L}}(p^{2}-\mu)a_{p}^{*}a_{p}$

and taking into account the fact that, for every momentum
$p\in\Lambda^{*}_{L}$, the spectrum of the harmonic oscillator
$a_{p}^{*}a_{p}$ consists of all natural numbers, we find

$Z(\beta,\mu)=\prod_{p\in\Lambda^{*}_{L}}\sum_{n=0}^{\infty}e^{-\beta
n(p^{2}-\mu)}=\prod_{p\in\Lambda^{*}_{L}}\frac{1}{1-e^{-\beta(p^{2}-\mu)}}$

under the assumption that $\mu<0$ (to make sure that the sum converges, for
all $p\in\Lambda_{L}^{*}$). The expected number of particles is then given by

$\langle\mathcal{N}\rangle=\frac{1}{Z(\beta,\mu)}{\rm
tr}_{\mathcal{F}}\,\mathcal{N}e^{-\beta(H-\mu\mathcal{N})}=\frac{1}{\beta}\frac{d}{d\mu}\log
Z(\beta,\mu)=\sum_{p\in\Lambda_{L}^{*}}\frac{1}{e^{\beta(p^{2}-\mu)}-1}\,.$

Hence, the chemical potential $\mu<0$ can be fixed through the density

$\rho=\frac{\langle\mathcal{N}\rangle}{L^{3}}=\frac{1}{L^{3}}\sum_{p\in\Lambda_{L}^{*}}\frac{1}{e^{\beta(p^{2}-\mu)}-1}\,.$

Let us now write $\rho=\rho_{0}+\rho_{+}$, where

$\rho_{0}=\frac{1}{L^{3}}\frac{1}{e^{-\beta\mu}-1}$ (1.2)

is the density of particles with zero momentum. In the limit $L\to\infty$, we
can approximate

$\rho_{+}=\frac{1}{L^{3}}\sum_{p\in\Lambda_{L}^{*}\backslash\\{0\\}}\frac{1}{e^{\beta(p^{2}-\mu)}-1}\simeq\frac{1}{(2\pi)^{3}}\int\frac{1}{e^{\beta(p^{2}-\mu)}-1}dp=\frac{1}{(4\pi\beta)^{3/2}}\sum_{n=1}^{\infty}\frac{e^{\beta\mu
n}}{n^{3/2}}\,.$

The r.h.s. of the last equation is increasing in $\mu\in(-\infty;0)$ and
remains finite, as $\mu\to 0$. This implies that the density $\rho_{+}$ of
particles with momentum $p\not=0$ is always bounded by the critical density

$\rho_{c}(\beta)=\frac{1}{(4\pi\beta)^{3/2}}\sum_{n=1}^{\infty}\frac{1}{n^{3/2}}\,.$

If $\rho>\rho_{+}(\beta)$, we have to accommodate a macroscopic fraction of
the particles in the state with zero momentum. In other words, we have to
choose $\mu$ depending on $L$ with $\mu(L)\to 0$ so that, inserting in (1.2),
we find $\rho_{0}=\rho-\rho_{c}(\beta)$, in the limit $L\to\infty$ (we can
choose $\mu(L)=-1/[\beta L^{3}(\rho-\rho_{c}(\beta))]$).

The accumulation of a macroscopic fraction of the particles in a single
quantum state is known as Bose-Einstein condensation. For the ideal gas at
inverse temperature $\beta>0$, Bose-Einstein condensation takes place in the
zero-momentum state whenever $\rho>\rho_{c}(\beta)$. Alternatively, we can
invert the relation $\beta\to\rho_{c}(\beta)$ to define a critical temperature

$\beta_{c}(\rho)=\frac{1}{4\pi\rho^{2/3}}\Big{[}\sum_{n=1}^{\infty}\frac{1}{n^{3/2}}\Big{]}^{2/3}.$

For fixed density $\rho$, the ideal gas exhibits Bose-Einstein condensation at
low temperature, when $\beta>\beta_{c}(\rho)$. At zero temperature (for
$\beta=\infty$), we have complete condensation, ie. all particles are in the
same zero-momentum state.

Condensation for interacting gases in the thermodynamic limit. So far, we
considered ideal gases. Establishing the existence of Bose-Einstein
condensation for interacting Bose gases is much more challenging. In fact, the
only available proof of condensation for interacting systems in the
thermodynamic limit has been given by Dyson, Lieb and Simon in [22], for
three-dimensional hard-core bosons on a lattice at half-filling, mapping the
Bose gas to a spin system and using the tool of reflexion positivity that was
previously introduced in [30] (the approach of [22] has been later extended by
Kennedy, Lieb and Shastry in [33] to any dimension greater than one and, more
recently, by Aizenman, Lieb, Seiringer, Solovej and Yngvason in [3] to certain
models describing optical lattices). There has been, however, partial progress
towards a better mathematical understanding of condensation for more general
Bose gases in the thermodynamic limit through a renormalization group
approach; see [4] for a review of recent results.

The Gross-Pitaevskii regime. Since 1995, Bose-Einstein condensates have been
realized in labs; see [17, 34]. In typical experiments, a dilute Bose gas is
initially trapped by strong magnetic fields. Then, the system is cooled down
until it thermalizes at very low temperatures (the experimental realization of
Bose-Einstein condensation was achieved through the development of cooling
techniques, like laser cooling and evaporative cooling, which allowed
physicists to reach temperatures in the range of nano-Kelvin; see [50] and
references therein). Afterwards, the traps are switched off so that the gas
starts to evolve and the momentum distribution of the particles can be
measured. At sufficiently low temperature, it turns out that most particles
occupy the same one-particle orbital, showing condensation.

From the point of view of mathematics, Bose-Einstein condensates produced in
these experiments can be described as systems of $N$ bosons, trapped by an
external potential with a characteristic length scale $L$ and interacting
through a repulsive potential with scattering length $\mathfrak{a}\ll L$. A
particularly important scaling limit is the so called Gross-Pitaevskii regime,
where $N\mathfrak{a}=L$. In this regime, we have
$\rho\mathfrak{a}^{3}=N\mathfrak{a}^{3}/L^{3}=N^{-2}$; the Gross-Pitaevskii
limit describes therefore very dilute Bose gases.

In the following, we are going to choose units so that $L=1$. We consider $N$
bosons moving in the three-dimensional space $\mathbb{R}^{3}$. The Hamilton
operator is given by

$H_{N}=\sum_{j=1}^{N}\big{[}-\Delta_{x_{j}}+V_{\text{ext}}(x_{j})\big{]}+\sum_{i<j}^{N}N^{2}V(N(x_{i}-x_{j})$
(1.3)

and acts on the Hilbert space $L^{2}_{s}(\mathbb{R}^{3N})$, defined as the
subspace of $L^{2}(\mathbb{R}^{3N})$ consisting of functions that are
symmetric with respect to permutations of the $N$ particles. In (1.3),
$V_{\text{ext}}:\mathbb{R}^{3}\to\mathbb{R}$ is a confining potential (ie.
$V_{\text{ext}}(x)\to\infty$, as $|x|\to\infty$) and
$V:\mathbb{R}^{3}\to\mathbb{R}$ is an interaction potential, which we assume
to be radial, non-negative and to decay sufficiently fast at infinity (at
least as $|x|^{-3-\varepsilon}$, for an $\varepsilon>0$).

The scattering length. We denote by $\mathfrak{a}$ the scattering length of
the potential $V$. Let us recall that the scattering length is a physical
quantity measuring the effective range of $V$. It is defined through the
solution of the zero-energy scattering equation

$\Big{[}-\Delta+\frac{1}{2}V\Big{]}f=0$ (1.4)

with the boundary condition $f(x)\to\infty$, as $|x|\to\infty$. Asymptotically
for $|x|\gg 1$, we find

$f(x)\simeq 1-\frac{\mathfrak{a}}{|x|}$

where the constant $\mathfrak{a}>0$ is the scattering length of $V$. For
integrable potentials, it can be equivalently determined by the identity

$8\pi\mathfrak{a}=\int V(x)f(x)dx$ (1.5)

which in particular shows that, for repulsive interactions,
$8\pi\mathfrak{a}<\widehat{V}(0)$. By scaling, it is then simple to check that
the scattering length of the interaction potential $N^{2}V(N.)$ appearing in
(1.3) is given by $\mathfrak{a}/N$, which characterises the Gross-Pitaevskii
regime.

Leading order estimate for ground state energy. The ground state energy
$E_{N}(V_{\text{ext}},V)$ of the Hamilton operator (1.3) has been first
estimated by Lieb, Seiringer and Yngvason in [40]. They proved that

$\lim_{N\to\infty}\frac{E_{N}(V_{\text{ext}},V)}{N}=\inf_{\begin{subarray}{c}\varphi\in
L^{2}(\mathbb{R}^{3}):\\\
\|\varphi\|=1\end{subarray}}\mathcal{E}_{\text{GP}}(\varphi)$ (1.6)

with the Gross-Pitaevskii energy functional

$\mathcal{E}_{\text{GP}}(\varphi)=\int\Big{[}|\nabla\varphi|^{2}+V_{\text{ext}}|\varphi|^{2}+4\pi\mathfrak{a}|\varphi|^{4}\Big{]}dx\,.$
(1.7)

In particular, (1.6) implies that, to leading order, the ground state energy
only depends on the interaction potential through its scattering length
$\mathfrak{a}$ (more recently, a different proof of (1.6) has been given in
[48]).

The proof of (1.6) obtained in [40] is based on matching upper and lower
bounds for $E_{N}(V_{\text{ext}},V)$. The upper bound is achieved through an
appropriate modification of the trial state used by Dyson in [21] to estimate
the energy of an homogeneous, dilute gas of hard spheres, in the thermodynamic
limit. As for the lower bound, it relies on the techniques introduced by Lieb
and Yngvason in [46], extended to the non-homogeneous setting.

Dyson’s Lemma. A key ingredient in the proof of the lower bound is a lemma,
originally due to Dyson, that can be used to replace the singular and short
range interaction appearing in (1.3) by a more regular potential, having
smaller size and longer range. Since we believe it is interesting to compare
Dyson’s lemma with more recent tools that will be discussed in the next
section, we state it precisely and we reproduce its short proof, in the form
proposed by Lieb and Yngvason in [46].

######  Lemma 1.1.

Let $v$ be a non-negative function on $[0;\infty)$, with finite range
$R_{0}>0$. Let $U$ be a non-negative function on $[0;\infty)$, with $U(r)=0$
for all $r<R_{0}$ and with $\int U(r)r^{2}dr\leq 1$. Let
$\mathcal{B}\subset\mathbb{R}^{3}$ be star shaped with respect to $0$. Then,
we have

$\int_{\mathcal{B}}\Big{[}\mu|\nabla\phi(x)|^{2}+\frac{1}{2}v(|x|)|\phi(x)|^{2}\Big{]}dx\geq\mu\mathfrak{a}\int_{\mathcal{B}}U(|x|)|\phi(x)|^{2}dx$
(1.8)

where $\mathfrak{a}$ is the scattering length (as defined in (1.4)) of the
potential $\mathbb{R}^{3}\ni x\to v(|x|)/\mu$.

###### Proof.

It is enough to prove the claim for radial $\phi$. The general case follows
because the total kinetic energy is always larger than its radial component.
We consider first the case $U(r)=\delta(r-R)/R^{2}$ for an $R>R_{0}$ (this
choice satisfies $\int U(r)r^{2}dr\leq 1$). Writing $\phi(x)=u(|x|)/|x|$, we
need to prove that

$\int_{0}^{R_{1}}\Big{[}\mu\big{|}u^{\prime}(r)-u(r)/r\big{|}^{2}+\frac{1}{2}v(r)|u(r)|^{2}\Big{]}dr\geq\left\\{\begin{array}[]{ll}0&\text{if
}R_{1}<R\\\ \mu\mathfrak{a}\frac{|u(R)|^{2}}{R^{2}}&\text{if
}R_{1}>R\end{array}\right.\,.$ (1.9)

By homogeneity in $u$, we can impose the normalization $u(R)=R-\mathfrak{a}$.
For $R_{1}<R$, the inequality holds trivially. If $R_{1}>R$, on the other
hand, we can minimize the l.h.s. of (1.9) over all functions $u$, with
$u(0)=0$ and $u(R)=R-\mathfrak{a}$. We find that the minimum is attained if
$u=u_{0}$, with $u_{0}$ solving the zero-energy scattering equation

$-\mu u^{\prime\prime}_{0}(r)+\frac{1}{2}v(r)u_{0}(r)=0\,.$

The normalization $u_{0}(R)=R-\mathfrak{a}$ implies that
$u_{0}(r)=r-\mathfrak{a}$ for all $r>R_{0}$. Integrating by parts, we conclude
that

$\begin{split}\int_{0}^{R_{1}}\Big{[}&\mu\big{|}u^{\prime}(r)-u(r)/r\big{|}^{2}+\frac{1}{2}v(r)|u(r)|^{2}\Big{]}dr\\\
&\geq\int_{0}^{R}\Big{[}\mu\big{|}u^{\prime}_{0}(r)-u_{0}(r)/r\big{|}^{2}+\frac{1}{2}v(r)|u_{0}(r)|^{2}\Big{]}dr\\\
&=\mu u_{0}(R)u^{\prime}_{0}(R)-\mu
u_{0}(R)^{2}/R=\mu\mathfrak{a}\big{(}1-\mathfrak{a}/R\big{)}\geq\mu\mathfrak{a}\big{(}1-\mathfrak{a}/R\big{)}^{2}=\mu\mathfrak{a}|u(R)|^{2}/R^{2}\,.\end{split}$

The lemma follows by noticing that a general $U$ can be written as linear
combination of interactions having the form $\delta(r-R)/R^{2}$, ie.

$U(r)=\int\frac{\delta(r-R)}{R^{2}}U(R)R^{2}dR$

where $\int U(R)R^{2}dR\leq 1$ by assumption. ∎

Already Dyson [21] applied a version of Lemma 1.1 to derive a lower bound for
the ground state energy of a dilute hard-sphere Bose gas. In his work,
however, he sacrificed the full kinetic energy to replace the potential. As a
result, his lower bound was off, approximately by a factor 14. Lieb and
Yngvason, on the other hand, realized in [46] that it is more convenient to
choose $\mu=1-\varepsilon$ in (1.8). Therefore, through Lemma 1.1, they ended
up with a nice potential and still with a bit of kinetic energy available.
After dividing the volume in small boxes, this procedure allowed them to
consider the regularized potential as a perturbation of the remaining kinetic
energy and to apply rigorous perturbation theory (in the form of Temple’s
inequality) to achieve the correct lower bound.

Condensation in the Gross-Pitaevskii limit. With standard tools in analysis,
Lieb, Seiringer and Yngvason proved in [40] the existence of a unique,
positive and continuously differentiable minimizer $\varphi_{\text{GP}}$ for
the Gross-Pitaevskii energy functional (1.7). The convergence (1.6) of the
ground state energy per particle towards the minimum of (1.7) suggests that
the ground state of the Hamiltonian (1.3) (and, more generally, many-body
states with sufficiently small energy) should exhibit Bose-Einstein
condensation in $\varphi_{\text{GP}}$.

In contrast with the Hamilton operator for the ideal gas, eigenfunctions of
(1.3) are not products of one-particle orbitals. For this reason, some care is
required when defining the meaning of condensation. It turns out that Bose-
Einstein condensation can be formulated in terms of reduced density matrices.
We define the reduced one-particle density matrix $\gamma_{N}$ associated with
a many-body wave function $\psi_{N}\in L^{2}_{s}(\mathbb{R}^{3N})$ as the non-
negative trace-class operator on the one-particle Hilbert space
$L^{2}(\mathbb{R}^{3})$ with the integral kernel

$\gamma_{N}(x;y)=N\int dx_{2}\dots
dx_{N}\,\psi_{N}(x,x_{2},\dots,x_{N})\overline{\psi}_{N}(y,x_{2},\dots,x_{N})\,.$
(1.10)

Here, we chose the normalization ${\rm tr}\,\gamma_{N}=N$. With this
definition, we can express the expectation of the number of particles in a
state $\varphi\in L^{2}(\mathbb{R}^{3})$ as

$\langle\psi_{N},a^{*}(\varphi)a(\varphi)\psi_{N}\rangle=\langle\varphi,\gamma_{N}\varphi\rangle\,.$
(1.11)

Therefore, we say that $\psi_{N}\in L_{s}^{2}(\mathbb{R}^{3N})$ exhibits Bose-
Einstein condensation in the state $\varphi\in L^{2}(\mathbb{R}^{3})$ if
$\langle\varphi,\gamma_{N}\varphi\rangle$ remains of order $N$, in the limit
$N\to\infty$ (in this sense, Bose-Einstein condensation is actually a property
of a sequence of wave functions $\\{\psi_{N}\\}_{N\in\mathbb{N}}$, rather than
of a single $\psi_{N}$).

In [37], Lieb and Seiringer proved that the one-particle density matrix
$\gamma_{N}$ associated with the ground state of the Hamilton operator (1.3)
is such that

$\frac{1}{N}\langle\varphi_{\text{GP}},\gamma_{N}\varphi_{\text{GP}}\rangle\to
1$ (1.12)

in the limit $N\to\infty$. According to (1.11), this means that the ground
state of (1.3) exhibits complete Bose-Einstein condensation in the minimizer
$\varphi_{\text{GP}}$ of the Gross-Pitaevskii functional (1.7); all particles,
up to a fraction vanishing in the limit $N\to\infty$, are described by the
same one-particle orbital $\varphi_{\text{GP}}$. In [39, 48], the convergence
(1.12) was extended, with different techniques, to any sequence of approximate
ground states, ie. any sequence of wave functions $\psi_{N}$ satisfying

$\lim_{N\to\infty}\frac{1}{N}\langle\psi_{N},H_{N}\psi_{N}\rangle\to\inf_{\begin{subarray}{c}\varphi\in
L^{2}(\mathbb{R}^{3}):\\\
\|\varphi\|_{2}=1\end{subarray}}\mathcal{E}_{\text{GP}}(\varphi)\,.$ (1.13)

Observe that (1.12) does not imply that the product state
$\psi_{N}^{\text{prod}}=\varphi_{\text{GP}}^{\otimes N}$ is a good
approximation for the ground state of (1.3). In fact, a simple computation
shows that

$\frac{1}{N}\langle\psi_{N}^{\text{prod}},H_{N}\psi_{N}^{\text{prod}}\rangle\simeq\int\Big{[}|\nabla\varphi_{\text{GP}}|^{2}+V_{\text{ext}}|\varphi_{\text{GP}}|^{2}+\frac{1}{2}\widehat{V}(0)|\varphi_{\text{GP}}|^{4}\Big{]}dx>\mathcal{E}_{\text{GP}}(\varphi_{\text{GP}})$

where we used the remark following (1.5). Thus, the energy of the product
state exceeds the ground state energy, even at leading order. Correlations
among particles are crucial to approach the ground state energy (1.6); only at
the level of reduced density matrices, correlations are integrated out and
convergence (1.12) can be established.

Proof of condensation. The proof of (1.12) obtained in [37] is based on two
main ingredients. First of all, it relies on accurate energy estimates, which
imply that the part of the ground state kinetic energy contributing, in the
limit of large $N$, to the quartic term in the Gross-Pitaevskii functional
(1.7) (ie. the difference between the many-body kinetic energy and the kinetic
energy of the Gross-Pitaevskii minimizer $\phi_{\text{GP}}$) is localized in
small regions of the configuration space, where two particles are close (much
closer than they typically are). Denoting by $\psi_{N}\in
L^{2}_{s}(\Lambda^{N})$ the normalized ground state wave function, we define,
for fixed ${\bf X}=(x_{2},\dots,x_{N})$,

$f_{\bf X}(x)=\frac{1}{\varphi_{\text{GP}}(x)}\psi_{N}(x,{\bf X})\,.$

Introducing the set

$\Omega_{\bf X}=\big{\\{}x\in\mathbb{R}^{3}:\min_{k=2,\dots,N}|x-x_{k}|\geq
N^{-1/3-\delta}\big{\\}}\,,$

Lieb and Seiringer show in [37] that, for $\delta>0$ small enough,

$\lim_{N\to\infty}\int d{\bf X}\int_{\Omega_{\bf
X}}dx\,|\varphi_{\text{GP}}(x)|^{2}\,|\nabla_{x}f_{\bf X}(x)|^{2}=0\,.$ (1.14)

Thus, for every fixed ${\bf X}=(x_{2},\dots,x_{N})\in\mathbb{R}^{3(N-1)}$,
most of the mass of $\varphi_{\text{GP}}(x)\nabla_{x}f_{\bf X}(x)$ is
contained in the set $\Omega_{\bf X}^{c}$, given by the union of small balls
around the points $x_{j}$, $j=2,\dots,N$, whose total volume is of the order
$N^{-3\delta}$ and converges to $0$, as $N\to\infty$.

The second important ingredient is a generalized Poincaré inequality, which
can be used, together with (1.14), to show that $f_{\bf X}$ is almost constant
and thus to conclude Bose-Einstein condensation, as stated in (1.12). A non-
standard Poincaré inequality is needed here because, despite the fact that
$\Omega_{\bf X}^{c}$ has a small volume, the set $\Omega_{\bf X}$ may be quite
strange, it does not even need to be connected. In general, smallness of the
gradient $\nabla_{x}f_{\bf X}$ on non-connected sets with small complements
does not imply that $f_{\bf X}$ is approximately constant (counterexamples can
be easily constructed). It does, however, if the total $L^{2}$-norm of
$\nabla_{x}f_{\bf X}$ is bounded; more details on this part of the proof can
be found in [43].

In the next section, we will discuss, in some details, recent improvements of
the estimate (1.6) on the ground state energy and of the bound (1.12) on the
number of particles in the condensate, in the Gross-Pitaevskii limit. Before
doing so, let us briefly mention some other mathematical results concerning
the Bose gas in the Gross-Pitaevskii regime that have been established in the
last two decades (of course, the list is not complete).

The Gross-Pitaevskii functional for rotating gases. Working in the moving
coordinate system, a Bose gas rotating with angular velocity
$\Omega\in\mathbb{R}^{3}$ in the Gross-Pitaevskii regime can be described by
the many-body Hamiltonian

$H^{\text{rot}}_{N}=\sum_{j=1}^{N}\big{[}(i\nabla_{x_{j}}+A(x_{j}))^{2}+W(x_{j})\big{]}+\sum_{i<j}^{N}N^{2}V(N(x_{i}-x_{j}))$
(1.15)

with $A(x)=(\Omega\wedge x)/2$ and $W(x)=V_{\text{ext}}(x)-(\Omega\wedge
x)^{2}/4$. Mathematically, the rotation of the gas modifies the Hamilton
operator like a constant magnetic field $\Omega$, with vector potential $A$.
In order for (1.15) to be bounded below, the external confining potential must
grow fast enough, as $|x|\to\infty$, to compensate the centrifugal potential.
Under the assumption that $W(x)\to\infty$ (more precisely, $W(x)\geq
c\log|x|-C$, for constants $c,C>0$), Lieb and Seiringer proved in [39] that,
in the same spirit as in (1.6), the ground state energy
$E_{N}^{\text{rot}}(\Omega,V_{\text{ext}})$ of (1.15), defined minimizing the
expectation of (1.15) over all normalized bosonic wave functions, is such that

$\lim_{N\to\infty}\frac{E^{\text{rot}}_{N}(\Omega,V_{\text{ext}},V)}{N}=\inf_{\begin{subarray}{c}\varphi\in
L^{2}(\mathbb{R}^{3}):\\\
\|\varphi\|_{2}=1\end{subarray}}\int\Big{[}|(i\nabla+A)\varphi|^{2}+W|\varphi|^{2}+4\pi\mathfrak{a}|\varphi|^{4}\Big{]}dx\,.$
(1.16)

In contrast with the case $\Omega=0$, the functional on the r.h.s. of (1.16)
does not always have a unique minimizer. For this reason, Bose-Einstein
condensation has to be stated in a slightly more complicated form, compared
with (1.12). Denote by $\Gamma$ the set of reduced one-particle density
matrices obtained from sequences of approximate ground states of (1.15), in
the limit $N\to\infty$. Complete Bose-Einstein condensation, as established in
[39], means here that the extremal points in $\Gamma$ coincide with the rank-
one orthogonal projections into minimizers of the Gross-Pitaevskii functional
(1.16) and that every element of $\Gamma$ can be expressed as a convex
combination of these projections.

The behavior of the Gross-Pitaevskii functional (1.16) depends on the size of
the angular velocity $\Omega$. Assume the confining potential $V_{\text{ext}}$
to be cylindrically symmetric. Then, if $|\Omega|$ is sufficiently small,
(1.16) has a unique cylindrically symmetric minimizer. In a fixed reference
frame, its velocity is zero. As already remarked in [42], this can be
interpreted as a manifestation of superfluidity (for sufficiently small
$\Omega$, the gas does not react to the rotation). For larger values of
$|\Omega|$, on the other hand, the minimizers of (1.16) form quantized
vortices, breaking the axial symmetry and therefore leading to a continuous
family of minimizers; this was shown in [56] (remark that there is no symmetry
breaking in the ground state, if we considered (1.15) over the full Hilbert
space $L^{2}(\mathbb{R}^{3N})$, neglecting the bosonic statistics). In fact,
the formation of quantized vortices in minimizers of (1.16) and of related
functionals describing superfluids and superconductors has generated a
substantial mathematical literature; for a review of this interesting subject,
see [2, 8, 18, 53, 60].

The case of a rotating Bose gas in an harmonic trap is particularly
interesting. Here, there is a critical angular velocity $\Omega_{c}$, above
which the gas is no longer confined. Sufficiently close to (but below)
$\Omega_{c}$, the system is effectively projected to the lowest Landau level
and can be described there by a Hamilton operator, modelling the bosonic
analogue of the fractional quantum Hall effect; this was proven in [36]. Also
for this simplified Hamiltonian, convergence to the minimum of an
appropriately defined Gross-Pitaevskii energy functional can be established in
appropriate scaling regimes, see [44].

The Gross-Pitaevskii limit in 2 dimensions. In two dimensions, for Bose gases
trapped in a volume of order one by a confining potential, the Gross-
Pitaevskii limit is characterised by the condition that

$g=\frac{N}{|\log(N\mathfrak{a}^{2})|}$ (1.17)

remains fixed (or tends to a finite, non-vanishing, limit, as $N\to\infty$).
This implies that the Gross-Pitaevskii regime can only be approached if the
scattering length of the interaction potential $\mathfrak{a}$ is exponentially
small in the number of particles. For this reason, from the point of view of
physics, the Gross-Pitaevskii limit in two dimensions is probably not as
relevant as it is in three dimensions. From the point of view of mathematics,
however, it is a well-defined regime and it is certainly the source of
interesting and challenging questions.

In [41], Lieb, Seiringer and Yngvason considered a two-dimensional Bose gas
confined by a trapping potential $V_{\text{ext}}$ and interacting through a
repulsive two-body potential $V$ with scattering length $\mathfrak{a}$.
Assuming (1.17) to be fixed (positive and finite), they proved the analogue of
(1.6), showing that the many-body ground state energy is such that, as
$N\to\infty$,

$\frac{E_{N}(V_{\text{ext}},V)}{N}\to\min_{\begin{subarray}{c}\varphi\in
L^{2}(\mathbb{R}^{3}):\\\
\|\varphi\|=1\end{subarray}}\int\big{[}|\nabla\varphi|^{2}+V_{\text{ext}}|\varphi|^{2}+4\pi
g|\varphi|^{4}\big{]}dx\,.$

Also in this case, the techniques of [37, 39, 48] can be used to show that the
ground state of the Hamiltonian exhibits complete Bose-Einstein condensation.

Dynamics of Bose gases. Instead of considering equilibrium properties of the
Hamilton operator (1.3) describing Bose gases in the Gross-Pitaevskii regime,
it is also intersting to approach the problem from a dynamical point of view.
Physically, studying dynamics means that we consider the time-evolution
resulting from a change of the external fields. Motivated by the experiments
described above, where the evolution of initially confined gases is observed
after traps are switched off, it seems important to consider the solution of
the time-dependent many-body Schrödinger equation

$i\partial_{t}\psi_{N,t}=\Big{[}\sum_{j=1}^{N}-\Delta_{x_{j}}+\sum_{i<j}^{N}N^{2}V(N(x_{i}-x_{j}))\Big{]}\psi_{N,t}$
(1.18)

with initial data $\psi_{N,0}\in L^{2}_{s}(\mathbb{R}^{3N})$ describing the
ground state of the trapped Hamiltonian (1.3). As explained in the previous
paragraphs, $\psi_{N,0}$ exhibits Bose-Einstein condensation in the minimizer
$\varphi_{\text{GP}}$ of the Gross-Pitaevskii energy functional (1.7). A
natural question is therefore the following: Is Bose-Einstein condensation
preserved by the time-evolution? In other words, does $\psi_{N,t}$ continue to
exhibit condensation also for $t\not=0$ and, if yes, in which one-particle
state?

This question was first addressed (for Bose gases in the Gross-Pitaevskii
regime) in [24, 25, 26], where it was proven that, for every fixed
$t\in\mathbb{R}$, the solution $\psi_{N,t}=e^{-iH_{N}t}\psi_{N,0}$ of (1.18)
does indeed exhibit complete Bose-Einstein condensation and that the evolution
of the condensate wave function $\varphi_{t}$ is determined by the time-
dependent Gross-Pitaevskii equation

$i\partial_{t}\varphi_{t}=-\Delta\varphi_{t}+V_{\text{ext}}\varphi_{t}+8\pi\mathfrak{a}|\varphi_{t}|^{2}\varphi_{t}$
(1.19)

with the initial data $\varphi_{t=0}=\varphi_{\text{GP}}$. This result shows
that Gross-Pitaevskii theory can also be used to predict non-equilibrium
properties of Bose gases at low energy. Different proofs of the convergence
towards the solution of the nonlinear Gross-Pitaevskii equation (1.19) have
been later obtained in [51, 6]. More recently, convergence with optimal rates
has been established in [14].

The Gross-Pitaevskii regime at positive temperature. Recently, the results
(1.6) and (1.12) have been extended to positive temperature in [20, 19]. In
the homogenous setting, with $N$ particles moving in the unit torus
$\Lambda=[0;1]^{3}$ (so that the density of the gas is $\rho=N$) and
interacting through a repulsive potential with scattering length
$\mathfrak{a}_{N}=\mathfrak{a}/N$, for temperatures comparable with the
critical temperature of the ideal Bose gas (ie. $T\simeq\rho^{2/3}=N^{2/3}$),
the free energy has been shown to have the form (for some $\varepsilon>0$
small enough)

$F(\beta,N)=F_{0}(\beta,N)+4\pi\mathfrak{a}_{N}\big{(}2\rho^{2}-\rho^{2}_{0}\big{)}+\mathcal{O}(N^{1-\varepsilon})$
(1.20)

where $F_{0}(\beta,N)$ and $\rho_{0}=\rho_{0}(\beta,N)$ are the free energy
and the condensate density for the ideal gas (so that the first term on the
r.h.s. of (1.20) is of the order $N^{5/3}$ and the second of order $N$).
Moreover, the one-particle reduced density matrix associated with the thermal
state at inverse temperature $\beta$ (or to any approximate sequence of
thermal states) has been proven to approach (in the trace norm topology) the
reduced one-particle density matrix associated with the free Gibbs state. In
particular, this shows the existence of a phase transition for Bose-Einstein
condensation in the Gross-Pitaevskii regime.

Additional references. There are several review articles and lecture notes
covering in more details most of the subjects discussed in this section. The
interested reader can consult, for example, [7, 38, 52, 54, 55, 57, 58, 62].

##  2 Bogoliubov theory in the Gross-Pitaevskii regime

In the last 5 years there has been some progress in the mathematical
understanding of the spectral properties of Bose gases in the Gross-Pitaevskii
limit, beyond the leading order estimates (1.6), (1.12). In this section, we
are going to review some of these new results, which are based on a rigorous
version of Bogoliubov theory, developed in [9, 10, 11, 12].

To simplify our discussion, we will focus here on the homogeneous, translation
invariant setting. We will therefore consider a system of $N$ particles moving
in the three-dimensional unit torus $\Lambda$, described by the Hamilton
operator

$H_{N}=\sum_{j=1}^{N}-\Delta_{x_{j}}+\sum_{i<j}^{N}N^{2}V(N(x_{i}-x_{j}))$
(2.1)

acting on $L^{2}_{s}(\Lambda^{N})$. From (1.6), the ground state energy
$E_{N}(V)$ of (2.1) is such that

$\lim_{N\to\infty}\frac{E_{N}(V)}{N}=4\pi\mathfrak{a}\,.$ (2.2)

Moreover, from (1.12) we conclude that every sequence $\psi_{N}\in
L^{2}_{s}(\Lambda^{N})$ of approximate ground states of (2.1), satisfying
$\langle\psi_{N},H_{N}\psi_{N}\rangle/N\to 4\pi\mathfrak{a}$ exhibits complete
Bose Einstein condensation, in the sense

$\frac{1}{N}\langle\varphi_{0},\gamma_{N}\varphi_{0}\rangle\to 1$ (2.3)

into the zero-momentum state $\varphi_{0}\in L^{2}(\Lambda)$ defined by
$\varphi_{0}(x)=1$ for all $x\in\Lambda$. In this section, we are going to
sketch some of the main ideas in the proof of the following theorem, taken
from [11, 12], improving on the estimates (2.2), (2.3), for a class of
integrable interaction potentials.

######  Theorem 2.1.

Let $V\in L^{3}(\mathbb{R}^{3})$, non-negative, radial and with compact
support.

  * i)

Optimal rate for condensation. Let $\psi_{N}\in L^{2}_{s}(\Lambda^{N})$ be a
sequence of approximate ground states, such that

$\langle\psi_{N},H_{N}\psi_{N}\rangle\leq 4\pi\mathfrak{a}N+\zeta$

for $\zeta\ll N$. Then, denoting by $\gamma_{N}$ the one-particle reduced
density matrix of $\psi_{N}$, defined as in (1.10), we find

$N-\langle\varphi_{0},\gamma_{N}\varphi_{0}\rangle\leq C(\zeta+1)$ (2.4)

for a constant $C>0$.

  * ii)

Precise estimate for ground state energy. We have

$\begin{split}E_{N}(V)=\;&4\pi\mathfrak{a}(N-1)+e_{\Lambda}\mathfrak{a}^{2}\\\
&-\frac{1}{2}\sum_{p\in
2\pi\mathbb{Z}^{3}\backslash\\{0\\}}\left[p^{2}+8\pi\mathfrak{a}-\sqrt{|p|^{4}+16\pi\mathfrak{a}p^{2}}-\frac{(8\pi\mathfrak{a})^{2}}{2p^{2}}\right]+\mathcal{O}(N^{-1/4})\,\end{split}$
(2.5)

where

$e_{\Lambda}=2-\lim_{M\to\infty}\sum_{\begin{subarray}{c}p\in\mathbb{Z}^{3}\backslash\\{0\\}:\\\
|p_{1}|,|p_{2}|,|p_{3}|\leq M\end{subarray}}\frac{\cos(|p|)}{p^{2}}$ (2.6)

and, in particular, the limit exists.

  * iii)

Excitation spectrum. The spectrum of $H_{N}-E_{N}(V)$, below a threshold
$\zeta>0$, consists of eigenvalues having the form

$\begin{split}\sum_{p\in
2\pi\mathbb{Z}^{3}\backslash\\{0\\}}n_{p}\sqrt{|p|^{4}+16\pi\mathfrak{a}p^{2}}+\mathcal{O}(N^{-1/4}\zeta^{3})\,\end{split}$
(2.7)

with $n_{p}\in\mathbb{N}$ for all momenta $p\in
2\pi\mathbb{Z}^{3}\backslash\\{0\\}$.

Remarks.

  * 1)

The estimate (2.4) shows that the number of excitations of the Bose-Einstein
condensate in a many-body state $\psi_{N}$ can be controlled by the difference
between the energy of $\psi_{N}$ and the ground state energy. In fact, for the
ground state vector $\psi^{\text{gs}}_{N}$, we obtain the precise estimate on
the condensate depletion (number of excitations of the condensate)

$N-\langle\varphi_{0},\gamma_{N}^{\text{gs}}\varphi_{0}\rangle=\sum_{p\in
2\pi\mathbb{Z}^{3}\backslash\\{0\\}}\Big{[}\frac{p^{2}+8\pi\mathfrak{a}-\sqrt{|p|^{4}+16\pi\mathfrak{a}p^{2}}}{2\sqrt{|p|^{4}+16\pi\mathfrak{a}p^{2}}}\Big{]}+\mathcal{O}(N^{-1/8})\,.$

Non-optimal estimates establishing Bose-Einstein condensation have been also
recently established in regimes interpolating between the Gross-Pitaevskii and
the thermodynamic limit, in [27].

  * 2)

The correction $e_{\Lambda}\mathfrak{a}^{2}$, appearing in (2.5), is a finite
volume effect, arising because $\mathfrak{a}$ is defined through the
scattering equation (1.4) on the whole space $\mathbb{R}^{3}$, rather than on
the unit torus $\Lambda$.

  * 3)

The expression (2.5) is the analogue, in the Gross-Pitaevskii regime, of the
Lee-Huang-Yang formula for the energy of a dilute Bose gas in the
thermodynamic limit. The validity of the Lee-Huang-Yang formula has been
recently established through matching upper [61, 5] and lower bounds [28, 29].

  * 4)

Eq. (2.7) states that excited eigenvalues are determined, in good
approximation, by the sum of the energies of quantized excitations, labelled
by their momentum and characterized by the dispersion law
$\varepsilon(p)=\sqrt{|p|^{4}+16\pi\mathfrak{a}p^{2}}$.

  * 5)

Theorem 2.1 has been recently extended to Bose gases in the Gross-Pitaevskii
regime, trapped by an external potential in [47, 49, 15, 16]. In particular,
the approach of [47, 49] shows how the assumption $V\in L^{3}(\mathbb{R}^{3})$
appearing in Theorem 2.1 can be relaxed to $V\in L^{1}(\mathbb{R}^{3})$.

Below, we sketch some of the main steps in the proof of Theorem 2.1. We follow
here the approach of [9, 10] (an alternative approach has been recently
proposed in [31, 32]). The proof of the optimal bound (2.4) for the number of
excitations and the proof of the estimates (2.5) and (2.7) for the low-energy
spectrum of (2.1) follow a similar strategy. We will discuss in more details
the proof of (2.4). At the end, we will briefly comment on how (2.4) can be
used to deduce (2.5) and (2.7).

Factoring out the Bose-Einstein condensate. The first step to show Theorem 2.1
consists in factoring out the Bose-Einstein condensate and focussing instead
on its orthogonal excitations. Following [35], we remark that every
$\psi_{N}\in L^{2}_{s}(\Lambda^{N})$ can be written as

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

with $\alpha_{j}\in L^{2}_{\perp}(\Lambda)^{\otimes j}$, for $j=0,\dots,N$.
Here $L^{2}_{\perp}(\Lambda)$ denotes the orthogonal complement in
$L^{2}(\Lambda)$ of the condensate wave function $\varphi_{0}$. This remark
allows us to introduce a unitary operator $U_{N}$, defined by
$U_{N}\psi_{N}=\\{\alpha_{0},\alpha_{1},\dots,\alpha_{N}\\}$, mapping the
$N$-particle Hilbert space $L^{2}_{s}(\Lambda^{N})$ onto the truncated Fock
space

$\mathcal{F}_{+}^{\leq N}=\bigoplus_{j=0}^{N}L^{2}_{\perp}(\Lambda)^{\otimes
j},$

where we describe orthogonal excitations of the condensate.

With $U_{N}$, we can define the excitation Hamiltonian
$\mathcal{L}_{N}=U_{N}H_{N}U_{N}^{*}$, acting on $\mathcal{F}^{\leq N}_{+}$.
To compute $\mathcal{L}_{N}$, we switch to momentum space and introduce the
formalism of second quantization. For momenta
$p,q\in\Lambda^{*}_{+}=2\pi\mathbb{Z}^{3}\backslash\\{0\\}$, we find that

$\begin{split}U_{N}a_{p}^{*}a_{q}U_{N}^{*}&=a_{p}^{*}a_{q},\\\
U_{N}a_{0}^{*}a_{0}U_{N}^{*}&=N-\mathcal{N}_{+}\\\
U_{N}a_{p}^{*}a_{0}U_{N}^{*}&=a_{p}^{*}\sqrt{N-\mathcal{N}_{+}}=:\sqrt{N}b_{p}^{*},\\\
U_{N}a_{0}^{*}a_{p}U_{N}&=\sqrt{N-\mathcal{N}_{+}}a_{p}=:\sqrt{N}b_{p}\end{split}$
(2.8)

where $\mathcal{N}_{+}$ denotes the number of particles operator on
$\mathcal{F}_{+}^{\leq N}$ (measuring the number of excitations of the
condensate) and where we defined modified creation and annihilation operators
$b_{p}^{*},b_{p}$ creating and annihilating excitations with momentum $p$ (on
states with few excitations of the condensate, we expect $b_{p}^{\sharp}\simeq
a_{p}^{\sharp}$; in contrast to the standard creation and annihilation
operators, $b_{p},b_{p}^{*}$ preserve the total number of particles and thus
map $\mathcal{F}_{+}^{\leq N}$ back to itself). Writing

$H_{N}=\sum_{p\in
2\pi\mathbb{Z}^{3}}p^{2}a_{p}^{*}a_{p}+\frac{1}{2N}\sum_{p,q,r\in
2\pi\mathbb{Z}^{3}}\widehat{V}(r/N)a_{p+r}^{*}a_{q}^{*}a_{q+r}a_{p}$

and applying (2.8), we obtain the following lemma.

######  Lemma 2.2.

The excitation Hamiltonian $\mathcal{L}_{N}$ has the form

$\begin{split}\mathcal{L}_{N}=\;&\frac{(N-1)}{2N}\widehat{V}(0)(N-\mathcal{N}_{+})+\frac{\widehat{V}(0)}{2N}\mathcal{N}_{+}(N-\mathcal{N}_{+})\\\
&+\sum_{p\in\Lambda^{*}_{+}}p^{2}a_{p}^{*}a_{p}+\sum_{p\in\Lambda_{+}^{*}}\widehat{V}(p/N)a_{p}^{*}\frac{N-\mathcal{N}_{+}-1}{N}a_{p}+\frac{1}{2}\sum_{p\in\Lambda^{*}_{+}}\widehat{V}(p/N)(b_{p}^{*}b_{-p}^{*}+\text{h.c.})\\\
&+\frac{1}{\sqrt{N}}\sum_{p,q\in\Lambda_{+}^{*},p+q\not=0}\widehat{V}(p/N)\left[b_{p+q}^{*}a_{-p}^{*}a_{q}+a_{q}^{*}a_{-p}b_{p+q}\right]\\\
&+\frac{1}{2N}\sum_{p,q\in\Lambda^{*}_{+},r\not=p,-q}\widehat{V}(r/N)a_{p+r}^{*}a_{q}^{*}a_{p}a_{q+r}\,.\end{split}$
(2.9)

Conjugation with the unitary map $U_{N}$ can be interpreted as a rigorous
version of Bogoliubov c-number substitution, where operators
$a^{*}_{0},a_{0}$, creating and, respectively, annihilating a particle in the
condensate are replaced by a factor $\sqrt{N}$.

Renormalized excitation Hamiltonian. Following Bogoliubov’s approach, we
should now approximate the excitation Hamiltonian (2.9) by a quadratic
operator, neglecting cubic and quartic contributions appearing in the last two
lines. While this step leads to the correct energy in mean-field type regimes
[59, 35], it cannot be justified in the Gross-Pitaevskii limit. Cubic and
quartic terms in (2.9) still contain important contributions to the energy,
which cannot be neglected (in fact, quartic terms even contribute to (2.2) at
leading order).

The point here is that conjugation with $U_{N}$ factors out products of
$\varphi_{0}$ but keeps correlations in the excitation vector
$U_{N}\psi_{N}=\\{\alpha_{0},\dots,\alpha_{N}\\}$. To extract relevant
contributions from cubic and quartic terms in (2.9), we need to factor out the
correlation structure characterizing low-energy states.

To model correlations we fix $\ell_{0}>0$ (small but independent of $N$) and
we consider the ground state solution of the Neumann problem

$\Big{[}-\Delta+\frac{N^{2}}{2}V(Nx)\Big{]}f_{N}(x)=\lambda_{N}f_{N}(x)$
(2.10)

on the ball $|x|\leq\ell_{0}$, normalized so that $f_{N}(x)=1$, if
$|x|=\ell_{0}$. We extend $f_{N}(x)=1$, for $|x|\geq\ell_{0}$, and we define
$\check{\eta}(x)=-N(1-f_{N}(x))$. To understand the choice of $\check{\eta}$,
we remark that, heuristically,

$\begin{split}\prod_{i<j}^{N}f_{N}(x_{i}-x_{j})&=\prod_{i<j}^{N}\big{[}1+\frac{1}{N}\check{\eta}(x_{i}-x_{j})\big{]}\\\
&\simeq\prod_{i<j}^{N}e^{\frac{1}{N}\check{\eta}(x_{i}-x_{j})}=e^{\frac{1}{N}\sum_{i<j}\check{\eta}(x_{i}-x_{j})}\\\
&\simeq\Big{[}U^{*}_{N}\,e^{\frac{1}{2}\sum_{p\in\Lambda^{*}_{+}}\eta_{p}b^{*}_{p}b_{-p}^{*}}\Omega\Big{]}(x_{1},\dots,x_{N})\end{split}$
(2.11)

where $\\{\eta_{p}\\}_{p\in\Lambda^{*}_{+}}$ are the Fourier coefficients of
$\check{\eta}$. To justify the last step in (2.11), we expand the exponential
map, we use (2.8) to compute the action of $U^{*}_{N}$, and we switch between
momentum and position space. For technical reasons it is more convenient,
instead of working with
$\exp(\frac{1}{2}\sum_{p\in\Lambda^{*}_{+}}\eta_{p}b^{*}_{p}b_{-p}^{*})$, to
use unitary operators and to introduce a cutoff $\kappa_{H}>0$, restricting
the sum in the exponent to large momenta. Thus, we define the generalized
Bogoliubov transformation

$T_{\eta}=\exp\Big{[}\frac{1}{2}\sum_{|p|>\kappa_{H}}\eta_{p}\big{(}b_{p}^{*}b_{-p}^{*}-b_{p}b_{-p}\big{)}\Big{]}\,.$
(2.12)

Motivated by (2.11), we expect that conjugation with $T_{\eta}$ factors out
two-body correlations described by the solution $f_{N}$ of (2.10). For this
reason, we define the renormalized excitation Hamiltonian
$\mathcal{G}_{N}=T_{\eta}^{*}\mathcal{L}_{N}T_{\eta}=T^{*}_{\eta}U_{N}H_{N}U^{*}_{N}T_{\eta}$.

On states with few excitations of the condensate, we have
$b^{\sharp}_{p}\simeq a^{\sharp}_{p}$ and we can expect that the action of
$T_{\eta}$ is close to action of a standard Bogoliubov transformation, ie.

$\begin{split}T_{\eta}^{*}a_{p}^{*}T_{\eta}\simeq\cosh(\eta_{p})\,a_{p}^{*}+\sinh(\eta_{p})\,a_{-p}\\\
T_{\eta}^{*}a_{p}T_{\eta}\simeq\cosh(\eta_{p})\,a_{p}+\sinh(\eta_{p})\,a^{*}_{-p}\end{split}$
(2.13)

for all $p\in\Lambda^{*}_{+}$ with $|p|>\kappa_{H}$. With these formulas, we
can approximately compute the renormalized excitation Hamiltonian
$\mathcal{G}_{N}$.

######  Lemma 2.3.

Introducing the notation

$\mathcal{K}=\sum_{p\in\Lambda^{*}_{+}}p^{2}a_{p}^{*}a_{p},\qquad\mathcal{V}_{N}=\frac{1}{2N}\sum_{p,q\in\Lambda^{*}_{+},r\not=-p,-q}\widehat{V}(r/N)a_{p+r}^{*}a_{q}^{*}a_{q+r}a_{p}$
(2.14)

for the kinetic and potential energy operators, we find that

$\begin{split}\mathcal{G}_{N}=\;&4\pi\mathfrak{a}(N-\mathcal{N}_{+})+\big{[}\widehat{V}(0)-4\pi\mathfrak{a}\big{]}\mathcal{N}_{+}\frac{(N-\mathcal{N}_{+})}{N}\\\
&+\mathcal{K}+\sum_{p\in\Lambda^{*}_{+}}\widehat{V}(p/N)a_{p}^{*}a_{p}(1-\mathcal{N}_{+}/N)+4\pi\mathfrak{a}\sum_{|p|\leq\kappa_{H}}\big{[}b_{p}^{*}b_{-p}^{*}+b_{p}b_{-p}\big{]}\\\
&+\frac{1}{\sqrt{N}}\sum_{p,q\in\Lambda^{*}_{+}:p+q\not=0}\widehat{V}(p/N)\big{[}b_{p+q}^{*}a_{-p}^{*}a_{q}+\text{h.c.}\big{]}+\mathcal{V}_{N}+\mathcal{E}_{\mathcal{G}_{N}}\end{split}$
(2.15)

where the error operator $\mathcal{E}_{\mathcal{G}_{N}}$ is such that

$\pm\mathcal{E}_{\mathcal{G}_{N}}\leq
C\kappa_{H}^{-1}(\mathcal{K}+\mathcal{V}_{N})+C\kappa_{H}\,.$ (2.16)

Comparing (2.15) with the corresponding expression (2.9) for the excitation
Hamiltonian $\mathcal{L}_{N}$, we observe two main differences. First of all,
the vacuum expectation of $\mathcal{G}_{N}$ is given by $4\pi\mathfrak{a}N$.
To leading order, we know from (2.2) that this is the correct ground state
energy. Secondly, the off-diagonal quadratic term has been renormalized,
meaning that the singular potential $\widehat{V}(p/N)$, decaying only for
$|p|$ of order $N$, has been now replaced by
$4\pi\mathfrak{a}\chi(|p|\leq\kappa_{H})$, which decays already on momenta of
order one (the cutoff $\kappa_{H}$ will be chosen large but fixed, independent
of $N$).

Sketch of proof of Lemma 2.3. Let us briefly explain the mechanism leading
from (2.9) to (2.15). With (2.13) and approximating $\cosh\eta_{p}\simeq 1$,
$\sinh\eta_{p}\simeq\eta_{p}$ for all $|p|>\kappa_{H}$ (higher order terms
decay faster in $p$, hence they produce smaller errors), the conjugation of
the kinetic energy operator gives

$\begin{split}T_{\eta}^{*}\Big{[}\sum_{p\in\Lambda^{*}_{+}}p^{2}a_{p}^{*}a_{p}\Big{]}T_{\eta}&\simeq\sum_{p\in\Lambda_{+}^{*}}p^{2}(\cosh(\eta_{p})a_{p}^{*}+\sinh(\eta_{p})a_{-p})(\cosh(\eta_{p})a_{p}+\sinh(\eta_{p})a_{-p}^{*})\\\
&\simeq\sum_{p\in\Lambda^{*}_{+}}p^{2}a_{p}^{*}a_{p}+\sum_{|p|>\kappa_{H}}p^{2}\eta_{p}(a_{p}^{*}a_{-p}^{*}+a_{p}a_{-p}\big{)}+\sum_{|p|>\kappa_{H}}p^{2}\eta_{p}^{2}\end{split}$
(2.17)

(in this heuristic discussion, we exchange freely modified creation and
annihilation operators with standard creation and annihilation operators).
Analogously, we can compute the action of $T_{\eta}$ on the off-diagonal
quadratic term in the second line on the r.h.s of (2.9). We find

$T_{\eta}^{*}\Big{[}\frac{1}{2}\sum_{p\in\Lambda^{*}_{+}}\widehat{V}(p/N)\big{(}a_{p}^{*}a_{-p}^{*}+\text{h.c.}\big{)}\Big{]}T_{\eta}\simeq\frac{1}{2}\sum_{p\in\Lambda^{*}_{+}}\widehat{V}(p/N)(a_{p}^{*}a_{-p}^{*}+\text{h.c.}\big{)}+\frac{1}{2}\sum_{|p|\leq\kappa_{H}}\widehat{V}(p/N)\eta_{p}$
(2.18)

up to terms that can be included in the error operator
$\mathcal{E}_{\mathcal{G}_{N}}$ in (2.15). Finally, we consider the
conjugation of the quartic term on the r.h.s. of (2.9). In this case, the
computation is a bit longer. At the end we find, up to irrelevant error terms,

$\begin{split}T_{\eta}^{*}\Big{[}\frac{1}{2N}&\sum_{p,q,r\in\Lambda^{*}_{+}}\widehat{V}(r/N)a_{p+r}^{*}a_{q}^{*}a_{q+r}a_{p}\Big{]}T_{\eta}\\\
\simeq\;&\frac{1}{2N}\sum_{p,q,r\in\Lambda^{*}_{+}}\widehat{V}(r/N)a_{p+r}^{*}a_{q}^{*}a_{q+r}a_{p}\\\
&+\frac{1}{2N}\sum_{\begin{subarray}{c}q,r\in\Lambda^{*}_{+}:\\\
|q+r|>\kappa_{H}\end{subarray}}\widehat{V}(r/N)\eta_{q+r}\big{(}a^{*}_{q}a^{*}_{-q}+a_{q}a_{-q}\big{)}+\frac{1}{2N}\sum_{\begin{subarray}{c}q,r\in\Lambda^{*}_{+}:\\\
|q|,|q+r|>\kappa_{H}\end{subarray}}\widehat{V}(r/N)\eta_{q+r}\eta_{q}\end{split}$
(2.19)

Combining the quadratic terms generated in (2.17), (2.18) and (2.19) we
obtain, separating contributions associated with $|p|>\kappa_{H}$ and with
$|p|\leq\kappa_{H}$,

$\begin{split}\sum_{|p|>\kappa_{H}}\Big{[}p^{2}\eta_{p}&+\frac{1}{2}\widehat{V}(p/N)+\frac{1}{2N}\sum_{\begin{subarray}{c}r\in\Lambda^{*}:\\\
|p+r|>\kappa_{H}\end{subarray}}\widehat{V}(r/N)\eta_{p+r}\Big{]}\big{(}a_{p}^{*}a_{-p}^{*}+a_{-p}a_{p}\big{)}\\\
&+\frac{1}{2}\sum_{|p|\leq\kappa_{H}}\Big{[}\widehat{V}(p/N)+\frac{1}{N}\sum_{\begin{subarray}{c}r\in\Lambda^{*}:\\\
|p+r|>\kappa_{H}\end{subarray}}\widehat{V}(r/N)\eta_{p+r}\Big{]}\big{(}a_{p}^{*}a^{*}_{-p}+a_{-p}a_{p}\big{)}\,.\end{split}$
(2.20)

In the first line, the expression in the parenthesis (after extending the sum
over $r$ to complete the convolution of $\widehat{V}$ and $\eta$, at the
expenses of a small error) reconstructs exactly the l.h.s. of (2.10); using
the identity (2.10), we can prove that this term is small and can be
neglected. On the second line, on the other hand, we obtain, after completing
the sum and switching to position space, the integral (1.5) (using the fact
that the difference between $f_{N}$, defined in (2.10), and the solution of
the zero energy scattering equation (1.4) is small). This explains the
emergence of the renormalized off-diagonal quadratic term in (2.15).
Similarly, combining the constant terms on the r.h.s. of (2.17), (2.18),
(2.19) with the constant term in (2.9), we obtain, using again the eigenvalue
equation (2.10),

$\begin{split}\frac{N}{2}\widehat{V}(0)+\sum_{|p|<\kappa_{H}}\Big{[}p^{2}\eta_{p}^{2}+\widehat{V}(p/N)\eta_{p}&+\frac{1}{2N}\sum_{\begin{subarray}{c}r\in\Lambda^{*}:\\\
\ |p+r|>\kappa_{H}\end{subarray}}\widehat{V}(r/N)\eta_{p+r}\Big{]}\\\
&\simeq\frac{N}{2}\int V(x)\big{[}1+\frac{1}{N}\check{\eta}(x)\big{]}dx\simeq
4\pi\mathfrak{a}N,\end{split}$

which explains the constant term on the r.h.s. of (2.15). More details on the
proof of Lemma 2.3 can be found in [12, Sect. 7].

Cubic renormalization. Although the renormalized excitation Hamiltonian (2.15)
looks simpler to handle, compared with (2.9), it is not yet coercive enough to
prove Bose-Einstein condensation (it would be, if we assumed the interaction
potential $V$ to be small enough, because then we could control the r.h.s. of
(2.15) with kinetic and potential energy operators, which are positive; this
approach has been considered in [9]). The main problem appears to be the cubic
term on the r.h.s. of (2.15), which has been left unchanged by the action of
the Bogoliubov transformation $T_{\eta}$. To regularize the cubic term, we
need a second renormalization of the excitation Hamiltonian, this time through
the exponential of a cubic operator. We define

$A=\frac{1}{\sqrt{N}}\sum_{|r|>\kappa_{H},|v|<\kappa_{L}}\eta_{r}\big{[}b_{r+v}^{*}a_{-r}^{*}a_{v}-\text{h.c.}\big{]}$
(2.21)

with an additional cutoff $\kappa_{L}<\kappa_{H}$ and we introduce the unitary
operator $S=e^{A}$. With $S$, we construct the (twice) renormalized excitation
Hamiltonian $\mathcal{J}_{N}=S^{*}\mathcal{G}_{N}S$.

######  Lemma 2.4.

We have

$\begin{split}\mathcal{J}_{N}=\;&4\pi\mathfrak{a}(N-\mathcal{N}_{+})+4\pi\mathfrak{a}\frac{(N-\mathcal{N}_{+})}{N}\\\
&+\mathcal{K}+8\pi\mathfrak{a}\sum_{|p|\leq\kappa_{H}}a_{p}^{*}a_{p}\frac{(N-\mathcal{N}_{+})}{N}+4\pi\mathfrak{a}\sum_{|p|\leq\kappa_{H}}\big{(}b_{p}^{*}b_{-p}^{*}+b_{p}b_{-p}\big{)}\\\
&+\frac{8\pi\mathfrak{a}}{\sqrt{N}}\sum_{|p|\leq\kappa_{H},q\in\Lambda^{*}_{+}}\big{[}b_{p+q}^{*}a_{-p}^{*}a_{q}+\text{h.c.}\big{]}+\mathcal{V}_{N}+\mathcal{E}_{\mathcal{J}_{N}}\end{split}$
(2.22)

where $\mathcal{E}_{\mathcal{J}_{N}}$ satisfies
$\pm\mathcal{E}_{\mathcal{J}_{N}}\leq(1/4)(\mathcal{K}+\mathcal{V}_{N})+C$, if
the cutoffs $\kappa_{L}<\kappa_{H}$ are chosen appropriately.

Sketch of proof of Lemma 2.4. We briefly discuss the derivation of (2.22). In
contrast with the generalized Bogoliubov transformation $T_{\eta}$ (whose
action can be approximately computed using the formulas (2.13)), there is no
explicit expression for the action of $S$. For this reason, we compute the
renormalized excitation Hamiltonian $\mathcal{J}_{N}$ through a commutator
expansion having the form

$\mathcal{J}_{N}=e^{-A}\mathcal{G}_{N}e^{A}\simeq\mathcal{G}_{N}+[\mathcal{G}_{N},A]+\frac{1}{2}[[\mathcal{G}_{N},A],A]+\dots\,.$

Fortunately, it turns out that only the first and the second commutator
produce relevant contributions, everything else can be neglected. Being $A$
cubic in creation and annihilation operators, its commutator with the kinetic
energy operator, ie. $[\mathcal{K},A]$, is going to be cubic in creation and
annihilation operators (because of the canonical commutation relation
$[a_{p},a_{q}^{*}]=\delta_{pq}$ and of the similar relation satisfied by the
modified creation and annihilation operators). On the other hand, the
commutator with the potential energy operator, $[\mathcal{V}_{N},A]$ is
quintic in creation and annihilation operators. Some of the contributions to
this last commutator, however, are not in normal order. When we rearrange them
into normal order, we produce cubic terms. The important observation, then, is
that with the correct choice of the cubic operator $A$, the cubic
contributions emerging from the commutators of $A$ with kinetic and potential
energy operators combine with the cubic term, let us denote it by
$\mathcal{C}_{N}$, in $\mathcal{G}_{N}$ and produce the renormalized cubic
term appearing in (2.22). The algebra here is quite similar to the one
discussed in (2.20) for the cancellation in the off-diagonal quadratic terms
(this is the reason why the coefficients in the cubic operator (2.21) are the
same as the coefficients appearing in the Bogoliubov transformation
$T_{\eta}$; physically, this reflects the idea that correlations are just
produced by two-body scattering events).

While $[\mathcal{K},A]$ and $[\mathcal{V}_{N},A]$ combine with the cubic term
$\mathcal{C}_{N}$ in $\mathcal{G}_{N}$ to produce the renormalized cubic term
in (2.22), the commutator $[\mathcal{C}_{N},A]$ and the double commutator
$[[\mathcal{K},A],A]$ are quartic in creation and annihilation operators.
Similarly, the double commutator $[[\mathcal{V}_{N},A],A]$ is of order six (in
this heuristic argument, we do not distinguish between standard and modified
creation and annihilation operators). When we rearrange these terms in normal
order, we obtain new quadratic and constant terms, which renormalize quadratic
and constant terms on the r.h.s. of (2.15) to produce quadratic and constant
terms in (2.22). More details on the proof of Lemma 2.4 can be found in [12,
Sect. 8].

Proof of condensation. Observing the expression on the r.h.s. of (2.22), we
remark that $\mathcal{J}_{N}$ has essentially the same form as the original
excitation Hamiltonian (2.9), but with the singular interaction
$\widehat{V}(p/N)$ (decaying only for very large momenta of order $N$)
replaced by the regularized potential
$\widehat{\nu}_{p}=8\pi\mathfrak{a}\,\chi(|p|\leq\kappa_{H})$. The only
difference is that, in (2.22), the quartic term $\mathcal{V}_{N}$ still
depends on the original interaction. To complete the renormalization, we could
conjugate $\mathcal{J}_{N}$ with a last unitary operator, given this time by
the exponential of a quartic expression in creation and annihilation
operators; this strategy has been considered in [1].

In [12], on the other hand, a simpler strategy has been developed to prove
Bose-Einstein condensation directly from (2.22), without quartic
renormalization. This strategy is based on localization in the number of
particles, a technique first introduced by Lieb and Solovej in [45] (and then
adapted in [35] to a setting similar to the one we are considering here). The
idea is as follows. To prove a lower bound for $\mathcal{J}_{N}$ we can
discard in (2.22) the positive quartic term $\mathcal{V}_{N}$ (after using
some of it to control part of the error $\mathcal{E}_{\mathcal{J}_{N}}$).
Furthermore, we can insert, by hand, the quartic term

$\frac{1}{2N}\sum_{p,q\in\Lambda^{*}_{+},r\not=-p,-q}\widehat{\nu}_{r}\,a_{p+r}^{*}a_{q}^{*}a_{q+r}a_{p}$
(2.23)

associated with the regularized potential; this produces an error that can be
bounded (since the sum in (2.23) runs only over $|r|<\kappa_{H}$, with
$\kappa_{H}$ independent of $N$) by $C\mathcal{N}_{+}^{2}/N$. Comparing with
(2.9,) we find

$\mathcal{J}_{N}\geq
U_{N}H_{N}^{\text{ren}}U_{N}^{*}-\frac{C}{N}\mathcal{N}_{+}^{2}-\frac{1}{2}\mathcal{K}-C$
(2.24)

with the renormalized many-body Hamiltonian (here $\nu$ is the function on
$\Lambda$, with Fourier coefficients $\widehat{\nu}_{p}$ defined above)

$H_{N}^{\text{ren}}=\sum_{j=1}^{N}-\Delta_{x_{j}}+\frac{1}{N}\sum_{i<j}^{N}\nu(x_{i}-x_{j})\,.$
(2.25)

Using the positivity of $\widehat{\nu}$, we have

$\begin{split}0&\leq\int_{\Lambda^{2}}dxdy\,\nu(x-y)\Big{[}\sum_{j=1}^{N}\delta(x-x_{j})-N\Big{]}\Big{[}\sum_{i=1}^{N}\delta(y-x_{j})-N\Big{]}\\\
&=\sum_{i,j=1}^{N}\nu(x_{i}-x_{j})-N^{2}\widehat{\nu}_{0}=2\sum_{i<j}^{N}\nu(x_{i}-x_{j})+N\nu(0)-N^{2}\widehat{\nu}_{0}\end{split}$

which implies (with $\widehat{\nu}_{0}=4\pi\mathfrak{a}$ and $\nu(0)|\leq C$)
that

$H_{N}^{\text{ren}}\geq 4\pi\mathfrak{a}N+\sum_{j=1}^{N}(-\Delta_{x_{j}})-C$

and thus, from (2.24), that

$\mathcal{J}_{N}\geq
4\pi\mathfrak{a}N+\frac{1}{2}\mathcal{K}-\frac{C}{N}\mathcal{N}_{+}^{2}-C\,.$
(2.26)

In order to control the negative term, quadratic in $\mathcal{N}_{+}$, we
choose smooth $g,h:\mathbb{R}\to[0;1]$ with $g(s)=1$ for all $s\leq 1$,
$g(s)=0$ for $s>2$, such that $g^{2}+h^{2}=1$. For $\delta>0$ small enough, we
introduce the notation $g_{\delta}=g(\mathcal{N}_{+}/\delta N)$,
$h_{\delta}=h(\mathcal{N}_{+}/\delta N)$. Up to errors that are small in the
limit $N\to\infty$, we can approximate

$\mathcal{J}_{N}\simeq
g_{\delta}\mathcal{J}_{N}g_{\delta}+h_{\delta}\mathcal{J}_{N}h_{\delta}\,.$
(2.27)

On the range of $g_{\delta}$, we have $\mathcal{N}_{+}<2\delta N$. From
(2.26), we find, for $\delta>0$ small enough,

$\begin{split}g_{\delta}\mathcal{J}_{N}g_{\delta}&\geq
g_{\delta}\big{[}4\pi\mathfrak{a}N+\frac{1}{2}\mathcal{K}-C\delta\mathcal{N}_{+}-C\big{]}g_{\delta}\\\
&\geq\big{[}4\pi\mathfrak{a}N+\frac{1}{4}\mathcal{K}-C\big{]}g_{\delta}^{2}\geq\big{[}4\pi\mathfrak{a}N+c\mathcal{N}_{+}-C\big{]}g_{\delta}^{2}\end{split}$
(2.28)

for a $c>0$ sufficiently small. On the range of $h_{\delta}$, on the other
hand, we have $\mathcal{N}_{+}>\delta N$; as a consequence, here we cannot
have complete Bose-Einstein condensation. Applying (1.12), in the form proven
in [39, 48], we conclude by contradiction that there exists $c>0$ such that

$h_{\delta}\mathcal{J}_{N}h_{\delta}\geq(4\pi\mathfrak{a}+c)Nh_{\delta}^{2}\geq\big{[}4\pi\mathfrak{a}N+c\,\mathcal{N}_{+}\big{]}h_{\delta}^{2}.$

From (2.27) and (2.28), this implies that, for $c>0$ small enough,

$\mathcal{J}_{N}\geq 4\pi\mathfrak{a}N+c\,\mathcal{N}_{+}-C\,.$

Suppose now that $\psi_{N}\in L^{2}_{s}(\Lambda^{N})$ is a sequence of
approximate ground states, satisfying

$\langle\psi_{N},H_{N}\psi_{N}\rangle\leq 4\pi\mathfrak{a}N+\zeta$ (2.29)

and let $\xi_{N}=ST_{\eta}U_{N}\psi_{N}$ denote the corresponding
(renormalized) excitation vectors. Then, using (2.26) and the inequality
$\mathcal{N}_{+}\leq C\mathcal{K}$ (due to the spectral gap of the kinetic
energy), we find

$\begin{split}\langle\xi_{N},\mathcal{N}_{+}\xi_{N}\rangle&\leq
C\langle\xi_{N},\mathcal{K}\,\xi_{N}\rangle\\\ &\leq
C\langle\xi_{N},(\mathcal{M}_{N}-4\pi\mathfrak{a}N)\xi_{N}\rangle+C\\\
&=C\langle\psi_{N},(H_{N}-4\pi\mathfrak{a})\psi_{N}\rangle+C\leq
C(\zeta+1)\,.\end{split}$

Since conjugation with $T_{\eta}$ and $S$ can only increase the expected
number of excitations by a constant factor, we conclude that

$C(\zeta+1)\geq\langle
U_{N}\psi_{N},\mathcal{N}_{+}U_{N}\psi_{N}\rangle=N-\langle\psi_{N},a_{0}^{*}a_{0}\psi_{N}\rangle=N-\langle\varphi_{0},\gamma_{N}\varphi_{0}\rangle$
(2.30)

which implies Bose-Einstein condensation with optimal rate, as stated in
(2.4).

It is interesting to compare the renormalization procedure based on
conjugation with unitary operators that we used to show (2.30) with Dyson’s
Lemma (Lemma 1.1). Both techniques can be used to replace a singular
interaction (large size, small range) with a regularized potential (smaller
size, larger range) having the same scattering length. A clear advantage of
Dyson’s lemma is its simplicity. Also the renormalization procedure has some
advantages, which play an important role in the proof of (2.30). First of all,
with the renormalization procedure, there is almost no loss of kinetic energy
(some kinetic energy is needed to bound the error
$\mathcal{E}_{\mathcal{J}_{N}}$; what is important, however, is that although
we only use a little bit of kinetic energy, we nevertheless construct the full
renormalized potential, producing the energy $4\pi\mathfrak{a}N$ on the r.h.s.
of (2.26)). Another advantage of the renormalization procedure is that it
leads to a genuine many-body interaction, as the one appearing in (2.25).
Dyson’s lemma, on the other hand, works on the one-particle level and
generates only an interaction between nearest neighbours.

A-priori bounds. Eq. (2.30) gives optimal control on the number of excitations
of the Bose-Einstein condensate in approximate ground states satisfying
(2.29). In turn, inserting in (2.22), this implies a bound for the energy of
the excitations. If we strengthen (2.29), assuming that

$\psi_{N}=\chi(H_{N}\leq 4\pi\mathfrak{a}N+\zeta)\psi_{N}\,,$ (2.31)

we can also establish a-priori estimates for the expectation of higher powers
of the number of particles operator and of their product with kinetic and
potential energy operators. More precisely, for a sequence $\psi_{N}\in
L^{2}_{s}(\Lambda^{N})$ satisfying (2.31), it is possible to show that, for
every $k\in\mathbb{N}$, there exists $C>0$ such that the excitation vector
$\xi_{N}=T_{\eta}U_{N}\psi_{N}$ satisfies

$\big{\langle}\xi_{N},(\mathcal{K}+\mathcal{V}_{N}+1)(\mathcal{N}_{+}+1)^{k}\xi_{N}\rangle\leq
C(\zeta+1)^{k+1}\,.$ (2.32)

In fact, the same estimate holds true if we replace $\xi_{N}$ by
$S\xi_{N}=ST_{\eta}U_{N}\psi_{N}$, since cubic renormalization can only
increase number and energy of excitations by constant factors, independent of
$N$ (on the contrary, conjugation with $T_{\eta}$ changes the energy by order
$N$). The proof of (2.32) can be found in [11, Prop. 4.1]; we skip here the
details.

Ground state energy and excitation spectrum. With the help of the strong
a-priori bounds (2.32), we can go back to the renormalized excitation
Hamiltonian (2.15), defined after (2.12). As we did in (2.15), we can compute
again $\mathcal{G}_{N}$, this time keeping track of all terms, whose
expectation does not vanish in the limit $N\to\infty$ (in (2.15), on the other
hand, order one contributions are included in the error $\mathcal{E}$). We
find

$\mathcal{G}_{N}=C_{N}+Q_{N}+\mathcal{C}_{N}+\mathcal{V}_{N}+\delta_{\mathcal{G}_{N}}$

where $C_{N}$ is a constant term, $Q_{N}$ is quadratic, $\mathcal{C}_{N}$ is
cubic in creation and annihilation operators, and where
$\delta_{\mathcal{G}_{N}}$ is an error term, satisfying

$\pm\delta_{\mathcal{G}_{N}}\leq\frac{C}{\sqrt{N}}(\mathcal{H}_{N}+1)(\mathcal{N}_{+}+1)^{2}\,.$
(2.33)

From the a-priori estimates we know that the contribution of the error
$\delta_{\mathcal{G}_{N}}$ is negligible, in the limit of large $N$, on low-
energy states.

As already discussed in the proof of condensation, the contribution of the
cubic term $\mathcal{C}_{N}$ is relevant, it cannot be neglected. To get rid
of the cubic term, we proceed as in the proof of condensation and perform a
cubic renormalization. We define
$\mathcal{J}^{\prime}_{N}=e^{-A^{\prime}}\mathcal{G}_{N}e^{A^{\prime}}$, for a
cubic operator $A^{\prime}$, similar to (2.21) (but not identical; for
technical reasons, we find it convenient to choose $A^{\prime}$ slightly
different from the operator $A$ that we used in the proof of condensation).
Computing $\mathcal{J}^{\prime}_{N}$ up to errors that vanish in the limit
$N\to\infty$, we find

$\mathcal{J}^{\prime}_{N}=C^{\prime}_{N}+Q^{\prime}_{N}+\mathcal{V}_{N}+\delta_{\mathcal{J}^{\prime}_{N}}$
(2.34)

where $C^{\prime}_{N}$ and $Q^{\prime}_{N}$ are new (renormalized) constant
and cubic terms and where, similarly to (2.33),

$\pm\delta_{\mathcal{J}^{\prime}_{N}}\leq\frac{C}{N^{1/4}}(\mathcal{H}_{N}+1)(\mathcal{N}_{+}+1)^{2}\,.$
(2.35)

A part from the positive quartic term $\mathcal{V}_{N}$, the Hamiltonian
$\mathcal{J}^{\prime}_{N}$ is essentially quadratic in creation and
annihilation operators. Thus, it can be (approximately) diagonalized by a
(generalized) Bogoliubov transformation $T_{\tau}$ (defined similarly as in
(2.12), but of course with appropriate coefficients $\tau_{p}$). We define the
final diagonalized excitation Hamiltonian
$\mathcal{M}_{N}=T_{\tau}^{*}\mathcal{J}^{\prime}_{N}T_{\tau}$. We find

$\begin{split}\mathcal{M}_{N}=\;&4\pi\mathfrak{a}(N-1)+e_{\Lambda}\mathfrak{a}^{2}\\\
&-\frac{1}{2}\sum_{p\in\Lambda^{*}_{+}}\left[p^{2}+8\pi\mathfrak{a}-\sqrt{|p|^{4}+16\pi\mathfrak{a}p^{2}}-\frac{(8\pi\mathfrak{a})^{2}}{2p^{2}}\right]+\sum_{p\in\Lambda^{*}_{+}}\sqrt{|p|^{4}+16\mathfrak{a}p^{2}}\,a_{p}^{*}a_{p}\\\
&+\mathcal{V}_{N}+\delta_{\mathcal{M}_{N}}\end{split}$ (2.36)

where the error operator $\delta_{\mathcal{M}_{N}}$ satisfies a bound similar
to (2.35). To prove (2.5) and (2.7) we need to show matching lower and upper
bounds. For the lower bounds, the quartic interaction $\mathcal{V}_{N}$ is
positive and can be neglected. For the upper bounds, it turns out that the
expectation of $\mathcal{V}_{N}$ is small, on the appropriate trial states
(determined by the quadratic part of (2.36)). Hence, in (2.36) we can
completely forget about $\mathcal{V}_{N}$. Therefore, the estimates (2.5),
(2.7) follow easily from (2.36) and from the observation that the spectrum of
the harmonic oscillator $a_{p}^{*}a_{p}$ consists of all natural numbers. This
concludes the sketch of the proof of Theorem 2.1.

Acknowledgements. The author gratefully acknowledges support from the European
Research Council through the ERC Advanced Grant CLaQS. Additionally, he
acknowledges partial support from the NCCR SwissMAP and from the Swiss
National Science Foundation through the Grant “Dynamical and energetic
properties of Bose-Einstein condensates”.

## References

  * [1] A. Adhikari, C. Brennecke, B. Schlein. Bose-Einstein condensation beyond the Gross-Pitaevskii regime. Ann. H. Poincaré 22 (2021), 1163–1233.
  * [2] A. Aftalion. Vortex patterns in Bose Einstein condensates. Perspectives in nonlinear partial differential equations. Contemp. Math. 446, Amer. Math. Soc., Providence, RI, 2007, pp. 1–18
  * [3] M. Aizenman, E. H. Lieb, R. Seiringer, J. P. Solovej, and J. Yngvason. Bose-Einstein quantum phase transition in an optical lattice model. Phys. Rev. A 70 (2004), 023612.
  * [4] T. Balaban, J. Feldman, H. Knörrer, E. Trubowitz. Complex Bosonic Many-Body Models: Overview of the Small Field Parabolic Flow. Ann. Henri Poincaré  18 (2017), 2873–2903.
  * [5] G. Basti. S. Cenatiempo, B. Schlein. A new second order upper bound for the ground state energy of dilute Bose gases. Forum Math. Sigma 9 (2021), Paper No. e74.
  * [6] N. Benedikter, G. de Oliveira and B. Schlein. Quantitative derivation of the Gross-Pitaevskii equation. _Comm. Pure Appl. Math._ (2014).
  * [7] N. Benedikter, M. Porta and B. Schlein. Effective Equations from Quantum Dynamics. Springer Briefs in Mathematical Physics 7, Springer, 2016.
  * [8] F. Béthuel, H. Brézis and F. Hélein. Ginzburg-Landau vortices. Progress in Nonlinear Differential Equations and their Applications, 13, Birkhäuser Boston, Inc., Boston, MA, 1994.
  * [9] C. Boccato, C. Brennecke, S. Cenatiempo, B. Schlein. Complete Bose-Einstein condensation in the Gross-Pitaevskii regime. Commun. Math. Phys. 359 (2018), no. 3, 975-1026.
  * [10] C. Boccato, C. Brennecke, S. Cenatiempo, B. Schlein. The excitation spectrum of Bose gases interacting through singular potentials. J. Eur. Math. Soc. 22 (2020), no. 7, 2331–2403.
  * [11] C. Boccato, C. Brennecke, S. Cenatiempo, B. Schlein. Bogoliubov Theory in the Gross-Pitaevskii limit. Acta Mathematica 222(2) (2019), 219-335.
  * [12] C. Boccato, C. Brennecke, S. Cenatiempo, B. Schlein. Optimal rate for Bose-Einstein condensation in the Gross-Pitaevskii regime. Commun. Math. Phys 376 (2020), 1311–1395 .
  * [13] S. N. Bose. “Plancks Gesetz und Lichtquantenhypothese”. Zeitschrift für Physik 26 (1924), no. 1, 178–181.
  * [14] C. Brennecke, B. Schlein. Gross-Pitaevskii dynamics for Bose-Einstein condensates. Anal. PDE 12 (2019), no. 6, 1513-1596.
  * [15] C. Brennecke, B. Schlein, S. Schraven. Bose-Einstein Condensation with Optimal Rate for Trapped Bosons in the Gross-Pitaevskii Regime. Math. Phys. Anal. Geom. 25 (2022), 12.
  * [16] C. Brennecke, B. Schlein, S. Schraven. Bogoliubov Theory for Trapped Bosons in the Gross-Pitaevskii Regime. Ann. Henri Poincaré 23 (2022), no. 5, 1583–1658.
  * [17] E. A. Cornell, C. E. Wieman. Bose-Einstein condensation in a dilute gas, the first 70 years and some recent experiments. Rev. Mod. Phys. 74 (2002), pp. 875–893.
  * [18] M. Correggi, F. PInsker, N. Rougerie and J. Yngvason. Rotating superfluids in anharmonic traps: From vortex lattices to giant vortices. Phys. Rev. A 84 (2011), 053614.
  * [19] A. Deuchert, R. Seiringer. Gross-Pitaevskii Limit of a Homogeneous Bose Gas at Positive Temperature. Arch. Rat. Mech. An. 236 (2020), no. 3, 1217.
  * [20] A. Deuchert, R. Seiringer, J. Yngvason. Bose-Einstein Condensation in a Dilute, Trapped Gas at Positive Temperature. Comm. Math. Phys. 368 (2019), 723.
  * [21] F.J. Dyson. Ground-State Energy of a Hard-Sphere Gas. Phys. Rev. 106 (1957), 20–26.

  * [22] F.J. Dyson, E.H. Lieb, B. Simon. Phase Transitions in Quantum Spin Systems with Isotropic and Nonisotropic Interactions. J. Stat. Phys. 18 (1978), 335–383.
  * [23] A. Einstein. Quantentheorie des einatomigen idealen Gases. Sitzungsberichte der Preussische Akademie der Wissenschaften, 1924 (261-267) and 1925 (3-14).
  * [24] L. Erdős, B. Schlein and H.-T. Yau.  Derivation of the cubic nonlinear Schrödinger equation from quantum dynamics of many-body systems.  _Inv. Math._ 167 (2006), 515–614.
  * [25] L. Erdős, B. Schlein and H.-T. Yau.  Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate,  _Ann. of Math. (2)_ 172 (2010), no. 1, 291–370.
  * [26] L. Erdős, B. Schlein and H.-T. Yau.  Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential.  _J. Amer. Math. Soc._ 22 (2009), 1099–1156.
  * [27] S. Fournais. Length scales for BEC in the dilute Bose gas. In Partial differential equations, spectral theory, and mathematical physics – the Ari Laptev anniversary volume, pp. 115–133, EMS Ser. Congr. Rep., EMS Press, Berlin, 2021.
  * [28] S. Fournais, J.P. Solovej. The energy of dilute Bose gases. Ann. Math. 192(3) (2020), 893–976.
  * [29] S. Fournais, J.P. Solovej. The energy of dilute Bose gases II: The general case. Preprint arXiv:2108.12022.
  * [30] J. Fröhlich, B. Simon, T. Spencer. Infrared Bounds, Phase Transitions and Continuous Symmetry Breaking. Comm. Math. Phys. 50 (1976), 79–95.
  * [31] C. Hainzl. Another proof of BEC in the GP-limit. J. Math. Phys. 62 (2021), no. 5, Paper No. 051901.
  * [32] C. Hainzl, B. Schlein, A. Triay. Bogoliubov theory in the Gross-Pitaevskii limit: a simplified approach. Preprint arXiv:2203.03440.
  * [33] T. Kennedy, E. H. Lieb, and B. S. Shastry. The XY model has long-range order for all spins and all dimensions greater than one. Phys. Rev. Lett. 61 (1988), 2582–2584.
  * [34] W. Ketterle. When atoms behave as waves: Bose-Einstein condensation and the atom laser. Rev. Mod. Phys. 74 (2002), pp. 1131–1151.
  * [35] M. Lewin, P. T. Nam, S. Serfaty, J.P. Solovej. Bogoliubov spectrum of interacting Bose gases.  _Comm. Pure Appl. Math._ 68 (2014), no. 3, 413 - 471
  * [36] M. Lewin and R. Seiringer. Strongly correlated phases in rapidly rotating Bose gases. J. Stat. Phys. 137 (2009), 1040.
  * [37] E. H. Lieb and R. Seiringer.  Proof of Bose-Einstein condensation for dilute trapped gases.  _Phys. Rev. Lett._ 88 (2002), 170409.
  * [38] E. H. Lieb, R. Seiringer, J. P. Solovej, J. Yngvason. The mathematics of the Bose gas and its condensation. Oberwolfach Seminars 34 (2005), Birkhäuser Basel.
  * [39] E. H. Lieb and R. Seiringer.  Derivation of the Gross-Pitaevskii equation for rotating Bose gases.  _Comm. Math. Phys._ 264:2 (2006), 505-537.
  * [40] E. H. Lieb, R. Seiringer, and J. Yngvason.  Bosons in a trap: A rigorous derivation of the Gross-Pitaevskii energy functional.  _Phys. Rev. A_ 61 (2000), 043602.
  * [41] E. H. Lieb, R. Seiringer, and J. Yngvason. A Rigorous Derivation of the Gross-Pitaevskii Energy Functional for a Two-Dimensional Bose Gas. Comm. Math. Phys. 224 (2001), 17.
  * [42] E.H. Lieb, R. Seiringer, J. Yngvason. Superfluidity in dilute trapped Bose gases. Phys. Rev. B 66 (2002), 134529.
  * [43] E. H. Lieb, R. Seiringer, and J. Yngvason. Poincaré inequalities in punctured domains. Ann. Math 158 (2003), 1067–1080.
  * [44] E. H. Lieb, R. Seiringer, and J. Yngvason. Yrast line of a rapidly rotating Bose gas: Gross-Pitaevskii regime. Phys. Rev. A 79 (2009), 063626.
  * [45] E. H. Lieb, J. P. Solovej. Ground state energy of the one-component charged Bose gas. Comm. Math. Phys. 217 (2001), 127–163. Errata: Comm. Math. Phys. 225 (2002), 219-221.
  * [46] E. H. Lieb, J. Yngvason. Ground State Energy of the low density Bose Gas. Phys. Rev. Lett. 80 (1998), 2504–2507.
  * [47] P. T. Nam, M. Napiórkowski, J. Ricaud, A. Triay. Optimal rate of condensation for trapped bosons in the Gross–Pitaevskii regime. Preprint arXiv:2001.04364.
  * [48] P. T. Nam, N. Rougerie, R. Seiringer. Ground states of large bosonic systems: The Gross-Pitaevskii limit revisited. _Analysis and PDE._ 9 (2016), no. 2, 459-485
  * [49] P. T. Nam, A. Triay. Bogoliubov excitation spectrum of trapped Bose gases in the Gross-Pitaevskii regime. Preprint arXiv:2106.11949.
  * [50] W. D. Phillips. Laser cooling and trapping of neutral atoms. Rev. Mod. Phys. 70 (1998), no. 3.
  * [51] P. Pickl. Derivation of the time dependent Gross–Pitaevskii equation with external fields. Rev. Math. Phys. 27, No. 01, 1550003 (2015)
  * [52] N. Rougerie. Scaling limits of bosonic ground states, from many-body to nonlinear Schrödinger. EMS Surveys in Mathematical Sciences 7 (2020), 253–408.
  * [53] E. Sandier and S. Serfaty. Vortices in the magnetic Ginzburg-Landau model. Progress in Nonlinear Differential Equations and their Applications, 70, Birkhäuser Boston, Inc., Boston, MA, 2007.
  * [54] B. Schlein. Derivation of effective evolution equations from microscopic quantum dynamics. Evolution Equations, CMI Summer School, ETH 2008. Clay Mathematics Proceedings, 17, 2013. Edited by D. Ellwood, I. Rodnianski, G. Staffilani, J. Wunsch. 511–572.
  * [55] B. Schlein. Derivation of effective evolution equations from many body quantum dynamics. XVIth International Congress on Mathematical Physics. World Sci. Publ., Hackensack, NJ, 2010, 406–416.
  * [56] R. Seiringer. Gross-Pitaevskii Theory of the Rotating Bose Gas. Commun. Math. Phys. 229 (2002), 491–509.
  * [57] R. Seiringer. Dilute, trapped Bose gases and Bose-Einstein Condensation. Large Coulomb Systems, H. S. E. J. Derezinski, ed., vol. 695 of Lecture Notes in Physics, Springer, 2006, 249–274.
  * [58] R. Seiringer. Hot topics in cold gases. Proceedings of the XVI-th International Congress on Mathematical Physics, P. Exner, ed., World Scientific, 2010, 231–245.
  * [59] R. Seiringer. The excitation spectrum for weakly interacting bosons. Comm. Math. Phys. 306 (2011), 565-578.
  * [60] I. M. Sigal. Magnetic Vortices, Abrikosov Lattices and Automorphic Functions. In Mathematical and computational modeling, pp. 19–58, Pure Appl. Math. (Hoboken), Wiley, Hoboken, NJ, 2015.
  * [61] H.-T. Yau, J. Yin. The second order upper bound for the ground state energy of a Bose gas. J. Stat. Phys. 136(3) (2009), 453–503.
  * [62] J. Yngvason. The interacting Bose gas: A continuing challenge. Phys. Particles Nuclei 41 (2010), 880–884.