Large deviations for the ground state of weakly interacting Bose gases

We consider the ground state of a Bose gas of N particles on the three-dimensional unit torus in the mean-field regime that is known to exhibit Bose-Einstein condensation. Bounded one-particle operators with law given through the interacting Bose gas' ground state correspond to dependent random variables due to the bosons' correlation. We prove that in the limit $N \rightarrow \infty$ bounded one-particle operators with law given by the ground state satisfy large deviation estimates. We derive a lower and an upper bound on the rate function that match up to second order and that are characterized by quantum fluctuations around the condensate.


INTRODUCTION
We consider N bosons on the three dimensional unit torus Λ = [0, 1] 3 in the mean-field regime described by the Hamiltonian acting on L 2 s Λ N , the symmetric subspace of L 2 Λ N .We consider two-particles interaction potentials with Fourier transform v given by v(x) = p∈Λ * v(p)e ip•x for Λ * = 2πZ 3 with v ≥ 0, v ∈ ℓ 1 (Λ * ) . (1.2) At zero temperature the bosons relax to the unique ground state ψ N of H N realizing ψ, H N ψ = ψ N , H N ψ N . (1. 3) The ground state ψ N exhibit Bose-Einstein condensation, i.e. a macroscopic fraction of the N particles occupies the same quantum state, called the condensate.Mathematically, ψ N is said to satisfy the property of Bose-Einstein condensation if its corresponding one-particle reduced density given by γ (k) for k = 1 converges in trace norm to γ (1) where ϕ 0 ∈ L 2 (Λ) denotes the condensate wave function.In fact the convergence (1.5) holds true not only for the one-but also in for general k-particle reduced density densities.However due to particle's correlation, the ground state ψ N is not a purely factorized state of the condensate's wave function.
Both, the computation of the ground state energy (1.3) and the ground state's property of BEC (1.5) and beyond are widely studied in the literature (see for example [3,10,11,13,14,15,16,20,24]).In fact, [20] proves besides BEC that the Bose gas' excitation spectrum is well described by Bogoliubov theory.Consequently, the fluctuations around the condensate, namely the particles orthogonal to the condensate can be effectively described as a quasi-free state (namely a Gaussian Date: June 21, 2023. 1 quantum state) on an appropriate Fock space.This characterization of the condensates' excitations will be important for our analysis.
1.1.Probabilistic approach.Recently the characterization of Bose-Einstein condensation through probabilistic concepts became of interest.In fact, the property of Bose-Einstein condensation (1.5) implies a law of large numbers for bounded one particle operators [1].To be more precise, let O denote a bounded one particle operator on L 2 (R 3 ) for which we define the N -particle operator O (i)  by i.e. as operator acting as O on the i-th particle and as identity elsewhere.We consider O (i) as a random variable with law given by where χ A denotes the characteristic function of A ⊂ R. We remark that factorized states lead to i.i.d.random variables in this picture [19] and thus a law of large numbers and a large deviation principle hold true from basic theorems of probability theory.Random variables with law given by the ground state ψ N (known to not be a factorized state) of H N , satisfy a law of large numbers, too (though they are not independent random variables).To be more precise, the averaged centered (w.r.t. to the condensate's expectation value ϕ 0 , Oϕ 0 ) sum with O (i) given by (1.6) satisfies for any δ > 0 lim The law of large numbers, in fact, is a consequence of the property of Bose-Einstein condensation [1,19] namely of the trace norm convergence of the one-and two-particle reduced density matrix.
Here, we are interested in the precise decay of the probability distribution (1.9) in probability theory described through the rate function (1.10) if the limit exists.In case of i.i.d.random variables (i.e.factorized states ϕ ⊗N ) Cramer's theorem shows that the rate functions exists and is given in terms of the Legendre-Fenchel transform through where Λ ϕ ⊗N (λ) equals for i.i.d.random variables the logarithmic moment generating function Λ ϕ ⊗N (λ) = log ϕ, e λ(O (1) − ϕ,Oϕ ) ϕ . (1.12) In our main theorem we show that for the ground state ψ N of H N , known to be not factorized due to particles' correlation, still large deviation estimates hold true.
Then, there exist C 1 , C 2 > 0 (independent of O) such that (i) for all 0 ≤ x ≤ 1/(C 1 |||O|||) lim sup (1.17) . (1.18) We remark that for sufficiently small x ≤ min{ f 4 ℓ 2 (Λ * + ) /(C 2 |||O||| 3 ), 1/C 1 |||O|||}, Theorem 1.1 characterizes the rate function up to second order.Namely Theorem 1.1 shows that in the regime of large deviations, i.e. x = O(1), we have Regime of large deviations.The present result in Theorem 1.1 provides a first characterization of the regime of large deviations (i.e.x = O(1)) for fluctuations around the condensate of bounded one-particle operators in the ground state.We remark that the variance f ℓ 2 (Λ * + ) differs from the variance of factorized state ϕ ⊗sN 0 and is, in particular, fully characterized by the ground state's Bogoliubov approximation (for more details see (2.15) and subsequent discussions resp.Lemma 4.3 in Section 4) representing the particles' correlation.
Up to now, results in the regime of large deviations are available for the dynamics in the meanfield regime only.For factorized initial data, the rate function characterizing the fluctuations of bounded one-particles operators around the condensate's Hartree dynamics, were proven to satisfy a upper bound of the form of Theorem 1.1 (i) first [12], and a lower bound of the form of Theorem 1.1 (ii) later [22].
Regime of standard deviations.In the regime of standard deviations, i.e. x = O(N −1/2 ), Theorem 1.1 furthermore implies lim thus a central limit theorem where the limiting Gaussian random variable's variance is given by f ℓ 2 (Λ * + ) agreeing with earlier results [21].In fact [21] proves a central limit theorem for fluctuations around the condensate for the ground state in the Gross-Pitaevski regime.The Gross-Pitaevski scaling regime considers instead of v, the N -dependent two-body interaction potential v β N = N 3β v N (N β •) with β = 1 (for more details and recent progress on results in the Gross-Pitaevski regime see [6,5,7,9,17]).However, (1.20) follows from adapting the analysis in [21] to the mathematically easier accessible mean-field scaling regime (corresponding to β = 0).
Recently, [2] refined the characterization of the regime of standard deviations and derived an edge-worth expansion.
Central limit theorems were proven first for the mean-field dynamics of Bose gases.Fluctuations of bounded one-particle operators around the Hartree equations were proven to have Gaussian behavior [1], though they do not correspond to i.i.d.random variables.These results were later generalized to multivariate central limit theorem [8], dependent random variables (i.e.k-particle operators) [19] and singular particles interaction in the intermediate scaling regime (for v β N with β ∈ (0, 1))) [20].Theorem 1.1 follows (similarly to [12,22]) from estimates on the logarithmic moment generating function given in the following theorem.
Idea of the proof.The rest of this paper is dedicated to the proof of Theorem 1.2, thus on estimates on the moment generating function.We recall that for the result of Theorem 1.2 we are interested in the leading order of the exponential of the moment generating function that is o(N λ 2 ) in the limit of small λ and large N .We will show that for the leading order fluctuations around the condensate are crucial that we describe by the excitation vector U N ψ N (for a precise definition of the unitary map U N to the Fock space of excitations see (2.2)).As a first step we prove that we can replace the moment generating function E ψ N e λO N with the expectation value paying a price exponentially O(N λ 3 ) and thus subleading (see Lemma 4.1).Here we introduced the notation where a, a * denote the creation and annihilation operators on the bosonic Fock space and N + the number of excitations (for a precise definition see Section 2).Note that the operator φ + , in contrast to its asymptotic limit for N → ∞, does not increase the number of excitations which will be crucial for our analysis.We remark that the excitation vector U N ψ N is the ground state of the excitation Hamiltonian Its quadratic (in modified creation and annihilation operators) part Q and the remainder term R N are given in (2.15) resp.(2.16).In the second step we show that replacing U N ψ N with the ground state ψ Q of the quadratic operator Q leads to an error exponentially O(N λ 3 ) ( and thus subleading).While the first step follows strategies presented in [12] on the dynamical problem, the second step uses novel techniques.The proof is based on the Heymann-Feynmann theorem and Gronwall's inequality applied for s ∈ [0, 1] to the family of ground states ψ G N (s) that corresponds to the Hamiltonians G N (s) = Q + sR N and thus interpolates between the excitation vector U N ψ N and ψ Q (for more details see Proposition 2.1 and Lemma 4.2).We remark that the ground state of operators quadratic in standard creation and annihilation operators is well known and given by a quasi-free state, i.e. by where µ is given by (1.16) and vacuum vector Ω.Note that the operator Q is quadratic in modified creation and annihilation operators.However, we will prove that its ground state ψ Q is approximately given by a generalized quasi-free state, i.e. by (1.28) A crucial property of a Bogoliubov transform (1.27) is that its action on creation and annihilation operators is explicitly known.In particular, we have for the asymptotic limit of φ + that Though the explicit action of the generalized Bogoliubov transform (1.28) on the operator φ + is not known we show in the third step that we still have with an error exponentially O(N λ 3 ).This argument will be based again on the Hellmann-Feynmann theorem together with Gronwall's inequality applied to the family of ground states where D is a quadratic, diagonal operator.Thus Q(s) interpolates between the ground state e B(µ) ψ Q and the vacuum vector (see Lemma 4.3).
In the last step we then compute the remaining expectation value Ω, e λφ + (f )/2 e λκN + e λφ + (f )/2 Ω (1.31) with f given by (1.15).A comparison with the asymptotic limit φ + shows that the exponential of λN + contributes exponentially O(N λ 3 ), and thus subleading, leading to Theorem 1.2.For the true operator φ + this holds still true in the limit N → ∞ and follows from arguments given in [12] (see Lemma 4.4).
Structure of the paper.The paper is structured as follows: In Section 2 we introduce the description of the fluctuations (called excitations) around the condensate in the Fock space of excitations.In particular, we prove properties of the excitations' Hamiltonian G N and the quadratic approximation Q and their corresponding ground states (see Propositions 2.1, 2.2).In Section 3 we recall preliminary results from [12,22] and prove further auxiliary Lemmas (in particular for generalized Bogoliubov transforms (1.28)) that we will use later for the proof of Theorem 1.2 in Section 4.

FLUCTUATIONS AROUND THE CONDENSATE
2.1.Fock space of excitations.On the unit torus the condensate wave function ϕ 0 is given by the constant function.To study the fluctuations around the condensate, we need to factor out the condensates contributions.For this we use an observation from [13] that any N -particle wave function ψ N ∈ L 2 (Λ N ) can be decomposed as where the excitation vectors η j are elements of L 2 ⊥ϕ 0 (Λ j ), the orthogonal complement in L 2 (Λ j ) of the condensate wave function ϕ 0 and ⊗ s denotes the symmetric tensor product.Furthermore, we define the unitary mapping any N -particle wave function ψ N onto its excitation vector {η 1 , . . ., η N } that is an element of the Fock space of excitations A crucial property of elements of the Fock space of excitations ξ N ∈ F ≤N ⊥ϕ 0 is that the number of particles operator N = p∈Λ * a * p a p is bounded, i.e. ξ N , N ξ N ≤ N ξ N 2 .Here we introduced the standard creation and annihilation operators a * p , a p in momentum space defined through the following relation by the well known creation and annihilation operators in position space ǎ(f ), ǎ * (f ) that satisfy the canonical commutation relations a * p , a q = δ p,q , and [a p , a q ] = a * p , a * q = 0 . (2.5) Contrarily, on the full bosonic Fock space built over L 2 (Λ j ) (instead of L 2 ⊥ϕ 0 (Λ j )) and given by the number of particles N = p∈Λ a * p a p is an unbounded operator.For our analysis it will be useful to work on the Fock space of excitations that is equipped with modified creation and annihilation operators b * p , b q that leave (in contrast to the standard ones a * p , a q ) F ≤N ⊥ϕ 0 invariant and were first introduced in [4].They are given by We remark that in the limit of N → ∞, the commutation relations of b * p , b q agree with the canonical commutation relations (2.5).However the corrections that are O(N −1 ) lead to difficulties in the analysis later.
The operator b * p , b q arise from the unitary U N applied for p, q = 0 to products of creation and annihilation operators, namely Moreover, U N satisfies the property (2.12) If ψ N denotes the ground state of H N , then the excitation vector U N ψ N =: ψ G N denotes the ground state of the excitation Hamiltonian that can be explicitly computed using the properties of the unitary (2.10), (2.11) and is of the form where Q denotes an operator quadratic in (standard) creation and annihilation operators and is given with the notation Λ * + = Λ + \ {0} by whereas the remainder terms collected in R N and given by will be shown to contribute to our analysis sub-leading only.In fact, in the proof of Theorem 1.2 in Section 4 it turns out that Q resp.its corresponding ground state ψ Q is approximately given by with µ p given by (1.16),i.e.ψ Q is a generalized Bogoliubov transform e B(µ) applied to the vacuum Ω, fully determines the variance (i.e.f 2 ℓ 2 (Λ * + ) in Theorem 1.1).We remark that the approximation of G N (s) by Q is often referred to as Bogoliubov approximation.
Furthermore, we introduce the family of Hamiltonians {G N (s)} s∈[0,1] given by interpolating between the excitation Hamiltonian G N and its quadratic approximation Q.In the following Proposition we collect useful properties of {G N (s)} s∈[0,1] .For this we introduce the following notation for the particles' kinetic energy for k = 1, 2 and the spectrum of the Hamiltonian G N (s) has a spectral gap above the ground state E N (s) independent of s, N .Moreover, there exists C > 0 such that for any Fock space vector ξ ∈ F ≤N ⊥ϕ 0 we have Proof.The proof uses well known ideas and techniques introduced to prove results on the properties of G N and its corresponding ground state ψ G N showing that the remainder R N contributes subleading only (see for example [13,15,20]).Since G N (s) differs from G N by a multiple of the remainder only, these techniques apply for G N (s) as we shall show in the following.
The strategy is as follows: First we show that G N (s) is bounded from below by a multiple of N + − C yielding the estimate (2.64) for k = 1 and with further arguments for k = 2, too.Then the remainder R N can be proven to be sub-leading, and the existence of a spectral gap of the spectrum of G N (s) independent of N, s follows from the spectral properties of Q. Finally we prove (2.65) from the previously proven properties.
Proof of lower bound for G N (s).First we shall prove that there exists constants (2.22) To this end, we recall that by definition (2.18) we have Thus assuming that there exists sufficiently small ε 1 > 0 with then (2.22) follows from (2.18) and (2.23).We are left with proving (2.24).For this we use that the contribution of R N quartic in creation and annihilation operator is non-negative, i.e. that we can write and V N ≥ 0 following from v ≥ 0. Therefore to prove (2.24) it suffices to show that for sufficiently small ε 1 , ε 2 > 0. We estimate the single contributions of R N given in (2.16) separately using the bounds where a * (h) = p∈Λ * h p a * p for any h ∈ ℓ 2 (Λ * + ), resp.for the modified creation and annihilation operators By definition (2.16), the operator R N is cubic in creation and annihilation operators and can be bounded with (2.28) by

.29)
We switch to position space for the first factor and find and therefore which proves (2.64) for k = 1.To prove (2.64) for k = 2 we remark that (2.22) furthermore implies With spectral calculus we find that We recall the definition of G N (s) and compute the commutators for every term separately.We have and thus The commutator with R N follows similarly using that [N + , a p ] = −a p resp.[N + , a * p ] = a * p and analogous estimates as in (2.31)) (with and with (2.33) furthermore at With (2.32) we can now refine the estimates on the remainder.We shall prove that For the contribution of R N in (2.16) cubic in (modified) creation and annihilation operators, we switch to position space and compute with the commutation relations for any vector (2.41) We proceed similarly for the contribution of R N cubic in creation and annihilation operators V N (see (2.16) resp.(2.25)).With the commutation relation we find for any and thus we arrive with (2.43) Summarizing (??), (2.41) and (2.43) we thus arrive at the desired estimate (2.39).Therefore with (2.39) we find that for any ψ ∈ F ≤N ⊥ϕ 0 in the limit and with the min max principle it follows that the low energy states are determined through the quadratic Hamiltonian Q.In particular the spectrum of G N (s) has a spectral gap independent of N, s (given in leading order by the spectral gap of Q (for more details see for example [13]).Furthermore with similar arguments as in [11] it follows that for every s ∈ [0, 1] there exists a ground state ψ N (s) approximated by the ground state of Q.
Proof of (2.65).With (2.22) we find In order to use (2.22) once more, we write the r.h.s. as For the first term we find similarly to (2.34) with spectral calculus and thus with similar estimates as in (2.34)-(2.37)and (2.22) (2.48) The second term of the r.h.s. of (2.45) can be estimated similarly and we find with (2.34)-(2.37),(2.22) that (2.49) Thus we conclude with (2.48), (2.49) from (2.45) with the operator inequality that finally leads to the desired first bound of (2.65).
The second bound follows with similar arguments from (2.47).
2.3.Generalized Bogoliubov transform.We note that the quadratic Hamiltonian Q is formulated w.r.t. to modified creation and annihilation operators.For operators quadratic in standard creation and annihilation operators, the corresponding unique ground state is explicitly known and given by a quasi-free state.However, here we do not have an explicit expression for the ground state ψ Q , but we will use that it is approximately given by the generalized quasi-free state e B(µ) Ω as defined in (1.28).In contrast to the standard Bogoliubov transform (1.27) formulated w.r.t. to standard creation and annihilation operators, there is no exact formula for the action of e B(µ) on modified creation and annihilation operators.However, we have where we write σ p = sinh(µ p ) and γ p = cosh(µ p ).The remainders d p , d * p are small on states with a small number of excitations.More precisely, [5,Lemma 2.3] shows (since µ ∈ ℓ 2 (Λ * + )) that for any k ∈ Z there exists C k > 0 such that for all p ∈ Λ * + and In particular this leads to and with their (double commutator) with N + ; we have resp.
and similarly for the other operators (note that (2.56), (2.57) follow from the proof of [6, Corollary 3.5]).Also, we know the generalized Bogoliubov transform approximate action on the kinetic term that is given by where the remainder R K satisfies and also similar bounds for its (double) commutator as formulated before (2.56), (2.57).Note that since p 2 µ p ∈ ℓ 1 (Λ) and σ p , γ p ∈ ℓ ∞ (Λ * + ), this is a consequence of (2.54), (2.55) (for more details see Lemma 3.10 below).Consequently, conjugating the quadratic Hamiltonian Q with the generalized Bogoliubov transform e B(µ) almost diagonalizes Q.More precisely, we have where the diagonal operator D is given by and the remainder is (2.62) Though we do not have an explicit form of e B(µ) ψ Q , the ground state of the diagonal operator D is explicitly known and given by the vacuum Ω.For this reason we will study for s ∈ [0, 1] the family of Hamiltonians Similarly to Proposition 2.1 we have the following properties.
Proposition 2.2.Let s ∈ [0, 1], then there exists a ground state ψ Q(s) of the Hamiltonian Q(s) defined in (2.18).Furthermore, there exists a constant for k = 1, 2 and the spectrum of the Hamiltonian Q(s) has a spectral gap above the ground state E(s) independent of s, N .Moreover, for k = 1, 2 there exists C k > 0 (independent of N, s) such that for any Fock space vector ξ ∈ F ≤N ⊥ϕ 0 we have (2.65) Proof.We proceed similarly as in the proof of Proposition 2.1.First note that from Proposition 2.1 we have (since for some C 1 , C 2 > 0. The generalized Bogoliubov transform approximately preserves the number of particles.More precisely it follows from [5,Lemma 2.4] that for some positive constants C 3 , C 4 > 0. Since D ≥ CN + for some positive C > 0 and for some positive constants c 0 , C > 0. This implies that for any normalized ξ With formula 2.34 we write the commutator as it follows from (2.28) and (2.57) that we can bound the double commutator in form by the number of particles.Thus we arrive for any ξ ∈ F ≤N ⊥ϕ 0 at ξ, (N + 1) 2 ξ ≤Cζ ξ, (N + + 1)ξ + C (N + 1)ξ ξ (2.71) and we conclude by ξ, (N + 1) 2 ξ ≤ C. The spectral gap and the bound on the resolvent follow with similar arguments as in the proof of Proposition 2.1 using again the estimates on the double commutator.

PRELIMINARIES
The proof of Theorem 1.2 is based on closed formulas derived in [12] for the conjugation of operators of the form b p , b * p and for any bounded operator H on ℓ 2 (Λ * + ) with the exponential of N + (given by (2.8)) and the symmetric operator with h ∈ ℓ 2 (Λ * + ).For this we furthermore define for any and (in abuse of notation) the shorthand notation γ s = cosh(s) and σ s = sinh(s) We recall the closed formulas from [12] that are formulated in position space and easily translate with (2.4) to momentum space relevant for the present analysis.Lemma 3.1 (Proposition 2.2,2.4 in [12]).With the shorthand notation Furthermore for any self-adjoint h, Hh a * (h)a(h) .
A similar formula as (3.46) for e follows when replacing h with its negative −h and taking the hermitian conjugate of (3.46).
Furthermore the following closed formulas hold for the conjugation with respect to the exponential of the number of particles operator on the excitation Fock space N + .Lemma 3.2 (Proposition 2.5 [12]).Let N + be given by (2.8) and h ∈ ℓ 2 (Λ + + ).Then for every s ∈ R we have with the short hand notation (3.4) Moreover we shall use the following Lemma proven in [12].
Lemma 3.3 (Proposition 2.6 [12]).Let h • : R → ℓ 2 (Λ * + ), t → h t be a differentiable.For ξ 1 , ξ 2 ∈ F ≤N ⊥ϕ 0 we find with the short hand notation (3.4) In the proof of the main theorem we consider operators conjugated w.r.t. to both exponentials e λ √ N φ + (h) e λN + where the parameter λ ∈ [0, 1] is considered to be small.The previous Lemma yield in the following Corollary for the first order contributions.

.24)
(iv) There is exactly one term having the form (3.21) with k = 0 and such that all Λ-operators are factors of (N − N + )/N or of (N + 1 − N )/N .It is given by if n is even, and by or the form for some r ∈ N, j 1 , . . ., j k ∈ N \ {0}.If it is of order k = 0, then it is either given by µ 2r p b p or by µ 2r+1 p b * −p for some r ∈ N. (vi) For every non-normally ordered term of the form appearing either in the Λ-operators or in the Π (1) -operator in (3.21), we have i ≥ 2.
As a consequence of Lemma 3.5, for µ small enough we have and the series converge absolutely (see [6,Lemma 3.3]).From this, we also get an explicitly define the remainder operators (2.51) by if m is odd.This representation allows to prove the following improved error estimates on the remainder terms d p using Lemma 3.5 and Lemma 3.1.We start with the conjugation w.r.t. to e λN + first.
Next we prove similar estimates for the conjugation of d p , d * p with the two exponentials e κλN + e λ √ N φ + (g) .To this end we first prove the following auxiliary estimates.Lemma 3.7.Under the same assumptions and notations of Lemma 3.5, let |κλ|, |λ| ≤ 1, g ∈ ℓ 2 (Λ * + ).Then for sufficiently small µ there exists C > 0 (independent of κ, λ) such that Proof.The first bound follows with similar arguments as in the proof of Lemma 3.4.For the second we note that from definition (3.19) it follows a ♯t βtpt a ♭t αtp t+1 a ♯n βnpn a ♭n (g) On the one hand, it follows from Lemma 3.1 that We recall that from the estimates (2.28), any term is O( √ N λ) for small λ and large N .On the other hand, from Lemma 3.1 for ♯ j = * and and, similarly to Lemma 3.4, any term is either bounded by multiples of √ N λ (N + + 1) 1/2 ψ or O(λ 2 N ).Note that the case ♯ j = • and ♭ j = * follows in the same way using the commutation relations.Moreover, (3.46), (3.46) show that terms appearing in (3.45) of the form . Since the number of particles operator can be easily commuted through a and Lemma 3.7 follows.
From these estimates, we derive the following estimates for (3.16).
Lemma 3.8.Under the same assumptions and notations as in Lemma 3.5, g ∈ ℓ 2 (Λ * + ) and |λ|, |λκ| ≤ 1 and µ small enough.Then there exists C > 0 (independent of κ, λ) such that and Thus the first term of the r.h.s. of (3.57) can be estimated by Lemma 3.7 distinguishing the case ℓ 1 = 0 and ℓ 1 > 0 as in the proof of Lemma 3.6 by For the second term of the r.h.s. of (3.57) we proceed similarly as in the proof of Lemma 3.7 by Lemma 3.1 distinguishing again the case ℓ 1 = 0 and ℓ 1 > 0 and thus finally get Plugging these estimates into (3.50)we arrive for sufficiently small µ at Lemma 3.8.The second estimate of Lemma 3.8 follows similarly.Lemma 3.9.Under the same assumptions as in Lemma 3.8, there exists C > 0 (independent of λ, κ) such that and similarly Proof.We start with ♯ 1 = ♯ 2 = * .We observe that from (2.51) we have and thus With (2.52) and Lemma 3.8 we find for all |λ|, |κλ| ≤ 1 The remaining bounds follow similarly with (2.28) and Lemmas 3.1, 3.8.
Additionally, we consider the conjugation of the kinetic energy with the generalized Bogoliubov transform that we write as where the remainder R K satisfies the following properties.
Lemma 3.10.Under the same assumptions as in Lemma 3.5, 3.8, let p 2 µ ∈ ℓ 1 (Λ * + ) and µ small enough.Then there exists C > 0 such that (3.67) Ffurthermore for |λ|, |κλ| ≤ 1 there exists C > 0 (independent of λ, κ) such that Proof.We compute where We recall that it follows with the same arguments as in Lemma 3.8 (see for example [6,Lemma 3.4] that by assumption, the first estimate (3.67) follows.This estimates remains true for the double commutator, too (see [6,Lemma 3.4]) and thus (3.68) follows.For the second estimate (3.69), we recall that in the proof of Lemma 3.8 we more precisely prove that In this section we prove Theorem 1.2, thus we estimate the logarithmic moment generating function.For this we define the centered (w.r.t. to the condensate's expectation value) operator and recall that we need to compute the moment generating function We consider the embedding of ψ N ∈ L 2 s (R 3N ) in the full bosonic Fock space where we have the identity where O p,q denotes the Fourier coefficients of O, i.e.O p,q = ´Λ×Λ dxdy O(x; y)e i(px+qy) .By definition of U N in (2.2) we observe that we can write ψ N as where ψ G N denotes the ground state of the excitation Hamiltonian G N defined in (2.14).The properties (2.10), (2.11) of the unitary U N show that where we recall the notation 3.2.Furthermore we introduce the notation and thus arrive at In the following we will compute the expectation value of the r.h.s. of (4.7).First we will show that the operator B contributes to our analysis sub-leading only (see Lemma 4.1).This will be based on ideas introduced in [12,22].Second we will show that the ground state ψ G N of the excitation Hamiltonian G N (defined in (2.14)) approximately behaves as the ground state ψ Q of the excitation Hamiltonian's quadratic approximation Q (defined in (2.15)) (Lemma 4.2).Then we show that ψ Q effectively acts as a Bogoliubov transformation on the observable φ + (g).We remark that this would be an immediate consequence if the operator φ + defined in (3.2) would be formulated w.r.t. to standard creation and annihilation operators.However, φ + is formulated w.r.t. to modified creation and annihilation operators that lead to more involved calculations (see Lemma 4.3).Finally, in the last step, we compute the remaining expectation value (see Lemma 4.4).
While the first and the forth step are based on ideas presented in [12] for the dynamical problem, the second and third step use novel ideas and techniques based on the Hellmann-Feynmann theorem and Gronwall's inequality.

4.1.
Step 1.In this step we show that the operator B defined in (4.6) contributes to the expectation value (4.7) exponentially cubic in λ only.This Lemma follows closely the proof of [22,Lemma 3.3] resp.[12, Lemma 3.1] considering a similar result for the dynamics in the mean-field regime (β = 0).The results [12,22] are formulated in position space, however the proofs and results easily carried over to momentum space.Lemma 4.1.Under the same assumptions as in Theorem 1.2 there exists C > 0 such that for all 0 ≤ λ ≤ 1/ O we have Proof.We start with the lower bound (i.e. the first inequality of Lemma 4.1) and define similarly to [22,Lemma 3.3] for s ∈ [0, 1] and κ > 0 the Fock space vector ξ(s) := e −(1−s)λκN + /2 e (1−s)λ We remark that by construction ξ N (s) is an element of the Fock space of excitations F ≤ϕ 0 ⊥ϕ N , thus the number of particles of ξ(s) is at most N .This observation will be crucial for our analysis later.Since we have for s = 0 and for s = 1 it suffices to control the difference of (4.10) and (4.11) to get the desired estimate.We aim to control their difference through the derivative with the operator M s given by With the bounds (2.28) for any Fock space vector ξ ∈ F ≤N ⊥ϕ 0 and h(s for all λ O ≤ 1, we observe that that all but the terms of the first line of the r.h.s. of (4. as operator inequality on the Fock space of excitations F ≤N ⊥ϕ 0 .Summarizing, we arrive at For the contribution of (4.40) linear in λ we find that is bounded for ψ ∈ F ≤N ⊥ϕ 0 and λκ s ≤ 1 by that is again of the desired form.For the second term of (4.27) we first observe that by the commutation relations we can write R and thus, we find with [12, Proposition 2.2-2.4](resp.Lemma 3.1, 3.2) that where and thus bounded for ψ ∈ F ≤N ⊥ϕ 0 and λκ s ≤ 1 by For the linear contributions of (4.47) we find We recall that from (4.45) and (4.52), we similarly have for all λκ s ≤ 1 for any ψ ∈ F ≤N ⊥ϕ 0 .Thus from (4.63) we find and we find with Lemma 2.1 The first two summands of the r.h.s. of (4.64) is already of the desired form as it can be bounded by terms O(λ 3 N ) resp.terms of the form λN + .For the last term, we however have to estimate more carefully.We use once more the resolvent formula and write with the notation With N + a p = a p (N + − 1) it follows from (4.44), (4.51) and (4.60) that for any ψ ∈ F ≤N ⊥ϕ 0 , and therefore where we used Lemma 2.1 and (2.39).Similarly we find with Lemma 2.1 that is of the desired form.Together with (4.53) we thus get for (4.38) that The upper bound follows with similar ideas, replacing the lower with upper bounds and κ s with −κ s .

4.3.
Step 3. We recall that we are left with computing the expectation value w.r.t. to the ground state ψ Q of the quadratic Hamiltonian Q given by (2.15).In this step we will show that the ground state is approximately given by e B(µ) Ω, that is a an generalized Bogoliubov transform applied to the vacuum vector.Furthermore, we show that e B(µ) acts on the observable φ + (h) as a Bogoliubov transform, i.e. that e −B(µ) φ + (h)e B(µ) can be approximated by φ + (f ) with f given by (1.15).The main difficulty in this step here is that all quantities are formulated w.r.t. to modified creation and annihilation operators for which the action of the Bogoliubov transform is not explicitly given.However we use (2.51) and (2.52),(2.53)and Lemmas 3.5-3.9 to prove the following Lemma.
where f is given by (1.15).
Proof.We start with the proof of the lower bound.The upper bound then follows with similar arguments as in the previous steps.As the generalized Bogoliubov transform (1.28) is a unitary operator we write The final goal is to compare the r.h.s. with the r.h.s. of the lower bound (4.74).To this end we perform three steps: we first show that we can replace the exponent −κe B(µ) N + e −B(µ) with κ 2 N + for sufficiently large κ 2 > 0. Second, we show that the exponent e B(µ) φ + (g)e −B(µ) can be effectively replaced by φ + (f ) with f given by (1.15), again paying a sub-leading price.As a third and last step we then show that we can replace e B(µ) ψ Q with an interpolation argument (similarly as in the proof of Lemma 4.2) by Ω.
To that end we observe first that from (2.51) we have and consequently Q(s) − E(s) ψ ≤ C(N 1/2 λ|||O||| + 1) q ψ Q(s) ψ .(4.107) We can prove a similar bound not only for the square root but for the number of particle operator.
For that we write with the resolvent identity

. 14 )
Note that the bound linear in λ depends on κ (in contrast to (3.13)) as b * p b * −p does not commute with e λκN + .