Central limit theorem for Bose gases interacting through singular potentials

We consider a system of $N$ bosons in the limit $N \rightarrow \infty$, interacting through singular potentials. For initial data exhibiting Bose-Einstein condensation, the many-body time evolution is well approximated through a quadratic fluctuation dynamics around a cubic non-linear Schr\"odinger equation of the condensate wave function. We show that these fluctuations satisfy a (multi-variate) central limit theorem.


Introduction
We consider a system of N bosons with Hamilton operator acting on L 2 s (R 3N ), the subspace of L 2 (R 3N ) consisting of functions which are symmetric with respect to permutations. The N -dependent two-body interaction potential is given through In the following, we assume V to be smooth, spherically symmetric and compactly supported. For β = 0, the Hamiltonian (1.1) describes the mean-field regime characterized by a large number of weak collisions, whereas β = 1 corresponds to the Gross-Pitaevskii regime where the collisions of the particles are rare but strong. Here, we study intermediate regimes β ∈ (0, 1) in the limit N → ∞ where the particles interact through singular potentials.
Besides the convergence of the one-particle reduced density γ N,t associated with ψ N,t , the norm approximation of ψ N,t has been studied for different settings of β ∈ (0, 1) in [10,22,26,27]. Our result is based on the norm approximation obtained in [10] covering β < 1 whose ideas we explain in the following.
Truncated Fock space. As first step towards the norm approximation in [10], the contribution of the Bose-Einstein condensate is factored out. This is realized through the unitary U ϕ N,t : F → F ≤N ⊥ϕ N,t . It maps the bosonic Fock space into the truncated Fock space defined over the orthogonal complement L 2 ⊥ϕ N,t R 3 of the subspace of L 2 (R 3 ) spanned by the condensate wave function ϕ N,t . This unitary has first been used in [25] in the mean-field regime. Its definition is based on the observation that every ψ N ∈ L 2 s (R 3N ) has a unique decomposition where α (n) ∈ L 2 ⊥ϕ N,t (R 3 ) ⊗sn for all n = 1, · · · , N . Then, U ϕ N,t ψ N = {α (0) , α (1) , · · · , α (n) }.
This unitary satisfies the following properties proven in [25] U ϕ N,t a * (ϕ N,t )a(ϕ N,t )U * ϕ N,t =N − N + (t) U ϕ N,t a * (ϕ N,t )a(f )U * ϕ N,t = N − N + (t)a(f ) U ϕ N,t a * (f )a(ϕ N,t )U * ϕ N,t =a * (f ) N − N + (t)a(f ) U ϕ N,t a * (f )a(g)U * ϕ N,t =a * (f )a(g), (1.6) for all f, g ∈ L 2 ⊥ϕ N,t (R 3 ). Here a * (f ), a(f ) denote the standard creation and annihilation operators on the bosonic Fock space F. On the truncated Fock space, we define modified creation and annihilation operators The modified creation operator b * (f ) excites one particle from the condensate into its complement while b(f ) annihilates an excitation into the condensate. We define the vector ξ N,t := U ϕ N,t ψ N,t representing the fluctuation outside the condensate and observe From the truncated Fock space to the bosonic Fock space. We approximate the generator L N,t acting on the truncated Fock space only with a modified generator L N,t defined on the whole bosonic Fock space. We consider regimes with a small number of excitations N + (t). For this reason, we realize the approximation of L N,t through L N,t by replacing is the Taylor series of √ 1 − t expanded at the point t 0 = 0. For a precise definition see [10, eq. (54)].
Correlation structure through Bogoliubov transformation. In the intermediate regime correlations are important (at least if β > 1/2). For their implementation, we consider for fixed ℓ > 0 the ground state of the scattering equation with Neumann boundary conditions on the ball B ℓ (0). We fix f N (x) = 1 for all |x| = ℓ and extend f N to R 3 by setting f N (x) = 1 for all |x| ≥ ℓ.
In [9], the non linear Schrödinger equation (1.3) is replaced by the N -dependent Hartree equation with initial data ϕ N,0 = ϕ 0 (the condensate wave function at time t = 0) to approximate the time evolved condensate wave function. The well-posedness of (1.10) is shown in [9,Appendix B]. The correlation structure is implemented through the Bogoliubov transformation Here, ω N and ϕ N,t are as defined in (1.9) resp. (1.10) and q N,t = 1 − |ϕ N,t ϕ N,t |. The Bogoliubov transformation acts on the creation and annihilation operators as for all f ∈ L 2 (R 3 ). The operators sinh η N,t and cosh η N,t are defined through the absolutely convergent series Let G N,t be the generator given through (1.14) Note that G N,t consists of terms which quadratic in creation and annihilation operators and of terms of higher order. Nevertheless, in [10,Lemma 5] it is shown that G N,t can be approximated through the generator G 2,N,t containing quadratic terms only.
Limiting quadratic dynamics. We are interested in the limit N → ∞ of G 2,N,t . For this reason, we define for |x| ≤ ℓ and ω ∞ (x) = 0 otherwise. Here, we used the notation b 0 = dx V (x). Furthermore, the solution ϕ N,t of the modified Hartree equation (1.10) with initial data ϕ 0 ∈ H 4 (R 3 ) can be approximated with the solution ϕ t of (1.3) with initial data ϕ 0 . To be more precise, [9,Proposition B.1] shows that there exists a constant C > 0 (depending on ϕ 0 H 4 ) such that with γ = min{β, 1 − β}. Standard arguments (see for example [9, Proposition B.1]) imply, that there exists a constant C > 0 such that We define the limiting Bogoliubov transformation In fact, (1.15) and (1.16) yield that there exists a constant C > 0 such that where γ = min{β, 1 − β}.
In order to define the limiting dynamics we introduce some more notation. We use the shorthand notation j x (·) = j(·, x) for any j ∈ L 2 (R 3 × R 3 ). Furthermore, we decompose sh ηt = η t + r t , ch ηt = 1 + p t and for all x, y ∈ R 3 . A slight modification of the arguments in [9, Appendix C] shows some properties of the kernels. For these, we consider initial data ϕ 0 ∈ H 4 (R 3 ) of (1.3). There exist a constant C > 0 (depending only on ϕ 0 H 4 (R 3 ) and on V ) such that on one hand ch ηt ≤ C, and k t 2 , η t 2 , sh ηt 2 , p t 2 , r t 2 , µ t 2 ≤ C. (1.21) where · denotes the operator norm. On the other hand, denoting with ∇ 1 k t and ∇ 2 k t the operator with the kernel ∇ x k t (x; y) Furthermore, let ∆ 1 r t resp. ∆ 2 r t be the operator having the kernel ∆ x r t (x; y) resp. ∆ y r t (x; y), then for all i = 1, 2 In order to simplify notation, we write in the following sh ηt = sh, ch ηt = ch resp. r t = r, k t = k, p t = p. Now, we define the limiting dynamics U 2 (t; s) satisfying i∂ t U 2 (t; s) = G 2 (t)U 2 (t; s) and U 2 (s; s) = 1 (1.24) where G 2,t is given by and and Here, we used the notation K = dx a * x (−∆ x ) a x and K 1,t = q t K 1,t q t , K 2,t = (q t ⊗ q t ) K 2,t where K 1,t is the operator with integral kernel and K 2,t is the function given through Norm approximation. We consider the solution ψ N,t of the Schrödinger equation (1.2) with initial data Ψ N,0 = U * ϕ 0 1 ≤N T * N,0 Ω. It is proven in [10, Theorem 2] that for all α < min{β/2, (1 − β)/2} there exists a constant C > 0 such that for all N sufficiently large and all t ∈ R.
The limiting dynamics U 2 (t; s) defined in (1.24) is quadratic in creation and annihilation operators. As the following Proposition shows, it gives rise to a Bogoliubov transformation. For this, we define (1.30) On one hand On the other hand, the commutation relations imply for preserving the relations (1.31) and (1.32), i.e. ν * Sν = S and J ν = νJ . It turns out that a Bogoliubov transformation ν is of the form The following Proposition is proven in Section 2.2.
Proposition 1.1. Let U 2 (t; s) be the dynamics defined in (1.25). For every t, s ∈ R there exists a bounded linear map , where J and S are defined in (1.31) resp. (1.32). The Bogoliubov transformation Θ(t; s) can be written as (1.34) Central limit theorem. From a probabilistic point of view (1.5) implies a law of large numbers, in the sense that for a one-particle self-adjoint operator O on L 2 (R 3 ) and for every Here O (j) denotes the operator on L 2 (R 3N ) acting as O on the j-th particle and as identity elsewhere. The proof of (1.35) follows from Markov's inequality (see [12]). As a next step, we are interested in a central limit theorem. For this, we consider the rescaled random variable where ϕ N,t denotes the solution of (1.10) with initial data ϕ N,0 = ϕ 0 . We consider initial data ψ N,0 of the form ψ N,0 = U * ϕ 0 1 ≤N T * N,0 Ω exhibiting Bose-Einstein condensation [10, Theorem 3]. As a consequence, such a initial data satisfy a law of large numbers in the sense of (1.35). Moreover, such initial data obeys a central limit theorem in the sense that for every −∞ < a < b < ∞. Here, G 0 denotes the centered Gaussian random variable with variance σ 0 2 2 where following from Theorem 1.2 for time t = 0. Note that initial data of the form ψ N,0 = U * ϕ 0 1 ≤N T * N,0 Ω describe approximate ground states of trapped systems [8]. In experiments such initial data are prepared by trapping particles through external fields and by cooling them down to extremely low temperatures so that the system essentially relaxes to its ground state.
The validity of a central limit theorem for the ground state of trapped systems has already been adressed in [30]. To be more precise, [30] considers the ground state of (1.1) for β = 1, i.e. in the Gross-Pitaevskii regime. The ground state is known to exhibit Bose-Einstein condensation. It is proven that the ground state satisfies a central limit theorem. The arguments of the proof can be adapted to the intermediate regime β < 1 using the norm approximation for the ground state obtained in [8]. Now, we consider the time evolution of the initial data ψ N,0 = U * ϕ 0 1 ≤N T * N,0 Ω with respect to the Schrödinger equation (1.2) and show the validity of a (multi-variate) central limit theorem.
Theorem 1.2. Let β ∈ (0, 1) and assume V to be radially symmetric, smooth, compactly supported and point-wise non-negative. Furthermore fix ℓ > 0 (independent of N ). Let ϕ t denote the solution of (1.3) and ϕ N,t the solution of (1.10) both with initial data ϕ 0 ∈ H 4 (R 3 ). Moreover, we denote by Ψ N,t the solution of the Schrödinger equation Assume Σ t ∈ C k×k , given through . . k} and let O j,N,t denote the random variable (1.36) associated to O j for all j ∈ {1, . . . , k}. For every α < min{β/2, (1 − β)/2}, there exists C > 0 such that A similiar result has been established in [6,12] for the mean-field regime characterized through weak interaction of the particles. It is shown that fluctuations around the non-linear Hartree equation of bounded self-adjoint one-particle operators satisfy a (multi-variate) central limit theorem. We show that this result is true in the intermediate regime, where the interaction is singular, too. In particular, the correlation structure which becomes of importance in the intermediate regime does not affect the validity of a central limit theorem. Though, it affects the covariance matrix (1.39) through the Bogolioubiv transform T t .
Similarily as in [12, Corollary 1.3 ], Theorem 1.2 implies a Berry-Esséen type central limit theorem. To be more precise, we consider a bounded self-adjoint operator O on L 2 (R 3 ) and the random variable For every α < min{β/2,

Preliminaries
The proof of Theorem 1.2 is based on the norm approximation (1.29) from [10]. In the following we collect useful properties of the unitaries used therein.
As proven in [10,Proposition 8], any moments of the number of particles operator are approximately preserved with respect to conjugation with the Bogoliubov transformation T N,t . To be more precise for every fixed k ∈ N and δ > 0, there exists C > 0 such that As the following Lemma shows, the moments of number of particles operator are approximately preserved along the limiting dynamics U 2 (t; 0), too.
Proof. The proof of the lemma is structured in two steps. First, we show that the generator G 2,t of the limiting dynamics U 2 (t; s) is of the form with H (1) t ≤ Ce C|t| and H (2) t 2 ≤ Ce C|t| for constants c, C > 0. Secondly, we prove Lemma 2.1 with respect the dynamics U (t; s) having a generator of the form of the r.h.s. of (2.2).
Step 1. The observation (2.2) follows from the definition (1.25) of G 2,t . We split and consider each of the summands separately. First, we consider G V 2,t defined in (1.26), which is again split into four terms. The first one, G V, 1 2,t of the r.h.s. of (1.26) satisfies assumption (2.2) since on one hand following from (1.17) and (1.21). For the same reasons, the second term G V,2 2,t of the r.h.s. of (1.26) satisfies assumption (2.2), too. For the third term G V, 3 2,t , the definition of K 2,t implies again from (1.17) and (1.21) . The forth term G V,4 2,t satisfies the assumption (2.2) due to (1.17).
Furthermore, the interpolation and the Sobloev inequality imply Therefore, G λ 2,t is of form (2.2). Moreover, for the term G K 2,t − K we observe with (1.21) and (1.23) The remaining bounds follow in the same way. Note that (1.22) implies the bound Moreover, by definition (1.15) of the limiting kernel, ω ∞ ∈ L p (R 3 ) for all p < 3. Hence, the remaining terms of G K 2,t satisfy the assumptions, too.
Step 2. We prove Lemma 2.1 with respect to the dynamics U * (t; s) with generator We compute the derivative Using the commutation relations and the definition of (2.4), we find For the first term of the right hand side, the commutation relations yield where C depends on k ∈ N. The second of the r.h.s. of (2.5) follows in the same way. Hence, there exists C > 0 such that d dt ψ, U * (t; s)(N + 1) k U (t; s)ψ ≤ Ce C|t| ψ, U * (t; s)(N + 1) k U (t; s)ψ .

Proof of Proposition 1.1
It follows from the first step of the proof of Lemma 2.1 that it is enough to prove Proposition 1.1 with respect the dynamics U (t; s) with generator L 2,t of the form (2.4). First, we prove that for f ∈ L 2 (R 3 ) the Fock space vectors U * (t; s)a * (f )U (t; s)Ω and U * (t; s)a(f )U (t; s)Ω are elements of the one-particle sector. The following Lemma is a generalization of [ where either a ♯ (f ) = a(f ) or a ♯ (f ) = a * (f ) and where P 1 denotes the projection onto the one particle sector of the Fock space F.
Proof. The proof follows the arguments of the proof of [13, Lemma 8.1]. For m ∈ N, m = 1, we define for arbitrary ψ ∈ F with ψ = 1 and ψ = 1 N =m ψ the function We observe that F (s) = 0 and furthermore e iKt a(f )e −iKt = a(e −i∆t f ) = a(f t ), using the notation f t = e −it∆ f . Since e −i∆t is a unitary operator, we find [a(f t ), The assumption (2.4) implies on one hand and on the other hand Hence, and analogously Note that these bounds are independent of f ∈ L 2 (R 3 ). Thus, Using the bounds a ♯ (f )ψ ≤ f 2 (N + 1) 1/2 ψ , we obtain F (t) ≤ 2 (N + 1) 1/2 U (t; s)Ω ≤ C exp (exp(C|t − s|)) ψ, (N + 1)ψ Here, we used Lemma 2.1 (resp. step 2 of its proof) for the last estimate. Since F (s) = 0, the Gronwall inequality implies F (t) = 0 for all t ∈ R.
Proof of Proposition 1.1. . We prove the Proposition with respect to the dynamics U (t; s) with generator L 2,t defined in (2.4). Then, Proposition 1.1 follows from the first step of the proof of Lemma 2.1.
The proof follows the arguments of the proof of [6, Theorem 2.2]. Let P k denotes the projection onto the k-th sector F k of the Fock space. As for all f ∈ L 2 (R 3 ) and k = 1 following from Lemma 2.2 . Thus, there exists linear operators We define the bounded operator Θ on Then g)) Ω, (2.8) for all f, g ∈ L 2 (R 3 ). For fixed ψ ∈ D(K + N ), g ∈ L 2 (R 3 ), s ∈ R and any bounded operator M on F with MD(K + N ) ⊂ D(K + N ), we define furthermore Here, a ♯ , a ♭ are either creation or annihilation operators. Since e −iKt a ♯ (f )e iKt = a ♯ (e it∆ f ) and e it∆ f 2 = f 2 for allf ∈ L 2 (R 3 ), we can write The commutation relations imply that F (s) = 0. Furthermore, using the notation f t = e −i∆t f . Analogous calculations as in the proof of Lemma 2.2 show that The assumption (2.2) implies h i,t 2 ≤ Ce C|t| f 2 for i = 1, 2. Thus, for all f ∈ L 2 (R 3 ) and therefore Since F (s) = 0, the Gronwall inequality implies F (t) = 0 for all t ∈ R. Hence, for every f 1 , f 2 , h 1 , h 2 ∈ L 2 (R 3 ) and every bounded operator M on the Fock space F such that MD (K + N ) ⊂ D (K + N ). We claim, that for all ψ ∈ D (K + N ) with ψ = 1. Combining (2.8) with (2.10), we find where S is defined in 1.32 . It follows that h 1 )). On the one hand, (2.8) shows that RΩ = 0 and on the other hand it follows from (2.11), that R commutes with any creation and annihilation operator. Since states of the form a * (f 1 ) . . . a * (f n )Ω build a basis of the Fock space F, we conclude Now, we are left with proving (2.10). For this, note that (2.9) implies where P ψ resp. P Ω denote the projection on the subspace of F spanned by ψ resp. Ω. Therefore, on one hand Assuming that ψ, Ω = 0, the claim (2.10) follows. If ψ, Ω = 0, we repeat the same arguments with ψ = 1 √ 2 (ψ + Ω). This leads to (2.10). It remains to prove the properties (1.33). Since for all f, g ∈ L 2 (R 3 ) =A(Θ(t; s)(Jf, Jh)), the first property follows. Furthermore, from we deduce the second property.

Proof of Theorem 1.2
The proof uses ideas introduced in [30]. We consider the expectation value The norm approximation (1.29) from [10] implies, that for every α < min{β/2, (1 − β)/2} there exists C > 0 such that (2.12) We are hence left with computing the expectation value We split this computation in several steps.
Step 1: Action of the unitary U ϕ N,t . Let ξ N,t = T * N,t U 2 (t; 0)Ω. The goal of this step is to prove and denote dΓ(A) = k j=1 A j for a bounded operator A on L 2 (R 3 ). We observe that where p N,t = |ϕ N,t ϕ N,t | and q N,t = 1 − p N,t . The properties (1.6) of the unitary U ϕ N,t imply Hence, We compute Using the fundamental theorem of calculus, we can write the difference as an integral The estimate dΓ(A)ψ ≤ A N ψ leads to for α ≥ 1. Recall that ξ N,t = T * N,t U 2 (t; 0)Ω. It follows from (2.1) and Lemma 2.1 that for a constant C > 0 uniform in N . Hence, Step 2: Replace modified creation and annihilation operators with standard ones. Recall that φ a (f ) = a * (f ) + a(f ) with the standard creation and annihilation operators a * (f ), a(f ) while with the modified creation and annihilation operators defined in (1.7). The goal of this step is to prove To this end, we compute By definition of the modified creation and annihilation operators (1.7) we obtain Since Step 3: Replace modified Hartree equation with non-linear Schrödinger equation. By linearity of the operator φ a (f ), we compute We conclude again with Lemma (2.6), Lemma (2.1) and Lemma 2.1 exp (exp (C|t|)) .
Step 4: Action of T N,t . Let ξ t = T N,t ξ N,t (recall that by definition of ξ N,t , the fluctuation vector ξ t = U 2,t Ω is in fact independent of the number of particles). The goal of this step is to show ξ N,t , e is 1 φa(qtO 1 ϕt) . . . e is k φa(qtO k ϕt) ξ N,t − ξ t , e is 1 φa(h 1,t ) . . . e is k φa(h k,t) ξ t with h j,t = cosh(η t )q t Oϕ t + sinh(η t )q t O j ϕ t and η t as defined in (1.18). For this, we compute using the properties (1.12) of the Bogoliubov transformation with η N,t as defined in (1.18). In the following we denote h j,N,t = cosh(η N )q t Oϕ t +sinh(η N )q t O j ϕ t . Since ξ N,t , e is 1 φa(qtO 1 ϕt) . . . e is k φa(qtO k ϕt) ξ N,t = ξ t , e is 1 φa(h 1,N,t) . . . e is k φa(h k,N,t) ξ t we need to consider ξ t , e is 1 φa(h 1,N,t) . . . e is k φa(h k,N,t) ξ t − ξ t , e is 1 φa(h 1,t ) . . . e is k φa(h k,t) ξ t .
Note, that the Baker Campbell Hausdorff formula implies on one hand e iφa(f ) e iφa(g) = e iφa(f +g) e −iIm f,g , for f, g ∈ L 2 (R 3 ), i.e. k j=1 e is j φa(ν j,t ) = e iφa(νt) k i<j e −is i s j Im ν i,t ,ν j,t with ν t = k j=1 ν j,t . On the other hand, the Baker Campbell Hausdorff formula applied to the creation and annihilation operator implies k j=1 e is j φa(ν j,t ) = e − νt 2 2 /2 e a * (νt) e a(νt) k i<j e −is i s j Im ν i,t ,ν j,t .