Gibbs measures based on 1D (an)harmonic oscillators as mean-field limits Mathieu Lewin, Phan Thành Nam, Nicolas Rougerie
To cite this version:
Mathieu Lewin, Phan Thành Nam, Nicolas Rougerie. Gibbs measures based on 1D (an)harmonic oscillators as mean-field limits. Journal of Mathematical Physics, American Institute of Physics (AIP), 2018, 59, pp.041901. 10.1063/1.5026963. hal-01496137v2
HAL Id: hal-01496137 https://hal.archives-ouvertes.fr/hal-01496137v2 Submitted on 15 Feb 2018
HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. GIBBS MEASURES BASED ON 1D (AN)HARMONIC OSCILLATORS AS MEAN-FIELD LIMITS
MATHIEU LEWIN, PHAN THANH` NAM, AND NICOLAS ROUGERIE
Abstract. We prove that Gibbs measures based on 1D defocusing nonlinear Schr¨odinger functionals with sub-harmonic trapping can be obtained as the mean- field/large temperature limit of the corresponding grand-canonical ensemble for many bosons. The limit measure is supported on Sobolev spaces of negative regularity and the corresponding density matrices are not trace-class. The general proof strategy is that of a previous paper of ours, but we have to complement it with Hilbert-Schmidt estimates on reduced density matrices.
Contents 1. Introduction...... 1 2. Main result...... 3 3. Gibbs measures based on NLS functionals...... 6 4. Hilbert-Schmidt estimate...... 10 5. Proof of the main theorem...... 13 5.1. Convergence of the relative partition function ...... 14 5.2. Strong convergence of density matrices ...... 16 References ...... 20
1. Introduction Gibbs measures based on nonlinear Schr¨odinger energy functionals play a central role in constructive quantum field theory (CQFT) [26, 44, 16, 53] and in the low-regularity probabilistic Cauchy theory of nonlinear Schr¨odinger (NLS) equations [13, 4, 5, 6, 7, 8, 11, 30, 49, 50, 52]. They also are the natural long-time asymptotes for nonlinear dissipative stochastic PDEs [15, 14, 40, 51]. Recently, we have shown that, at least in the most well-behaved cases, they can be derived from the linear many-body quantum mechanical problem. Namely, many-body bosonic thermal equilibrium states converge in a certain mean-field/large-temperature limit [34, 32, 43] to nonlinear Gibbs measures (see the recent [20] for a corresponding time-dependent statement). The goal of this note is to extend this result to the case of somewhat less well-behaved measures, e.g. those based on the 1D harmonic oscillator studied in [10, 11, 15]. Consider the NLS flow on Rd+1 i∂ u = ∆u + V u + w u 2 u, (1.1) t − ∗ | | Date: February, 2018. 1 2 M.LEWIN,P.T.NAM,ANDN.ROUGERIE with V a trapping potential and w an interaction potential (say a delta function). A natural candidate for an invariant measure under (1.1) can be defined formally in the manner
1 1 2 2 µ(du)= exp u(x) w(x y) u(y) dxdy µ0(du) z −2 Rd Rd | | − | | r ZZ × with zr a normalization constant, and
2 2 µ0(du) = exp u + V u du − Rd |∇ | | | Z the free Gibbs (gaussian) measure associated1 with ∆+ V . The program of defining and studying the Schr¨odinger flow on the support of µ has− been initiated in [30], then pursued by many authors and extended to other nonlinear dispersive equations. The first result of measure invariance for a NLS equation is in [4]. It is well-known that the free Gibbs measure µ0 is supported on function spaces of low regularity. This is the main source of difficulty in the definition of the interacting measure µ and the proof of its invariance under the NLS flow. This is also an important issue as regards the derivation of nonlinear Gibbs measures from many-body quantum mechanics. In [34] we were able to fully control the mean-field limit only when (a) the gaussian measure is supported at least on L2(Rd); (b) its reduced density matrices are trace-class operators on L2(Rd); (c) consequently, the construction of the interacting Gibbs measure is straightforward. 2 s Essentially this limited us to the 1D case d = 1 with ∆+ V = ∂x + x ,s> 2 (the problem set on a bounded interval is included as the formal− case s =− ). In| higher| dimen- sions, we were able to derive nonlinear Gibbs measures only for very∞ smooth interaction operators. Multiplication operators by w(x y) as above, a fortiori by δ0(x y), were not allowed. − − In dimensions d > 2, properties (a) and (b) fail and a replacement for (c) necessitates a renormalization scheme, a minima a Wick ordering. This has been carried out decades ago in CQFT, see [26, 44, 16] for general references. More recently, the corresponding renormalized measures have been shown to be invariant under the (properly renormalized) NLS flow [5, 6, 49]. The derivation of these renormalized measures from many-body quantum mechanics is an open problem. The state of the art in this direction is contained in [19] where it has been shown that suitable modifications of bosonic Gibbs states based on renormalized Hamiltonians do converge to the desired measure. Completing the same program for the true Gibbs states remains an important challenge. In this note we address a particular case where (d) the gaussian measure is not supported on L2(Rd); (e) its reduced density matrices are not trace-class operators; (f) nevertheless, no renormalization is needed to make sense of the interacting measure. In fact, the gaussian measures we shall consider live on some Lp(Rd), for some p > 2. That their reduced density matrices are not trace-class has to do with a lack of decay at infinity, rather than a lack of local regularity.
− 1I.e., with covariance (−∆+ V ) 1 ANHARMONICOSCILLATORBASEDMEASURES 3
This situation is somewhat intermediate between the ideal “trace-class case”, solved in [34], and the “Wick renormalized case”, partially solved in [19]. That the 1D harmonic 2 2 oscillator case ∆+V = ∂x + x satisfies (d) and (f) has been observed in [11] and used to develop a low-regularity− − probabilistic| | Cauchy theory for the 1D nonlinear Schr¨odinger 2 s equation. Here we expain that (d) and (f) in fact hold in the case ∆+V = ∂x+ x ,s> 1 and derive the corresponding measures from many-body quantum− mechanics.− The| | main point to adapt the strategy of [34] is to overcome the problem posed by (e). Indeed, the trace-class topology of reduced density matrices (related to moments of the particle number) is the most natural one to pass to the mean-field limit in a many-body quantum problem. The main addition of the present paper is that we are able to work in weaker topologies (namely, the Hilbert-Schmidt and local trace class topologies), to pass to the limit and complete the program of [34]. Acknowledgments: We received financial support from the French ANR project ANR- 13-JS01-0005-01 (N. Rougerie).
2. Main result We consider the N-body quantum Hamiltonian N HN = hj + λ w(xi xj) (2.1) 6 6 − Xj=1 1 i
2The assumption of bosonic symmetry is essential. Without it, the mean-field limit of Gibbs states is very different [32, Section 3]. 4 M.LEWIN,P.T.NAM,ANDN.ROUGERIE appropriately in dependence of the particle number N to make the interaction sufficiently weak for the mean-field approximation to become asymptotically exact. Our starting point is the grand-canonical Gibbs state at temperature T > 0 −1 exp T Hλ Γλ,T := − −1 (2.5) TrF [exp ( T H )] − λ where Hλ is the second quantized version of (2.1): ∞ Hλ = HN (2.6) NM=0 acting on the bosonic Fock space F = C H H . . . H . . . ⊕ ⊕ 2 ⊕ ⊕ N ⊕ = C L2(R) L2 (R2) . . . L2 (RN ) . . . (2.7) ⊕ ⊕ sym ⊕ ⊕ sym ⊕ The Gibbs state is the unique minimizer over mixed grand canonical states (self-adjoint positive operators on F having trace 1) of the free energy functional
[Γ] = TrF [H Γ] + T TrF [Γ log Γ] (2.8) Fλ,T λ and the minimum equals −1 F = T log Z ,Z = TrF exp T H . λ,T − λ,T λ,T − λ The method of [34] that we adapt here is variational, based on this minimization principle. To see that Γλ,T is indeed the unique solution, observe that for any other state Γ
[Γ] = [Γ ]+ T TrF [Γ (log Γ log Γ )] . (2.9) Fλ,T Fλ,T λ,T − λ,T The last quantity in the right-hand side is the von Neumann relative entropy. It is positive, and equals zero if and only if Γ = Γλ,T , see e.g. [37, 54]. We are going to consider the mean-field limit: T (corresponding roughly to a large particle number limit) and → ∞ λ = T −1. The objects that will have a natural limit for large T are the reduced density matrices (k) Γλ,T , i.e. the operators on the k-particles space Hk defined by n Γ(k) = Tr Gn . (2.10) λ,T k k+1→n λ,T > nXk n Here Gλ,T is the projection of Γλ,T on the n-particle sector Hn and Trk+1→n is the partial trace taken over the symmetric space of n k 1 variables. Equivalently, we have − − (k) n ⊗(n−k) n TrH A Γ = TrH A 1 G (2.11) k k λ,T k n k sym λ,T > ⊗ h i nXk h i for every bounded operator Ak on Hk, where −1 1⊗n−k n Ak sym = (Ak)i1...ik (2.12) ⊗ k 6 6 1 i1 The limiting object is the nonlinear Gibbs measure 1 dµ(u)= exp ( FNL[u]) dµ0(u) (2.13) zr − with the nonlinear interaction term 1 2 2 FNL[u]= u(x) w(x y) u(y) dxdy, 2 R R | | − | | ZZ × the relative partition function z = exp ( F [u]) dµ (u), r − NL 0 Z and the gaussian measure µ0 associated with h. We refer to Section 3 for details, the main points being that t t µ0 can be defined as a measure over t<1/2−1/s H , where H is the Sobolev-like • space ∞ T ∞ Ht := u = α u λt α 2 < (2.14) n n n| n| ∞ ( n=0 n=0 ) X X for t R, and the spectral decomposition of h reads3 ∈ ∞ h = λ u u (2.15) n| nih n| n=0 X u FNL[u] is finite µ0-almost surely, so that µ is well-defined as a probability • measure.7→ To state our main result, we recall a convenient convention from [34], namely that, for ⊗n n a one-body operator A on H, we denote A the operator on Hn = sym H acting as ⊗n A (ϕ1 sym . . . sym ϕn)= Aϕ1 sym . . . symNAϕn. ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ The goal of this note is to prove the following: Theorem 2.1 (Derivation of Gibbs measures based on (an)harmonic oscillators). Let λ = T −1 and T . Then, we have the convergence of the relative partition function →∞ −1H Zλ,T TrF exp T λ = − −1 zr > 0. (2.16) Z TrF [exp ( T H )] → 0,T − 0 Moreover, for any k > 1, k! Γ(k) u⊗k u⊗k dµ(u) (2.17) T k λ,T → | ih | Z in the Hilbert-Schmidt norm, namely k! 2 Tr Γ(k) u⊗k u⊗k dµ(u) 0. T k λ,T − | ih | → Z Note that the limiting measure µ is uniquely characterized by the collection of the right-hand sides of (2.17) for all k N. Before turning to the proof, we make a few comments: ∈ 3 Using Dirac’s bra-ket notation |unihun| for the orthogonal projector onto un. 6 M.LEWIN,P.T.NAM,ANDN.ROUGERIE Remark 2.2 (Comparison with the trace-class case). In [34, Section 5.1] we had already proved this result in the case where Assumption (2.3) is strengthened to V (x) > C−1 x s,s > 2. Then, the convergence (2.17) is in fact strong in the trace-class and the proof| is| simpler, for this topology is more easily related to the many-body problem. In the case under consideration here, the right-hand side of (2.17) in fact belongs to 4 p the Schatten class S (Hk) for any p > 1/s + 1/2. The cases p = 1 and p = 2 corre- spond to the trace-class and the Hilbert-Schmidt class, respectively. We conjecture that p the convergence (2.17) is in fact strong in any S (Hk) with p> 1/s + 1/2. Note finally that, if V does not increase faster than x 2 at infinity, the expected particle number of the grand-canonical Gibbs state has to grow| | much faster than T in the limit T . It is then not obvious that choosing λ = T −1 should lead to a well-defined mean-field→ ∞ limit, but we prove it does. ⋄ 3. Gibbs measures based on NLS functionals In this section we briefly recall how to construct the interacting Gibbs measure µ. This has been done for s> 2 in [34]. The case s = 2 is covered by [11] (alternative constructions can be based on estimates for Hermite eigenfunctions from e.g. [28, 29, 55]). Here we give a softer argument allowing to define the defocusing measure for any s> 1, without resorting to local smoothing estimates or eigenfunction bounds. We start with well-known facts on the gaussian measure µ0. Proposition 3.1 (Free Gibbs measure: definition). Let h be as in (2.2) with V satisfying (2.3). Recall the spectral decomposition (2.15). Define a probability measure µ0,K on VK = span(u0, . . . , uK ) by setting K λ dµK(u) := j exp λ u, u 2 d u, u 0 π − j|h ji| h j i j=0 O where d u, uj = dajdbj and aj, bj are the real and imaginary parts of the scalar product. h i t There exists a unique probability measure µ0 over the space t<1/2−1/s H such that the measure µ0,K is the cylindrical projection of µ0 on VK for all K > 1. The corresponding k-particle density matrix T γ(k) := u⊗k u⊗k dµ (u)= k! (h−1)⊗k (3.1) 0 | ih | 0 Z belongs to Sp(H ) for all 1/s + 1/2 1 defines a unique measure µ0 on H if the tightness condition { } lim sup µ0,K u VK : u Ht > R = 0 (3.2) R→∞ K { ∈ k k } 4I.e. the sequence of its eigenvalues belongs to ℓp(N), see [45]. ANHARMONICOSCILLATORBASEDMEASURES 7 holds true. This is satisfied if Tr(ht−1) < since ∞ K −2 2 −2 t−1 −2 t−1 µ u V : u Ht > R 6 R u t dµ (u)= R λ 6 R Tr[h ]. 0,K { ∈ K k k } || ||H 0,K j ZVK j=1 X Applying the Lieb-Thirring inequality in [17, Theorem 1] to h = ∆+ V (x), we have − −p p −p p dx dk Tr h 6 2 Tr(h + λ0) 6 2 p R R 2πk 2 + V (x)+ λ Z Z | | 0 where λ > 0 is the lowest eigenvalue of h. Using V (x) > C−1 x s, we conclude that 0 | | Tr h−p < for all p> 1/s + 1/2. (3.3) ∞ Thus (3.2) holds true for all t < 1/2 1/s, and hence µ0 is well-defined (uniquely) over t − t<1/2−1/s H . The formula (3.1) follows from a direct calculation: T u⊗k u⊗k dµ (u) | ih | 0 Z k 1 u u u u = k! | i1 ⊗s ···⊗s ik ih i1 ⊗s ···⊗s ik | = k! (h−1)⊗k, λ 2 6 6 6 iℓ ! ui1 s s uik i1 iX2 ··· ik ℓY=1 || ⊗ ···⊗ || see [34, Lemma 3.3] for details. In order to make sense of the interacting measure, we need to prove that the gaussian measure is in fact supported on Lp spaces. Lemma 3.2 (Free Gibbs measure: support). r The gaussian measure µ0 constructed in Proposition 3.1 is supported on L (R) for every max(2, 4/s) 0 satisfying χ2 Lq(R) with 1/p + 1/q = 1, the operator χh−1χ is trace class and ∈ 2 −1 −1/2 2 Tr χh χ = h χ 6 C χ q R . (3.6) S2(H) L ( ) −1/2 Let us estimate the Hilbert-Schmidt norm of h χ. We pick some 0 <α< 1/2, write h−1/2χ = hα−1/2 h−α(1 ∂2)α (1 ∂2)−αχ (3.7) − x − x 8 M.LEWIN,P.T.NAM,ANDN.ROUGERIE and estimate the three factors separately. First, returning to (3.3) we have 1 1 1 hα−1/2 S2p(H) for 2p α > + . (3.8) ∈ 2 − s 2 Second, h > C−1(1 ∂2) as operators, for some constant C > 0. Indeed − x 1 h = ∂2 + V > ∂2 + λ − x 2 − x 0 with λ0 > 0 the lowest eigenvalue of h. Thus, using the operator-monotonicity [3, Theo- rem V.1.9] of x x2α for 0 < α 6 1/2, we deduce that 7→ h2α > C−2α(1 ∂2)2α, − x and thus h−α(1 ∂2)2αh−α 6 C2α. (3.9) − x −α 2 α In particular, h (1 ∂x) is a bounded operator for every α 6 1/2. Third, we aply the− Kato-Seiler-Simon inequality [45, Theorem 4.1] to get 1 dk 2q 2 −α 6 (1 ∂x) χ S2q H χ L2q(R) (3.10) − ( ) R (1 + 2πk 2)2αq || || Z | | when q > 1 and 4αq > 1. Combining (3.7) with (3.8), (3.9) and (3.10) we infer from H¨older’s inequality [45, Theorem 2.8] that5 −1/2 α−1/2 −α 2 α ∞ 2 −α h χ 6 h S2p h (1 ∂x) S (1 ∂x) χ S2q 6 C χ L2q (R) (3.11) S2(H) k k k − k k − k || || for 1/p + 1 /q = 1. The two constraints that 2p(1/2 α) > 1/s + 1/2 and 4αq > 1 require − 1 1 1 1 1 1 1 1 = α + α> + + = + , 2 2 − 2p s 2 4q 2ps 4 or equivalently 2 p> . s Thus (3.11), and hence (3.6), holds true for all p > max(1, 2/s). Note that (3.6) implies that h−1 is locally trace-class, which ensures that h−1(x; x) L1 (R) and ∈ loc 2 −1 2 −1 −1/2 2 h (x; x)χ (x)dx = Tr χh χ = h χ 6 C χ q R . R S2(H) L ( ) Z By duality, we conclude that x h−1(x; x) Lp (R) for all p> max(1 , 2 /s). 7→ ∈ r d Step 2. We deduce from the above that µ0 is supported on L (R ) for r> max(2, 4/s). We will use an interpolation argument in the spirit of Khintchine’s inequality (see, e.g. [12, Lemma 4.2]). Formally, when r = 2k is an even integer, by considering the diagonal of the kernels of operators in (3.1), we have u(x) 2kdµ (u)= k![h−1(x; x)]k. (3.12) | | 0 Z 5 || . ||S∞ stands for the operator norm. ANHARMONICOSCILLATORBASEDMEASURES 9 Then by interpolation, we get r −1 r u(x) dµ (u) 6 C [h (x; x)] 2 | | 0 r Z for all r > 2. The right side is integrable when r> max(2, 4/s) by Step 1. Now we go to the details with full rigor. Let PK be the projection onto VK = span(u0, ..., uK ). Using u , u dµ (u) = 0, u , u 2dµ (u)= λ−1 h j i 0 |h j i| 0 j Z Z we obtain 2 K K u (x) 2 P u(x) 2dµ (u)= u , u u (x) dµ (u)= | j | 6 h−1(x; x). | K | 0 h j i j 0 λ Z Z j=0 j=0 j X X More generally, when r = 2k is an even integer (k = 1, 2, 3, ...), by Wick’s theorem we can compute 1 2k k 2 K r P u(x) rdµ (u) = u , u u (x) dµ (u) | K | 0 h j i j 0 Z Z j=0 X K 2 uj(x) −1 6 Cr | | 6 Crh (x; x). (3.13) λj Xj=0 p By H¨older’s inequality in L spaces associated with the measure µ0, we can extend (3.13) to all r > 2. Then we rewrite this inequality as r −1 r P u(x) dµ (u) 6 C [h (x; x)] 2 | K | 0 r Z and integrate over x R. This gives ∈ r r −1 2 PKu Lr(R) dµ0(u) 6 Cr [h (x; x)] dx || || R Z Z where the right side is finite for r > max(2, 4/s). Passing to the limit K , we find → ∞ that u r R is finite µ -almost surely and || ||L ( ) 0 r r −1 2 u Lr(R) dµ0(u) 6 Cr [h (x; x)] dx. || || R Z Z Then, by Fernique’s theorem [18], there must exist a number αr > 0 such that (3.4) holds. As regards the interacting measure we deduce the following. Corollary 3.3 (Interacting Gibbs measure). Let h be as in (2.2) with V satisfying (2.3) and w be as in (2.4). Then the functional 1 2 2 u FNL[u]= u(x) w(x y) u(y) dxdy > 0 7→ 2 R R | | − | | ZZ × 10 M.LEWIN,P.T.NAM,ANDN.ROUGERIE 1 is in L (dµ0), F [u]dµ (u) < . NL 0 ∞ Z In particular, FNL[u] is finite µ0-almost surely. Thus, the measure defined by (2.13) makes t sense as a probability measure on t<1/2−1/s H and z = Texp ( F [u]) dµ (u) > 0. r − NL 0 Z Proof. Since w > 0 we have FNL[u] > 0 and it is sufficient to show that its integral with respect to µ0 is finite. Writing w = w1 +w2 as in (2.4), this follows immediately from (3.4) since 4 4 F [u] 6 w u 4 R + w p R u r R NL || 1||M || ||L ( ) || 2||L ( ) || ||L ( ) by Young’s inequality, with 4/r + 1/p = 2. 4. Hilbert-Schmidt estimate k We shall henceforth denote points in R in the manner Xk = (x1,...,xk) and denote dXk the corresponding Lebesgue measure. Very often we identify a Hilbert-Schmidt op- 2 k erator Ak on L (R ) with its integral kernel Ak(Xk; YK) (AkΨk)(Xk)= Ak(Xk; Yk)Ψk(Yk)dYk. Rk Z The main new estimate we need to put the proof strategy of [34] to good use is the following Proposition 4.1 (Bounds in Hilbert-Schmidt norm). (k) −1 Let the reduced density matrices Γλ,T be defined as in (2.11), with λ = T . Then we have the integral kernel estimate 6 (k) 6 (k) 0 Γλ,T (Xk; Yk) C Γ0,T (Xk; Yk). (4.1) Consequently 2 2 (k) 6 2 (k) 6 2 2k −2 k TrHk Γλ,T C TrHk Γ0,T C T Tr(h ) (4.2) for all k N. ∈ Note that the density matrices of the non-interacting Gibbs state Γ0,T are given by [34, Lemma 2.1] 1 ⊗k Γ(k) = 6 T k(h−1)⊗k. (4.3) 0,T eh/T 1 − Therefore, the second inequality in (4.2) follows immediately from the fact that h−1 S2(H), see Proposition 3.1. The first inequality in (4.2) follows from (4.1) and the well-∈ known fact that the L2-norm of the kernel is equivalent to the Hilbert-Schmidt norm of the operator, see e.g [38, Theorem VI.23]. It remains to prove (4.1). This is very much in the spirit of [9, Theorem 6.3.17], which is proved using a Feynman-Kac representation of reduced density matrices originating in [23, 24, 25] (see also [21, 22]). We certainly could obtain such a representation, in the ANHARMONICOSCILLATORBASEDMEASURES 11 spirit of [9, Theorem 6.3.14]. However, we do not need to go that far to obtain the desired bound: the Trotter product formula is sufficient for our purpose. Our proof of (4.1) is based on two useful lemmas. The first is essentially taken from [34, Lemma 8.1]. Lemma 4.2 (Bounds on partition functions). Let the partition function be defined as −1 Z = TrF exp T H . (4.4) λ,T − λ Then, for λ = T −1, we have Z 1 6 0,T 6 C (4.5) Zλ,T where the constant C > 0 is independent of T . Proof. Using w > 0, we have Hλ > H0, and hence −1 −1 Z = TrF exp T H 6 TrF exp T H = Z . λ,T − λ − 0 0,T On the other hand, since Γ minimizes the free energy functional (Γ) in (2.8), λ,T Fλ,T T log Z = (Γ ) 6 (Γ )= T log Z + λ Tr[wΓ(2) ]. − λ,T Fλ,T λ,T Fλ,T 0,T − 0,T 0,T Inserting (4.3) and λ = T −1 into the latter estimate, we conclude that Zλ,T −1 (2) −1 −1 log 6 λT Tr[wΓ0,T ] 6 Tr[w h h ] < . − Z0,T ⊗ ∞ Here the last estimate is taken from Corollary 3.3. The second lemma is a well-known comparison result for the heat kernels of Schr¨odinger operators on L2(Rn) (with no symmetrization). Lemma 4.3 (Heat kernel estimate). 2 n Consider two Schr¨odinger operators K = ∆Rn + W on L (R ), j = 1, 2, with W > j − j 1 W2 > 0. Then for all t> 0, we have the integral kernel estimate 0 6 exp( tK )(X ; Y ) 6 exp( tK )(X ; Y ) (4.6) − 1 n n − 2 n n for almost every (X ; Y ) Rn Rn. n n ∈ × Proof. This follows e.g. from the considerations of [47, Sec. II.6]. According to the Trotter product formula (see e.g. [38, Theorem VIII.30] or [47, Theorem 1.1]), we have, for any 2 n Ψn, Φn L (R ), ∈ m t∆Rn tWj Ψn exp( tKj) Φn = lim Ψn exp exp Φn . h | − | i m→∞ m − m In terms of integral kernels this means Ψ (X ) exp( tK )(X ; Y )Φ (Y )dX dY n n − j n n n n n n Z 1 t∆Rn 1 tWj(Zn) = lim Ψn(Xn)exp (Xn; Z )exp . . . m→∞ m n − m Z t∆Rn tW (Y ) exp (Zm−1; Y )exp j n Φ (Y )dX dZ1 . . . dZm−1dY m n n − m n n n n n n 12 M.LEWIN,P.T.NAM,ANDN.ROUGERIE k k k Rn where the Zn = (z1 ,...,zn) are auxiliary sets of variables in that we integrate over. Therefore, we can specialize to nonnegative functions Ψn, Φn and obtain 0 6 Ψ (X ) exp( tK )(X ; Y )Φ (Y )dX dY n n − 1 n n n n n n Z 6 Ψ (X ) exp( tK )(X ; Y )Φ (Y )dX dY . (4.7) n n − 2 n n n n n n Z t n Here we have used the fact that the heat kernel exp( m ∆R )(Xn; Yn) is positive and tW tW 0 6 exp 1 6 exp 2 pointwise. − m − m There remains to let Ψn, Φn converge to delta functions in (4.7) to conclude the proof. Now we can give the Proof of Proposition 4.1. Our bosonic state Γλ,T can be written in the unsymmetrized Fock space in the manner ∞ 1 Γ = P n exp T −1H (4.8) λ,T Z sym − n λ,T n=0 M with the symmetric projector 1 P n = U . sym n! σ σX∈Sn Here the sum is over the permutation group Sn and Uσ is the unitary operator permuting n −1 2 Rn variables according to σ. We consider Psym exp T Hn as an operator on L ( ). Note that P n commutes with H and, in terms of integral− kernels, sym n 1 P n exp T −1H (X ; Y )= exp T −1H (σ X ; Y ) sym − n n n n! − n · n n σ∈Sn X where σ Xn = (xσ(1), ..., xσ(n)) are the permuted variables. By applying· Lemma 4.3 to the potentials n n W1(Xn)= V (xj)+ λ w(xi xj) > V (xj)= W2(Xn) 6 6 − Xj=1 1 Xi n exp T −1H (X ; Y ) 6 exp T −1 h (X ; Y ). − n n n − j n n j=1 X Since the kernel estimate remains unchanged by the symmetrization 6, we have n P n exp T −1H (X ; Y ) 6 P n exp T −1 h (X ; Y ). (4.9) sym − n n n sym − j n n j=1 X 6 n Which would not be true if we were dealing with fermions, i.e. Psym was replaced by the antisymmetric projector. ANHARMONICOSCILLATORBASEDMEASURES 13 (k) Finally, by Definition (2.10), the integral kernel of Γλ,T is given by 1 n Γ(k) (X ; Y )= P n exp T −1H (X ,Z ; Y ,Z )dZ λ,T k k Z k sym − n k n−k k n−k n−k λ,T n>k Z X with Z = (z ,...,z ) Rn−k. Inserting (4.6) into the latter formula, we thus obtain n−k k+1 n ∈ Z 6 (k) 6 0,T (k) 6 (k) 0 Γλ,T (Xk; Yk) Γ0,T (Xk; Yk) CΓ0,T (Xk; Yk). Zλ,T Here the last estimate follows from Lemma 4.2. 5. Proof of the main theorem As in [34], our strategy is based on Gibbs’ variational principle, which states that Γλ,T minimizes the free energy functional [Γ] in (2.8). It follows from a simple computation Fλ,T that Γλ,T is also the unique minimizer for the relative free energy functional: Z (Γ ) (Γ ) log λ,T = Fλ,T λ,T − F0,T 0,T − Z0,T T −2 = inf (Γ, Γ0,T )+ T Tr[wΓλ,T ] . (5.1) Γ>0 H TrF Γ=1 Here ′ ′ (Γ, Γ ) = TrF Γ(log Γ log Γ ) > 0 H − is called the relative entropy [37, 54] of two states Γ and Γ′. We will relate the quantum problem (5.1) to its classical version: The interacting Gibbs measure µ is the unique minimizer for the variational problem 1 ⊗2 ⊗2 log zr = inf cl(ν,µ0)+ u , wu dν(u) (5.2) − ν probability measure H 2 h i Z where7 dν dν ′ ′ > cl(ν,ν ) := ′ (u) log ′ (u) dν (u) 0 H s dν dν ZH is the classical relative entropy of two probability measures ν and ν′. Note that cl(ν,µ0)=+ unless ν is absolutely continuous with respect to µ0, and the other term ofH the functional∞ is positive. Thus the minimization above is amongst measures 4 of the form dν(u) = f(u)dµ0(u) that all live over L (R) as per Lemma 3.2. Hence the variational problem makes sense. To see that µ is the unique minimizer, one argues exactly as in (2.9). ′ 7Positivity of this quantity follows from Jensen’s inequality. It is zero if and only if ν = ν . 14 M.LEWIN,P.T.NAM,ANDN.ROUGERIE 5.1. Convergence of the relative partition function. Let us prove (2.16). We recall the following result from [34, Lemma 8.3]. Lemma 5.1 (Free-energy upper bound ). Let h> 0 satisfy h−1 Sp(H) for some 1 6 p< and let w > 0 satisfy ∈ ∞ −1 −1 TrH w h h < . 2 ⊗ ∞ Then we have h i Z lim sup log λ,T 6 log z . (5.3) − Z − r T →∞ 0,T Note that −1 −1 −1 −1 TrH2 w h h = w(x y) h (x; x) h (y; y) dx dy ⊗ R×R − h i ZZ is finite by the proof of Proposition 3.1, under our assumptions on h and w. Therefore, the upper bound 5.3 holds true. The main difficulty is to establish the matching lower bound. To do this, we need two tools from [34]. The first one is a variant of the quantum de Finetti Theorem in Fock space [34, The- orem 4.2] (whose proof goes back to the analysis of [1, 33], see [41, 42] for a general presentation). Theorem 5.2 (Quantum de Finetti theorem in Schatten classes). Let Γn be a sequence of states on the bosonic Fock space F, namely Γn is a self-adjoint { } + operator with Γn > 0 and TrF Γn = 1. Assume that there exists a sequence εn 0 such that → p pk (k) (ε ) TrH Γ 6 C < , (5.4) n k n k ∞ for some 1 6 p< and for all k > 1.h Let h> i0 be a self-adjoint operator on H with ∞ −p TrH[h ] < (5.5) ∞ and H1−p the associated Sobolev space (2.14). Then, up to a subsequence of Γ , there exists a Borel probability measure ν on H1−p { n} (invariant under multiplication by a phase factor), called the de Finetti measure of Γn at scale εn, such that k (k) ⊗k ⊗k k!(εn) Γn ⇀ u u dν(u) (5.6) 1−p | ih | ZH weakly- in Sp(H ) for every k > 1. ∗ k Proof. This follows straightforwardly from [34, Theorem 4.2]. Using (5.4), (5.5) and the H¨older inequality in Schatten spaces, one readily checks that Assumption (4.7) of [34, Theorem 4.2] is satisfied for all integer s. Convergence of density matrices, along a subse- quence, to the right-hand side of (5.6) in a weaker topology is then Statement (4.9) of [34, Theorem 4.2]. Passing to a further subsequence, (5.4) allows to get weakly- convergence p ∗ in S (Hk). The second tool is a link between the quantum relative entropy and the classical one, taken from [34, Theorem 7.1] (this is a Berezin-Lieb-type inequality, its proof goes back to the techniques in [2, 35, 46]). ANHARMONICOSCILLATORBASEDMEASURES 15 Theorem 5.3 (Relative entropy: quantum to classical). ′ Let Γn and Γn be two sequences of states on the bosonic Fock space F. Assume that { } { } + they satisfy the assumptions of Theorem 5.2 with the same scale εn 0 and the same power p > 1. Let µ and µ′ be the corresponding de Finetti measures.→ Then ′ ′ lim inf (Γn, Γ ) > cl(µ,µ ). n→∞ H n H Now we are ready to prove a lower bound to the relative free energy matching the upper bound of Lemma 5.1. Lemma 5.4 (Free-energy lower bound ). With the notation and assumptions of Theorem 2.1 we have Zλ,T lim inf log > log zr. (5.7) T →∞ − Z − 0,T Proof. We pass to the liminf first in the relative entropy and then in the interaction energy. Step 1. From the Hilbert-Schmidt estimate (4.2) in Proposition 4.1, we can apply The- −1 orem 5.2 to the sequence Γλn,Tn for any Tn , with scale εn = Tn . Thus, up to { } → ∞ −1 a subsequence of Γλn,Tn , there exists is a Borel probability measure ν on H (the de Finetti measure for{ Γ } ) such that { λn,Tn } k! Γ(k) ⇀ u⊗k u⊗k dν(u) (5.8) k λn,Tn T −1 | ih | n ZH 2 weakly in S (Hk) for every k > 1. Next, from (4.3) and (3.1), by Lebesgue’s Dominated Convergence Theorem we find that k! Γ(k) u⊗k u⊗k dµ (u) T k 0,T → | ih | 0 Z 2 strongly in S (Hk) for every k > 1. In particular, the free Gibbs measure µ0 is the de −1 Finetti measure for the sequence Γ0,Tn with scale εn = Tn . Therefore, Lemma 5.3 implies that { } lim inf (Γλ ,T , Γ0,T ) > cl(ν,µ0). (5.9) n→∞ H n n n H Consequently, cl(ν,µ0) is finite and thus ν is absolutely continuous with respect to µ0. In particular, νHis supported on L4(R) by Lemma 3.2. Step 2. From Lemma 5.1 and the variational principle, it follows that −2 1/2 1/2 Tn Tr w Γλn,Tn w 6 C h i and thus the positive operator T −2w1/2Γ w1/2 has a trace-class weak- limit along a n λn,Tn ∗ subsequence. Using (5.8) with k = 2 to identify the limit and Fatou’s lemma for operators8, we get −2 1 ⊗2 ⊗2 lim inf T Tr [wΓλ ,T ] > u , w u dν(u). (5.10) n→∞ n n n 2 h i Z Note that on the right side of (5.10), u⊗2, w u⊗2 is finite when u Lr(R) for max(2, 4/s) Putting (5.9) and (5.10) together, then combining with (5.1) and (5.2), we arrive at Zλn,Tn −2 lim inf log = lim inf (Γλ ,T , Γ0,T )+ T Tr wΓλ ,T n→∞ − Z n→∞ H n n n n n n 0,Tn 1 h i > (ν,µ )+ u⊗2, wu⊗2 dν(u) > log z . (5.11) Hcl 0 2 h i − r Z From (5.11) and the upper bound (5.3), we conclude that Zλn,Tn 1 ⊗2 ⊗2 lim inf log = cl(ν,µ0)+ u , wu dν(u)= log zr. (5.12) n→∞ − Z H 2 h i − 0,Tn Z Since the interacting Gibbs measure µ is the unique minimizer for (5.2), we deduce from (5.12) ν = µ. Moreover, we can remove the dependence of the subsequence Tn in (5.12) and (5.8) since the limiting objects are unique, and thus obtain the corresponding convergences for the whole family, namely Zλ,T lim log = log zr, (5.13) T →∞ − Z − 0,T which is equivalent to (2.16), and k! Γ(k) ⇀ u⊗k u⊗k dµ(u) (5.14) T k λ,T | ih | Z 2 weakly in S (Hk) for every k > 1. To complete the proof of Theorem 2.1 we now onmy need to upgrade the last convergence from weak to strong. 5.2. Strong convergence of density matrices. There remains to upgrade the weak convergence in (5.14) to the strong convergence. Case k = 1. For the one-body density matrix, the strong convergence follows from the Dominated Convergence Theorem (for operators), the weak convergence in (5.8) and the following estimate in [34, Lemma 8.2] (whose proof is based on a Feynman-Hellmann argument). Lemma 5.5 (Operator bound on the one-particle density matrix). Let h> 0 satisfy h−1 Sp(H) for some p > 1 and let w > 0 satisfy ∈ −1 −1 Tr 2 w h h < . H ⊗ ∞ Then we have h i 6 (1) 6 −1 0 Γλ,T CT h . (5.15) Case k > 2. In this case an analogue of (5.15) is not available. Instead, we will use kernel estimates. Recall that from Proposition 4.1 we know that (k) (k) Γ (Xk; Yk) Γ (Xk; Yk) 0 6 λ,T 6 C 0,T (5.16) T k(k!) k T k(k!) ANHARMONICOSCILLATORBASEDMEASURES 17 −k (k) −1 ⊗k pointwise. Moreover, since T Γ0,T converges strongly to h in the Hilbert-Schmidt norm, its kernel converges strongly in L2. It easily follows, using the Cauchy-Schwarz inequality, that (k) 2 Γ0,T (Xk; Yk) −1 ⊗k 2 2k 2 h (Xk; Yk) dXkdYk 0. (5.17) Rk Rk T (k!) − T−→→∞ Z × The function h−1 ⊗k ( X ; Y )2 is in L1(Rk Rk): it is positive and we easily check k k × h−1(x; y)2dxdy = Tr h−2 < R R ∞ ZZ × −k (k) by Proposition 3.1. Therefore, if we can show that the kernel T Γλ,T (Xk; Yk) converges pointwise, then it converges strongly in L2 by Lebesgue’s Dominated Convergence Theo- −k (k) rem (see the remark following [36, Theorem 1.8]). Then the operator T Γλ,T will converge strongly in the Hilbert-Schmidt norm, as desired. −k (k) To prove that the kernel T Γλ,T (Xk; Yk) converges pointwise, it suffices to show that −k ⊗k (k) ⊗k the operator T χ Γλ,T χ converges strongly in the Hilbert-Schmidt norm when χ is a characteristic function of a ball. Indeed, we will prove a stronger statement Lemma 5.6 (Local trace class convergence of density matrices). −k ⊗k (k) ⊗k Let χ be the characteristic function of a ball. Then T χ Γλ,T χ converges strongly in the trace class for all k > 1. Proof. From the kernel estimate (5.16), we have k T −k Tr χ⊗kΓ(k) χ⊗k 6 CT −k Tr χ⊗kΓ(k) χ⊗k 6 C Tr χh−1χ < . λ,T 0,T ∞ h i h i Recall that we have shown during the proof of Lemma 3.2 that Tr χh−1χ < for χ ∞ −k ⊗k (k) ⊗k a characteristic function. Thus T χ Γλ,T χ is bounded in trace class, and hence the weak convergence in (5.8) implies that k! ⊗k (k) ⊗k ⊗k ⊗k k χ Γλ,T χ ⇀ (χu) (χu) dµ(u) (5.18) T 1−p | ih | ZH weakly- in trace-class norm9. There∗ remains to show that the convergence in (5.18) is strong in the trace class. In the case k = 1, the strong convergence again follows from the Dominated Convergence Theorem (for operators) and the operator bound from Lemma 5.5: 0 6 T −1χΓ(1) χ 6 Cχh−1χ S1(H). λ,T ∈ In the case k > 2, we use a general observation which has its own interest, Lemma 5.7 below. We postpone the proof of this result and finish that of Lemma 5.6. Using the Fock space isomorphism (L2(R)) = χL2(R) (1 χ)L2(R) χL2(R) (1 χ)L2(R) F F ⊕ − ≃ F ⊗ F − 9 2 4 On the right side of (5.18), χu ∈ L when u ∈ Supp µ ⊂ Supp µ0 ⊂ L (R). 18 M.LEWIN,P.T.NAM,ANDN.ROUGERIE we can define the localized state Γ on (χL2(R)) by taking the partial trace of Γ λ,T F λ,T over ((1 χ)L2(R)). The density matrices of the localized state Γ are given by F − λ,T (ke) ⊗k (k) ⊗k Γλ,T = χ Γ χ , k > 1. e λ,T ∀ This localization procedure is well-known for many-particle quantum systems; see for e instance [27, Appendix A], [31] or [41, Chapter 5] for more detailed discussions. Applying Lemma 5.7 with (εn, Γn) replaced by (1/T, Γλ,T ), we obtain the desired con- clusion of Lemma 5.6. e The general lemma we used above is as follows: Lemma 5.7 (Strong convergence of higher density matrices). Let H be a separable Hilbert space and let Γ be a sequence of states on the bosonic Fock { n} space F(H). Assume that there exists a sequence 0 < ε 0 and operators γ(k) such that n → k (k) (k) (εn) Γn ⇀ γ (5.19) weakly- in trace class on k H for all k N. If the convergence (5.19) holds strongly ∗ sym ∈ in trace class for k = 1, then it holds strongly in trace class for all k N. N ∈ The equivalent of this lemma for states with a fixed number of particles is a straight- forward consequence of the weak quantum de Finetti theorem [33, Section 2]. Proof. The strong convergence in (5.19) follows from the fact that k (k) (k) lim sup(εn) Tr[Γn ] 6 Tr γ . (5.20) n→∞ We will show that if (5.20) holds for k = 1, then it holds for all k > 2. Let 0 6 P 6 1 be a finite rank projection on H and let Q = 1 P . We can decompose − 1⊗k = Q 1⊗k−1 + P 1⊗k−1 ⊗ ⊗ = Q 1⊗k−1 + P Q 1⊗k−2 + P ⊗2 1⊗k−2 = ⊗ ⊗ ⊗ ⊗ · · · = Q 1⊗k−1 + P Q 1⊗k−2 + P ⊗2 Q 1⊗k−3 + + P ⊗k−1 Q + P ⊗k ⊗ ⊗ ⊗ ⊗ ⊗ · · · ⊗ 6 Q 1⊗k−1 + 1 Q 1⊗k−2 + 1⊗2 Q 1⊗k−3 + + 1⊗k−1 Q + P ⊗k. ⊗ ⊗ ⊗ ⊗ ⊗ · · · ⊗ Therefore, (ε )k Tr Γ(k) 6 (ε )k Tr P ⊗kΓ(k) + k(ε )k Tr Q 1⊗k−1 Γ(k) . (5.21) n n n n n ⊗ n Now we estimateh thei right side ofh (5.21).i The weak convergench e in (5.19) i implies that k ⊗k (k) ⊗k (k) (k) lim (εn) Tr P Γ = Tr P γ 6 Tr γ . (5.22) n→∞ n To estimate the second term on theh right sidei ofh (5.21), wei use the definition m Γ(k) = Tr Gm n k k+1→m n > mXk m m with Gn the projection of Γn onto sym H, namely Γ =NG0 G1 G2 , n n ⊕ n ⊕ n ⊕··· ANHARMONICOSCILLATORBASEDMEASURES 19 and Tr Gm is the partial trace of Gm with respect to m k variables10. In particular, k+1→m n n − m (ε )k Tr Q 1⊗k−1 Γ(k) = (ε )k Tr [(Q 1) Tr Gm] n n n k 2→m n ⊗ > ⊗ h i mXk 6 (ε m)k Tr [(Q 1) Tr Gm] . n ⊗ 2→m n m>2 X Let M > 1 and divide the sum into two parts: εnm 6 M and εnm > M. Then, using k−1 M (εnm) if εnm 6 M, (ε m)k 6 n −1 k+1 (M (εnm) if εnm > M, we can estimate (ε )k Tr (Q 1⊗k−1)Γ(k) n ⊗ n 6 M k−1h (ε m) Tr [(Qi 1) Tr Gm]+ M −1 (ε m)k+1 Tr [Gm] n ⊗ 2→m n n n m>2 m>2 X X 6 M k−1ε Tr QΓ(1) + M −1 Tr (ε )k+1 Γ . n n nN n h i h i (1) Here is the usual number operator on the Fock space H. Since εnΓn converges strongly in traceN class, we get (1) (1) lim εn Tr QΓ = Tr Qγ . n→∞ n ℓ (ℓ) h i h i On the other hand, since (εn) Γn converges weakly- in trace class, its trace is bounded uniformly in n. Combining with the identity ∗ (ℓ) Tr Γ = Tr H N Γ , ℓ > 1 n F( ) ℓ n ∀ we find that ℓ lim sup Tr (εn ) Γn 6 Cℓ, ℓ > 1 n→∞ N ∀ h i for a constant Cℓ independent of n. Thus we have shown that k 1⊗k−1 (k) k−1 (1) Ck lim sup(εn) Tr (Q )Γn 6 M Tr Qγ + . (5.23) n→∞ ⊗ M h i h i In summary, inserting (5.22) and (5.23) into (5.21) we obtain k (k) (k) k−1 (1) kCk lim sup(εn) Tr Γn 6 Tr γ + kM Tr Qγ + n→∞ M h i h i for all projections Q, all M > 1 and all k > 2. It remains to take P 1, then M , to conclude that → → ∞ k (k) (k) lim sup(εn) Tr[Γn ] 6 Tr γ . n→∞ The proof is complete. 10No matter which, by bosonic symmetry. 20 M.LEWIN,P.T.NAM,ANDN.ROUGERIE By the same proof, we can show that if (5.19) holds weakly- in trace class for all 1 6 k 6 κ and strongly in trace class for k = 1, then it holds strongly∗ in trace class for all 1 6 k 6 κ 1. − References [1] Z. Ammari and F. Nier, Mean field limit for bosons and infinite dimensional phase-space analysis, Ann. Henri Poincar´e, 9 (2008), pp. 1503–1574. [2] F. A. Berezin, Convex functions of operators, Mat. Sb. (N.S.), 88(130) (1972), pp. 268–276. [3] R. Bhatia, Matrix analysis, vol. 169, Springer, 1997. [4] J. Bourgain, Periodic nonlinear Schr¨odinger equation and invariant measures, Comm. Math. Phys., 166 (1994), pp. 1–26. [5] , Invariant measures for the 2D-defocusing nonlinear Schr¨odinger equation, Comm. Math. Phys., 176 (1996), pp. 421–445. [6] , Invariant measures for the Gross-Pitaevskii equation, Journal de Math´ematiques Pures et Appliqu´ees, 76 (1997), pp. 649–02. [7] J. Bourgain and A. Bulut, Almost sure global well posedness for the radial nonlinear Schr¨odinger equation on the unit ball I: the 2D case, Annales I. H. Poincare (C), 31 (2014), pp. 1267–1288. [8] , Almost sure global well posedness for the radial nonlinear Schr¨odinger equation on the unit ball II: the 3D case, Journal of the European Mathematical Society, 16 (2014), pp. 1289–1325. [9] O. Bratelli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 2: Equi- librium States. Models in Quantum Statistical Mechanics, Texts and Monographs in Physics, Springer, 2nd ed., 2002. [10] N. Burq, L. Thomann, and N. Tzvetkov, Gibbs measures for the non linear harmonic oscillator, 1 in Journ´ees EDP ¨ı¿ 2 vian 2009., 2009. [11] , Long time dynamics for the one dimensional non linear Schr¨odinger equation, Ann. Inst. Fourier., 63 (2013), pp. 2137–2198. [12] N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations. I. Local theory, Invent. Math., 173 (2008), pp. 449–475. [13] F. Cacciafesta and A.-S. de Suzzoni, Invariant measure for the Schr¨odinger equation on the real line, J. Func Anal, 269 (2015), pp. 271–324. [14] G. da Prato and A. Debbussche, Strong solutions to the stochastic quantization equations, Ann. Probab., 32 (2003), pp. 1900–1916. [15] A. de Bouard, A. Debussche, and R. Fukuizumi, Long time behavior of Gross-Pitaevskii equation at positive temperature. arXiv:1708.01961, 2017. [16] J. Derezinski´ and C. Gerard´ , Mathematics of Quantization and Quantum Fields, Cambridge Uni- versity Press, Cambridge, 2013. [17] J. Dolbeault, P. Felmer, M. Loss, and E. Paturel, Lieb-Thirring type inequalities and Gagliardo-Nirenberg inequalities for systems, J. Funct. Anal., 238 (2006), pp. 193–220. [18] X. Fernique, Int´egrabilit´edes vecteurs gaussiens, C. R. Acad. Sci. Paris S´er. A-B, 270 (1970), pp. A1698–A1699. [19] J. Frohlich,¨ A. Knowles, B. Schlein, and V. Sohinger, Gibbs measures of nonlinear Schr¨odinger equations as limits of quantum many-body states in dimensions d 6 3. arXiv:1605.07095, 2016. [20] , A microscopic derivation of time-dependent correlation functions of the 1D cubic nonlinear Schr¨odinger equation. arXiv:1703.04465, 2017. [21] J. Frohlich¨ and Y. M. Park, Correlation Inequalities and the Thermodynamic Limit for Classical and Quantum Continuous Systems, Commun. Math. Phys., 59 (1978), pp. 235–266. [22] , Correlation Inequalities and the Thermodynamic Limit for Classical and Quantum Continuous Systems II. Bose-Einstein and Fermi-Dirac Statistics, J. Stat. Phys., 23 (1980), pp. 701–753. [23] J. Ginibre, Reduced density matrices for quantum gases I. Limit of infinite volume, J. Math. Phys., 6 (1965), pp. 238–251. [24] , Reduced density matrices for quantum gases II. Cluster Property, J. Math. Phys., 6 (1965), pp. 252–262. ANHARMONICOSCILLATORBASEDMEASURES 21 [25] , Reduced density matrices for quantum gases II. Hard-core potentials, J. Math. Phys., 6 (1965), pp. 1432–1446. [26] J. Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View, Springer-Verlag, 1987. [27] C. Hainzl, M. Lewin, and J. P. Solovej, The thermodynamic limit of quantum Coulomb systems. Part II. Applications, Advances in Math., 221 (2009), pp. 488–546. [28] H. Koch and D. Tataru, Lp eigenfunction bounds for the Hermite operator, Duke Math. J., 128 (2005), pp. 369–392. [29] H. Koch, D. Tataru, and M. Zworski, Semiclassical Lp estimates, Ann. Henri Poincar´e, 8 (2007), pp. 885–916. [30] J. L. Lebowitz, H. A. Rose, and E. R. Speer, Statistical mechanics of the nonlinear Schr¨odinger equation, J. Statist. Phys., 50 (1988), pp. 657–687. [31] M. Lewin, Geometric methods for nonlinear many-body quantum systems, J. Funct. Anal., 260 (2011), pp. 3535–3595. [32] M. Lewin, P. Nam, and N. Rougerie, Bose gases at positive temperature and non-linar Gibbs measures, Preprint (2016) arXiv:1602.05166. [33] , Derivation of Hartree’s theory for generic mean-field Bose systems, Adv. Math., 254 (2014), pp. 570–621. [34] , Derivation of nonlinear Gibbs measures from many-body quantum mechanics, Journal de l’Ecole Polytechnique, 2 (2016), pp. 553–606. [35] E. H. Lieb, The classical limit of quantum spin systems, Comm. Math. Phys., 31 (1973), pp. 327–340. [36] E. H. Lieb and M. Loss, Analysis, vol. 14 of Graduate Studies in Mathematics, American Mathe- matical Society, Providence, RI, 2nd ed., 2001. [37] M. Ohya and D. Petz, Quantum entropy and its use, Texts and Monographs in Physics, Springer- Verlag, Berlin, 1993. [38] M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional analysis, Academic Press, 1972. [39] , Methods of Modern Mathematical Physics. II. Fourier analysis, self-adjointness, Academic Press, New York, 1975. [40] M. Rockner, R. Zhu, and X. Zhu, Ergodicity for the stochastic quantization problems on the 2D- torus. arXiv:1606.02102, 2016. [41] N. Rougerie, De Finetti theorems, mean-field limits and Bose-Einstein condensation. arXiv:1506.05263, 2014. LMU lecture notes. [42] , Th´eor`emes de de Finetti, limites de champ moyen et condensation de Bose-Einstein. arXiv:1409.1182, 2014. Lecture notes for a cours Peccot. [43] , From bosonic grand-canonical ensembles to nonlinear Gibbs measures, 2014-2015. S´eminaire Laurent Schwartz. [44] B. Simon, The P (Φ)2 Euclidean (quantum) field theory, Princeton University Press, Princeton, N.J., 1974. Princeton Series in Physics. [45] , Trace ideals and their applications, vol. 35 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1979. [46] , The classical limit of quantum partition functions, Comm. Math. Phys., 71 (1980), pp. 247–276. [47] , Functional integration and quantum physics, AMS Chelsea Publishing, Providence, RI, sec- ond ed., 2005. [48] A. Skorokhod, Integration in Hilbert space, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, 1974. [49] L. Thomann and T. Oh, Invariant Gibbs measures for the 2D defocusing nonlinear Schr¨odinger equations. arXiv:1509.02093, 2015. [50] L. Thomann and N. Tzvetkov, Gibbs measure for the periodic derivative nonlinear Schr¨odinger equation, Nonlinearity, 23 (2010), p. 2771. [51] P. Tsatsoulis and H. Weber, Spectral gap for the stochastic quantization equation on the 2- dimensional torus. arXiv:1609.08447, 2016. [52] N. Tzvetkov, Invariant measures for the defocusing nonlinear Schr¨odinger equation, Ann. Inst. Fourier (Grenoble), 58 (2008), pp. 2543–2604. 22 M.LEWIN,P.T.NAM,ANDN.ROUGERIE [53] G. Velo and A. Wightman, eds., Constructive quantum field theory: The 1973 Ettore Majorana international school of mathematical physics, Lecture notes in physics, Springer-Verlag, 1973. [54] A. Wehrl, General properties of entropy, Rev. Modern Phys., 50 (1978), pp. 221–260. 1 [55] K. Yajima and G. Zhang, Smoothing property for Schr¨ı¿ 2 dinger equations with potential su- perquadratic at infinity, Comm. Math. Phys., 221 (2001), pp. 537–590. CNRS & CEREMADE, Universite´ Paris-Dauphine, PSL Research University, Place de Lat- tre de Tassigny, F-75016 PARIS, France E-mail address: [email protected] Masaryk University, Department of Mathematics and Statistics, Kotla´rskˇ a´ 2, 61137 Brno, Czech Republic E-mail address: [email protected] Universite´ Grenoble 1 & CNRS, LPMMC (UMR 5493), B.P. 166, F-38042 Grenoble, France E-mail address: [email protected]