A microscopic derivation of Gibbs measures for the 1D focusing cubic nonlinear Schrödinger equation

Abstract In this paper, we give a microscopic derivation of Gibbs measures for the focusing cubic nonlinear Schrödinger equation on the one-dimensional torus from many-body quantum Gibbs states. Since we are not making any positivity assumptions on the interaction, it is necessary to introduce a truncation of the mass in the classical setting and of the rescaled particle number in the quantum setting. Our methods are based on a perturbative expansion of the interaction, similarly as in [1]. Due to the presence of the truncation, the obtained series have infinite radius of convergence. We treat the case of bounded, L 1 and delta function interaction potentials, without any sign assumptions. Within this framework, we also study time-dependent correlation functions. This is the first such known result in the focusing regime.

Here, dϕ denotes Lebesgue measure on h (which is ill-defined when h is infinitedimensional) and z Gibbs is the partition function, i.e. the normalisation constant which makes dP Gibbs into a probability measure on h.The problem of the rigorous construction of measures of type (1.1) was first considered in the constructive quantum field theory literature.For an overview, see the classical works [23,40,41,49] and the later works [2,11,12,14,25,31,37,38,44], as well as the references therein.When H is not positive-definite, it is sometimes not possible to define (1.1) with finite normalisation constant z Gibbs , as formally one can have e −H(ϕ) dϕ = ∞.Instead, one considers a modification of (1.1) given by where f is a suitable cut-off function in C ∞ c (R), and z f Gibbs is a normalisation constant that makes (1.2) a probability measure on h.In general, when H is not positive-definite, we say that we are in the focusing (or thermodynamically unstable) regime.
In this paper, we fix the spatial domain T ≡ T 1 ≡ R/Z ≡ [− 1  2 , 1 2 ) to be the one-dimensional torus 1 .We henceforth consider the Hilbert space h := L 2 (T; C) ≡ L 2 (T).Let us now define the precise Hamiltonian.We make the following assumption.
Assumption 1.1 (The interaction potential).We consider an interaction potential which is of one of the following types.
(i) w : T → R is even and belongs to L 1 (T).
(ii) w = −δ, where δ is the Dirac delta function.Let us note that, in Assumption 1.1, we do not assume any conditions on the sign of w or the sign of ŵ (pointwise almost everywhere).
1 Some of our results generalise to other domains; see Remark 1.7 (3) below.For simplicity, we work on T 1 .
2 When one has suitable positivity (in other words defocusing) assumptions on w, the analysis of the problem we are considering for (1.5) has already been done in [17]; see Section 1.5 below for an overview.Our main interest lies in the case when these assumptions are relaxed, which we refer to as the focusing regime.
The main goal of this paper is to show that one can obtain (1.8) as a mean-field limit of corresponding many-body quantum objects, which we henceforth refer to as a microscopic derivation.We do this in two steps. (i) Step 1: Analysis of the time-independent problem, i.e. when Step 2: Analysis of the time-dependent problem.This is the general case.The precise results are stated in Section 1. 4. In Section 1.2, we define the objects with which we work in the classical setting.In Section 1.3, we define the objects with which we work in the quantum setting.
1.2.The Classical Problem.The one-particle space on which we work is h = L 2 (T).We use the following convention for the scalar product.g 1 , g 2 h := dx g 1 (x) g 2 (x) .
We consider the one-body Hamiltonian given by where κ > 0 is as in (1.3).This is a positive self-adjoint densely defined operator on h.We can write h spectrally as where λ k := 4π 2 |k| 2 + κ (1.11) are the eigenvalues of h and u k := e 2πikx (1.12) are the normalised eigenvalues of h on h.Since we are working on T, we have where the trace is taken over h.
For each k ∈ N, we define µ k to be a standard complex Gaussian measure.In other words, µ k := 1 π e −|z| 2 dz, where dz is the Lebesgue measure on C. Let (C N , G, µ) be the product probability space with µ := k∈N µ k . (1.14) We denote elements of the probability space C N by ω = (ω k ) k∈N .Let the classical free field ϕ ≡ ϕ ω be defined by Note that (1.13) implies (1.15) converges almost surely in H 1 2 −ε (T) for ε > 0 arbitrarily small.Here H s (T) denotes the L 2 -based Sobolev space of order s on T with norm given by We take the following convention for the Fourier transform on L 1 (T).ĝ(k) := dx g(x)e −2πikx , k ∈ Z . (1.16) The measure µ satisfies the following Wick theorem; see for example [19,Lemma 2.4] for a self-contained summary.
We note that, by gauge invariance, for all (i, j) ∈ Π Therefore, each non-zero factor arising on the right-hand side of (1.17) can be computed using Here, the Green function h −1 is the covariance of µ.We note that, under a suitable pushforward, we can identify µ with a probability measure on H s ; see e.g.[17,Remark 1.3].As in [17], we work directly with the measure µ as above and do not use this identification.
Given w as in Assumption 1.1, we define the classical interaction as The free classical Hamiltonian is given by The interacting classical Hamiltonian is given by The mass is defined as At this stage, we have to introduce the cut-off f that appears in (1.2).We now state the precise assumptions on f that we use in the sequel.
for some K > 0.
All of our estimates depend on K in (1.23), but we do not track this dependence explicitly.
We define the classical state where X is a random variable.Let the classical partition function z ≡ z f Gibbs be defined as z := dµ e −W f (N ) . (1.25) Note that both ρ and z are well defined by Lemma 2.1 and Corollary 2.4 below.We characterise ρ(•) through its moments.Namely, we define the classical p-particle correlation function γ p ≡ γ f p , which acts on h (p) through its kernel γ p (x 1 , . . ., x p ; y 1 , . . ., y p ) := ρ(ϕ(y 1 ) . . .ϕ(y p ) ϕ(x 1 ) . . .ϕ(x p )) . (1.26) 1.3.The Quantum Problem.We use the same conventions as in [17,Section 1.4].For more details and motivation, we refer the reader to the aforementioned work.In the quantum setting, we work on the bosonic Fock space, which is defined as Let us denote vectors of F by Ψ = (Ψ (p) ) p∈N .For g ∈ h, let b * (g) and b(g) denote the bosonic creation and annihilation operators, defined respectively as These are closed, densely-defined operators which are each other's adjoints.The creation and annihilation operators satisfy the canonical commutation relations, i.e. [b(g for all g 1 , g 2 ∈ h.Furthermore, we define the rescaled creation and annihilation operators for g ∈ h.Here, we think of ϕ * τ and ϕ τ as operator valued distributions, and we denote their distribution kernels by ϕ * τ (x) and ϕ τ (x), respectively.Formally, ϕ * τ (x) and ϕ τ (x) correspond to taking g = δ x (the Dirac delta function centred at x) in (1.28).In analogy to (1.15), we call ϕ τ the quantum field.
1.4.Statement of the results.We can now state our main results.In Section 1.4.1, we state the time-independent results.In Section 1.4.2,we state their timedependent generalisations.In all of the results, we will consider the limit τ → ∞, which we interpret as being the mean-field or semiclassical limit, with semiclassical parameter 1/τ → 0. Physically, this corresponds to taking a high-density limit, where we let the mass of the bosonic particles or the temperature tend to infinity.For a precise justification of this terminology and the choice of parameters, we refer the reader to [19, Section 1.1] for a detailed discussion.By applying an approximation argument, we prove results for w as in Assumption 1.1.Throughout the sequel, any object with a superscript ε is the corresponding object defined by taking the interaction potential to be w ε , which will be a suitable bounded approximation of w.In what follows, we always assume that all the approximating interaction potentials w ε are real-valued and even, without mentioning this explicitly.We can now state the result for L 1 (T) interaction potentials.Theorem 1.5 (Convergence for w ∈ L 1 (T)).Let w be as in Assumption 1.1 (i).Suppose that (w ε ) is a sequence of interaction potentials which are in L ∞ (T) such that w ε → w in L 1 (T).Then there exists a sequence and such that lim Before considering w = −δ as in Assumption 1.1 (ii), we need to define the sequence more w ε precisely.We fix U : R → R to be a continuous even function, with supp U ⊂ T satisfying (1.42) For ε ∈ (0, 1), we define where [x] is defined to be the unique element in (x + Z) ∩ T. In particular, w ε ∈ L ∞ (T) and w ε converges to −δ weakly, with respect to continuous functions.Remark 1.7.We make the following observations about Theorems 1.4, 1.5, and (1) For a pointwise almost everywhere non-negative, bounded, even interaction potential w, Theorem 1.4 holds without the need for a cut-off function f .This is the content of [17,Theorem 1.8].Moreover, by working with the non-normal ordered quantum interaction W ′ τ defined in (B.1), for a bounded, real-valued, even interaction potential w of positive type (i.e.ŵ pointwise almost everywhere non-negative), the same proof as [17,Theorem 1.8] again shows that Theorem 1.4 holds without the need for a cut-off function f .We include the details of the proof of this claim in Appendix B.1.
(2) We conjecture that the results hold for f a characteristic function of an interval.The method that we apply in Lemma 3.13 of Section 3.6 requires suitable smoothness assumptions on f .This is a technical assumption.(3) For an individual w ∈ L ∞ , Theorem 1.4 holds with a cut-off function of the form f (x) = e −cx 2 , for c > 0 sufficiently large depending on w L ∞ .This is also proved by working with a non-normal ordered quantum interaction.The details are given in Appendix B.2.We note this c cannot be chosen uniformly in the L ∞ norm of the interaction potential.So we cannot treat the unbounded interactions as in Theorems 1.5 and 1.6 using this kind of truncation.(4) One could consider the questions from Theorems 1.4 and 1.10 in the nonperiodic setting when the spatial domain is R for the one-body Hamiltonian h = −∆ + κ + v, where v : R → [0, ∞) is a positive one-body potential such that h has compact resolvent and Tr h −1 < ∞ holds (as in (1.13)).The analysis that we present in the periodic setting would carry through to this case, provided that we know that the time evolution S t given in (1.7) is well-defined on the support of the Gibbs measure.We do not address this question further in our paper.
( where S t is the flow map defined in (1.7).This is well defined since ϕ ∈ h almost surely and S t is norm preserving on h.Definition 1.9.Suppose A : F → F .Define the quantum time evolution of A as Ψ t τ A := e itτ Hτ A e −itτ Hτ .
We also recall the quantities ρ τ and ρ defined as in (1.34) and (1.24) respectively.
Theorem 1.10 (Convergence for w ∈ L ∞ (T)).Let w be as in Theorem 1.4.Given m ∈ N * , p i ∈ N * , ξ i ∈ L(h (pi) ), and t i ∈ R, we have Theorem 1.11 (Convergence for w ∈ L 1 (T)).Let w, w ε be as in the assumptions of Theorem 1.5.Then, there exists a sequence ), and t i ∈ R, we have Theorem 1.12 (Convergence for w = −δ).Let w, w ε be as in the assumptions of Theorem 1.6.Then, there exists a sequence (ε τ ) satisfying ε τ → 0 as τ → ∞ such that, given m ∈ N * , p i ∈ N * , ξ i ∈ L(h (pi ) ), and t i ∈ R, we have Remark 1.13.Theorems 1.10-1.12can indeed be seen as generalisations of the results given in Theorems 1.38-1.6respectively (the latter of which correspond to setting m = 1 and t 1 = 0).Namely, we use Remark 1.7 (3) above and noting that the proofs show that the convergence is uniform in ξ 1 ≤ 1.
1.5.Previously known results.In the context of the NLS, Gibbs measures (1.1)-(1.2) were relevant to study a substitute for a conservation law at low regularity.Namely, one can show that they are invariant under the flow and that they are supported on Sobolev spaces of low regularity.Consequently, it is possible to construct global solutions for random rough initial data.This was first rigorously obtained in the work of Bourgain [4][5][6].Some preliminary results were previously known by Zhidkov [58].This is an active area of research in nonlinear dispersive PDEs.We refer the reader to the expository works [13,39,45] for further explanations and background.For more recent developments, we refer the reader to [8-10, 15, 16, 44] and the references therein.
The focusing problem is more challenging.In one dimension, it was addressed in the earlier works [4,31].The one-dimensional problem was revisited recently in [1,14,44].For recent results on the fractional NLS, see [30].
It also makes sense to consider the higher-dimensional problem, i.e. when the spatial domain is T d , with d = 2, 3. Here, one needs to renormalise the interaction by means of Wick ordering.Formally, this refers to replacing (1.19) by

Finally, zf
Gibbs denotes a normalisation constant.For precise definitions of these objects, we refer the reader to [6, (12)].The invariance of (1.48) under the corresponding (Wick-ordered, focusing) NLS flow was first shown in [6] for w satisfying appropriate decay conditions on its Fourier coefficients (or under appropriate integrability conditions on w) .It was noted in [12] that, when d = 2, Wick ordering and truncation as in (1.48) do not yield a well-defined probability measure when w = −δ.Gibbs measures for the focusing NLS and related models were also studied in [7,42,43,57].
The first result showing how Gibbs measures for the NLS arise as limits of manybody quantum Gibbs states was proved by Lewin, Nam, and Rougerie [32].More precisely, the authors show that the quantum Gibbs state (as in (1.34), with f = 1, which we henceforth take throughout this subsection) converges to the classical Gibbs state (as in (1.24)) in the sense of partition functions and correlation functions as τ → ∞, as in Section 1.4.In [32], the authors studied the full defocusing problem in one dimension, as well as systems in d = 2, 3 with suitably chosen non translation-invariant interactions (which do not require Wick ordering) .Their method is based on the Gibbs variational principle and the quantum de Finetti theorem.The techniques from [32] were later applied to the regime of one-dimensional sub-harmonic traps in [33].
In [17], Fröhlich, Knowles, Schlein, and the second author developed an alternative approach based on a series expansion of the classical and quantum state in terms of the interaction, combined by a comparison of the explicit terms of the obtained series, and a Borel resummation.In doing so, they could give an alternative proof of the one-dimensional result obtained in [32] and consider (Wick-ordered) Gibbs measures obtained from translation-invariant interaction potentials for d = 2, 3, under a suitable modification of the quantum Gibbs state.The results for d = 2, 3 in [17] (under the same modification of the quantum Gibbs state) were originally stated for interaction potentials w ∈ L ∞ (T d ) of positive type.In [18], the results from [17] were used to study time-dependent correlations for d = 1.Moreover, the methods from [17] were later extended to w ∈ L q (T d ), d = 2, 3 with optimal q in [53].The optimal range of q was observed in [6].
In [34], and in [36], Lewin, Nam, and Rougerie obtained the derivation of the (Wick-ordered) Gibbs measures obtained from translation-invariant Gibbs measures when d = 2, 3 without the modification of the quantum Gibbs state from [17].Their methods are based on a non-trivial extension of the ideas from [32].An expository summary of the results of Lewin, Nam, and Rougerie can be found in [35].
Independently, and simultaneously with [36], Fröhlich, Knowles, Schlein, and the second author [19] obtained a derivation of the (Wick-ordered) Gibbs measure when d = 2, 3 based on a functional integral representation, and an infinite-dimensional saddle-point argument.The fundamental tool for setting up the functional integral representation in [19] is the Hubbard-Stratonovich transformation.In [19], convergence in the L ∞ norm of Wick-ordered correlation functions was shown.
The result of [19] was shown for continuous interaction potentials of positive type.In recent work [22], the same group of authors obtained the result with w = δ when d = 2. Here, one takes a limit in which the range of the interaction potential varies appropriately.The limiting object corresponds to the (complex) Euclidean Φ 4  2 theory.The proof in [22] is based on the combination quantitative analysis of the infinite-dimensional saddle point argument from [19] and a Nelsontype estimate for a general nonlocal theory in two dimensions (analogous to [40]) .
Related results were proved for systems on the lattice [21,29,51].We refer to [20] for an expository summary of some of the aforementioned results.We emphasise that all of the results mentioned in this subsection are proved in the defocusing (or thermodynamically stable) regime.We also refer the interested reader to subsequent applications of the methods in this paper to the three-body setting [48].
1.6.Main ideas of the proofs.The starting point of our analysis of the timeindependent problem with w ∈ L ∞ (i.e. of the proof of Theorem 1.4) is the perturbative expansion of the interaction e −H # in the interaction, similarly as in [17, Section 2.2] for the quantum and [17, Section 3.2] for the classical setting.Due to the presence of the truncation f (N # ) in (1.24) and (1.34), the resulting series have infinite radius of convergence; see Propositions 3.7 and 3.10 below.Thus, we avoid the need to apply Borel resummation techniques as in [17].
When analysing the remainder term in the quantum setting, we apply the Feynman-Kac formula and use the truncation property from Assumption 1.3.This analysis is possible since we are not Wick-ordering the interaction in one dimension; see Lemma 3.6.The truncation is likewise crucially used in the analysis of the classical remainder term; see Lemma 3.9.
When studying the convergence of the explicit terms of the obtained series, we use complex analytic methods as in [18,Section 3.1] to perform an expansion of the truncation f (N # ) and thus reduce to the study of the problem with a shifted chemical potential, but without a truncation.It is important that at this step, where we no longer have the control coming from the truncation, the analysis does not depend on the sign of the interaction.The details of this step are given in Lemmas 3.11 and 3.13 below.
The proofs of Theorems 1.5 and 1.6 are based on the application of Theorem 1.4 for appropriate w ε and on a diagonal argument.At this step, we have to crucially use [4, Lemma 3.10], which is recalled in Lemma 2.1 below.Even when we are working with L 1 interaction potentials, it is important that we apply the local version of this result (instead of Corollary 2.4).For details, see the proof of Lemma 4.1, in particular see steps (4.6)-(4.7).
For the time-dependent problem, we apply a Schwinger-Dyson expansion, similarly as in [18,.For the precise statements, see Lemmas 5.1-5.2below.Note that, due to the presence of the truncation, we do not need to consider the large particle number regime as in [18, Section 4] (whose analysis, in turn, relies crucially on the defocusing assumption).With this setup, we can easily deduce Theorem 1.10 from Theorem 1.4.
In order to prove Theorems 1.11 and 1.12, we need to apply an approximation argument.In particular, we want to estimate the difference of the flow map of the NLS with interaction potential w and of the NLS with interaction potential w ε .For the precise statement, see Lemma 5.4 when w ∈ L 1 and Lemma 5.6 when w = −δ.We prove these results by working in X s,b spaces; see Definition 5.5.1.7.Organisation of the paper.In Section 2, we set up some more notation and recall several auxiliary results from analysis and probability theory.Section 3 is devoted to the analysis of the time-independent problem with bounded interaction potential.Here, we prove Theorem 1.4.In Section 4, we study the time-independent problem with unbounded interaction potential and prove Theorems 1.5 and 1.6.Section 5 is devoted to the time-dependent problem and the proofs of Theorems 1.10-1.12.In Appendix A, we recall the proof of Lemma 2.1, which was originally given in [4, Lemma 3.10].In Appendix B, we give a detailed proof of the comments on the cut-off f given in Remark 1.7 (1) and (2) above.

Notation and auxiliary results
2.1.Notation.Throughout the paper, we use C > 0 to denote a generic positive constant that can change from line to line.If C depends on a finite set of parameters α 1 , . . ., α n , we indicate this dependence by writing C(α 1 , . . ., α n ).Sometimes, we also write a ≤ Cb as a b.We denote by N = {0, 1, 2, . . ., } the set of nonnegative integers and by N * = {1, 2, 3, . ..} the set of positive integers.
We write 1 to denote the identity operator on a Hilbert space.For a separable Hilbert space H and q ∈ [1, ∞], we define the Schatten space S q (H) to be the set of A ∈ L(H) satisfying A S q (H) , where and |A| = √ A * A. We usually omit the argument H where there is no confusion.We also have the following notation 2.2.Auxiliary Results.We recall several auxiliary results that we use in the paper.
Gibbs measures for the focusing local NLS.When analysing Gibbs measures for the focusing cubic NLS with w ∈ L ∞ (T), it is straightforward to make rigorous sense of (1.2) due to the presence of the truncation as in Assumption 1.3; see Lemma 3.1 (1) below.
For unbounded potentials, we will need to make use of the following result of Bourgain, found in [4, Lemma 3.10], whose proof is recalled in Appendix A.
Remark 2.2.When p = 6, the optimal value of B in Lemma 2.1 was recently determined in [44, Theorem 1.1 (ii)].We do not need to use this precise result since we work with p = 4 in the remainder of the paper.
Remark 2.3.When p = 6, an upper bound for the choice of B is determined by the constant c > 0. For details, see (A.6) below.
Corollary 2.4.Let (C N , G, µ) be the probability space defined in (1.14), and let We note that Corollary 2.4 follows from Lemma 2.1 with p = 4 by the same argument as estimate (3.2) below.
Hölder's inequality for Schatten spaces.We have the following version of Hölder's inequality for Schatten spaces (2.1), found in [50].

Lemma 2.5 (Hölder's Inequality
The Feynman-Kac Formula.In our analysis, we make use of the Feynman-Kac formula.To this end, let τ > 0 and let Ω τ denote the space of continuous paths ω : [0, τ ] → T. Given x, x ∈ T, we let Ω τ x,x denote the set of all elements of Ω τ such that ω(0) = x and ω(τ ) = x.Given t > 0, we define (2.4) to be the periodic heat kernel on T. For x, x ∈ T, we characterise the Wiener measure W τ x,x on Ω τ x,x by its finite-dimensional distribution.Namely for 0 < t 1 < . . .< t n < τ and f : Then we have the following result, see for example [47,Theorem X.68].
Proposition 2.6 (Feynman-Kac Formula).Let V : T → C be continuous and bounded below.For t > 0 3. The time-independent problem with bounded interaction potential.Proof of Theorem 1.4.
In this section, we study the time-independent problem with bounded interaction potential.In Section 3.1, we state some basic estimates which will be used throughout the rest of the paper.In Section 3.2, we set up the Duhamel expansion in the quantum setting.For this expansion, bounds on the explicit term are shown in Section 3.3 and bounds on the remainder term are shown in Section 3.4.The analogous expansion in the classical setting is analysed in Section 3.5.In Section 3.5, we prove convergence of the explicit terms.The proof of Theorem 1.4 is given in Section 3.7.(1) For w ∈ L ∞ (T), we have (2) For w ∈ L 1 (T), we have Proof.For (1), we note that For (2), we apply Cauchy-Schwarz and Young's inequality to get (3.2).
For the remainder of this section, we fix p ∈ N * .Unless otherwise specified, we consider ξ ∈ L(h (p) ).Moreover, • denotes the operator norm.The following lemma follows from the definition of Θ(ξ) in (1.18).Lemma 3.2.We have Let us note that with Θ τ as in (1.29), we have where 1 (q) denotes the identity map on h (q) and P + is the orthogonal projection onto the subspace of symmetric tensors.More details of the above equality can be found in [29, (3.88)].We also have the quantum analogue of Lemma 3.2, which follows from (3.3).
Lemma 3.3.For all n ∈ N * , we have 3.2.Duhamel Expansion.Throughout this section, we take w ∈ L ∞ (T).Note that with ρ τ defined as in (1.34), we have where We also define 3.3.Bounds on the explicit terms.Throughout the following proofs, we will use without mention that for any function g : C → C, g(N τ ) commutes with all operators on F that commute with N τ , which is clear from the definition of g(N τ ).Namely, g(N τ ) acts on the n th sector of Fock space as multiplication by g(n/τ ).
In particular, all of the operators appearing in the integrands of a ξ τ,m and R ξ τ,M commute with g(N τ ).
Lemma 3.5.For m ∈ N, we have Proof.Lemma 2.5 implies S ∞ e −tmHτ,0 Since e −sHτ,0 is a positive operator for s ∈ [0, 1], we have e −sHτ,0 S 1/s = (Z τ,0 ) s .So it follows from (3.9) that From Lemma 3.3, for fixed n we have where the final inequality follows from Assumption 1.3.It follows from (3.11) that, when viewed as an operator on F we note that W τ acts on h (n) as multiplication by In particular, arguing as in (3.12), it follows that Then for any ζ ∈ C, Proof.Define where the trace on the left hand side of (3.16) is taken over Fock space, whereas on the right hand side for each term it is taken over the nth sector of Fock space.
e −tM (Hτ,0+ζWτ ) f where W t x,y (dω) := n i=1 W t xi,yi (dω i ).Here we used that where we have used Proposition 2.6 in the second line.We have Integrating (3.15) in the variables t ∈ A, as defined in (3.7), implies We note that this converges to 0 as M → ∞ for any fixed ζ ∈ C.Moreover, since the radius of convergence of a ξ τ,m is infinite by Lemma 3.5, we conclude the following proposition.Then, for M ∈ N where Lemma 3.8.For each m ∈ N, we have Proof.We have From Lemma 3.2 and Assumption 1.3, we have Moreover, Lemma 3.1 (1) and Assumption 1.3 imply Note that Lemma 3.1 implies that Applying the same arguments as the proof of Lemma 3.8, we have the following lemma.
Lemma 3.9.For any M ∈ N, we have for suitable ψ ∈ C ∞ c (C).For the precise setup, see (3.47)-(3.48)below.Using (3.32), we use that 1 for Re ζ < 0, which leads us to analyse analogues of (3.6) and (3.25) without the truncation f (N ) and with chemical potential shifted by ν > 0.More precisely, we note the following boundedness and convergence result.We recall that in this section, we are considering w ∈ L ∞ .
Then, the following results hold Let us first consider the case when ξ ∈ B p .We define Then the deformed classical state defined by satisfies a Wick theorem with Green function given by G ν := 1 h ν .This follows directly from Proposition 1.2, since all we have done is shift the chemical potential by ν.
Moreover, the deformed quasi-free state defined by .
We also need the following result.
Proof.For an operator A : F → F , we define A (n) := P (n) A P (n) , where P (n) is the projection of an operator on Fock space to the nth component of Fock space.We also define Â := ⊕ n≥0 Â(n) .We have Proof.For ζ ∈ C\[0, ∞), we define We prove that α ξ τ,m and α ξ m are analytic in ζ ∈ C\[0, ∞).We first deal with α ξ m .Note that Using Lemma 3.2, Lemma 3.1 (1), and that dµ ϕ 2p h ≤ C(p) by Remark 3.14, we have Arguing similarly to (3.40), it follows that so by the dominated convergence theorem, we can differentiate under the integral sign in (3.39) and conclude that α ξ m is analytic in C\[0, ∞).To show α ξ τ,m is analytic in C\[0, ∞), we first note that 1 Nτ −ζ acts as multiplication by 1 (n/τ )−ζ on the nth sector of Fock space.By using Lemma 3.12 we get for any triangle First, for Re ζ < 0, we recall (3.33).Therefore,  for some F ∈ L 1 (C).By (3.49), we can use Fubini's theorem to write Using β ξ τ,m → 0 as τ → ∞ almost everywhere in C uniformly in ξ and (3.49), the dominated convergence theorem implies (3.38).

3.7.
Convergence of correlation functions.Proof of Theorem 1.4.We recall the class C p , defined in (3.34).The following convergence result holds.
Here we have used Lemma 3.13, Lemma 3.5, and the dominated convergence theorem.We also recall the notation (2.2).
Before proceeding to the proof of Theorem 1.4, we first need to prove the following technical lemma.Lemma 3.17.Recalling (1.26) and (1.37), we have γ #,p ≥ 0 in the sense of operators.
In the quantum setting, we use (1.37) and linearity to compute By (1.29) and the definition of Π η we deduce that that the expression in (3.51) equals ρ τ (Θ τ (Π η )), thus showing (3.50) in the quantum setting.Similarly in the classical setting, we use (1.26) and (1.18) to compute as was claimed.
We now show that the expression on the right-hand side of (3.50) is non-negative.Let us first show this in the quantum setting.By (1.29), we note that Θ τ (Π η ) is a positive operator.Furthermore f (N τ ) is a positive operator which commutes with Θ τ (Π η ).In particular, their composition is a positive operator.Recalling (1.33), we know that is a quantum state.In particular, when applied to positive operators it is nonnegative, so we obtain that Since P τ and f (N τ ) commute, by using (3.52), and recalling (1.33) as well as Assumption 1.3, it follows that We deduce the claim in the quantum setting from (3.50) and (3.53).
In the classical setting, we use (1.18) to write For the last inequality in (3.54), we recalled (1.24).We deduce the claim in the classical setting from (3.50) and (3.54).
To prove Theorem 1.4, we argue similarly as in [17, Sections 4.2-4.3],and use the following result.Lemma 3.19.Let p ∈ N * be fixed.Suppose that for all τ > 0, γ τ,p ∈ S 1 (h (p) ) are positive and that γ p ∈ S 1 (h (p) ) is positive (both in the sense of operators).Furthermore, suppose that The result of Lemma 3.19 is based on [50, Lemma 2.20], and proved in this form in [17,Lemma 4.10].We refer the reader to the latter reference for the details of the proof.
Proof of Theorem 1.4.We first prove (1.38).Let p ∈ N * be given.We verify the conditions of Lemma 3.19.Using the fact that S 2 (h (p) ) ∼ = S 2 (h (p) ) * and recalling (2.2), we have as τ → ∞ by Corollary 3.16.For the second equality in (3.56), we used the identity for all ξ ∈ L(h (p) ).One directly verifies (3.57) from (1.29), (1.34), (1.37) in the quantum setting, and from (1.18), (1.24), (1.26) in the classical setting.The proof of (1.39) is similar.Namely we start from (3.5) with A = 1 and repeat the previous argument (in which we formally set p = 0).We note that in this case, we do not need to use Lemma 3.19 above.
4. The time-independent problem with unbounded interaction potentials.Proofs of Theorems 1.5 and 1.6.
In this section, we analyse the time-dependent problem for general w as in Assumption 1.1.In particular, we no longer assume that w is bounded, as in Section 3. In Section 4.1, we consider w satisfying Assumption 1.1 (i) and prove Theorem 1.5.In Section 4.2, we consider w satisfying Assumption 1.1 (ii) and prove Theorem 1.6.As before, we fix p ∈ N * throughout the section.4.1.L 1 interaction potentials.Proof of Theorem 1.5.We first consider the case where w satisfies Assumption 1.1 (i), i.e. when it is taken to be an even and realvalued function in L 1 (T).To do this, we approximate w with bounded potentials w ε , which are even and real-valued.For instance, we can take w ε := wχ {|w|≤1/ε} .We then use the results of the previous section combined with a diagonal argument.
Let us first note the following result.
Lemma 4.1.Let w be as in Assumption 1.1 (i), and suppose w ε ∈ L ∞ is a sequence of even, real-valued interaction potentials satisfying w ε → w in L 1 (T) as ε → 0.
Then there exists a sequence (ε τ ) converging to 0 as τ → ∞ such that for all p ∈ N * lim τ →∞ uniformly in ξ ∈ C p .We recall that C p is given by (3.34).
Proof.Using a standard diagonal argument, it suffices to prove that for each fixed ε > 0 lim uniformly in ξ ∈ C p , and uniformly in ξ ∈ C p .The convergence in (4.2) holds by Corollary 3.16 because w ε ∈ L ∞ (T).To show (4.3), we first note that by Lemma 3.1 (2) and the Sobolev embedding theorem Since ϕ ∈ H and for ε sufficiently small By Lemma 2.1 and Assumption 1.3, we know that By Lemma 3.2, we have that Using (4.6)-(4.8)and the dominated convergence theorem, it follows that The same argument implies lim Noting that ) follows from (4.9) and (4.10).
We can now prove Theorem 1.5.
Proof of Theorem 1.5.We deduce (1.40) from Lemma 4.1 by arguing analogously as in the proof of (1.38).The proof of (1.41) is similar to that of (1.40).Instead of (4.2), we use lim τ →∞ Z ε τ = z ε , for fixed ε > 0, which follows from (1.39).Instead of (4.3), we use (4.10).4.2.The Delta Function.Proof of Theorem 1.6.We now deal with the case w = −δ.Let us first recall the definition (1.43) of w ε .Let us note that since U is even, it is not necessary to take U to be non-positive, since we can argue as in [18, (5.33)] using |U | (note that in [18], one writes w for U ).In what follows, we again denote objects corresponding to the interaction potential w ε by using a superscript ε.Again, by following Section 3.7, to prove Theorem 1.6, it suffices to prove the following proposition.Lemma 4.2.Let w := −δ, and let w ε be defined as in (1.43).Then there is a sequence (ε τ ) satisfying ε τ converging to 0 as τ → ∞ such that uniformly in ξ ∈ C p , where C p is given by (3.34).
Proof.As in the proof of Lemma 4.1, it suffices to prove for fixed ε that lim uniformly in ξ ∈ C p , and uniformly in ξ ∈ C p .Since w ε ∈ L ∞ (T), (4.12) follows from Lemma 4.1.To prove (4.13), we note that Lemma 3.1 (2) now implies Since dx U = −1 and U is even, We can then follow the argument in [18, (5.49) in the proof of Theorem 1.6.] to conclude that W ε → W . (4.16)We omit the details.Arguing as in the proof of Lemma 4.1, we obtain (4.13), and thus (4.11).We emphasise that, in order to apply the dominated convergence theorem as in the proof of Lemma 4.1, it is important that the upper bound (4.14) is uniform in ε.

The time-dependent problem
In this section, we consider the time-dependent problem.The analysis for bounded w and the proof of Theorem 1.10 are given in Section 5.1.The case when w is unbounded is analysed in Section 5.2.Here, we prove Theorems 1.11 and 1.12.Throughout the section, we fix p ∈ N * and ξ ∈ L(h (p) ).In particular, we have the following two lemmas.Lemma 5.1.Given K > 0, ε > 0, and t ∈ R, there exists L = L(K, ε, t, ξ , p) ∈ N, a finite sequence (e l ) L l=0 , with e l = e l (ξ, t) ∈ L(h (p) ) and τ 0 = τ 0 (K, ε, t, ξ ) > 0 such that < ε , for all τ ≥ τ 0 .Here we recall the definition of h (≤p) from (3.42).
In other words, for large τ and restricted numbers of particles, we can approximate the evolution of the lift of an arbitrary operator with finitely many unevolved lifts.We also have the corresponding classical result.Lemma 5.2.Given K > 0, ε > 0, and t ∈ R, then there exist L = L(K, ε, t, ξ , p) ∈ N, τ 0 = τ 0 (K, ε, t, ξ ) > 0 both possibly larger than in Lemma 5.1, and for the same choice of e l = e l (ξ, t) as in Lemma 5.1, we have Θ(e l ) χ {N ≤K} < ε , for all τ ≥ τ 0 .
We note that the proofs of Lemmas 5.1 and 5.2, respectively [18, Lemmas 3.9 and 3.12], do not use the sign of the interaction potential, so still hold in our case.The proofs of both results also require a compactly supported cut-off function, demonstrating the cut-off function of the form f (x) = e −cx 2 discussed in Remark 1.7 (3) would not suffice here.
Remark 5.3.Recalling the proof of [18, Proposition 2.1], it follows that the convergence in Theorem 1.10 is uniform on the set of parameters 5.2.Unbounded interaction potentials.Proofs of Theorems 1.11 and 1.12.Before proceeding, we need to prove a technical result concerning the flow of the NLS.Lemma 5.4.Let w ∈ L 1 (T) and s ≥ 3 8 be given, and suppose ϕ ∈ H s .Consider the Cauchy problem on T given by (5.1) In addition, given ε > 0 and letting w ε ∈ L ∞ be a sequence satisfying w ε → w in L 1 , we consider (5.2) Since s > 3/8 ≥ 0, the flow map defined in (1.7) is globally well defined.Denote by u and u ε the solutions of (5.1) and (5.2) respectively.Then for T > 0 We need to recall the dispersive Sobolev X s,b spaces.
denotes the spacetime Fourier transform of f .
In the following, we always take b = 1 2 + ν, for ν > 0 small.Proof of Lemma 5.4.We recall the details of proof of [18,Proposition 5.1].Firstly, we can take κ = 0 by considering ũ := e iκt u.So, we construct global mild solutions to (5.1) and (5.2) in the following way.
Let ζ, ψ : R → R be smooth functions with (5.3) (5.4) We also define ζ δ (t) := ζ(t/δ) and ψ δ (t) := ψ(t/δ).We consider where v δ (x, t) := ψ δ (t)v(x, t).By proving L and L ε are both contractions on appropriate function spaces for δ > 0 sufficiently small, we are able to find local mild solutions to (5.1) and (5.2).The arguments used to prove (5.5) and (5.6) are contractions in [18,Proposition 5.1] still hold if we can show that To show (5.7), we define V δ as the function satisfying To prove (5.7), it remains to show We have Parseval's identity implies where θ > 0.Here the final inequality follows from [18, Lemma 5.3 (iv)].Combining (5.9) with (5.10) and (5.11) yields (5.8).So, for a time of existence δ that depends only on the L 2 norm of the initial data, we are able to construct local mild solutions, v (n) and v ε (n) on [nδ, (n + 1)δ].We then piece these solutions together to create mild solutions u and u ε to (5.1) and (5.2) respectively.Using v and v ε to denote v (0) and v ε (0) respectively, we have (5.12) For the first term of (5.12), we have where the δ 1−2b 2 comes from the estimates for local X s,b spaces proved in [27] and [28].For a summary of these local X s,b spaces, we direct the reader to [18, Appendix A].
Arguing as in (5.7), we have The bound on the second term in (5.12) follows by the same argument as in the proof of [18, Proposition 5.1], although we note that since w ε L 1 is only bounded rather than equal to 1, we may get a larger constant times a positive power of ε, which is not a problem.The third term in (5.12) then follows for the same reasons combined with [18,Proposition 5.1].
Following the remainder of the argument from [18, Proposition 5.1] and noting that there we gain no negative powers of ε, we have The corresponding negative time estimates follow from an analogous argument.
We also have the corresponding result for the focusing local NLS.Lemma 5.6.Let s ≥ 3 8 be given, and suppose ϕ ∈ H s (T).Consider the Cauchy problem on T given by (5.13) In addition, given ε > 0, let w ε be as in (1.43).We consider Since s > 3/8 ≥ 0, the flow map defined in (1.7) is globally well defined.Denote by u and u ε the solutions of (5.13) and (5.14) respectively.Then for T > 0 We can follow exactly the proof of [18, Proposition 5.1], recalling (1.42), and noting that the function w ε defined in (1.43) is even and to deduce similarly as in [18, (5.27)].We also have the same point about w ε L 1 not necessarily equal to 1 as in the proof of Lemma 5.4, which does not affect the argument.
Proposition 5.7.Let (Z k ) k∈N be an increasing sequence of sets in the sense that Z k ⊂ Z k+1 .Let us define Z := ∪ k∈N Z k .For ε, τ > 0, suppose that g, g ε , g ε τ : Z → C are functions with the following properties.
Proof of Theorem 1.11.Throughout this proof we use X ε or X to denote an object defined using w ε or w respectively.Define where in (5.15)-(5.16),we take i ∈ {1, . . ., m}.Let us also define By Theorem 1.10, Remark 5.3, and Proposition 5.7, we note that it suffices to show that for fixed k ∈ N, we have uniformly in Z k .Let S t and S ε t denote the flow maps for (5.1) and (5.2) respectively.Let ϕ 0 ∈ H 1 2 − ⊂ h be the classical free field defined in (1.15).Recalling (1.18), it follows that for ξ ∈ L(h (k) ), we have almost surely.By (4.5) and (5.24), we have almost surely.Using conservation of mass for (5.1)-(5.2),as well as Lemma 3.1 (2) and Lemma 3.2, we have (5.26) Using Lemma 2.1 and Assumption 1.3, both of the bounding functions in (5.26) are L 1 (dµ).Furthermore, by construction of w ε , the quantity w ε L 1 is bounded uniformly in ε.Therefore, the first function is in L 1 (dµ) uniformly in ε.Consequently, we deduce (5.21) follows from (5.25), (5.26), and the dominated convergence theorem.The claim now follows.
Proof of Theorem 1.12.We argue analogously as in the proof of Theorem 1.11, with the same definitions of Z, Z k , g ε τ , g ε , g as in (5.15)-(5.18)above, except that now w ε is chosen as in (1.43).We recall that again w ε L 1 is bounded uniformly in ε.The proof is analogous to that of Theorem 1.11.The only difference is that instead of (4.5) and Lemma 5.4, we use their local analogues (4.16) and Lemma 5.6 respectively.
Appendix A. Proof of Lemma 2.1 In this appendix, we prove Lemma 2.1, which was originally proved in [4,Lemma 3.10].For the convenience of the reader, we present the full details of the proof in a self-contained way.For an alternative summary, see also [44,Section 2].Before proceeding with the proof, we recall in Section A.1 several auxiliary results concerning Fourier multipliers in the periodic setting and concentration inequalities.In Section A.2, we recall the notion of an norming set, which we use to prove duality results in L p spaces.The proof of Lemma 2.1 is given in Section A.3.
A.1.Auxiliary Results.We will need the following result about Fourier multipliers on the torus, the full statement and proof of which can be found in full generality in [54,VII,Theorem 3.8].We recall our convention (1.16) for the Fourier transform.We also recall the definition of a sub-gaussian random variable.Definition A.2. Let (Ω, A, P) be a probability space.We say a random variable X is sub-gaussian if there exist constants C, v > 0 such that for all t > 0 we have We will use the following inequality about sub-gaussian random variables.For a proof, see [56,Proposition 5.10].
Lemma A.3 (Hoeffding's Inequality).Suppose that X 1 , . . ., X N are all independent, centred sub-gaussian random variables.Let Q := max i X i ψ2 for X ψ2 := sup p≥1 p −1/2 (E|X| p ) 1/p and let a ∈ R N .Then, for any t > 0, we have A.2. Norming Sets.To prove Lemma 2.1, we need the following result about duality in L p spaces.We emphasise that this is a known result, but whose proof we could not find in the literature, so we write out the proof for the convenience of the reader.Then there is some subset Ξ of the unit sphere of L p ′ satisfying the following properties.
Remark.This result can be extended to finite dimensional subsets of normed vector spaces, but we do not need the result in full generality.
Definition A.5.Let X be a Banach space, Y ⊂ X linear subspace, and θ ∈ (0, 1].We denote by X * the (continuous) dual space of X.We say that a set F ⊂ X * is θ-norming over Y if sup g∈F \{0} g(y) g ≥ θ y for all y ∈ Y .
Definition A.6.Let X be a Banach space.Given x ∈ X and ε > 0, we write B ε (x) = {y ∈ X : x − y < ε} for the ball in X of radius ε around x. Let Y ⊂ X be a subset of X.Given ε > 0, we call We write S X := {x ∈ X : x = 1} for the unit sphere of X.
We want to relate norming sets to ε-nets.To do this, we take inspiration from the following result, the proof of which comes from [26, Section 17.2.4,Theorem 1].
Lemma A.7. Suppose X is a Banach space, Y ⊂ X is a linear subspace, and G ⊂ S X * a set that is 1-norming over Y .Let ε ∈ (0, 1), and suppose that N ε is an ε-net on the unit sphere of Y .For each element x ∈ N ε , fix a functional g x ∈ G such that g x (x) > 1 − ε (which we can do since N ε ⊂ S Y and G is 1-norming over Y ).Then the set F = {g x } x∈Nε is θ-norming over S Y for θ = 1 − 2ε.Letting ε → 0 then yields (B.2) ≥ 0 for w smooth of positive type.B.2.General L ∞ interaction potentials.For a general bounded, even, realvalued interaction potential w we show we could have used a Gaussian cut-off rather than a compactly supported one.Notice that since w ∈ L ∞ , there is some c such that w c := w + c ≥ 0 pointwise.Throughout this section, for an object X # , we use X c # to denote X # defined using w c rather than w.Notice that

. 31 )
Like in the quantum case, for each ζ ∈ C, R ξ M (ζ) converges to 0 as M → ∞ and a ξ m has infinite radius of convergence, so we have the following result.Proposition 3.10.The function A ξ (ζ) = ∞ m=0 a ξ m ζ m is analytic in C. 3.6.Convergence of the Explicit Terms.When analysing the convergence of the explicit terms, we argue similarly as in [18, Section 3.1] and rewrite f (N # ) as an integral of the form

Definition 5 . 5 .
Given f : T × R → C and s, b ∈ R we define

Lemma A. 1 (
Mikhlin Multiplier Theorem in the periodic setting).Let p ∈ [1, ∞] and T ∈ (L p (R), L p (R)) be a Fourier multiplier operator.Let û be the multiplier corresponding to T and suppose that û is continuous at every point of Z.For k ∈ Z, let λ(k) := û(k).Then there is a unique periodised lattice operator T defined byT f (x) ∼ k∈Z λ(k) f (k)e 2πikxsuch that T ∈ (L p (T), L p (T)) and T L p →L p ≤ T L p →L p .

Lemma A. 4 .
Suppose that M ⊂ Z has cardinality m, and let S := Span C e 2πikx : k ∈ M .

Proof.
Let y ∈ S Y .By definition, there is some x y ∈ N ε satisfying y − x y < ε.Then, by definition of F , linearity, the definition of x y , and G ⊂ S X * , we havesup g∈F |g(y)| = sup x∈Nε |g x (y)| ≥ |g xy (y)| = |g xy (x y ) − g xy (y − x y )| > 1 − ε − y − x y > 1 − 2ε = θ .A.2.2.Conclusion of the proof of Lemma A.4.We begin by bounding the size of an ε-net of C m .For M ⊂ Z with |M| = m, we consider the following norm on C m |||(a) k∈M ||| := k∈M a k e 2πikx L p ′.where we recall (1.30) and(1.32).We consider this non-normal ordered interaction since W ′ τ acts on the nth sector of Fock space as multiplication by1 i − x j ) .(B.2)The key difference from (3.13) is (B.2) includes the diagonal terms of the sum.The remark follows from showing that if w is of positive type, (B.2) ≥ 0 almost everywhere, since we can apply Proposition 2.6 as in the proof of[17, Proposition  4.5].We can further reduce this to showing (B.2) ≥ 0 for w ∈ C ∞ of positive type by taking w ε := w * ϕ ε for a standard approximation to the identity ϕ ε of positive type, since then w ε → w pointwise almost everywhere.To see this, recall that for g ∈ L 2 , Parseval's theorem impliesg, w * g ∼ k∈Z |ĝ(k)| 2 ŵ(k) ≥ 0 (B.3)since w is of positive type.Taking g ε ∈ C ∞ with g ε → n j=1 δ(• − x j ) weakly with respect to continuous functions, for w ∈ C ∞ we have, by (B.3)0 ≤ g ε , w * g ε → n i,j=1w(x i − x j ) .