We prove that Gibbs measures of nonlinear Schrödinger equations arise as high-temperature limits of thermal states in many-body quantum mechanics. Our results hold for defocusing interactions in dimensions d = 1, 2, 3. The many-body quantum thermal states that we consider are the grand canonical ensemble for d = 1 and an appropriate modification of the grand canonical ensemble for d = 2, 3. In dimensions d = 2, 3, the Gibbs measures are supported on singular distributions, and a renormalization of the chemical potential is necessary. On the many-body quantum side, the need for renormalization is manifested by a rapid growth of the number of particles. We relate the original many-body quantum problem to a renormalized version obtained by solving a counterterm problem. Our proof is based on ideas from field theory, using a perturbative expansion in the interaction, organized by using a diagrammatic representation, and on Borel resummation of the resulting series.

Abstract. In this paper, we present a uniqueness result for solutions to the Gross-Pitaevskii hierarchy on the three-dimensional torus, under the assumption of an a priori spacetime bound. We show that this a priori bound is satisfied for factorized solutions to the hierarchy which come from solutions of the nonlinear Schrödinger equation. In this way, we obtain a periodic analogue of the uniqueness result on R 3 previously proved by Klainerman and Machedon [76], except that, in the periodic setting, we need to assume additional regularity. In particular, we need to work in the Sobolev class H α for α > 1. By constructing a specific counterexample, we show that, on T 3 , the existing techniques from the work of Klainerman and Machedon approach do not apply in the endpoint case α = 1. This is in contrast to the known results in the non-periodic setting, where the these techniques are known to hold for all α ≥ 1.In our analysis, we give a detailed study of the crucial spacetime estimate associated to the free evolution operator. In this step of the proof, our methods rely on lattice point counting techniques based on the concept of the determinant of a lattice. This method allows us to obtain improved bounds on the number of lattice points which lie in the intersection of a plane and a set of radius R, depending on the number-theoretic properties of the normal vector to the plane. We are hence able to obtain a sharp range of admissible Sobolev exponents for which the spacetime estimate holds.

We study the Gross-Pitaevskii hierarchy on the spatial domain T 3 . By using an appropriate randomization of the Fourier coefficients in the collision operator, we prove an averaged form of the main estimate which is used in order to contract the Duhamel terms that occur in the study of the hierarchy. In the averaged estimate, we do not need to integrate in the time variable. An averaged spacetime estimate for this range of regularity exponents then follows as a direct corollary. The range of regularity exponents that we obtain is α > 3 4 . It was shown in our previous joint work with Gressman (J Funct Anal 266(7): 2014) that the range α > 1 is sharp in the corresponding deterministic spacetime estimate. This is in contrast to the non-periodic setting, which was studied by Klainerman and Machedon (Commun Math Phys 279(1): [169][170][171][172][173][174][175][176][177][178][179][180][181][182][183][184][185] 2008), where the spacetime estimate is known to hold whenever α ≥ 1. The goal of our paper is to extend the range of α in this class of estimates in a probabilistic sense. We use the new estimate and the ideas from its proof in order to study randomized forms of the Gross-Pitaevskii hierarchy. More precisely, we consider hierarchies similar to the Gross-Pitaevskii hierarchy, but in which the collision operator has been randomized. For these hierarchies, we show convergence to zero in low regularity Sobolev spaces of Duhamel expansions of fixed deterministic density matrices. We believe that the study of the randomized collision operators could be the first step in the understanding of a nonlinear form of randomization.