On the uniqueness of solutions to the periodic 3D Gross-Pitaevskii hierarchy

TL;DR

This paper proves a uniqueness result for solutions to the 3D periodic Gross-Pitaevskii hierarchy, requiring additional regularity compared to the non-periodic case, and introduces lattice point counting techniques for sharper bounds.

Contribution

It extends the uniqueness results to the periodic setting with a detailed analysis of spacetime estimates using lattice point counting, highlighting the regularity requirements and limitations at the endpoint case.

Findings

Uniqueness holds for solutions in Sobolev class H^α with α>1 on the 3D torus.

Introduces lattice point counting methods to improve bounds on spacetime estimates.

Counterexample shows techniques don't apply at the endpoint α=1 in the periodic setting.

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 Schr\"{o}dinger equation. In this way, we obtain a periodic analogue of the uniqueness result on $\mathbb{R}^3$ previously proved by Klainerman and Machedon, except that, in the periodic setting, we need to assume additional regularity. In particular, we need to work in the Sobolev class $H^{\alpha}$ for $\alpha>1$. By constructing a specific counterexample, we show that, on $\mathbb{T}^3$, the existing techniques don't apply in the endpoint case $\alpha=1$. This is in contrast to the known results in the non-periodic setting, where the these techniques are known to hold for all $\alpha \geq 1$. 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.