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

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 Schrodinger equation. In this way, we obtain a periodic analogue of the uniqueness result on 1113 previously proved by Klainerman and Machedon [75], except that, in the periodic setting, we need to assume additional regularity. In particular, we need to work in the Sobolev class H degrees for alpha > 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 alpha = 1. This is in contrast to the known results in the nonperiodic setting, where these techniques are known to hold for all alpha >= 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. (C) 2014 Elsevier Inc. All rights reserved.