On Strichartz estimates for many-body Schrödinger equation in the periodic setting

Abstract

In this paper, we prove Strichartz estimates for many body Schrödinger equations in the periodic setting, specifically on tori 𝕋 d {\mathbb{T}^{d}} , where d ≥ 3 {d\geq 3} . The results hold for both rational and irrational tori, and for small interacting potentials in a certain sense. Our work is based on the standard Strichartz estimate for Schrödinger operators on periodic domains, as developed in [J. Bourgain and C. Demeter, The proof of the l 2 l^{2} decoupling conjecture, Ann. of Math. (2) 182 2015, 1, 351–389]. As a comparison, this result can be regarded as a periodic analogue of [Y. Hong, Strichartz estimates for N-body Schrödinger operators with small potential interactions, Discrete Contin. Dyn. Syst. 37 2017, 10, 5355–5365] though we do not use the same perturbation method. We also note that the perturbation method fails due to the derivative loss property of the periodic Strichartz estimate.