On Strichartz estimate for many body Schr\"odinger equation in the waveguide setting

TL;DR

This paper establishes Strichartz estimates for many-body Schrödinger equations on waveguide manifolds, enabling analysis of scattering behavior in semiperiodic spaces with small interaction potentials.

Contribution

It extends existing Strichartz estimates to waveguide settings for N-body Schrödinger equations, combining methods from prior works and handling semiperiodic geometries.

Findings

Proved Strichartz estimates in waveguide manifolds for small potentials.

Derived scattering asymptotics for the N-body Schrödinger model.

Extended previous results to the semiperiodic waveguide setting.

Abstract

In this short paper, we prove Strichartz estimates for N-body Schr\"odinger equations in the waveguide manifold setting (i.e. on semiperiodic spaces $\mathbb{R}^m\times \mathbb{T}^n$ where $m\geq 3$), provided that interaction potentials are small enough (depending on the number of the particles and the universal constants, not on the initial data). The proof combines both the ideas of Tzvetkov-Visciglia \cite{TV1} and Hong \cite{hong2017strichartz}. As an immediate application, the scattering asymptotics for this model is also obtained. This result extends Hong \cite{hong2017strichartz} to the waveguide case.