Strichartz estimates for the Schroedinger equation on non-rectangular two-dimensional tori

TL;DR

This paper explores long-time Strichartz estimates for the Schrödinger equation on generic non-rectangular 2D tori, providing partial proofs using Weyl bounds and Diophantine problem solutions.

Contribution

It introduces a conjecture for these estimates and offers partial proof in two dimensions, advancing understanding of dispersive PDEs on complex geometric structures.

Findings

Proposes a conjecture for long-time Strichartz estimates on non-rectangular tori.

Provides partial proof in two dimensions using Weyl bounds.

Connects bounds on Diophantine solutions to PDE estimates.

Abstract

We propose a conjecture for long time Strichartz estimates on generic (non-rectangular) flat tori. We proceed to partially prove it in dimension 2. Our arguments involve on the one hand Weyl bounds; and on the other hands bounds on the number of solutions of Diophantine problems.