Diophantine approximation and the solubility of the Schroedinger equation

TL;DR

This paper investigates how number theory influences the existence of periodic solutions to the Schrödinger equation on a 2D torus, focusing on periods that cause obstructions and the frequency of resonances.

Contribution

It characterizes the set of periods obstructed by number theoretical issues and analyzes the asymptotic behavior of resonances in this context.

Findings

Identifies specific periods where solutions are blocked by number theory.

Analyzes the distribution and frequency of resonances asymptotically.

Provides a framework linking Diophantine approximation to Schrödinger solutions.

Abstract

We characterise the set of periods for which number theoretical obstructions prevent us from finding periodic solutions of the Schroedinger equation on a two dimensional torus as well as the asymptotic occurrence of possible resonances.