Spectral asymptotics of all the eigenvalues of Schr\"odinger operators on flat tori

TL;DR

This paper derives asymptotic expansions for all eigenvalues of Schr"odinger operators on flat tori, distinguishing between stable and unstable eigenvalues, using a novel iterative quasimode approach.

Contribution

It introduces a new spectral asymptotic expansion for all eigenvalues of Schr"odinger operators on flat tori, including a directional expansion for unstable eigenvalues.

Findings

Asymptotic expansion in rac{1}{\, ext{lambda}^{\, ext{elta}}} for most eigenvalues

Directional expansion for unstable eigenvalues

A new iterative quasimode method for spectral analysis

Abstract

We study Schr\"odinger operators with Floquet boundary conditions on flat tori obtaining a spectral result giving an asymptotic expansion of all the eigenvalues. The expansion is in $\lambda^{-\delta}$ with $\delta\in(0,1)$ for most of the eigenvalues $\lambda$ (stable eigenvalues), while it is a "directional expansion" for the remaining eigenvalues (unstable eigenvalues). The proof is based on a structure theorem which is a variant of the one proved in \cite{PS10,PS12} and on a new iterative quasimode argument.