Generic Simplicity of a Schr\"odinger-type Operator on the Torus

TL;DR

This paper investigates how generic perturbations of a Schrödinger-type operator on the n-dimensional torus lead to simple spectra, proving the existence of such potentials and analyzing the splitting of degeneracies.

Contribution

It introduces conditions on potentials that ensure the spectrum's simplicity and demonstrates the first-order splitting of degeneracies under perturbation.

Findings

Existence of perturbation potentials guaranteeing spectrum simplicity

Conditions for generic simplicity of the spectrum

Degeneracy splitting at first order of perturbation

Abstract

The generic simplicity of the spectrum of a Schr\"odinger-type operator on the n-dimensional torus is studied using the Rayleigh-Schr\"odinger perturbation theory. The existence of a perturbation potential of the Laplacian is proved and suitable conditions on the potential that guarantee the generic simplicity of the spectrum constructed. It is also proved that with the potential, the degeneracy of the spectrum of the Laplacian on the n-dimensional torus splits at first order of the perturbation.