On the asymptotically simplicity of periodic eigenvalues and Titchmarsh's formula

TL;DR

This paper investigates conditions on the Fourier coefficients of a potential function in a Sturm-Liouville problem that ensure the asymptotic simplicity of periodic eigenvalues, providing asymptotic estimates for spectral gaps.

Contribution

It establishes new conditions on Fourier coefficients that guarantee the asymptotic simplicity of eigenvalues in Sturm-Liouville problems, extending Titchmarsh's formula.

Findings

Conditions on Fourier coefficients ensure eigenvalue simplicity

Asymptotic estimates for spectral gaps are derived

Eigenvalues follow Titchmarsh's form asymptotically

Abstract

We consider Sturm-Liouville equation $y^{\prime\prime}+(\lambda-q)y=0$ where $q\in L^{1}[0,a]$. We obtain various conditions on the Fourier coefficients of q such that the periodic eigenvalues having the form given by Titchmarsh are asymptotically simple. Under these conditions, we give some asymptotic estimates for the spectral gaps.