Riesz bases generated by the spectra of Sturm-Liouville problems

TL;DR

This paper investigates whether systems of cosine and sine functions generated by the spectra of Sturm-Liouville problems form Riesz bases in L^2[0,π], finding that they almost always do, which has implications for spectral theory.

Contribution

It provides a nearly complete answer to the question of Riesz basis properties for systems generated by Sturm-Liouville spectra.

Findings

Systems of cosine functions form Riesz bases in L^2[0,π] almost always.

Systems of sine functions form Riesz bases in L^2[0,π] almost always.

The result applies broadly to spectra of Sturm-Liouville problems.

Abstract

Let $\{\lambda_n^2\}_{n = 0}^\infty$ be the spectra of a Sturm-Liouville problem on $[0,\pi ]$. We investigate the question: Do the systems $\{\cos(\lambda_n x)\}_{n= 0}^\infty$ or $\{\sin(\lambda_n x)\}_{n = 0}^\infty$ form Riesz bases in ${L^2}[0,\pi]$? The answer is almost always positive.