An application of the fixed point theorem to the inverse Sturm-Liouville problem

TL;DR

This paper presents a concise proof of the spectral data characterization theorem for Sturm-Liouville operators with potentials in L^p spaces, utilizing the fixed point theorem to address the inverse problem.

Contribution

It introduces a novel application of the fixed point theorem to simplify the proof of the spectral data characterization for inverse Sturm-Liouville problems.

Findings

Short proof of the spectral data characterization theorem

Applicable to potentials in L^p(0,1) for 1 ≤ p < ∞

Enhanced understanding of inverse Sturm-Liouville problems

Abstract

We consider Sturm-Liouville operators $-y''+v(x)y$ on $[0,1]$ with Dirichlet boundary conditions $y(0)=y(1)=0$. For any $1\le p<\infty$, we give a short proof of the characterization theorem for the spectral data corresponding to $v\in L^p(0,1)$.