Inverse Problem For Dirichlet Boundary Value Problems

TL;DR

This paper proves that if the spectrum of a Sturm-Liouville operator with Dirichlet boundary conditions contains the set {(nπ)^2}, then the potential q must be zero almost everywhere, addressing an inverse spectral problem.

Contribution

It establishes a uniqueness result for the inverse spectral problem of Sturm-Liouville operators with specific spectral data.

Findings

Spectrum containing {(nπ)^2} implies q=0 a.e.

Provides conditions for potential recovery in inverse problems

Advances understanding of spectral characterization of Sturm-Liouville operators.

Abstract

In this article we consider Sturm-Liouville operator with $q\in W_{1}^{2}[0,1]$ and Dirichlet boundary conditions. We prove that if the set $\{(n\pi)^{2}:n\in \mathbb{N}\}$ is a subset of the spectrum of the Sturm-Liouville operator with Dirichlet boundary conditions, then $q=0$ a.e.