About one inverse problem for the Sturm-Liouville operator

TL;DR

This paper investigates spectral problems for Sturm-Liouville operators under various boundary conditions, establishing conditions for spectrum coincidence and applying results to periodic cases.

Contribution

It provides necessary and sufficient conditions for spectrum coincidences between different boundary value problems for Sturm-Liouville operators.

Findings

Spectrum coincidence conditions for Dirichlet-Neumann and Neumann-Dirichlet problems.

Conditions for spectrum coincidence of Dirichlet and Neumann problems excluding zero.

Application of results to periodic and anti-periodic spectral problems.

Abstract

We consider the spectral problems for the Sturm-Liouville operator generated by the Dirichlet, Neumann, Dirichlet-Neumann and Neumann-Dirichlet conditions. The necessary and sufficient condition for the coincidence of the spectrum of the Dirichlet-Neumann and Neumann-Dirichlet problems is proved. Also the necessary and sufficient condition for the coincidence of the spectrum, except zero, of the Dirichlet and Neumann problems is proved. An application to periodic and anti-periodic problems is given.