Direct and inverse problems for a third order self-adjoint differential operator with periodic boundary conditions and nonlocal potential

TL;DR

This paper analyzes a third order self-adjoint differential operator with periodic boundary conditions, characterizes its spectrum and eigenfunctions, and solves inverse problems to recover the potential, including an Ambarzumyan-type theorem.

Contribution

It provides a detailed spectral analysis and solves inverse problems for a third order operator with nonlocal potential, including potential reconstruction from spectral data.

Findings

Spectrum consists of simple eigenvalues and finitely many double eigenvalues.

Explicit expressions for eigenfunctions and resolvent are derived.

Inverse problems are solved, allowing potential reconstruction from three spectra.

Abstract

A third order self-adjoint differential operator with periodic boundary conditions and an one-dimensional perturbation has been considered. For this operator, we first show that the spectrum consists of simple eigenvalues and finitely many eigenvalues of multiplicity two. Then the expressions of eigenfunctions and resolvent are described. Finally, the inverse problems for recovering all the components of the one-dimensional perturbation are solved. In particular, we prove the Ambarzumyan-type theorem and show that the even or odd potential can be reconstructed by three spectra.