Direct and inverse problems for a third-order self-adjoint differential operator with non-local potential functions

TL;DR

This paper studies a third-order self-adjoint differential operator with non-local potentials, analyzing eigenvalue multiplicities, deriving eigenfunctions and resolvent expressions, and solving the inverse problem to recover the potentials.

Contribution

It provides new results on eigenvalue multiplicities, explicit eigenfunction and resolvent formulas, and a solution to the inverse problem for non-local potentials.

Findings

Eigenvalues are mostly simple, with finitely many of multiplicity two or three.

Explicit formulas for eigenfunctions and resolvent are derived.

The inverse problem for recovering non-local potentials is successfully solved.

Abstract

The direct and inverse problems for a third-order self-adjoint differential operator with non-local potential functions are considered. Firstly, the multiplicity for eigenvalues of the operator is analyzed, and it is proved that the differential operator has simple eigenvalues, except for finitely many eigenvalues of multiplicity two or three. Then the expressions of eigenfunctions and resolvent are obtained. Finally, the inverse problem for recovering non-local potential functions is solved.