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

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.