Abstract: This paper focuses on the direct and inverse problems for a third-order self-adjoint differential operator with non-local potential and anti-periodic boundary conditions. Firstly, we obtain the expressions for the characteristic function and resolvent of this third-order differential operator. Secondly, by using the expression for the resolvent of the operator, we prove that the spectrum for this operator consists of simple eigenvalues and a finite number of eigenvalues with multiplicity 2. Finally, we solve the inverse problem for this operator, which states that the non-local potential function can be reconstructed from four spectra. Specially, we prove the Ambarzumyan theorem and indicate that odd or even potential functions can be reconstructed by three spectra.

Key words: Direct problem, Inverse problem, Non-local potential, Anti-periodic boundary condition