Asymptotic behavior for differences of eigenvalues of two Sturm–Liouville problems with smooth potentials

The asymptotics for the differences of the eigenvalues of two Sturm–Liouville problems defined on [ 0 , π ] with the same boundary conditions and different smooth potentials is considered. Under the assumptions of that both problems with a suite of boundary conditions have the same one full spectrum and both potential functions and their derivatives are the same at the endpoint x = π , the asymptotic expressions associated with other boundary conditions are provided.