Calculation of eigenvalues of elliptic differential operators using the theory of regularized series

The study of the spectral properties of perturbed differential operators is one of the significant problems of the spectral theory. In order to solve this problem it is necessary to determine the asymptotic behavior of the spectrum. But when investigating the asymptotic behavior, the improvement of remainder term is often impossible. Moreover, even the separation of the second term of the asymptotics from the remainder term is impossible. As a consequence it is necessary to come over to the study of deeper spectrum structure. A standard research tool is the derivation of formulas for regularized traces. The author makes a calculation of four amendments of the perturbation theory with the help of the theory of regularized series, followed by the access to the eigenvalues of elliptic differential operators with potential on a projective plane. In this case the projective plane is identified with the sphere by comparing opposite points and poles puncturing.