Zeta function of self-adjoint operators on surfaces of revolution

Abstract

In this article we analyze the zeta function for the Laplace operator on a surface of revolution. A variety of boundary conditions, separated and unseparated, are considered. Formulas for several residues and values of the zeta function as well as for the determinant of the Laplacian are obtained. The analysis is based upon contour integration techniques in combination with a WKB analysis of solutions of related initial value problems.