Zeta Function on Surfaces of Revolution

TL;DR

This paper applies contour integral and WKB methods to analyze the zeta function of the Laplacian on surfaces of revolution, deriving residues, values, and a closed-form determinant formula, confirming results with heat kernel expansion.

Contribution

It introduces a novel combination of contour integral and WKB methods to compute the zeta function and determinant for surfaces of revolution.

Findings

Residues and values of the zeta function computed at key points

Closed-form formula for the Laplacian determinant on surfaces of revolution

Results consistent with heat kernel expansion methods

Abstract

In this paper we applied the contour integral method for the zeta function associated with a differential operator to the Laplacian on a surface of revolution. Using the WKB expansion, we calculated the residues and values of the zeta function at several important points. The results agree with those obtained from the heat kernel expansion. We also obtained a closed form formula for the determinant of the Laplacian on such a surface.