The Spectral Zeta Function for Laplace Operators on Warped Product Manifolds of the type $I\times_{f} N$

TL;DR

This paper analyzes the spectral zeta function for Laplace operators on warped product manifolds, deriving its analytic continuation and applying it to compute functional determinants and heat kernel coefficients.

Contribution

It provides a detailed method for the analytic continuation of the spectral zeta function on warped product manifolds and computes related spectral invariants.

Findings

Derived explicit formulas for the spectral zeta function on warped product manifolds.

Computed the zeta regularized functional determinant.

Analyzed heat kernel asymptotic expansion coefficients.

Abstract

In this work we study the spectral zeta function associated with the Laplace operator acting on scalar functions defined on a warped product of manifolds of the type $I\times_{f} N$ where $I$ is an interval of the real line and $N$ is a compact, $d$-dimensional Riemannian manifold either with or without boundary. Starting from an integral representation of the spectral zeta function, we find its analytic continuation by exploiting the WKB asymptotic expansion of the eigenfunctions of the Laplace operator on $M$ for which a detailed analysis is presented. We apply the obtained results to the explicit computation of the zeta regularized functional determinant and the coefficients of the heat kernel asymptotic expansion.