Functional Determinants for Regular-Singular Laplace-type Operators

TL;DR

This paper verifies the applicability of the contour integral method for regular-singular Laplace-type operators with matrix coefficients and extends zeta determinant calculations to generalized Neumann boundary conditions, with applications to cone Laplacians.

Contribution

It explicitly confirms the contour integral method's validity in the regular-singular setting and extends zeta determinant computations to new boundary conditions.

Findings

Confirmed the contour integral method applies in the regular-singular setup.

Extended zeta determinant calculations to generalized Neumann boundary conditions.

Applied results to Laplacians on bounded generalized cones with relative ideal boundary conditions.

Abstract

We discuss a specific class of regular-singular Laplace-type operators with matrix coefficients. Their zeta determinants were studied by K. Kirsten, P. Loya and J. Park on the basis of the Contour integral method, with general boundary conditions at the singularity and Dirichlet boundary conditions at the regular boundary. We complete the arguments of Kirsten, Loya and Park by explicitly verifying that the Contour integral method indeed applies in the regular-singular setup. Further we extend the zeta determinant computations to generalized Neumann boundary conditions at the regular boundary and apply our results to Laplacians on a bounded generalized cone with relative ideal boundary conditions.