Functional determinants for general self-adjoint extensions of Laplace-type operators resulting from the generalized cone

TL;DR

This paper derives a closed-form expression for the zeta regularized determinant of Laplace-type operators on generalized cones for all self-adjoint extensions, linking the determinant to finite-dimensional endomorphisms.

Contribution

It provides a general formula for the determinant of Laplace-type operators on generalized cones applicable to any self-adjoint extension, including explicit cases like Friedrich's extension.

Findings

Closed-form determinant expression for arbitrary self-adjoint extensions.

Determinant expressed via a finite-dimensional endomorphism.

Application to full Laplace-type operators on generalized cones.

Abstract

In this article we consider the zeta regularized determinant of Laplace-type operators on the generalized cone. For {\it arbitrary} self-adjoint extensions of a matrix of singular ordinary differential operators modelled on the generalized cone, a closed expression for the determinant is given. The result involves a determinant of an endomorphism of a finite-dimensional vector space, the endomorphism encoding the self-adjoint extension chosen. For particular examples, like the Friedrich's extension, the answer is easily extracted from the general result. In combination with \cite{BKD}, a closed expression for the determinant of an arbitrary self-adjoint extension of the full Laplace-type operator on the generalized cone can be obtained.