Determinant of pseudo-laplacians

TL;DR

This paper derives comparison formulas for the zeta-regularized determinants of self-adjoint Laplacians on compact Riemannian manifolds, relating different boundary conditions and extensions, especially around a point P.

Contribution

It introduces new comparison formulas connecting determinants of Laplacians with various self-adjoint extensions on manifolds of dimension two or three.

Findings

Established formulas relating determinants of different Laplacian extensions

Extended understanding of spectral invariants on Riemannian manifolds

Provided tools for analyzing boundary condition effects on determinants

Abstract

Let X be a compact Riemannian manifold of dimension two or three and let P be a point of X. We derive comparison formulas relating the zeta-regularized determinant of an arbitrary self-adjoint extension of (symmetric) Laplace operator with domain, consisting of smooth functions with compact supports which does not contain P, to the zeta-regularized determinant of the self-adjoint Laplacian on X.