Krein formula and S-matrix for Euclidean Surfaces with Conical Singularities

TL;DR

This paper derives formulas for the zeta-regularized determinant of Laplacians on Euclidean surfaces with conical singularities using Krein's formula and S-matrix, linking the S-matrix to the Bergman projective connection.

Contribution

It introduces a novel formula connecting the S-matrix at zero to the Bergman projective connection for non-Friedrichs Laplacian extensions on conical surfaces.

Findings

Derived explicit formulas for determinants involving the S-matrix.

Connected the S-matrix at zero to the Bergman projective connection.

Extended analysis to non-Friedrichs Laplacian extensions.

Abstract

We use Krein formula and the S-matrix formalism to give formulas for the zeta-regularized determinant of non-Friedrichs extensions of the Laplacian on Euclidean surfaces with Conical Singularities. This formula involves S(0) and we show that the latter can be expressed using the Bergman projective connection on the underlying Riemann surface.