On Functional Determinants of Laplacians in Polygons and Simplices

TL;DR

This paper calculates the functional determinants of Laplacians with Dirichlet boundary conditions in polygons and simplices, providing explicit formulas for triangles and regular polygons, extending previous work on smooth surfaces.

Contribution

It derives explicit formulas for the functional determinants of Laplacians in piecewise flat polygons and simplices, including triangles and regular polygons, with corners on the boundary.

Findings

Explicit closed-form expressions for triangles.

Explicit formulas for regular polygons.

Complementary to smooth surface results.

Abstract

The functional determinant of an elliptic operator with positive, discrete spectrum may be defined as $e^{-Z'(0)}$, where $Z(s)$, the zeta function, is the sum $\sum_n^{\infty} \lambda_n^{-s}$ analytically continued to $s$ around the origin. In this paper $Z'(0)$ is calculated for the Laplace operator with Dirichlet boundary conditions inside polygons and simplices with the topology of a disc in the Euclidean plane. The domains we consider are hence piece--wise flat with corners on the boundary and in the interior. Our results are complementary to earlier investigations of the determinants on smooth surfaces with smooth boundaries. We have explicit closed integrated expressions for triangles and regular polygons.