Spectral invariants of integrable polygons

TL;DR

This paper investigates the spectral properties of integrable polygons, deriving new invariants and formulas for the Laplace spectrum, and explores connections with general polygonal and smooth domains.

Contribution

It introduces new spectral invariants and explicit formulas for the spectral zeta function and heat trace asymptotics specific to integrable polygons.

Findings

Derived new spectral invariants for integrable polygons.

Provided explicit expressions for the spectral zeta function and determinant.

Linked heat trace invariants of polygonal and smooth domains.

Abstract

An integrable polygon is one whose interior angles are fractions of $\pi$; that is to say of the form $\frac \pi n$ for positive integers $n$. We consider the Laplace spectrum on these polygons with the Dirichlet and Neumann boundary conditions, and we obtain new spectral invariants for these polygons. This includes new expressions for the spectral zeta function and zeta-regularized determinant as well as a new spectral invariant contained in the short-time asymptotic expansion of the heat trace. Moreover, we demonstrate relationships between the short-time heat trace invariants of general polygonal domains (not necessarily integrable) and smoothly bounded domains and pose conjectures and further related directions of investigation.