Casimir Effect in Hyperbolic Polygons

TL;DR

This paper derives a trace formula for quantum systems in hyperbolic polygons and uses it to calculate the Casimir energy, revealing how vacuum energy depends on the polygon's vertices.

Contribution

It introduces a novel trace formula for hyperbolic polygons and applies it to compute Casimir energy, linking geometry to quantum vacuum effects.

Findings

Casimir energy depends on the number of vertices in hyperbolic polygons.

Derived a trace formula for spectra in hyperbolic polygons.

Calculated vacuum energy for scalar fields in these domains.

Abstract

We derive a trace formula for the spectra of quantum mechanical systems in hyperbolic polygons which are the fundamental domains of discrete isometry groups acting in the two dimensional hyperboloid. Using this trace formula and the point splitting regularization method we calculate the Casimir energy for a scalar fields in such domains. The dependence of the vacuum energy on the number of vertexes is established.