Vacuum Energy as Spectral Geometry

TL;DR

This paper reviews quantum vacuum energy from a spectral theory perspective, solves some one-dimensional systems exactly, and discusses the relationships among various spectral densities, providing a foundation for higher-dimensional analysis.

Contribution

It introduces a spectral theory approach to vacuum energy, solves specific one-dimensional cases exactly, and links spectral densities to energy calculations, aiding higher-dimensional studies.

Findings

Exact solutions for one-dimensional systems using classical paths

Demonstration of relations among spectral and energy densities

Foundation for semiclassical methods in higher dimensions

Abstract

Quantum vacuum energy (Casimir energy) is reviewed for a mathematical audience as a topic in spectral theory. Then some one-dimensional systems are solved exactly, in terms of closed classical paths and periodic orbits. The relations among local spectral densities, energy densities, global eigenvalue densities, and total energies are demonstrated. This material provides background and motivation for the treatment of higher-dimensional systems (self-adjoint second-order partial differential operators) by semiclassical approximation and other methods.