Investigating the Spectral Geometry of a Soft Wall

TL;DR

This paper explores the spectral geometry of a soft wall model in quantum vacuum energy, using analytical and asymptotic methods to understand boundary effects with potential functions of the form z^ and z.

Contribution

It introduces a soft wall boundary model within spectral theory, providing exact solutions and asymptotic analysis for specific potential exponents, advancing understanding of boundary effects in quantum fields.

Findings

Exact analytical solutions for  potential case.

Asymptotic analysis for  potential case.

Insights into boundary effects on quantum vacuum energy.

Abstract

The idealized theory of quantum vacuum energy density is a beautiful application of the spectral theory of differential operators with boundary conditions, but its conclusions are physically unacceptable. A more plausible model of a reflecting boundary that stays within linear spectral theory confines the waves by a steeply rising potential function, which can be taken as a power of one coordinate, z^\alpha. We report investigations of this model with considerable student involvement. An exact analytical solution with some numerics for \alpha=1 and an asymptotic (semiclassical) analysis of a related problem for \alpha=2 are presented.