Vacuum energy in the background of delta potentials

TL;DR

This paper investigates the vacuum energy response to delta-function potentials on curved boundaries like spherical and cylindrical shells, using scattering theory and zeta function regularization, revealing sign dependencies on geometry and size.

Contribution

It introduces a method to compute and analyze vacuum energy in the presence of delta-function potentials on curved boundaries, extending Casimir effect studies.

Findings

Vacuum energy is always negative for cylindrical shells.

Sign of vacuum energy depends on radius for spherical shells and magnetic flux tubes.

Renormalized energy is numerically evaluated and plotted.

Abstract

The response of vacuum to the presence of external conditions is the subject of this work. We consider a generalization of the Casimir effect in the presence of curved boundaries on which a sharp potential is concentrated. The profile of the potential is a delta function, which has some features in common with a hard boundary and some with a smooth background field. The boundaries investigated are: i) a spherical shell, ii) a cylindrical shell, iii) a magnetic flux tube. The vacuum energy is calculated by means of the Jost function of the scattering problem related to the field equation. The energy is then renormalized by means of a zeta functional approach adopting the heat-kernel expansion. The heat kernel coefficients are calculated and a discussion of the UV-divergences of the model is presented. The renormalized vacuum energy $E^{ren}$ is then numerically studied and plotted. The sign of $E^{ren}$ is found to be always negative in the case of the cylindrical shell, while in the case of the spherical shell and of the magnetic flux tube the sign depends on the value of the radius.