Casimir energy for a scalar field with a frequency dependent boundary condition

TL;DR

This paper investigates the Casimir energy of a scalar field with a frequency-dependent boundary condition, using incomplete zeta functions and asymptotic expansions to analyze the dominant energy contributions, with potential applications in dielectric media and sonoluminescence.

Contribution

It introduces a novel approach to calculating Casimir energy with frequency-dependent boundaries using incomplete zeta functions and asymptotic analysis.

Findings

Dominant terms in Casimir energy identified via asymptotic expansion

Method applicable to dielectric media and sonoluminescence models

Provides a framework for frequency-dependent boundary condition analysis

Abstract

We consider the vacuum energy for a scalar field subject to a frequency dependent boundary condition. The effect of a frequency cut-off is described in terms of an {\it incomplete} $\zeta$-function. The use of the Debye asymptotic expansion for Bessel functions allows to determine the dominant (volume, area, >...) terms in the Casimir energy. The possible interest of this kind of models for dielectric media (and its application to sonoluminescence) is also discussed.