Singular perturbations with boundary conditions and the Casimir effect in the half space

TL;DR

This paper investigates the mathematical modeling of the Casimir effect using singular perturbations of the Laplacian with boundary conditions, providing explicit formulas for the partition function and Casimir force in a half-space setting.

Contribution

It introduces a novel framework linking self-adjoint extensions of multiplication operators to singular rank one perturbations of the Laplacian, applied to Casimir effect calculations.

Findings

Explicit formulas for the partition function.

Analytic extension of the relative zeta function.

Results for the Casimir force in the half-space model.

Abstract

We study the self adjoint extensions of a class of non maximal multiplication operators with boundary conditions. We show that these extensions correspond to singular rank one perturbations (in the sense of \cite{AK}) of the Laplace operator, namely the formal Laplacian with a singular delta potential, on the half space. This construction is the appropriate setting to describe the Casimir effect related to a massless scalar field in the flat space time with an infinite conducting plate and in the presence of a point like "impurity". We use the relative zeta determinant (as defined in \cite{Mul} and \cite{SZ}) in order to regularize the partition function of this model. We study the analytic extension of the associated relative zeta function, and we present explicit results for the partition function, and for the Casimir force.