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

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.

31 pages LaTex, to be published in JMP

References in corpus (4)

• Calculating Vacuum Energies in Renormalizable Quantum Field Theories: A New Approach to the Casimir Problem
• Spectral analysis of a flat plasma sheet model
• Zeta function regularization for a scalar field in a compact domain
• Zeta-function approach to Casimir energy with singular potentials

Cited by in corpus (6)

• Local Casimir Effect for a Scalar Field in Presence of a Point Impurity
• Vacuum polarization with zero-range potentials on a hyperplane
• The Casimir energy anomaly for a point interaction
• Spectrum of a family of spin-orbit coupled Hamiltonians with singular perturbation
• A note on "Algebraic approach to Casimir force between two $\delta$-like potentials" (K. Ziemian, Ann. Henri Poincaré, Online First, 2021)
• Resonances and resonance expansions for point interactions on the half-space