Casimir Energy for concentric $\delta$-$\delta'$ spheres

TL;DR

This paper investigates the Casimir energy between two concentric spheres with a combined delta and delta prime potential, revealing how different coupling values influence the energy's sign and magnitude, and connecting to various boundary conditions.

Contribution

It introduces a model with delta and delta prime potentials on spheres, analyzing how these affect Casimir energy and boundary condition limits, a novel approach in Casimir effect studies.

Findings

Energy sign depends on coupling parameters.

Different radii alter the Casimir energy behavior.

Potential models boundary conditions like Dirichlet and Robin.

Abstract

We study the vacuum interaction of a scalar field and two concentric spheres defined by a singular potential on their surfaces. The potential is a linear combination of the Dirac-$\delta$ and its derivative. The presence of the delta prime term in the potential causes that it behaves differently when it is seen from the inside or from the outside of the sphere. We study different cases for positive and negative values of the delta prime coupling, keeping positive the coupling of the delta. As a consequence, we find regions in the space of couplings, where the energy is positive, negative or zero. Moreover, the sign of the $\delta'$ couplings cause different behavior on the value of the Casimir energy for different values of the radii. This potential gives rise to general boundary conditions with limiting cases defining Dirichlet and Robin boundary conditions what allows us to simulate purely electric o purely magnetic spheres.