Casimir self-energy of a δ−δ′ sphere

Abstract

We extend previous work on the vacuum energy of a massless scalar field in the presence of singular potentials. We consider a single sphere defined by the so-called $\ensuremath{\delta}\text{\ensuremath{-}}{\ensuremath{\delta}}^{\ensuremath{'}}$ interaction. Contrary to the Dirac $\ensuremath{\delta}$-potential, we find a nontrivial one-parameter family of potentials such that the regularization procedure gives an unambiguous result for the Casimir self-energy. The procedure employed is based on the zeta function regularization and the cancellation of the heat kernel coefficient ${a}_{2}$. The results obtained are in agreement with particular cases, such as the Dirac $\ensuremath{\delta}$ or Robin and Dirichlet boundary conditions.