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 $\delta \text{−}{\delta }^{\prime }$ interaction. Contrary to the Dirac $\delta$-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 $\delta$ or Robin and Dirichlet boundary conditions.

• Received 31 August 2022
• Accepted 8 December 2022

DOI:https://doi.org/10.1103/PhysRevD.107.025002

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Published by the American Physical Society