The generalized Abel-Plana formula. Applications to Bessel functions and Casimir effect

TL;DR

This paper introduces a generalized Abel-Plana formula that broadens its applicability, enabling more efficient and regularized calculations of vacuum expectation values in the Casimir effect for complex geometries involving Bessel functions.

Contribution

The paper develops a generalized Abel-Plana formula applicable to series over zeros of Bessel function combinations, extending previous methods for Casimir effect calculations.

Findings

Derived new summation formulas for Bessel function zeros.

Applied formulas to Casimir effect in spherical and cylindrical geometries.

Achieved manifestly cutoff-independent regularization of physical observables.

Abstract

One of the most efficient methods to obtain the vacuum expectation values for the physical observables in the Casimir effect is based on the using the Abel-Plana summation formula. This allows to derive the regularized quantities by manifestly cutoff independent way and to present them in the form of strongly converging integrals. However the applications of Abel- Plana formula in usual form is restricted by simple geometries when the eigenmodes have a simple dependence on quantum numbers. The author generalized the Abel-Plana formula which essentially enlarges its application range. Based on this generalization, formulae have been obtained for various types of series over the zeros of some combinations of Bessel functions and for integrals involving these functions. It have been shown that these results generalize the special cases existing in literature. Further the derived summation formulae have been used to summarize series arising in the mode summation approach to the Casimir effect for spherically and cylindrically symmetric boundaries. This allows to extract the divergent parts from the vacuum expectation values for the local physical observables in the manifestly cutoff independent way. The present paper reviews these results. Some new considerations are added as well.