The generalized Abel-Plana formula with applications to Bessel functions and Casimir effect

TL;DR

This paper generalizes the Abel-Plana formula to handle complex geometries involving Bessel functions, enabling more efficient and convergent calculations of vacuum expectation values in the Casimir effect across various boundary conditions.

Contribution

The paper introduces a generalized Abel-Plana formula applicable to series over zeros of Bessel function combinations, expanding the method's applicability to complex geometries in Casimir effect calculations.

Findings

Derived summation formulas for series over Bessel function zeros.

Applied the generalized formula to Casimir effect in diverse geometries.

Simplified renormalization by separating boundary and bulk contributions.

Abstract

One of the most efficient methods for the evaluation of the vacuum expectation values for physical observables in the Casimir effect is based on using the Abel-Plana summation formula. This enables to derive the renormalized quantities in a manifestly cutoff independent way and to present them in the form of strongly convergent integrals. However, applications of the Abel-Plana formula, in its usual form, are 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 combinations of Bessel functions and for integrals involving these functions. It has 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 direct mode summation approach to the Casimir effect for spherically and cylindrically symmetric boundaries, for boundaries moving with uniform proper acceleration, and in various braneworld scenarios. This allows to extract from the vacuum expectation values of local physical observables the parts corresponding to the geometry without boundaries and to present the boundary-induced parts in terms of integrals strongly convergent for the points away from the boundaries. As a result, the renormalization procedure for these observables is reduced to the corresponding procedure for bulks without boundaries. The present paper reviews these results. We also aim to collect the results on vacuum expectation values for local physical observables such as the field square and the energy-momentum tensor in manifolds with boundaries for various bulk and boundary geometries.