Casimir energy for the scalar field: a global approach with cut-off exponential function

TL;DR

This paper introduces a global regularization method using cut-off exponential functions to accurately compute the Casimir energy of a massless scalar field around a spherical shell, addressing divergences systematically.

Contribution

It presents a novel regularization approach with two regulators for calculating Casimir energy, ensuring mathematical consistency and matching known results.

Findings

The method effectively handles divergences in Casimir energy calculations.

Results agree with existing literature, confirming the approach's validity.

The approach distinguishes contributions from inner and outer regions of the shell.

Abstract

A global approach with cut-off exponential functions previously proposed is used to obtain the Casimir energy of a massless scalar field in the presence of a spherical shell. The proposed method makes the use of two regulators, one of them to makes finite the sum of the orders of Bessel functions and the other, to regularizes the integral involving the zeros of Bessel function. This procedure ensures a consistent mathematical handling in the calculations of the Casimir energy for a scalar field and allows to show all types of divergences. We consider separately the contributions of the inner and outer regions of a spherical shell and show that the results obtained are in agreement with those known in the literature and this gives a confirmation for the consistence of the proposed approach.