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