A Dispersion Relation for the Density of States With Application to the Casimir Effect

TL;DR

This paper develops a dispersion relation approach to express the density of states and Casimir energy of a body in terms of its scattering matrix, overcoming divergence issues and removing symmetry constraints.

Contribution

It introduces a convergent dispersion relation for the scattering matrix determinant, enabling shape-independent Casimir energy calculations.

Findings

Derived a convergent expression for the scattering matrix determinant.

Expressed Casimir energy in terms of optical scattering matrix without symmetry assumptions.

Provided a method to handle divergences in density of states calculations.

Abstract

The trace of a function of a Schrodinger operator minus the same for the Laplacian can be expressed in terms of the determinant of its scattering matrix. The naive formula for this determinant is divergent. Using a dispersion relation, we find another expression for it which is convergent, but needs one piece of information beyond the scattering matrix. Except for this `anomaly', we can express the Casimir energy of a compact body in terms of its optical scattering matrix, without assuming any rotational symmetry for its shape.