The relative trace formula in electromagnetic scattering and boundary layer operators

TL;DR

This paper develops trace-formulae for operators related to electromagnetic scattering, providing a rigorous mathematical foundation for formulas used in physics to compute Casimir energies on Lipschitz domains.

Contribution

It establishes trace-formulae for boundary operators in electromagnetic scattering, extending the Birman-Krein formula to unbounded functions and connecting spectral theory with physical Casimir energy calculations.

Findings

Trace-formulae are established for boundary operators in electromagnetic scattering.

The formulas extend the Birman-Krein formula to unbounded functions.

Rigorous justification of Casimir energy determinant formulas in Lipschitz domains.

Abstract

This paper establishes trace-formulae for a class of operators defined in terms of the functional calculus for the Laplace operator on divergence-free vector fields with relative and absolute boundary conditions on Lipschitz domains in $\mathbb{R}^3$. Spectral and scattering theory of the absolute and relative Laplacian is equivalent to the spectral analysis and scattering theory for Maxwell equations. The trace-formulae allow for unbounded functions in the functional calculus that are not admissible in the Birman-Krein formula. In special cases the trace-formula reduces to a determinant formula for the Casimir energy that is being used in the physics literature for the computation of the Casimir energy for objects with metallic boundary conditions. Our theorems justify these formulae in the case of electromagnetic scattering on Lipschitz domains, give a rigorous meaning to them as the trace of certain trace-class operators, and clarifies the function spaces on which the determinants need to be taken.