The relative trace formula in electromagnetic scattering and boundary layer operators

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 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 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 clarify the function spaces on which the determinants need to be taken.1. Introduction 361 2. Function spaces on Lipschitz domains 370 3. Laplace operators on the interior domain 372 4. Laplace operators on the exterior domain 375 5.The combined relative operators and the Birman-Krein formula 377 6. Maxwell boundary layer operators 378 7. Estimates and low-energy expansions for the layer potential operators 385 8. Resolvent formulae and estimates 394 9.The function 396 10.Relative trace formula 399 11.Proof of the main theorems