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
.
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.
Keywords
Maxwell equations, layer potential, Casimir energy, trace
formula
