A relative trace formula for obstacle scattering

We consider the case of scattering by several obstacles in Rd for d?2. In this setting, absolutely continuous part Laplace operator ? with Dirichlet boundary conditions and free ?0 are unitarily equivalent. For suitable functions that decay sufficiently fast, we have difference g(?)?g(?0) is a trace-class its trace described Krein spectral shift function. article, study contribution to (and hence function) arises from assembling relative setting where completely separated. two obstacles, operators ?1 ?2 obtained imposing only on one objects. Our main result states then g(?)?g(?1)?g(?2)+g(?0) much larger class (including polynomial growth) may still be computed modification Birman–Krein formula. g(x)=x1 2, has physical meaning as vacuum energy massless scalar field expressible an integral involving layer operators. Such integrals been derived physics literature using nonrigorous path derivations our formula provides both rigorous justification well generalization.