Trace singularities in obstacle scattering and the Poisson relation for the relative trace

Abstract

We consider the case of scattering by several obstacles in {mathbb {R}}^d, d ge 2 for the Laplace operator Delta with Dirichlet boundary conditions imposed on the obstacles. In the case of two obstacles, we have the Laplace operators Delta _1 and Delta _2 obtained by imposing Dirichlet boundary conditions only on one of the objects. The relative operator g(Delta ) - g(Delta _1) - g(Delta _2) + g(Delta _0) was introduced in Hanisch, Waters and one of the authors in (A relative trace formula for obstacle scattering. arXiv:2002.07291, 2020) and shown to be trace-class for a large class of functions g, including certain functions of polynomial growth. When g is sufficiently regular at zero and fast decaying at infinity then, by the Birman–Krein formula, this trace can be computed from the relative spectral shift function xi _mathrm {rel}(lambda ) = -frac{1}{pi } {text {Im}}(Xi (lambda )), where Xi (lambda ) is holomorphic in the upper half-plane and fast decaying. In this paper we study the wave-trace contributions to the singularities of the Fourier transform of xi _mathrm {rel}. In particular we prove that {hat{xi }}_mathrm {rel} is real-analytic near zero and we relate the decay of Xi (lambda ) along the imaginary axis to the first wave-trace invariant of the shortest bouncing ball orbit between the obstacles. The function Xi (lambda ) is important in the physics of quantum fields as it determines the Casimir interactions between the objects.