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

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.