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

Abstract

AbstractWe consider the case of scattering by several obstacles in $${\mathbb {R}}^d$$ R d , $$d \ge 2$$ d ≥ 2 for the Laplace operator $$\Delta $$ Δ with Dirichlet boundary conditions imposed on the obstacles. In the case of two obstacles, we have the Laplace operators $$\Delta _1$$ Δ 1 and $$\Delta _2$$ Δ 2 obtained by imposing Dirichlet boundary conditions only on one of the objects. The relative operator $$g(\Delta ) - g(\Delta _1) - g(\Delta _2) + g(\Delta _0)$$ g ( Δ ) - g ( Δ 1 ) - g ( Δ 2 ) + g ( Δ 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 $$\xi _\mathrm {rel}(\lambda ) = -\frac{1}{\pi } {\text {Im}}(\Xi (\lambda ))$$ ξ rel ( λ ) = - 1 π Im ( Ξ ( λ ) ) , where $$\Xi (\lambda )$$ Ξ ( λ ) 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 $$\xi _\mathrm {rel}$$ ξ rel . In particular we prove that $${\hat{\xi }}_\mathrm {rel}$$ ξ ^ rel is real-analytic near zero and we relate the decay of $$\Xi (\lambda )$$ Ξ ( λ ) along the imaginary axis to the first wave-trace invariant of the shortest bouncing ball orbit between the obstacles. The function $$\Xi (\lambda )$$ Ξ ( λ ) is important in the physics of quantum fields as it determines the Casimir interactions between the objects.