Dynamical zeta functions for billiards

Abstract

Let D ⊂ R d , d 2, be the union of a finite collection of pairwise disjoint strictly convex compact obstacles. Let µ j ∈ C, Im µ j > 0 be the resonances of the Laplacian in the exterior of D with Neumann or Dirichlet boundary condition on ∂D. For d odd, u(t) = j e i|t|µj is a distribution in D ′ (R \ {0}) and the Laplace transforms of the leading singularities of u(t) yield the dynamical zeta functions η N , η D for Neumann and Dirichlet boundary conditions, respectively. These zeta functions play a crucial role in the analysis of the distribution of the resonances. Under the non-eclipse condition, for d2 we show that η N and η D admit a meromorphic continuation in the whole complex plane. In the particular case when the boundary ∂D is real analytic, by using a result of Fried [Fri95], we prove that the function η D cannot be entire. Following Ikawa [Ika88], this implies the existence of a strip {z ∈ C : 0 < Im z δ} containing an infinite number of resonances µ j for the Dirichlet problem.

Semiclassical formulae for Wigner distributions

“…An even easier alternative could be to work with Grassmanian extension techniques from [FT17] instead of the density bundles. This has very recently even been transformed to the setting of obstacle scattering [CP22]. Establishing a completely rigorous foundation for Z FT a (λ) should be subjected to further mathematical research.…”