Institute of Mathematics and Informatics

Колоквиум на МЦМН
2024-03-19 16:00 - 17:30 (Europe/Sofia) | Zoom
Дата и час: 19.03.2024 г., сряда, 16:00 ч.
Място: изцяло в Zoom
Лектор: Веселин Петков (Université Bordeaux)
Доклад: Dynamical zeta function for billiard flow
Абстракт: We will present briefly the connection between Riemann zeta function, Ruellle zeta function and dynamical zeta function. The last one is related to the billiard flow for the union (D subset R^d) of a finite collection of pairwise disjoint strictly convex compact obstacles. Let (mu_j in C), (Im mu_j > 0) be the resonances of the Laplacian in the exterior of (D) with Neumann or Dirichlet boundary condition on (partial D). For (d) odd, (u(t) = sum_j e^{i |t| mu_j}) is a distribution in ( mathcal{D}'(R setminus {0})) and the Laplace transforms of the leading singularities of (u(t)) yield the dynamical zeta functions (eta_{mathrm N}(s),: eta_{mathrm D}(s)) for Neumann and Dirichlet boundary conditions, respectively. These zeta functions play a crucial role in the analysis of the distribution of the resonances. In this talk we discuss two results. (1) Under a non-eclipse condition, we show that (eta_{mathrm N}) and (eta_mathrm D) admit a meromorphic continuation in the whole complex plane with simple poles and integer residues. (2) In the case when the boundary (partial D) is real analytic, we prove that the function (eta_mathrm{D}) cannot be entire and the modified Lax Phillips conjecture (MLPC) holds. This conjecture introduced in 1990 says that there exists of a strip ({z in C: : 0 < Im z leqslantalpha}) containing an infinite number of resonances (mu_j) for the Dirichlet problem. The above results are obtained in a joint work with Yann Chaubet.
