The International Center for Mathematical Sciences – Sofia (ICMS-Sofia)
at the
Bulgarian Academy of Sciences


ICMS Colloquium

19.03.2024, 16:00 (Sofia time)

Vesselin Petkov

University of Bordeaux

Vesselin Petkov is a professor emeritus ат the University of Bordeaux who has made important contributions to hyperbolic partial differential equations, spectral and scattering theory and dynamical systems.

Prof. Petkov graduated from Sofia University in 1967. In 1972 he obtained his PhD from Moscow State University under the direction of Olga Oleinik. After the completion of his PhD he was appointed a research fellow at the Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences (IMI-BAS). He became a full professor at IMI-BAS in 1988. Since 1991 he has been a Professor at the University of Bordeaux, France. In 2017 Prof. Petkov was awarded with the IMI-BAS Medal with Ribbon.

One of the main achievements of Prof. Petkov is that he supervised students in Bulgaria and in France who became accomplished mathematicians in the field of Partial Differential Equations and Dynamical Systems.

Dynamical zeta function for billiard flow

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 $\{z \in C: \: 0 < \Im z \leqslant\alpha\}$ containing an infinite number of resonances $\mu_j$ for the Dirichlet problem. The above results are obtained in a joint work with Yann Chaubet.