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DTSTART;TZID=Europe/Sofia:20240319T160000
DTEND;TZID=Europe/Sofia:20240319T173000
DTSTAMP:20260420T213841
CREATED:20240315T081347Z
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UID:16060-1710864000-1710869400@math.bas.bg
SUMMARY:Колоквиум на МЦМН
DESCRIPTION:Дата и час: 19.03.2024 г.\, сряда\, 16:00 ч. \nМясто: изцяло в Zoom \nЛектор: Веселин Петков (Université Bordeaux) \nДоклад: Dynamical zeta function for billiard flow \nАбстракт: 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 <em>meromorphic continuation</em> 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.
URL:https://math.bas.bg/event/%d0%ba%d0%be%d0%bb%d0%be%d0%ba%d0%b2%d0%b8%d1%83%d0%bc-%d0%bd%d0%b0-%d0%bc%d1%86%d0%bc%d0%bd-2/
LOCATION:Zoom
CATEGORIES:Редовен семинар
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