Dynamical zeta functions for billiards
| dc.contributor.author | CHAUBET, Yann | |
| hal.structure.identifier | Institut de Mathématiques de Bordeaux [IMB] | |
| dc.contributor.author | PETKOV, Vesselin | |
| dc.date.accessioned | 2024-04-04T02:41:35Z | |
| dc.date.available | 2024-04-04T02:41:35Z | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/191165 | |
| dc.description.abstractEn | Let $D \subset {\mathbb R}^d,\: d \geqslant 2,$ be the union of a finite collection of pairwise disjoint strictly convex compact obstacles. Let $\mu_j \in {\mathbb C},\: {\rm Im}\: \mu_j > 0,$ 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}'({\mathbb R} \setminus \{0\})$ and the Laplace transforms of the leading singularities of $u(t)$ yield the dynamical zeta functions $\eta_{\mathrm N},\: \eta_{\mathrm D}$ 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 (1.1), for $d \geqslant 2$ we show that $\eta_{\mathrm N}$ and $\eta_\mathrm D$ admit a meromorphic continuation in the whole complex plane. In the particular case when the boundary $\partial D$ is real analytic, by using a result of Fried (1995), we prove that the function $\eta_\mathrm{D}$ cannot be entire. Following the result of Ikawa (1988), this implies the existence of a strip $\{z \in {\mathbb C}: \: 0 < {\rm Im}\: z \leq\delta\}$ containing an infinite number of resonances $\mu_j$ for the Dirichlet problem. Moreover, for $\alpha \gg 1$ we obtain a lower bound for the resonances lying in this strip. | |
| dc.language.iso | en | |
| dc.title.en | Dynamical zeta functions for billiards | |
| dc.type | Document de travail - Pré-publication | |
| dc.subject.hal | Mathématiques [math]/Equations aux dérivées partielles [math.AP] | |
| dc.identifier.arxiv | 2201.00683v4 | |
| bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
| bordeaux.institution | Université de Bordeaux | |
| bordeaux.institution | Bordeaux INP | |
| bordeaux.institution | CNRS | |
| hal.identifier | hal-03651808 | |
| hal.version | 1 | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-03651808v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.au=CHAUBET,%20Yann&PETKOV,%20Vesselin&rft.genre=preprint |
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