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Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function. (English) Zbl 1251.37031
The authors investigate the dynamical zeta function through the spectral properties of transfer operators for certain open billiard flows. They assume certain regularity for the family of local unstable manifolds of the billiard flow over the non-wandering set and the Dolgopyat type estimates for the Ruelle transfer operators for some class of functions. The authors mainly proved the cut-off resolvent to be analytic in a certain strip. The estimate enables the authors to obtain a scattering expansion with an exponential decay rate of the remainder for the solutions of a certain Dirichlet problem. The analytic properties and the estimates play a crucial role in many problems related to local energy decay, distribution of the resonances etc. To build approximations of the resolvent of a boundary value problem based on infinite series which are not absolutely convergent, the authors have to deal with infinite series related to reflections of trapping rays. In this direction it appears that the present work is the first one where infinite series of this kind are used for a WKB construction.

MSC:
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
35P20 Asymptotic distributions of eigenvalues in context of PDEs
35P25 Scattering theory for PDEs
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
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