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Register a new userSemiclassical resolvent estimates in chaotic scattering
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Added by
Stephane Nonnenmacher
Publication date
2009
fields
Physics
and research's language is
English
Authors
Stephane Nonnenmacher
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We prove resolvent estimates for semiclassical operators such as $-h^2 Delta+V(x)$ in scattering situations. Provided the set of trapped classical trajectories supports a chaotic flow and is sufficiently filamentary, the analytic continuation of the resolvent is bounded by $h^{-M}$ in a strip whose width is determined by a certain topological pressure associated with the classical flow. This polynomial estimate has applications to local smoothing in Schrodinger propagation and to energy decay of solutions to wave equations.
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This paper is devoted to the study of a semiclassical black box operator $P$. We estimate the norm of its resolvent truncated near the trapped set by the norm of its resolvent truncated on rings far away from the origin. For $z$ in the unphysical sheet with $- h |ln h| < Im z < 0$, we prove that this estimate holds with a constant $h |Im z|^{-1} e^{C|Im z|/h}$. We also obtain analogous bounds for the resonances states of $P$. These results hold without any assumption on the trapped set neither any assumption on the multiplicity of the resonances.
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