Semiclassical resolvent estimates in chaotic scattering

2009/04/20 by Stéphane Nonnenmacher, Nonnenmacher, Stéphane, Maciej Zworski +1
Mathematics · Physics and Astronomy · #35B34 #37D20 #81Q50 #81U05 #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.AP #math.DS #math.MP #msc:35B34 #msc:37D20 #msc:81Q50 #msc:81U05

paper · pdf · doi:10.48550/arxiv.0904.2986

9 pages

arxiv created 2009/09/11 · arxiv updated 2009/12/01

Abstract

We prove resolvent estimates for semiclassical operators such as -h² Δ+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 Schrödinger propagation and to energy decay of solutions to wave equations.

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