Vol. 10, No. 4, 2017

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ISSN: 1948-206X (e-only)
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Distorted plane waves in chaotic scattering

Maxime Ingremeau

Vol. 10 (2017), No. 4, 765–816
Abstract

We provide a precise description of distorted plane waves for semiclassical Schrödinger operators under the assumption that the classical trapped set is hyperbolic and that a certain topological pressure (a quantity defined using thermodynamical formalism) is negative. Distorted plane waves are generalized eigenfunctions of the Schrödinger operator which differ from free plane waves, eix,ξh, by an outgoing term. Under our assumptions we show that they can be written as a convergent sum of Lagrangian states. That provides a description of their semiclassical defect measures in the spirit of quantum ergodicity and extends results of Guillarmou and Naud obtained for hyperbolic quotients to our setting.

Keywords
scattering theory, quantum chaos, semiclassical measures, distorted plane waves
Mathematical Subject Classification 2010
Primary: 35P20, 35P25, 81Q50
Milestones
Received: 7 January 2016
Revised: 19 November 2016
Accepted: 7 March 2017
Published: 9 May 2017
Authors
Maxime Ingremeau
Université Paris-Sud
91400 Orsay
France