Distorted plane waves in chaotic scattering

Ingremeau, Maxime

doi:10.2140/apde.2017.10.765

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Distorted plane waves in chaotic scattering

Maxime Ingremeau

Vol. 10 (2017), No. 4, 765–816

DOI: 10.2140/apde.2017.10.765

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, $e^{i 〈 x, ξ 〉 ∕ h}$ , 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

Vol. 10, No. 4, 2017