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PERTURBATION OF THE SEMICLASSICAL SCHRÖDINGER EQUATION ON NEGATIVELY CURVED SURFACES

Part of: General mathematical topics and methods in quantum theory Equations of mathematical physics and other areas of application Partial differential equations on manifolds; differential operators Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  27 August 2015

Suresh Eswarathasan  and
Gabriel Rivière
Suresh Eswarathasan
Affiliation:
Institut des Hautes Études Scientifiques, Le Bois-Marie 35, route de Chartres, 91440 Bures-sur-Yvette, France Department of Mathematics and Statistics, McGill University, Montréal, Canada (suresh@ihes.fr; suresh@math.mcgill.ca) School of Mathematics, Cardiff University, Senghennyd Road, Cardiff, Wales, CF244AG, UK
Gabriel Rivière
Affiliation:
Laboratoire Paul Painlevé (U.M.R. CNRS 8524), U.F.R. de Mathématiques, Université Lille 1, 59655 Villeneuve d’Ascq Cedex, France (gabriel.riviere@math.univ-lille1.fr)
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Abstract

We consider the semiclassical Schrödinger equation on a compact negatively curved surface. For any sequence of initial data microlocalized on the unit cotangent bundle, we look at the quantum evolution (below the Ehrenfest time) under small perturbations of the Schrödinger equation, and we prove that, in the semiclassical limit, and for typical perturbations, the solutions become equidistributed on the unit cotangent bundle.

Keywords

hyperbolic dynamical systemssemiclassical analysisquantum chaosgeodesic flowssemiclassical measures

MSC classification

Primary: 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 58J51: Relations between spectral theory and ergodic theory, e.g. quantum unique ergodicity 81Q50: Quantum chaos 35Q41: Time-dependent Schrödinger equations, Dirac equations
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 16 , Issue 4 , September 2017 , pp. 787 - 835
Copyright
© Cambridge University Press 2015 

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