Dispersion and controllability for the Schrödinger equation on negatively curved manifolds

Anantharaman, Nalini; Rivière, Gabriel

doi:10.2140/apde.2012.5.313

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Dispersion and controllability for the Schrödinger equation on negatively curved manifolds

Nalini Anantharaman and Gabriel Rivière

Vol. 5 (2012), No. 2, 313–338

DOI: 10.2140/apde.2012.5.313

Abstract

We study the time-dependent Schrödinger equation $ı \frac{\partial u}{\partial t} = - \frac{1}{2} Δ u$ , on a compact Riemannian manifold on which the geodesic flow has the Anosov property. Using the notion of semiclassical measures, we prove various results related to the dispersive properties of the Schrödinger propagator, and to controllability.

Keywords

Schrödinger equation, semiclassical analysis, control theory

Mathematical Subject Classification 2000

Primary: 35B37

Milestones

Received: 28 July 2010

Revised: 27 August 2011

Accepted: 31 August 2011

Published: 27 August 2012

Authors

Nalini Anantharaman
Laboratoire de Mathématiques (UMR 8628) Université Paris-Sud XI Bâtiment 425 91405 Orsay Cedex France

Gabriel Rivière
Laboratoire Paul Painlevé (UMR 8524) Université Lille 1 Bâtiment M3 59655 Villeneuve d’Ascq Cedex France

Vol. 5, No. 2, 2012