Semiclassical measures for inhomogeneous Schrödinger equations on tori

Burq, Nicolas

doi:10.2140/apde.2013.6.1421

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Semiclassical measures for inhomogeneous Schrödinger equations on tori

Nicolas Burq

Vol. 6 (2013), No. 6, 1421–1427

DOI: 10.2140/apde.2013.6.1421

Abstract

The purpose of this note is to investigate the high-frequency behavior of solutions to linear Schrödinger equations. More precisely, Bourgain (1997) and Anantharaman and Macià (2011) proved that any weak- $*$ limit of the square density of solutions to the time-dependent homogeneous Schrödinger equation is absolutely continuous with respect to the Lebesgue measure on $ℝ \times T^{d}$ . The contribution of this article is that the same result automatically holds for nonhomogeneous Schrödinger equations, which allows for abstract potential type perturbations of the Laplace operator.

Keywords

defect-measures, Schrödinger equations

Mathematical Subject Classification 2010

Primary: 35LXX

Milestones

Received: 19 September 2012

Accepted: 12 December 2012

Published: 18 November 2013

Authors

Nicolas Burq
Mathématiques Université Paris Sud Bâtiment 425 91405 Orsay Cedex France	UMR 8628 du CNRS and Ecole Normale Supérieure 45 rue d’Ulm 75005 Paris Cedex 05 France

Vol. 6, No. 6, 2013