Vol. 6, No. 6, 2013

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ISSN: 1948-206X (e-only)
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Semiclassical measures for inhomogeneous Schrödinger equations on tori

Nicolas Burq

Vol. 6 (2013), No. 6, 1421–1427

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 × Td. 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.

defect-measures, Schrödinger equations
Mathematical Subject Classification 2010
Primary: 35LXX
Received: 19 September 2012
Accepted: 12 December 2012
Published: 18 November 2013
Nicolas Burq
Université Paris Sud
Bâtiment 425
91405 Orsay Cedex
UMR 8628 du CNRS and Ecole Normale Supérieure
45 rue d’Ulm
75005 Paris Cedex 05