# Semiclassical measures for the Schrödinger equation on the torus

Nalini Anantharaman; Fabricio Macià

Journal of the European Mathematical Society (2014)

- Volume: 016, Issue: 6, page 1253-1288
- ISSN: 1435-9855

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topAnantharaman, Nalini, and Macià, Fabricio. "Semiclassical measures for the Schrödinger equation on the torus." Journal of the European Mathematical Society 016.6 (2014): 1253-1288. <http://eudml.org/doc/277418>.

@article{Anantharaman2014,

abstract = {In this article, the structure of semiclassical measures for solutions to the linear Schrödinger equation on the torus is analysed. We show that the disintegration of such a measure on every invariant lagrangian torus is absolutely continuous with respect to the Lebesgue measure. We obtain an expression of the Radon-Nikodym derivative in terms of the sequence of initial data and show that it satisfies an explicit propagation law. As a consequence, we also prove an observability inequality, saying that the $L^2$-norm of a solution on any open subset of the torus controls the full $L^2$-norm.},

author = {Anantharaman, Nalini, Macià, Fabricio},

journal = {Journal of the European Mathematical Society},

keywords = {semiclassical (Wigner) measures; linear Schrödinger equation on the torus; semiclassical limit; dispersive estimates; observability estimates; semiclassical (Wigner) measures; linear Schrödinger equation; torus; semiclassical limit; Radon-Nikodym derivative; dispersive estimates; observability estimates},

language = {eng},

number = {6},

pages = {1253-1288},

publisher = {European Mathematical Society Publishing House},

title = {Semiclassical measures for the Schrödinger equation on the torus},

url = {http://eudml.org/doc/277418},

volume = {016},

year = {2014},

}

TY - JOUR

AU - Anantharaman, Nalini

AU - Macià, Fabricio

TI - Semiclassical measures for the Schrödinger equation on the torus

JO - Journal of the European Mathematical Society

PY - 2014

PB - European Mathematical Society Publishing House

VL - 016

IS - 6

SP - 1253

EP - 1288

AB - In this article, the structure of semiclassical measures for solutions to the linear Schrödinger equation on the torus is analysed. We show that the disintegration of such a measure on every invariant lagrangian torus is absolutely continuous with respect to the Lebesgue measure. We obtain an expression of the Radon-Nikodym derivative in terms of the sequence of initial data and show that it satisfies an explicit propagation law. As a consequence, we also prove an observability inequality, saying that the $L^2$-norm of a solution on any open subset of the torus controls the full $L^2$-norm.

LA - eng

KW - semiclassical (Wigner) measures; linear Schrödinger equation on the torus; semiclassical limit; dispersive estimates; observability estimates; semiclassical (Wigner) measures; linear Schrödinger equation; torus; semiclassical limit; Radon-Nikodym derivative; dispersive estimates; observability estimates

UR - http://eudml.org/doc/277418

ER -

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