Semiclassical measures for the Schrödinger equation on the torus

Anantharaman, 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 -