Uncertainty principles for the Schrödinger equation on Riemannian symmetric spaces of the noncompact type

[Principes d’incertitude pour l’équation de Schrödinger sur les espaces riemanniens symétriques de type non-compact]

Pasquale, Angela ; Sundari, Maddala

Annales de l'Institut Fourier, Tome 62 (2012), p. 859-886 / Harvested from Numdam

Résumé

Soit $X$ un espace riemannien symétrique de type non-compact. On montre que la solution de l’équation de Schrödinger dépendante du temps sur $X$, avec condition initiale de carré intégrable $f$, est nulle en tout temps $t$ lorsque $f$ et la solution à un temps $t_0>0$ donné sont simultanément très rapidement décroissantes. La condition de décroissance rapide considérée est de type Beurling. Des conditions respectivement de types Gelfand-Shilov, Cowling-Price et Hardy en sont déduites.

Abstract

Let $X$ be a Riemannian symmetric space of the noncompact type. We prove that the solution of the time-dependent Schrödinger equation on $X$ with square integrable initial condition $f$ is identically zero at all times $t$ whenever $f$ and the solution at a time $t_0>0$ are simultaneously very rapidly decreasing. The stated condition of rapid decrease is of Beurling type. Conditions respectively of Gelfand-Shilov, Cowling-Price and Hardy type are deduced.

Détail

Publié le : 2012-01-01

MR 3013810 | Zbl 1253.43007

DOI : https://doi.org/10.5802/aif.2710
Classification:  43A85,  58Jxx
Mots clés: principe d’incertitude, équation de Schrödinger, transformée de Helgason-Fourier, théorème de Beurling, théorème de Hardy

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