no. 3
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, Angela1 ; Sundari, Maddala2
1 Université Paul Verlaine Laboratoire de Mathématiques et Applications (LMAM, UMR CNRS 7122) Bâtiment A, Ile du Saulcy 57045 Metz cedex 1 (France)
2 Chennai Mathematical Institute Plot No. H1, SIPCOT IT Park Padur P.O. Siruseri 603 103 (India)
Annales de l'Institut Fourier, Tome 62 (2012) no. 3, pp. 859-886.

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.

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.

DOI : 10.5802/aif.2710
Classification : 43A85, 58Jxx
Keywords: Uncertainty principle, Schrödinger equation, Helgason-Fourier transform, Beurling theorem, Hardy theorem
Mot clés : principe d’incertitude, équation de Schrödinger, transformée de Helgason-Fourier, théorème de Beurling, théorème de Hardy
Pasquale, Angela 1 ; Sundari, Maddala 2

1 Université Paul Verlaine Laboratoire de Mathématiques et Applications (LMAM, UMR CNRS 7122) Bâtiment A, Ile du Saulcy 57045 Metz cedex 1 (France)
2 Chennai Mathematical Institute Plot No. H1, SIPCOT IT Park Padur P.O. Siruseri 603 103 (India) 
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     journal = {Annales de l'Institut Fourier},
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Pasquale, Angela; Sundari, Maddala. Uncertainty principles for the Schrödinger equation on Riemannian symmetric spaces of the noncompact type. Annales de l'Institut Fourier, Tome 62 (2012) no. 3, pp. 859-886. doi : 10.5802/aif.2710. https://aif.centre-mersenne.org/articles/10.5802/aif.2710/

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