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

Pasquale, Angela; Sundari, Maddala

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.

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

[1] Anker, Jean-Philippe A basic inequality for scattering theory on Riemannian symmetric spaces of the noncompact type, Amer. J. Math., Volume 113 (1991) no. 3, pp. 391-398 | DOI | MR | Zbl

[2] Bagchi, S. C.; Ray, Swagato K. Uncertainty principles like Hardy’s theorem on some Lie groups, J. Austral. Math. Soc., Volume 65 (1998) no. 3, pp. 289-302 | DOI | MR | Zbl

[3] Ben Said, Salem; Thangavelu, Sundaram Uniqueness of solutions to the Schrödinger equation on the Heisenberg group (2010) (arXiv:1006.5310)

[4] Bonami, Aline; Demange, Bruno; Jaming, Philippe Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms, Rev. Mat. Iberoamericana, Volume 19 (2003) no. 1, pp. 23-55 | DOI | MR

[5] Chanillo, Sagun Uniqueness of solutions to Schrödinger equations on complex semi-simple Lie groups, Proc. Indian Acad. Sci. Math. Sci., Volume 117 (2007) no. 3, pp. 325-331 | DOI | MR

[6] Cowling, Michael G.; Escauriaza, Luis; Kenig, Carlos E.; Ponce, Gustavo; Vega, Luis The Hardy uncertainty principle revisited arXiv:1005.1543v1, to appear in Indiana Univ. Math. J., 59 (2010) | MR

[7] Eguchi, Masaaki On the Radon transform of the rapidly decreasing functions on symmetric spaces. II., Hiroshima Math. J., Volume 1 (1971), pp. 161-169 | MR | Zbl

[8] Eguchi, Masaaki; Hirai, T.; Schiffmann, G. Some properties of Fourier transform on Riemannian symmetric spaces, Lectures on harmonic analysis on Lie groups and other topics (Strasbourg, 1979) (Lecture Notes in Mathematics), Volume 14, Kyoto Univ., 1982, pp. 9-43 | MR | Zbl

[9] Escauriaza, Luis; Kenig, Carlos E.; Ponce, Gustavo; Vega, Luis Hardy’s uncertainty principle, convexity and Schrödinger evolutions, J. Euro. Math. Soc., Volume 10 (2008), pp. 883-907 | DOI | MR

[10] Escauriaza, Luis; Kenig, Carlos E.; Ponce, Gustavo; Vega, Luis The sharp Hardy uncertainty principle for Schrödinger evolutions, Duke Math. J., Volume 155 (2010) no. 1, pp. 163-187 | DOI | MR

[11] Escauriaza, Luis; Kenig, Carlos E.; Ponce, Gustavo; Vega, Luis Uncertainty principles of Morgan type and Schrödinger evolutions, J. London Math. Soc., Volume 83 (2011) no. 1, pp. 187-207 | DOI | MR

[12] Gangolli, Ramesh; Varadarajan, V. S. Harmonic analysis of spherical functions on real reductive groups, Springer Verlag, 1988 | MR | Zbl

[13] Helgason, Sigurdur Differential geometry, Lie groups, and symmetric spaces, Academic Press, 1978 | MR | Zbl

[14] Helgason, Sigurdur Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions, Academic Press, 1984 | MR | Zbl

[15] Helgason, Sigurdur Geometric analysis on symmetric spaces, Mathematical Surveys and Monographs, 39, American Mathematical Society, 2008 | MR | Zbl

[16] Hörmander, Lars The analysis of linear partial differential operators. I, Springer Verlag, 1990 | MR | Zbl

[17] Hörmander, Lars A uniqueness theorem of Beurling for Fourier transform pairs, Ark. Mat., Volume 29 (1991) no. 2, pp. 237-240 | DOI | MR | Zbl

[18] Ionescu, I. D.; Kenig, Carlos E. Uniqueness properties of solutions of Schrödinger equations, J. Funct. Anal., Volume 232 (2006), pp. 90-136 | DOI | MR

[19] Rauch, Jeffrey Partial differential equations, Springer-Verlag, 1991 | MR | Zbl

[20] Sarkar, Rudra P.; Sengupta, Jyoti Beurling’s theorem and characterization of heat kernel for Riemannian symmetric spaces of noncompact type, Canad. Math. Bull. (2007) no. 2, pp. 291-312 | DOI | MR

[21] Sarkar, Rudra P.; Sengupta, Jyoti Beurling’s theorem for Riemannian symmetric spaces. II, Proc. Amer. Math. Soc., Volume 136 (2008) no. 2, pp. 1841-1853 | MR

[22] Schwartz, Laurent Théorie des distributions, Hermann, Paris, 1966 | MR | Zbl

[23] Sengupta, Joyti The uncertainty principle on Riemannian symmetric spaces of the noncompact type, Proc. Amer. Math. Soc., Volume 128 (2000) no. 8, pp. 2493-2499 | DOI | MR

[24] Sengupta, Joyti An analogue of Hardy’s theorem for semi-simple Lie groups, Proc. Amer. Math. Soc., Volume 130 (2002) no. 4, pp. 1009-1017 | DOI | MR

[25] Thangavelu, Sundaram An introduction to the uncertainty principle. Hardy’s theorem on Lie groups, Birkhäuser, 2004 | MR