Abstract
In this note we study the time-dependent Schrödinger equation on complex semi-simple Lie groups. We show that if the initial data is a bi-invariant function that has sufficient decay and the solution has sufficient decay at another fixed value of time, then the solution has to be identically zero for all time. We also derive Strichartz and decay estimates for the Schrödinger equation. Our methods also extend to the wave equation. On the Heisenberg group we show that the failure to obtain a parametrix for our Schrödinger equation is related to the fact that geodesics project to circles on the contact plane at the identity.
Similar content being viewed by others
References
Beals R, Greiner P C and Stanton N, The heat equation on a CR manifold, J. Differential Geom. 20 (1984) 343–387
Folland GB and Sitaram A, The uncertainty principles: Amathematical survey, J. Fourier Analysis Appl. 3 (1997) 207–238
Gindikin S and Karpelevic F I, Plancherel measure of Riemannian symmetric spaces of non-positive curvature, Dokl. Akad. Nauk. SSSR 145 (1962) 252–255
Hardy G H, A theorem concerning Fourier transforms, J. London Math. Soc. 8 (1933) 227–231
Harish-Chandra, Spherical functions on a semi-simple Lie group I, Amer. J. Math. 80 (1958) 241–310
Harish-Chandra, Spherical functions on a semi-simple Lie group II, Amer. J. Math. 80 (1958) 553–613
Helgason S, Groups and geometric analysis — integral geometry, Invariant differential operators and spherical functions (Academic Press: New York) (1984)
Monti R, Some properties of Carnot-Caratheodory balls in the Heisenberg group, Rend. Math. Acc. Lincei, 11(9) (2000) 155–167
Sitaram A and Sundari M, An analogue of Hardy’s theorem for very rapidly decreasing functions, Pacific J. Math.. 177 (1997) 187–200
Strichartz R S, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions to wave equations, Duke Math. J 44(3) (1977) 705–714
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to U. B. Tewari on his retirement
Rights and permissions
About this article
Cite this article
Chanillo, S. Uniqueness of solutions to Schrödinger equations on complex semi-simple Lie groups. Proc Math Sci 117, 325–331 (2007). https://doi.org/10.1007/s12044-007-0028-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12044-007-0028-7