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UNIQUENESS OF SOLUTIONS TO SCHRÖDINGER EQUATIONS ON $H$-TYPE GROUPS.
- Source :
- Journal of the Australian Mathematical Society; Dec2013, Vol. 95 Issue 3, p297-314, 18p
- Publication Year :
- 2013
-
Abstract
- This paper deals with the Schrödinger equation $i{\partial }_{s} u(\mathbf{z} , t; s)- \mathcal{L} u(\mathbf{z} , t; s)= 0, $ where $ \mathcal{L} $ is the sub-Laplacian on the Heisenberg group. Assume that the initial data $f$ satisfies $\vert f(\mathbf{z} , t)\vert \lesssim {q}_{\alpha } (\mathbf{z} , t), $ where ${q}_{s} $ is the heat kernel associated to $ \mathcal{L} . $ If in addition $\vert u(\mathbf{z} , t; {s}_{0} )\vert \lesssim {q}_{\beta } (\mathbf{z} , t), $ for some ${s}_{0} \in \mathbb{R} \setminus \{ 0\} , $ then we prove that $u(\mathbf{z} , t; s)= 0$ for all $s\in \mathbb{R} $ whenever $\alpha \beta \lt { s}_{0}^{2} . $ This result holds true in the more general context of $H$-type groups. We also prove an analogous result for the Grushin operator on ${ \mathbb{R} }^{n+ 1} . $ [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 14467887
- Volume :
- 95
- Issue :
- 3
- Database :
- Complementary Index
- Journal :
- Journal of the Australian Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 94346624
- Full Text :
- https://doi.org/10.1017/S1446788713000311