Abstract
AbstractThis 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} . $
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