Published online by Cambridge University Press: 07 August 2013
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} . $
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