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Uniqueness of solutions to Schrödinger equations on 2-step nilpotent Lie groups

Authors: Jean Ludwig and Detlef Müller
Journal: Proc. Amer. Math. Soc. 142 (2014), 2101-2118
MSC (2010): Primary 43A80, 22E30, 22E25, 35B05
Published electronically: March 13, 2014
MathSciNet review: 3182028
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Abstract: Let $ \mathfrak{g}=\mathfrak{g}_1\oplus \mathfrak{g}_2,[\mathfrak{g},\mathfrak{g}] =\mathfrak{g}_2, $ a nilpotent Lie algebra of step 2, $ V_1,\cdots , V_m $ a basis of $ \mathfrak{g}_1 $ and $ L=\sum _{j,k}^{m}a_{jk}V_j V_k $ a left-invariant differential operator on $ G=\mathrm {exp} (\mathfrak{g}) $, where $ (a_{jk})_{jk}\in M_n(\mathbb{R}) $ is symmetric. It is shown that if a solution $ w(t,x) $ to the Schrödinger equation $ \partial _t w(t,g)=i Lw(t,g),$
$ g\in G, t\in \mathbb{R}, w(0,g)=f(g)$, satisfies a suitable Gaussian type estimate at time $ t= 0 $ and at some time $ t=T\ne 0 $, then $ w=0 $. The proof is based on Hardy's uncertainty principle, on explicit computations within Howe's oscillator semigroup and on methods developed by Fulvio Ricci and the second author. Our results extend work by Ben Saïd, Thangavelu and Dogga on the Schrödinger equation associated to the sub-Laplacian on Heisenberg type groups.

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Additional Information

Jean Ludwig
Affiliation: Laboratoire de Mathématiques et Applications de Metz UMR 7122, Université de Lorraine, île du Saulcy, 57045 Metz Cedex 01, France

Detlef Müller
Affiliation: Mathematisches Seminar, C.A.-Universität Kiel, Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany

Keywords: Schr\"odinger equation, uniqueness, uncertainty principle, 2-step nilpotent Lie group, oscillator semigroup, Heisenberg type groups
Received by editor(s): July 19, 2012
Published electronically: March 13, 2014
Additional Notes: The second author was supported by a one-month research invitation from the University Paul Verlaine-Metz in 2010-2011
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2014 American Mathematical Society

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