#### Vol. 9, No. 3, 2016

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Dispersive estimates for the Schrödinger operator on step-2 stratified Lie groups

### Hajer Bahouri, Clotilde Fermanian-Kammerer and Isabelle Gallagher

Vol. 9 (2016), No. 3, 545–574
##### Abstract

The present paper is dedicated to the proof of dispersive estimates on stratified Lie groups of step 2 for the linear Schrödinger equation involving a sublaplacian. It turns out that the propagator behaves like a wave operator on a space of the same dimension $p$ as the center of the group, and like a Schrödinger operator on a space of the same dimension $k$ as the radical of the canonical skew-symmetric form, which suggests a decay rate $|t{|}^{-\left(k+p-1\right)∕2}$. We identify a property of the canonical skew-symmetric form under which we establish optimal dispersive estimates with this rate. The relevance of this property is discussed through several examples.

##### Keywords
step-2 stratified Lie groups, Schrödinger equation, dispersive estimates, sublaplacian
Primary: 35B40
##### Milestones
Received: 22 March 2014
Revised: 24 November 2015
Accepted: 30 January 2016
Published: 17 June 2016
##### Authors
 Hajer Bahouri Université Paris Est Laboratoire d’Analyse et de Mathématiques Appliquées, UMR CNRS 8050 61, avenue du Général de Gaulle 94010 Créteil Cedex France Clotilde Fermanian-Kammerer Université Paris Est Laboratoire d’Analyse et de Mathématiques Appliquées, UMR CNRS 8050 61, avenue du Général de Gaulle 94010 Créteil Cedex France Isabelle Gallagher Institut de Mathématiques UMR 7586 Université Paris Diderot (Paris 7) Bâtiment Sophie Germain, Case 7012 75205 Paris Cedex 13 France