AN UNCERTAINTY PRINCIPLE FOR SOLUTIONS OF THE SCHRÖDINGER EQUATION ON -TYPE GROUPS

Abstract

In this paper we consider uncertainty principles for solutions of certain partial differential equations on $H$ -type groups. We first prove that, on $H$ -type groups, the heat kernel is an average of Gaussians in the central variable, so that it does not satisfy a certain reformulation of Hardy’s uncertainty principle. We then prove the analogue of Hardy’s uncertainty principle for solutions of the Schrödinger equation with potential on $H$ -type groups. This extends the free case considered by Ben Saïd et al. [‘Uniqueness of solutions to Schrödinger equations on H-type groups’, J. Aust. Math. Soc. (3)95 (2013), 297–314] and by Ludwig and Müller [‘Uniqueness of solutions to Schrödinger equations on 2-step nilpotent Lie groups’, Proc. Amer. Math. Soc.142 (2014), 2101–2118].