An uncertainty principle for solutions of the Schr{\"o}dinger equation on H-type groups

TL;DR

This paper explores uncertainty principles for solutions of PDEs on H-type groups, revealing differences from Euclidean cases and extending Hardy's Uncertainty Principle to these non-commutative groups.

Contribution

It establishes that the heat kernel on H-type groups is not uniquely characterized by decay at two times and extends Hardy's Uncertainty Principle to Schrödinger equations on these groups.

Findings

Heat kernel not uniquely characterized by decay at two times on H-type groups

Extended Hardy's Uncertainty Principle to Schrödinger equations with potential on H-type groups

Demonstrated differences from Euclidean setting in PDE behavior on H-type groups

Abstract

In this paper we consider uncertainty principles for solutions of certain PDEs on H-type groups. We first prove that, contrary to the euclidean setting, the heat kernel on H-type groups is not characterized as the only solution of the heat equation that has sharp decay at 2 different times. We then prove the analogue of Hardy's Uncertainty Principle for solutions of the Schr{\"o}dinger equation with potential on H-type groups. This extends the free case considered by Ben Sa\"id, Dogga and Thangavelu [BTD] and by Ludwig and M{\"u}ller [LM].