Uniqueness of solutions to the Schrodinger equation on the Heisenberg group

TL;DR

This paper proves a uniqueness result for solutions to the Schrödinger equation on the Heisenberg group, showing that under certain decay conditions, the solution must be identically zero.

Contribution

It establishes a new uniqueness theorem for Schrödinger equations on the Heisenberg group and H-type groups based on decay conditions of initial data and solutions.

Findings

Solutions vanish if decay conditions are met with ab < s_0^2

The result extends to H-type groups

Provides conditions for uniqueness of Schrödinger solutions

Abstract

This paper deals with the Schr{\"o}dinger equation $i\partial_s u({\bf z},t;s)-\cal L u({\bf z}, t;s)=0,$ where $\cal L$ is the sub-Laplacian on the Heisenberg group. Assume that the initial data $f$ satisfies $| f({\bf z},t)| \leq C q_a({\bf z},t),$ where $q_s$ is the heat kernel associated to $\cal L.$ If in addition $ |u({\bf z},t;s_0)|\leq C q_b({\bf z},t),$ for some $s_0\in \R^*,$ then we prove that $u({\bf z},t;s)=0$ for all $s\in \R $ whenever $ab<s_0^2.$ This result also holds true on $H$-type groups.