Uniqueness of solutions to the Schrodinger equation on the Heisenberg group

Abstract

This paper deals with the Schr{ö}dinger equation $i{\partial }_{s}u(\mathbf{z},t;s)-\mathcal{L}u(\mathbf{z},t;s)=0,$ where $\mathcal{L}$ is the sub-Laplacian on the Heisenberg group. Assume that the initial data $f$ satisfies $|f(\mathbf{z},t)|\le C{q}_a(\mathbf{z},t),$ where $q_s$ is the heat kernel associated to $\mathcal{L}.$ If in addition $|u(\mathbf{z},t;s_0)|\le C{q}_b(\mathbf{z},t),$ for some $s_0\in {\mathbb{R}}^{\ast },$ then we prove that $u(\mathbf{z},t;s)=0$ for all $s\in \mathbb{R}$ whenever $ab<s_0^2.$ This result also holds true on $H$-type groups.

12 pages

Cited by in corpus (1)

• On the Uniqueness of Solutions of the Schrödinger Equation on Riemannian Symmetric Spaces of the Noncompact Type