On the Uniqueness of Solutions of the Schr\"odinger Equation on Riemannian Symmetric Spaces of the Noncompact Type

TL;DR

This paper proves that solutions to the Schr"odinger equation on noncompact Riemannian symmetric spaces are uniquely determined by their rapid decay properties, extending classical uncertainty principles to this geometric setting.

Contribution

It establishes a uniqueness result for Schr"odinger solutions on symmetric spaces under Beurling-type decay conditions, generalizing known Euclidean results.

Findings

Solutions are identically zero if initial and evolved states decay rapidly.

Conditions of Gelfand-Shilov, Cowling-Price, and Hardy types are derived.

The results extend uncertainty principles to noncompact symmetric spaces.

Abstract

Let X be a Riemannian symmetric space of the noncompact type. We prove that the solution of the time-dependent Schr\"odinger equation on X with square integrable initial condition f is identically zero at all times t whenever f and the solution at a time t0 > 0 are simultaneously very rapidly decreasing. The stated condition of rapid decrease is of Beurling type. Conditions respectively of Gelfand-Shilov, Cowling-Price and Hardy type are deduced.