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

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.