  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.
