### Maximal degenerate representations of SL$(n+1,H)$.

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Let $X$ be a Riemannian symmetric space of the noncompact type. We prove that the solution of the time-dependent Schrödinger equation on $X$ with square integrable initial condition $f$ is identically zero at all times $t$ whenever $f$ and the solution at a time ${t}_{0}\>0$ 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.

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