On the Space-Time Behavior of Schrödinger Wavefunctions

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Abstract
For the Schrödinger equation in Rl, with a potential V(x1 … xl) of the type considered by Kato, the following problem is solved: Given a monomial M(x1 … xl) of degree n in the coordinates, find sufficient conditions on the initial state u such that Me−iH tu is continuous in t and increasing in norm not faster than |t|n as |t| → ∞. In the special case where V(x1 … xl) is a bounded C∞‐function with bounded derivatives, the result implies that (u, t) → e−iH tu is a continuous mapping of S(Rl) × R onto S(Rl), S(Rl) being the Schwartz space of rapidly decreasing functions in the usual topology.

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