We seek a sufficient condition on initial data in order that the solution of Schrödinger equation has both analytically smoothing effect and exponential decay in time.
Yuya Dan. Kunihiko Kajitani. "Smoothing effect and exponential time decay of solutions of Schrödinger equations." Proc. Japan Acad. Ser. A Math. Sci. 78 (6) 92 - 95, June 2002. https://doi.org/10.3792/pjaa.78.92