MPEJ Volume 3, No.2, 22pp Received: March 4, 1997, Revised: May 9, 1997, Accepted: May 19, 1997 Walter Craig Microlocal moments and regularity of solutions of Schroedinger's equation ABSTRACT: There is a connection between the smoothness of solutions of Schr\"odinger's equation and the moments of the initial data. This relationship is microlocal in character, and extends on asymptotically flat Riemannian manifolds to a connection between the global scattering behavior of the geodesic flow, the moments of initial data properly microlocalized along bicharacteristics, and the microlocal regularity of the solution. A proof of these results involves an interesting class of symbols of pseudodifferential operators. This article gives an outline of the above results and the microlocal analysis of these symbols. It also contains a study of the evolution operator for the Schr\"odinger equation on weighted Sobolev spaces, and presents a series of results for the non-selfadjoint case. This article is an extension of seminar talks on the linear Schr\"odinger equation given at the Ecole Polytechnique on 9 April 1996 (s\'eminaire `\'equations aux d\'eriv\'ees partielles') and at the Universit\"at Bonn on 2 May 1996 (`Oberseminar zur Analysis').