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').