Microlocal dispersive smoothing for the Schrödinger equation

Abstract

Abstract This paper establishes a connection between the microlocal smoothness of solutions of the initial value problem for Schrödinger's equation and the global behavior of bicharacteristics of the principal symbol. In particular, localized initial data gives rise to solutions which are microlocally smooth at all points which are not trapped backwards by the bicharacteristic flow. The origin of the phenomenon is in the dispersive nature of the equation. The results imply microlocal regularity properties of the fundamental solution. ©1995 John Wiley & Sons. Inc.