In this paper we study the propagation of singularity for Schrödinger-type equations with variable coefficients. We introduce a new notion of wave propagation set, the homogeneous wave front set, which propagates along straight lines with finite speed away from x≠0. Then we show that it is related to the wave front set in a natural way. These results may be considered as a refinement of the microlocal smoothing property of Craig, Kappeler, and Strauss under more general assumptions.
"Propagation of the homogeneous wave front set for Schrödinger equations." Duke Math. J. 126 (2) 349 - 367, 15 February 2005. https://doi.org/10.1215/S0012-7094-04-12625-9