Propagation of singularities for Schr\"odinger equations with modestly long range type potentials

In a previous paper by the second author, we discussed a characterization of the microlocal singularities for solutions to Schr\"odinger equations with long range type perturbations, using solutions to a Hamilton-Jacobi equation. In this paper we show that we may use Dollard type approximate solutions to the Hamilton-Jacobi equation if the perturbation satisfies somewhat stronger conditions. As applications, we describe the propagation of microlocal singularities for $e^{itH_0}e^{-itH}$ when the potential is asymptotically homogeneous as $|x|\to\infty$, where $H$ is our Schr\"odinger operator, and $H_0$ is the free Schr\"odinger operator, i.e., $H_0=-\frac12 \triangle$. We show $e^{itH_0}e^{-itH}$ shifts the wave front set if the potential $V$ is asymptotically homogeneous of order 1, whereas $e^{itH}e^{-itH_0}$ is smoothing if $V$ is asymptotically homogenous of order $\beta\in (1,3/2)$.