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

TL;DR

This paper advances the understanding of how microlocal singularities propagate in Schr"odinger equations with long-range potentials by employing Dollard type solutions, revealing specific conditions under which wave front sets are shifted or smoothed.

Contribution

It introduces the use of Dollard type approximate solutions for analyzing microlocal singularity propagation under stronger potential conditions.

Findings

Wave front set shifts when potential is asymptotically homogeneous of order 1.

Smoothing occurs for potentials asymptotically homogeneous of order between 1 and 1.5.

Provides a detailed description of singularity propagation for specific Schr"odinger operators.

Abstract

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 $\beta\in (1,3/2)$.