Analytic wave front set for solutions to Schroedinger equation

TL;DR

This paper characterizes the analytic wave front set of Schrödinger solutions with short-range perturbations, extending previous work on long-range cases and providing an analytic analogue of known results.

Contribution

It introduces a characterization of the analytic wave front set for solutions to Schrödinger equations with short-range perturbations, building on prior long-range analysis.

Findings

Analytic wave front set of solutions is characterized in terms of free solutions.

Results hold for both forward and backward nontrapping regions.

Extends previous long-range smoothing effects to short-range perturbations.

Abstract

This paper is a continuation of a previous paper by the same authors, where an analytic smoothing effect was proved for long-range type perturbations of the Laplacian $H_0$ on $\re^n$. In this paper, we consider short-range type perturbations $H$ of the Laplacian on $\re^n$, and we characterize the analytic wave front set of the solution to the Schr\"odinger equation: $e^{-itH}f$, in terms of that of the free solution: $e^{-itH_0}f$, for $t<0$ in the forward nontrapping region. The same result holds for $t>0$ in the backward nontrapping region. This result is an analytic analogue of results by Hassel and Wunsch and Nakamura.