Singularities of solutions to Schrodinger equation on scattering manifold

TL;DR

This paper investigates the microlocal singularities of solutions to Schrödinger equations on scattering manifolds, providing a detailed characterization of the wave front set based on initial conditions and classical scattering maps, using a novel Egorov-type approach.

Contribution

It introduces a more general model for scattering manifolds and employs a simplified, scattering-theoretic method avoiding Legendre distributions to analyze singularities.

Findings

Characterizes wave front set in terms of initial data and scattering maps

Employs Egorov-type argument within pseudodifferential calculus

Provides a microlocal smoothing property with radially homogeneous wave front set

Abstract

In this paper we study microlocal singularities of solutions to Schrodinger equations on scattering manifolds, i.e., noncompact Riemannian manifolds with asymptotically conic ends. We characterize the wave front set of the solutions in terms of the initial condition and the classical scattering maps under the nontrapping condition. Our result is closely related to a recent work by Hassell and Wunsch, though our model is more general and the method, which relies heavily on scattering theoretical ideas, is simple and quite different. In particular, we use Egorov-type argument in the standard pseudodifferential symbol classes, and avoid using Legendre distributions. In the proof, we employ a microlocal smoothing property in terms of the radially homogenous wave front set, which is more precise than the preceding results.