Characterization of spacetime singularities for the Schr\"odinger equation by initial state

TL;DR

This paper characterizes spacetime singularities of Schr"odinger equation solutions with metric perturbations and potentials, linking their wave front sets to initial data and classical scattering data.

Contribution

It provides a new characterization of spacetime singularities using the quasi-homogeneous wave front set and classical scattering data, extending previous work on spatial singularities.

Findings

Wave front set of solutions is characterized by free solution and scattering data.

In 1D, the wave front set reduces to the initial time-slice's homogeneous wave front set.

The proof uses Egorov-type formulas and a specialized partition of unity.

Abstract

We discuss spacetime singularities of a solution to the Schr\"odinger equation with a metric perturbation and a sublinear potential. The quasi-homogeneous wave front set, due to Lascar (1977), of a solution is characterized by that of the free solution, and a classical high-energy scattering data. In the one-dimensional case, it further reduces to the homogeneous wave front set, due to Nakamura (2005), of the initial time-slice. For the proof of the former result we implement an idea inspired by Nakamura (2009), which was originally devised for spatial singularities of the Schr\"odinger equation. As for the latter result, we use an exact Egorov-type formula for the free propagator, and a special partition of unity conforming with the classical flow.