Wave front set of solutions to the fractional Schr\"{o}dinger equation

TL;DR

This paper characterizes the wave front sets of solutions to fractional Schrödinger equations using wave packet transforms, revealing how the fractional order and potential growth influence singularity propagation.

Contribution

It introduces a novel characterization of wave front sets for fractional Schrödinger solutions and links their propagation mechanisms to those of classical wave equations.

Findings

Wave front sets are characterized via wave packet transform.

Propagation of singularities depends on fractional order and potential growth.

A theorem connects fractional Schrödinger and wave equation singularity propagation.

Abstract

In this paper, we characterize the wave front sets of solutions to fractional Schr\"{o}dinger equations \(i\partial_{t}u =(-\Delta)^{\theta/2}u + V(x)u\) with $0<\theta <2$ via the wave packet transform (short-time Fourier transform). We clarify the relationship between the order \(\theta\) of the fractional Laplacian and the growth rate of the potential in the problem of propagation of singularities. In particular, we present a theorem that bridges the propagation mechanisms of singularities for the Schr\"odinger and wave equations.