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

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.