Recurrence of singularities for second order isotropic pseudodifferential operators

TL;DR

This paper investigates how singularities in solutions to the Schrödinger equation evolve over time for isotropic elliptic pseudodifferential operators, linking wavefront sets of initial data to those at later times.

Contribution

It characterizes the wavefront set propagation for solutions of Schrödinger equations involving second order isotropic pseudodifferential operators, incorporating principal and subprincipal symbols.

Findings

Wavefront set of solution determined by initial wavefront set and operator symbols

Singularities propagate along specific geometric trajectories

Results extend understanding of singularity behavior in quantum evolution

Abstract

Let $H$ be a self-adjoint isotropic elliptic pseudodifferential operator of order $2$. Denote by $u(t)$ the solution of the Schr\"odinger equation $(i\partial_t - H)u = 0$ with initial data $u(0) = u_0$. If $u_0$ is compactly supported the solution $u(t)$ is smooth for small $t > 0$, but not for all $t$. We determine the wavefront set of $u(t)$ in terms of the wavefront set of $u_0$ and the principal and subprincipal symbol of $H$.