Propagation of singularities for semilinear Schr\"odinger equations

TL;DR

This paper investigates how singularities propagate in semilinear Schrödinger equations with quadratic Hamiltonians, revealing that propagation follows the Hamiltonian flow under certain Sobolev regularity conditions and using adapted paradifferential calculus.

Contribution

It extends the understanding of singularity propagation in semilinear Schrödinger equations by adapting the Sobolev-wave front set and employing a weighted paradifferential calculus approach.

Findings

Propagation occurs along Hamiltonian flow for Sobolev regularities

Propagation results are analogous to hyperbolic equations with wave front set

Uses a weighted paradifferential calculus adapted to Schrödinger equations

Abstract

We study the propagation of singularities for semilinear Schrodinger equations with quadratic Hamiltonians, in particular for the semilinear harmonic oscillator. We show that the propagation still occurs along the flow the Hamiltonian flow, but for Sobolev regularities in a certain range and provided the notion of Sobolev-wave front set is conveniently modified. The proof makes use of a weighted version of the paradifferential calculus, adapted to our situation. The results can be regarded as the Schrodinger counterpart of those known for semilinear hyperbolic equations, which hold with the usual wave front set.