On Threshold Solutions of equivariant Chern-Simons-Schr\"odinger Equation

TL;DR

This paper characterizes all threshold solutions of the equivariant self-dual Chern-Simons-Schr"odinger equation, showing they are essentially the ground state solutions up to symmetries, and provides partial results for the non-self-dual case.

Contribution

It proves that the only non-scattering solutions at the threshold charge are the ground states modulo symmetries, extending understanding of solution behavior at critical charge levels.

Findings

Threshold solutions are exactly the ground state up to symmetries.

Non-scattering solutions at threshold are characterized explicitly.

Partial results obtained for the non-self-dual system.

Abstract

We consider the self-dual Chern-Simons-Schr\"odinger model in two spatial dimensions. This problem is $L^2$-critical. Under equivariant setting, global wellposedness and scattering were proved in [Liu-Smith, 2016] for solution with initial charge below certain threshold given by the ground state. In this work, we show that the only non-scattering solutions with threshold charge are exactly the ground state up to scaling, phase rotation and the pseudoconformal transformation. We also obtain partial result for non-self-dual system.