On threshold solutions of the equivariant Chern–Simons–Schrödinger equation

Abstract

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