Decay and Scattering for the Chern-Simons-Schr\"odinger Equations

We consider the Chern-Simons-Schr\"odinger model in 1+2 dimensions, and prove scattering for small solutions of the Cauchy problem in the Coulomb gauge. This model is a gauge covariant Schr\"odinger equation, with a potential decaying like $r^{-1}$ at infinity. To overcome the difficulties due to this long range decay we start by performing $L^2$-based estimates covariantly. This gives favorable commutation identities so that only curvature terms, which decay faster than $r^{-1}$, appear in our weighted energy estimates. We then select the Coulomb gauge to reveal a genuinely cubic null structure, which allows us to show sharp decay by Fourier methods.