Decay and Scattering for the Chern–Simons–Schrödinger Equations

We consider the Chern–Simons–Schrödinger 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 Schrödinger equation, with a potential decaying like |r⁻¹| at infinity. To overcome the difficulties due to this long-range decay, we perform |L²|-based estimates covariantly. This procedure 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 establish scattering to linear solutions by Fourier methods.