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 Schrodinger 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.
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