Decay and Scattering for the Chern-Simons-Schrödinger Equations

2013/11/08 by Sung‐Jin Oh, Fabio Pusateri, Oh, Sung-Jin +1 · 2 citations
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1311.2088

openalex publication_date 2013/11/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 start by performing L²-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.

Citations

• Uniform decay estimates and the lorentz invariance of the classical wave equation

Cited by

• Dimensional reduction for anyons in the average-field approximation
• Equivariant stability of vortices in Manton's Chern-Simons-Schrödinger system on the hyperbolic plane

Related

• Large Data Well-posedness in the Energy Space of the Chern-Simons-Schrödinger System
• A note on the Chern-Simons-Dirac equations in the Coulomb Gauge
• Scattering and blow-up for Chern-Simons-Schrödinger equations in the mass supercritical cas
• On Threshold Solutions of equivariant Chern-Simons-Schrödinger Equation
• Small data global existence and decay for relativistic Chern-Simons equations