The $1$d nonlinear Schrödinger equation with a weighted $L^1$ potential

Abstract

We consider the $1d$ cubic nonlinear Schrödinger equation with a large external potential $V$ with no bound states. We prove global regularity and quantitative bounds for small solutions under mild assumptions on $V$. In particular, we do not require any differentiability of $V$, and make spatial decay assumptions that are weaker than those found in the literature (see for example \cite{Del,N,GPR}). We treat both the case of generic and non-generic potentials, with some additional symmetry assumptions in the latter case. Our approach is based on the combination of three main ingredients: the Fourier transform adapted to the Schrödinger operator, basic bounds on pseudo-differential operators that exploit the structure of the Jost function, and improved local decay and smoothing-type estimates. An interesting aspect of the proof is an "approximate commutation" identity for a suitable notion of a vectorfield, which allows us to simplify the previous approaches and extend the known results to a larger class of potentials. Finally, under our weak assumptions we can include the interesting physical case of a barrier potential as well as recover the result of \cite{MMS} for a delta potential.

46 pages. Minor Revisions. To appear in Analysis & PDE

References in corpus (3)

• Dispersive estimates for Schrodinger operators in dimensions one and three
• Scattering and small data completeness for the critical nonlinear Schroediger equation
• Decay and asymptotics for the 1D Klein-Gordon equation with variable coefficient cubic nonlinearities

Cited by in corpus (8)

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• Asymptotic stability of solitary waves for the 1D cubic-quintic Schrödinger equation with no internal mode
• Bilinear estimates in the presence of a large potential and a critical NLS in 3d
• Asymptotic stability near the soliton for quartic Klein-Gordon in 1D
• Decay and asymptotics for the 1D Klein-Gordon equation with variable coefficient cubic nonlinearities
• Long-time dynamics of small solutions to 1$d$ cubic nonlinear Schrödinger equations with a trapping potential
• Scattering and localized states for defocusing nonlinear Schrödinger equations with potential
• Modified scattering for nonlinear Schrödinger equations with long-range potentials