The $1$d nonlinear Schr\"odinger equation with a weighted $L^1$ potential

TL;DR

This paper proves global regularity and bounds for small solutions of the 1D cubic nonlinear Schrödinger equation with large, non-differentiable potentials that decay weakly, extending previous results to broader potential classes.

Contribution

It introduces a novel approach combining adapted Fourier analysis, pseudo-differential bounds, and decay estimates to handle less regular, larger potentials in the 1D NLS.

Findings

Established global regularity for small solutions with weak decay potentials.

Extended analysis to include barrier and delta potentials.

Developed an approximate commutation identity simplifying previous methods.

Abstract

We consider the $1d$ cubic nonlinear Schr\"odinger 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 Schr\"odinger 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.