The 1-dimensional nonlinear Schrödinger equation with a weighted L1 potential

Abstract

We consider the 1-dimensional 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 work of Delort (2016), Naumkin (2016) and Germain et al. (2018)). We treat both the case of generic and nongeneric 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 pseudodifferential 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 vector field, 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 Masaki et al. (2019) for a delta potential.