Sharp asymptotic behavior of solutions for cubic nonlinear Schrödinger equations with a potential (Q2814193)

This article is concerned with the cubic (focusing or defocusing) one-dimensional nonlinear Schrödinger equation in the presence of a linear potential \(V(x)\). Under precise assumptions on the smoothness and the decay of \(V\) -- namely, \((1+x^2)^{s/2}V(x)\in W^{1,1}(\mathbb{R})\) with \(s>5/2\) -- the author studies the long-time asymptotics of solutions having small initial data in \(H^1(\mathbb{R})\) with small variance. In this context, sharp asymptotic representation formulas are established by means of classical scattering methods. In particular, a clever use is made of the Jost function formalism. Albeit rather technical, the paper is well written and pleasant to read.