Scattering for the nonlinear Schrodinger equation with a general one-dimensional confinement

Overview & Synopsis

We consider the defocusing nonlinear Schrödinger equation in several space dimensions, in the presence of an external potential depending on only one space vari-able. This potential is bounded from below, and may grow arbitrarily fast at infinity. We prove existence and uniqueness in the associated Cauchy problem, in a suitable functional framework, as well as the existence of wave operators when the power of the nonlinearity is sufficiently large. Asymptotic completeness then stems from at least two approaches, which are briefly recalled.