Wave and scattering operators for the nonlinear matrix Schrödinger equation on the half-line with a potential. (February 2023) - Ballesteros, Miguel; Iniesta, Diego; Naumkin, Ivan; Peña, Clemente
Abstract: We consider the nonlinear matrix Schrödinger equation on the half-line with general selfadjoint boundary condition. We prove the existence of local solutions in the corresponding energy space. Moreover, we consider the scattering problem for this problem in the scattering-supercritical regime for both generic and exceptional potentials. We construct the wave, inverse wave and scattering operators in a certain neighborhood of the origin in the scattering space Σ = H 1 ∩ L 2, 1 for generic potentials in L 1, 2 + η, with η > 1 / 2, and for exceptional potentials in L 1, 3 under further assumptions on the input data. In particular, thanks to the general boundary conditions, we are able to obtain a scattering result for the nonlinear matrix Schrödinger equation on the line with a potential and point interactions.