On a class of nonlinear Schrödinger equations. II. Scattering theory,general case

This is the second of a series of papers devoted to the study of a class of non linear Schrödinger equations of the form $\text{i(}\text{du}\text{dt}\text{) = (?Δ + m)u + f(u)}$ in $\text{R}$ⁿ where m is a real constant and f a complex valued non linear function. Here we study the scattering theory for the pair of equations that consists of the previous one and of the equation $\text{i(}\text{du}\text{dt}\text{) = (?Δ + m)u}\text{for}\text{n ? 2}$. Under suitable assumptions of f we prove the existence of the wave operators and asymptotic completeness for a class of repulsive interactions. The assumptions of f that ensure asymptotic completeness cover the case of a single power $\text{f(u) = λ ¦ u ¦}{}^{\mathrm{p?1}}\text{u}\text{where}\text{λ ? 0}\text{and}\text{(n + 4)}\text{n}\text{< p <}\text{(n + 2)}\text{(n ? 2)}$.