Long range scattering and modified wave operators for some Hartree type equations, III. Gevrey spaces and low dimensions

We study the theory of scattering for a class of Hartree type equations wit h long range interactions in arbitrary space dimension n greater than or eq ual to 1, including the case of Hartree equations with time dependent poten tial V(t, x) = kt(mu-gamma) \x \ (-mu) with 0 < gamma less than or equal to 1 and 0 < mu < n. This includes the case of potential V(x) k \x \ (-gamma) and can be extended to the limiting case of nonlinear Schrodinger equation s with cubic nonlinearity kt(n-gamma) \u \ (2) u. Using Gevrey spaces of as ymptotic states and solutions, we prove the existence of modified local wav e operators at infinity with no size restriction on the data and we determi ne the asymptotic behaviour in time of solutions in the range of the wave o perators, thereby extending the results of previous papers which covered th e range 0 < gamma less than or equal to 1 but only 0 < mu less than or equa l to n-2 and were therefore restricted to space dimension n greater than or equal to 3. (C) 2001 Academic Press.