A remark on long-range scattering for the Hartree type equation (Q2769881)

The paper deals with the problem of the existence of stationary solutions to the nonlinear Schrödinger equation which is generated by the Hartree approximation in quantum mechanics of multi-electron atoms: NEWLINE\[NEWLINE iu_t = -(1/2)\Delta u + f(|u|^2)u,NEWLINE\]NEWLINE where \(\Delta\) is the Laplacian acting in an \(n\)-dimensional space, and the function \(f(|u|^2)\) is a nonlocal expression, NEWLINE\[NEWLINE f(|u|^2) = \int |x-y|^{-\gamma} |u|^2 dy.NEWLINE\]NEWLINE The case \(\gamma > 1\) corresponds to the short-range (quasi-local) interaction, while in the case of long-range interaction \(\gamma \leq 1\). The existence and uniqueness of the scattering solution for the Hartree equation can be proven easily in the short-range case, while the long-range one is more difficult. In this work, the existence of the solution and some estimates for it are proven, assuming weaker smoothness of the solution than it was done in earlier works.