Small Data Global Existence and Scattering for the Mass-Critical Nonlinear Schrödinger Equation with Power Convolution in ℝ³

Abstract

The main purpose of the present paper is to consider the well-posedness of the L²-critical nonlinear Schrödinger equation of a Hartree type

ð’¾âˆ‚_tψ + △ψ = (|x|^−1 ∗ |ψ|^8/3)ψ,      (t, x) ∈ â„₊ × â„³.

More precisely, we shall establish the local existence of solutions for initial data ψ₀ in L²(ℳ), as well as the existence of global solutions for small initial data. Moreover, we shall prove the existence of scattering operator.