On the modified scattering of 3-d hartree type fractional schrödinger equations with coulomb potential

In this paper, we study 3-d Hartree type fractional Schrödinger equations: i∂_tu-|∇| ^α u =λ(|x|^-γ *|u|²)u, 1 < α < 2, 0 < γ < 3 λ ∈ R \(0). In [7] it is known that no scattering occurs in L² for the long range (0 < γ ≤ 1). In [4,108] the short-range scattering (1 < γ < 3) was treated for the scattering in H^s. In this paper, we consider the critical case (γ=1) and prove a modified scattering in L^∞ on the frequency to the Cauchy problem with small initial data. For this purpose, we investigate the global behavior of xe ^it|∇|α u, x² e ^it|∇|α u and 〈 ξ 〉⁵ e^it|∇|αu. Due to the non-smoothness of |∇|α near zero frequency the range of α is restricted to (17/10, 2).