Modified scattering operator for the Hartree–Fock equation

We study the scattering problem for the Hartree–Fock equation equation(HRF)i∂tu+12Δu=f(u),(t,x)∈R×Rn,n≥2 where u=(u1,…,uN)tu=(u1,…,uN)t is a CN(N≥2)-valued unknown function and f(u)=(f1(u),…,fN(u))tf(u)=(f1(u),…,fN(u))t denotes a nonlinear term whose jjth-element is defined by fj(u)=∫RnV(x−y)∑k=1N{|uk(y)|2uj(x)−uj(y)ūk(y)uk(x)}dy, where V(x)=λ|x|−1(λ∈R) is called a Coulomb potential. We show that if 12<δ<α, then the modified scattering operator for the system (HRF) is well-defined from a neighborhood at the origin in the space H0,α to a neighborhood at the origin in the space H0,δ, where H0,k={ϕ∈L2;(1+|x|2)k2ϕ∈L2}.