The global Cauchy problem and scattering of solutions for nonlinear Schrödinger equations in Hˢ

In this paper, we shall prove that the scattering operator for nonlinear Schr\"{o}dinger equations iuₜ+(-Δ)ᵐ u=λ₁|u|p₁u+ λ₂|u|p₂u carries a band B(p₁,p₂,δ) in Hˢ into Hˢ for some δ>0, where B(p₁,p₂,δ)=\φ∈ Hˢ: \|φ\|_ Hs(p₁)∩ Hs(p₂)≤ δ\, s(pᵢ)=n/2-2m/pᵢ, s(p₂)≤ s≤ s(p₁)+1, 4m/n≤ p₁≤ p₂≤ 4m/(n-2s).