Analyticity in time and smoothing effect of solutions to nonlinear Schrödinger equations

In this paper we consider analyticity in time and smoothing effect of solutions to nonlinear Schrödinger equations {i∂_tu + 1/2△u = λ|u|^2pu, (t, x) ∈ ℝ × ℝⁿ, u(0, x) = φ, x ∈ ℝⁿ, (1) where λ ∈ ℂ, p ∈ ℕ. We prove that if φ satisfies ∥e^|x|2φ∥ _Η[n/2]+1 < ∞, (2) then there exists a unique solution u(t, x) of (1) and positive constants Τ, C₀, C₁ such that u(t, x) is analytic in time and space variables for t ∈ [-Τ,Τ] \ {0} and x ∈ Ω = {x; |x| < R} and has an analytic continuation U(z₀, z) on {z₀ = t + iτ; -C₀t² < τ < C₀t²,t ∈ [-Τ,Τ] \ {0}} and {z = x + iy; -C₁|t| < y < C₁(t),(t, x) ∈ [-Τ,Τ] \ {0} × Ω} . In the case n = 1,2,3 the condition (2) can be relaxed as follows: ∥e^|x|2φ∥ _Ηm < ∞, where m = 0 if n = 1, p = 1, m = 1 if n = 2. p ∈ ℕ and m = 1 if n = 3, p = 1.