Global, small radially symmetric solutions to nonlinear Schrödinger equations and a gauge transformation

This paper proves the global existence of small radially symmetric solutions to the nonlinear Schrödinger equations of the form (i∂_tu + 1/2Δu = F(u,∇u, u,∇u), (t, x) ∈ R x Rⁿ, u(0, x) = ε₀Φ(|x|), x ∈ Rⁿ, where n ≥ 3, ∈0 is sufficiently small, |x| = with λ_α,λ_α,β l₁, l₂, l₃ ∈ N, l₀ = 3 for n = 3, 4, and l₀ = 2 for n ≥ 5. The method depends on the combination of a gauge transformation and generalized energy estimtes and does not require the condition such that ∂_∇uF is pure imaginary which is needed for the classical energy method.