Analyticity of Solutions to the Non Gauge Invariant Schrödinger Equations

Abstract

We study the global Cauchy problem for the non gauge invariant Schrödinger equations i∂_tu + Δu/2 = λu^σ, (t,x) ∈ R × Rⁿ, u|_t=0 = φ, x ∈ Rⁿ, where σ = 1 + 4/n, n = 1,2,4. The application of the Galilei generator for the proof of the analytic smoothing effect of solutions to the Cauchy problem for non gauge invariant Schrödinger equations involves difficulties. In this paper we construct analytic solutions to the non gauge invariant Schrödinger equations in the case of analytic and sufficiently small initial data. We use the power like analytic spaces and the analytic Hardy spaces as auxiliary analytic spaces characterized by the Galilei generator. Also we show that if the initial data φ decay exponentially and are sufficiently small in an appropriate norm, then the solutions of the Cauchy problem for non gauge invariant Schrödinger equations exist globally in time and are analytic.