Type: Article

Analyticity of solutions to the non gauge invariant Schrodinger equations

Hoshino, Gaku; Naumkin, Pavel I.;

Abstract:

We study the global Cauchy problem for the non gauge invariant Schrodinger equations i partial derivative(t)u + Delta u/2 = lambda u(sigma), (t,x) is an element of R x R-n, u|(t=0) = phi, x is an element of R-n,R- where sigma = 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 Schrodinger equations involves difficulties. In this paper we construct analytic solutions to the non gauge invariant Schrodinger 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 phi decay exponentially and are sufficiently small in an appropriate norm, then the solutions of the Cauchy problem for non gauge invariant Schrodinger equations exist globally in time and are analytic.