Analytic smoothing effect for nonlinear Schrödinger equation with quintic nonlinearity

We prove the global existence of analytic solutions to nonlinear Schrödinger equation with quintic nonlinearity in n   space dimensions for sufficiently small Cauchy data with exponential decay. The smallness assumption on the data is imposed in terms of the critical Sobolev space H˙n/2−1/2. Moreover, a characterization of some class of analytic functions is given.