SMOOTHING EFFECT OF SMALL ANALYTIC SOLUTIONS TO NONLINEAR SCHRODINGER-EQUATIONS

We consider the initial value problem for nonlinear Schrodinger equations in R(n)(greater-than-or-equal-to 2): {i partial derivative(t) u + 1/2 DELTAu = F(u, del, u, uBAR DELTA u BAR, (t,x) is an element of R x R(n) u(0,x) = psi(x), x is an element of R(n), where F: C x C(n) x C x C(n) --> C is a polynomial of degree 3 satisfying \F(u, del u, uBAR, del uBAR)\ less-than-or-equal-to C.(Absolute value of u + Absolute value of del u)3 and F (omega u, omega del u, omega uBAR, omega del uBAR) = omega F (u, del u, uBAR, del uBAR), for any complex number omega with Absolute value of omega = 1. It is shown that global solutions of (*) have a smoothing property.