Quadratic derivative nonlinear Schrodinger equations in two space dimensions

We study the global in time existence of small classical solutions to the nonlinear Schrodinger equation with quadratic interactions of derivative type in two space dimensions {i partial derivative(t)u + 1/2 Delta u = N(del u, del u), t > 0, x is an element of R-2, u(0, x) = u(0) (x), x is an element of R-2, where the quadratic nonlinearity has the form N(del u, del uv) = Sigma(lambda kl)(k,l=1,2) (partial derivative(k)u)(partial derivative(l)v) with lambda is an element of C. We prove that if the initial data u(0) is an element of H-6 boolean AND H-3,H-3 satisfy smallness conditions in the weighted Sobolev norm, then the solution of the Cauchy problem ( 0.1) exists globally in time. Furthermore we prove the existence of the usual scattering states. The proof depends on the energy type estimates, smoothing property by Doi, and method of normal forms by Shatah.