Domain and range of the modified wave operator for Schrodinger equations with a critical nonlinearity

We study the final problem for the nonlinear Schrodinger equation i partial derivative(t)u+1/2 Delta u = lambda\u\(2/n)u, (t, x) is an element of R x R(n), where lambda is an element of R, n = 1, 2, 3. If the final data u(+) is an element of H(0,alpha) = {phi is an element of L(2) : (1 + \x\)(alpha) phi is an element of L(2)} with n/2 < alpha < min (n, 2, 1 + 2/n) and the norm parallel to(u(+)) over cap parallel to(L)infinity is sufficiently small, then we prove the existence of the wave operator in L(2). We also construct the modified scattering operator from H(0,alpha) to H(0,delta) with n/2 < delta < alpha.