GLOBAL ESTIMATES FOR THE SCHRODINGER-EQUATION

Let u = u(x, t) be a solution to the IVP for the Schrödinger equation iu1 = (-Δ + V(x))u ≡ Hu, u(x, 0) = u0(x) ε{lunate} PacL2(Rn) (Pac is the projection on the absolutely continuous subspace of H). Assume that for some ε > 0 the multiplication operator (1+|x|)1+iV(x):H1-ε(Rn) → L2(Rn) is bounded. Then u(x, t) = u1(x, t) + u2(x, t) where, for every s > 1 2, ∫ R ∫ Rn (1 + |x|2)-s|(1+H) 1 4u1(x,t)|2 dxdt ≤ C {norm of matrix}u0{norm of matrix}L2, and for every integer j, sup{norm of matrix}(I + H)ju2(·, t){norm of matrix}L ≤ Cj {norm of matrix} U0{norm of matrix}L tε{lunate}R. © 1992.