DECAY AND REGULARITY FOR THE SCHRODINGER-EQUATION

Consider the Schrodinger equation iu(t) = (-DELTA + V)u, u(x, 0) = u0(x) is-an-element-of L2(R(n)), n greater-than-or-equal-to 3 The following estimates are proved: (A) If V = 0 then for any 0 less-than-or-equal-to alpha < 1/2, integral-Rn+1 (\x\2alpha-2\\D(x)\(alpha)u(t,x)\2 dxdt less-than-or-equal-to C\\u0\\2, and for alpha = 1/2, s > 1/2, integral-Rn+1 (1 + \x\2)-s\\D(x)\1/2u(t,x)\2dxdt less-than-or-equal-to C\\u0\\2. (B) If \B(x)\ less-than-or-equal-to C(1 + \x\2)-1-delta, delta > 0, then (if 0 is neither an eigenvalue nor a resonance of -DELTA + V). Integral-Rn+1 (1 + \x\2)-1-delta\(I + (-DELTA + V)ac)1/4u(t,x)\2dxdt less-than-or-equal-to C\\P(ac)u0\\2.