Uniqueness properties of solutions of Schrödinger equations

Under suitable assumptions on the potentials V and a, we prove that if u ∈ C([0, 1],H1) is a solution of the linear Schrödinger equation (i t + x)u = Vu + a · ∇xu on Rd × (0, 1) and if u ≡ 0 in {|x|>R}×{0, 1} for some R 0, then u ≡ 0 in Rd ×[0, 1]. As a consequence, we obtain uniqueness properties of solutions of nonlinear Schrödinger equations of the form (i t + x)u =G(x, t, u, u,∇xu,∇xu) on Rd × (0, 1), where G is a suitable nonlinear term. The main ingredient in our proof is a Carleman inequality of the form e (x1)v L2x L2t + e (x1)|∇xv| B∞,2 x L2t C e (x1)(i t + x)v B 1,2 x Lfor any v ∈ C(R : H1) with v(., t) ≡ 0 for t /∈ [0, 1]. In this inequality, B∞,2 x and B 1,2 x are Banach spaces of functions on Rd , and e (x1) is a suitable weight. © 2005 Elsevier Inc. All rights reserved