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 + Δ_x)u = Vu + a · ∇_xu on ℝ^d × (0,1) and if u ≡ 0 in { x > R} × {0,1} for some R ≥ 0, then u ≡ 0 in ℝ^d × [0,1]. As a consequence, we obtain uniqueness properties of solutions of nonlinear Schrödinger equations of the form (i∂_t + Δ_x)u = G(x, t, u, ū, ∇_xu, ∇_xū) on ℝ^d × (0,1), where G is a suitable nonlinear term. The main ingredient in our proof is a Carleman inequality of the form ∥e^βφλ(x1) ν ∥_Lx2Lt2 + ∥ e^βφλ(x1) ∇_x ν ∥_Bx∞,2Lt2 ≤ C̄ ∥ e^βφλ(x1) (i∂_t + Δ_x) ν ∥ _Bx1,2Lt2 for any ν ∈ C(ℝ : H¹) with ν(., t) ≡ 0 for t ∉ [0,1]. In this inequality, B_x^∞,2 and B_x^1,2 are Banach spaces of functions on ℝ^d, and e^βφλ(x1) is a suitable weight.