Uniqueness properties of solutions of Schrödinger equations

Under suitable assumptions on the potentials VV and a  , we prove that if u∈C([0,1],H1)u∈C([0,1],H1) is a solution of the linear Schrödinger equation(i∂t+Δx)u=Vu+a·∇xu on Rd×(0,1)and if u≡0u≡0 in {|x|>R}×{0,1}{|x|>R}×{0,1} for some R⩾0R⩾0, then u≡0u≡0 in Rd×[0,1]Rd×[0,1]. As a consequence, we obtain uniqueness properties of solutions of nonlinear Schrödinger equations of the form(i∂t+Δx)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∥eβϕλ(x1)v∥Lx2Lt2+∥eβϕλ(x1)|∇xv|∥Bx∞,2Lt2⩽C¯∥eβϕλ(x1)(i∂t+Δx)v∥Bx1,2Lt2for any v∈C(R:H1)v∈C(R:H1) with v(.,t)≡0v(.,t)≡0 for t∉[0,1]t∉[0,1]. In this inequality, Bx∞,2 and Bx1,2 are Banach spaces of functions on RdRd, and eβϕλ(x1)eβϕλ(x1) is a suitable weight.