Uniqueness properties of solutions of Schrödinger equations (Q2368786)

The authors consider a certain linear homogeneous Schrödinger equation on \({\mathbb R}^{d}\times [0,1]\) with \(d\geq 1,\) and they prove that if the solution \(u(x,t)\) satisfies \(u(x,0)=u(x,1)=0\) for all \(| x| >r\) for some \(r\geq 0\) then \(u\equiv 0\) for all \((x,t)\in {\mathbb R}^{d}\times [0,1].\) As a consequence, it is shown that if two solutions \(u_ 1\) and \(u_ 2\) to an associated nonlinear Schrödinger equation coincide for \(| x| >r\) at \(t=0\) and \(t=1\) then \(u_ 1\equiv u_ 2\) for all \((x,t)\in {\mathbb R}^{d}\times [0,1].\) The proof uses a Carleman inequality.