Inverse source problem for the 1-D Schrödinger equation

We consider the inverse problem of determining a source in the dynamical Schrödinger equation $iu_t-u_{xx}+q(x)u=w(t)a(x)$, $0<x<1$, with Dirichlet boundary conditions and zero initial condition. From the measurement $u_x(0,t)$, $0<t<T$, we recover unknown $a(x)$ provided $q(x)$ and $w(t)$ are given. We describe also how to recover $a(x)$ and $q(x)$ from the measurements at the both boundary points.