This paper is concerned with the internal distributed control problem for the 1D
Schrödinger equation,
i ut(x,t) = −uxx+α(x) u+m(u) u,
that arises in quantum semiconductor models. Here m(u)
is a non local Hartree–type nonlinearity stemming from the coupling with the 1D Poisson
equation, and α(x) is a regular function with linear
growth at infinity, including constant electric fields. By means of both the Hilbert
Uniqueness Method and the contraction mapping theorem it is shown that for initial and
target states belonging to a suitable small neighborhood of the origin, and for
distributed controls supported outside of a fixed compact interval, the model equation is
controllable. Moreover, it is shown that, for distributed controls with compact support,
the exact controllability problem is not possible.