I was wondering whether there are any rigorous results about the optimal controllability of Schrödinger operators.

So my question is something like this:

Let $i \partial_t \psi(x,t) = H_0(x)\psi(x,t) + u(t)H_1(x)\psi(x,t)$ be a Schrödinger equation. $H_0,H_1$ are nice operators (as nice as there is theory available, so for instance: $H_0$ has compact resolvent and $H_1$ is bounded, both on a finite interval with dirichlet conditions).

now we want to solve the problem whether we can move from a state $\psi_0$ to a state $\psi_1$ and want to minimize the function $u$ with respect to some norm.

Is there any theory available?

I already read that in an infinite dimensional space, we can only expect approximate controllability in the sense that $\psi_0$ can be only moved to $\psi_1$ up to an error of $\epsilon.$

So all this is fine, but I would really like to know if there is any literature on when this problem has a solution?