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?



Quantum Mechanics and bilinear optimal control theory
The earliest reference is [On the controllability of quantum‐mechanical systems](http://scitation.aip.org/content/aip/journal/jmp/24/11/10.1063/1.525634) (1983). For recent developments, see [On the problem of quantum control in infinite dimensions](http://arxiv.org/abs/1004.3447) (2011) and [Finite Controllability of Infinite-Dimensional Quantum Systems](http://arxiv.org/abs/quant-ph/0608075) (2006):



> 
> For some quantum systems, such as spin systems, the quantum evolution
>  equation (the Schrödinger equation) is finite-dimensional and old
>  results on controllability of systems defined on on Lie groups and
>  quotient spaces provide most of what is needed insofar as
>  controllability of non-dissipative systems is concerned. However, in
>  an infinite-dimensional setting, controlling the evolution of quantum
>  systems presents both conceptual and technical difficulties, in essence because the set of all attainable states has dense complement in the sphere. We
>  analyze controllability for infinite-dimensional bilinear systems
>  under assumptions that make controllability possible using
>  trajectories lying in a nested family of pre-defined subspaces. This
>  result provides a set of sufficient conditions for controllability in
>  an infinite-dimensional setting.
> 
> 
> 


Treatments of the *optimality* of quantum control are mainly restricted to finite-dimensional systems, but they do include the effects of coupling to the environment (which adds a non-unitary time-dependence, instead of the Schrödinger equation you would have the Lindblad equation). This is the topic of the Ph.D. thesis [Optimal Control of Quantum Systems](https://me.ucsb.edu/~bamieh/theses/symeon-thesis.pdf) by Symeon Grivopoulos (2005). (The reference list contains many other contributions, in particular Refs. 30-40.) There is also this 2007 overview, [Quantum Optimal Control Theory](http://arxiv.org/abs/0707.1883).


