Controllability of quantum systems with relatively bounded control potentials

We study the regularity of solutions of bilinear quantum systems of the type x= (A+u(t)B)x where the state x belongs to some complex infinite dimensional Hilbert space, the (possibly unbounded) linear operators A and B are skew-adjoint and the control u is a real valued function of bounded variations. Under suitable regularity assumptions on the operators A and B, it is possible to extend the definition of solution for BV controls and infer continuity of the propagators. While the regularity of the propagators represents an obstacle to exact controllability, on the other hand it is used to present fine estimates on the convergence of finite dimensional approximation schemes. We will then prove exact controllability in projections for bilinear quantum systems with piecewise constant controls taking only two values. As a consequence we extend approximate controllability results to linear quantum systems of the type x=−iH(u(t))x with nonlinear dependence on the control.