  This paper examines the controllability for quantum control systems with
SU(1,1) dynamical symmetry, namely, the ability to use some electromagnetic
field to redirect the quantum system toward a desired evolution. The problem is
formalized as the control of a right invariant bilinear system evolving on the
Lie group SU(1,1) of two dimensional special pseudo-unitary matrices. It is
proved that the elliptic condition of the total Hamiltonian is both sufficient
and necessary for the controllability. Conditions are also given for small time
local controllability and strong controllability. The results obtained are also
valid for the control systems on the Lie groups SO(2,1) and SL(2,R).
