Physics – Quantum Physics
Scientific paper
2010-04-20
Journal of Physics A: Math. Theor. 44 (2011) 135302
Physics
Quantum Physics
10 pages Latex
Scientific paper
10.1088/1751-8113/44/13/135302
In the framework of bilinear control of the Schr\"odinger equation with bounded control operators, it has been proved that the reachable set has a dense complemement in ${\cal S}\cap {\cal H}^{2}$. Hence, in this setting, exact quantum control in infinite dimensions is not possible. On the other hand it is known that there is a simple choice of operators which, when applied to an arbitrary state, generate dense orbits in Hilbert space. Compatibility of these two results is established in this paper and, in particular, it is proved that the closure of the reachable set of bilinear control by bounded operators is dense in ${\cal S}\cap {\cal H}^{2}$. The requirements for controllability in infinite dimensions are also related to the properties of the infinite dimensional unitary group.
Man'ko Vladimir I.
Mendes Rui Vilela
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