Smooth Controllability of Infinite Dimensional Quantum Mechanical Systems

Manipulation of infinite dimensional quantum systems is important to controlling complex quantum dynamics with many practical physical and chemical backgrounds. In this paper, a general investigation is casted to the controllability problem of quantum systems evolving on infinite dimensional manifolds. Recognizing that such problems are related with infinite dimensional controllability algebras, we introduce an algebraic mathematical framework to describe quantum control systems possessing such controllability algebras. Then we present the concept of smooth controllability on infinite dimensional manifolds, and draw the main result on approximate strong smooth controllability. This is a nontrivial extension of the existing controllability results based on the analysis over finite dimensional vector spaces to analysis over infinite dimensional manifolds. It also opens up many interesting problems for future studies.