On Invariant Subspace In Quantum Control Systems and Some Concepts of Integrable Quantum Systems

Trajectories of some dynamical systems can be analysed by algebraic methods. In this paper we discuss certain applications of the so-called Shemesh criterion and its generalisations to analysis of properties of quantum control systems. In particular, some Hamiltonians with non-degenerated spectrum are considered, and also the case of a Hamiltonian with m1,...,mN degeneracies, where ∑i=1Nmi=n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\sum }^{N}_{i=1}m_{i}=n$\end{document}, is discussed.