Calculus and mechanics on two-point homogeneous Riemannian spaces
This monograph gives a short and concise introduction to classical and quantum mechanics on two-point homogeneous Riemannian spaces, with emphasis on spaces with constant curvature. The first four chapters provide the basic notations from differential geometry for studying two-body dynamics in these spaces. The natural spaces for the two-body problem are two-point homogeneous Riemannian spaces, because the distance between bodies in these spaces is the only invariant of their location with respect to the isometry group. In Chapter 1 the classification of these spaces is given. There are also models of compact two-point homogeneous spaces as submanifolds of Euclidean spaces or its factor spaces, different models of real hyperbolic spaces \({\mathbf H}(\mathbb R)\), \(n\geq 2\) and the description of the transition from compact to noncompact two-point homogeneous spaces in terms of the corresponding Lie algebras. Chapter 2 provides the necessary data on differential operators: basic notions of the theory of invariant differential operators on smooth manifolds, the expression of the Laplace-Beltrami operator, self-adjointness of differential operators (e.g. Schrödinger operators), the general scheme of the quantum-mechanical reduction. Chapter 3 deals with algebras \(\text{Diff}_I(Q_S)\) of invariant differential operators on unit sphere bundles \(Q_S\) over two-point homogeneous spaces \(Q\). The description of these algebras in terms of generators and relations are found, also some elements from centers of these algebras. Chapter 4 contains basic facts concerning Hamiltonian dynamical systems with symmetry and the correspondence between classical and quantum-mechanical systems. A special symplectomorphism between a reduced cotangent bundle of a homogeneous manifold and a certain factor space of a coadjoint orbit of a corresponding Lie group is constructed. Chapter 5 deals with the problem of finding an explicitly invariant expression for the two-body Hamiltonian on two-point homogeneous spaces \(Q\) through a radial differential operator and generators of the algebra \(\text{Diff}_I(Q_S)\). In Chapter 6 the one-body problem in a central potential is discussed. For spaces of constant curvature more detailed results are given both, in classical and quantum cases. The genesis of Bertrand potentials is discussed. There is a historical survey of one- and two-body motions in central potentials in spaces of constant curvature, containing relevant references to many papers from the early beginning of non-Euclidean geometry till the present time. In Chapter 7 the expression for the two-body quantum Hamiltonian is transformed into the two-body Hamiltonian function of the corresponding classical system, with respect to the center-of-mass problem in two-point homogeneous spaces. Reduced classical two-body systems in spaces \({\mathbf S}^n\) and \({\mathbf H}^n(\mathbb R)\) are classified. The last Chapter 8 deals with the quantum two-body problem in compact two-point homogeneous spaces. Applications for the Coulomb and oscillator potentials are investigated. There are four appendices. In the fourth of them some unsolved problems are listed arising from the book content. Also a complete bibliography is added.
scientific article; zbMATH DE number 5116431