Dmitrií A. Sadovskií

Monodromy, diabolic points, and angular momentum coupling

Abstract Monodromy, or the most basic obstruction to global action-angle coordinates is shown to be present in the well known problem of two coupled angular momenta. It is also shown that in the corresponding quantum problem monodromy manifests itself as the redistribution of energy levels between different multiplets of the quantum spectrum.

Monodromy in the hydrogen atom in crossed fields

Abstract We show that the hydrogen atom in orthogonal electric and magnetic fields has a special property of certain integrable classical Hamiltonian systems known as monodromy. The strength of the fields is assumed to be small enough to validate the use of a truncated normal form H snf which is obtained from a two step normalization of the original system. We consider the level sets of H snf on the second reduced phase space. For an open set of field parameters we show that there is a special dynamically invariant set which is a “doubly pinched 2-torus”. This implies that the integrable Hamiltonian H snf has monodromy. Manifestation of monodromy in quantum mechanics is also discussed.

Qualitative analysis of vibration-rotation hamiltonians for spherical top molecules

A new approach to the qualitative analysis of the rotational structure of a group of degenerate or quasidegenerate vibrational states is proposed. The approach is based on a quantum description of resonant vibrational states and on using classical methods when describing the rotational structure of these states. The approach is used for studying the rotational structure of two- and three-fold degenerate vibrational states of the tetrahedral AB 4 molecule as well as for studying the v 1, v 3 diad of 28SiH4. The general scheme of the analysis is realized by computer.

Manifestation of bifurcations and diabolic points in molecular energy spectra

The general scheme of the qualitative analysis of finite‐particle quantum problems is discussed. Theoretical methods for the qualitative analysis of energy spectra in quantum problems are applied to the study of the rovibrational energy levels of spherical top molecules. The existence of modifications in the rotational cluster structure and of redistributions of the energy levels between different branches under the rotational excitation is shown for the ν2 /ν4 dyad of the 12 CH4 and 28 SiH4 molecules. An interpretation of the qualitative features of energy spectra in terms of bifurcations and formations of conical intersection points (diabolic points) on the energy surfaces of the corresponding classical problem is proposed.

Counting levels within vibrational polyads: Generating function approach

Simple analytical formulas for the number of energy levels in the vibrational polyads are given. These formulas account for the resonances between the vibrational modes, and for the symmetry of the problem, so that the number of states of a particular symmetry type can be computed. The formulas are used to estimate the differential and integral densities of states from the minimum initial information about the molecule. Examples of the vibrational structure of triatomic molecules A3, tetrahedral molecules AB4, and linear molecules AB2 are considered. The analytical formulas are compared to the ab initio results for H3+ [J. R. Henderson et al., J. Chem. Phys. 98, 7191 (1993)].

On the Dynamical Meaning of the Diabolic Points

Using a simple exactly soluble quantum model, it is shown that the diabolic points may be associated with the qualitative phenomenon of the redistribution of the energy levels between different branches in the energy spectra.

Monodromy of the quantum 1:1:2 resonant swing spring

We describe the qualitative features of the joint spectrum of the quantum 1:1:2 resonant swing spring. The monodromy of the classical analogue of this problem is studied in Dullin et al. [Physica D 190, 15–37 (2004)]. Using symmetry arguments and numerical calculations we compute its three-dimensional (3D) lattice of quantum states and show that it possesses a codimension 2 defect characterized by a nontrivial 3D-monodromy matrix. The form of the monodromy matrix is obtained from the lattice of quantum states and depends on the choice of an elementary cell of the lattice. We compute the quantum monodromy matrix, that is the inverse transpose of the classical monodromy matrix. Finally we show that the lattice of quantum states for the 1:1:2 quantum swing spring can be obtained—preserving the symmetries—from the regular 3D-cubic lattice by means of three “elementary monodromy cuts.”

Fractional monodromy of resonant classical and quantum oscillators

Abstract We introduce fractional monodromy for a class of integrable fibrations which naturally arise for classical nonlinear oscillator systems with resonance. We show that the same fractional monodromy characterizes the lattice of quantum states in the joint spectrum of the corresponding quantum systems. Results are presented on the example of a two-dimensional oscillator with resonance 1:(−1) and 1:(−2). To cite this article: N.N. Nekhoroshev et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 985–988.

Quantum monodromy and its generalizations and molecular manifestations

Quantum monodromy is a non-trivial qualitative characteristic of certain non-regular lattices formed by the joint eigenvalue spectrum of mutually commuting operators. The latter are typically the Hamiltonian (energy) and the momentum operator(s) which label the eigenstates of the system. We give a brief review of known quantum systems with monodromy, which include such fundamental systems as the hydrogen atom in external fields, Fermi resonant vibrations of the CO2 molecule, and non-rigid triatomic molecules. We emphasize the correspondence between the classical Hamiltonian monodromy and its quantum analogue and discuss possible generalizations of this characteristic in classical integrable Hamiltonian dynamical systems and their quantum counterparts.

Critical Phenomena and Diabolic Points in Rovibrational Energy Spectra of Spherical Top Molecules

Abstract The qualitative modifications of the rovibrational energy-level structure under rotational excitation are studied for spherical top molecules. The ν 2 , ν 4 bands of 12 CD 4 and 120 SnH 4 and the ν 1 , ν 3 bands of 120 SnH 4 are treated as examples. Two types of qualitative changes are shown to exist: the critical phenomena corresponding to the modification of the cluster structure and the so-called diabolic points associated with the redistribution of energy levels between different branches as the rotational quantum number increases. A simple model interpretation of energy-level redistribution is given to show that such a phenomenon is typical for tetrahedral molecules. Manifestation of critical phenomena and diabolic point formation in high-resolution infrared spectra are discussed.