[Show abstract][Hide abstract] ABSTRACT: Rearrangement of rotation-vibration energy bands in isolated molecules within
semi-quantum approach is characterized by delta-Chern invariants associated to
a local semi-quantum Hamiltonian valid in a small neighborhood of a degeneracy
point for the initial semi-quantum Hamiltonian and also valid in a small
neighborhood of a critical point corresponding to the crossing of the boundary
between iso-Chern domains in the control parameter space. For a full quantum
model, a locally approximated Hamiltonian is assumed to take the form of a
Dirac operator together with a specific boundary condition. It is demonstrated
that the crossing of the boundary along a path with a delta-Chern invariant
equal to $\pm1$ corresponds to the transfer of one quantum level from a
subspaces of quantum states to the other subspace associated with respective
positive and negative energy eigenvalues of the local Dirac Hamiltonian.
[Show abstract][Hide abstract] ABSTRACT: Qualitatively different systems of molecular energy bands are studied on
example of a parametric family of effective Hamiltonians describing rotational
structure of triply degenerate vibrational state of a cubic symmetry molecule.
The modification of band structure under variation of control parameters is
associated with a topological invariant "delta-Chern". This invariant is
evaluated by using a local Hamiltonian for the control parameter values
assigned at the boundary between adjacent parameter domains which correspond to
qualitatively different band structures.
Physics Letters A 07/2013; 377(38). · 1.63 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Recent developments associated with old technique of generating functions and invariant theory which I have started to apply to molecular problems due to my collaboration with Yu.F. Smirnov about 25 years ago are discussed. Three aspects are presented: the construction of diagonal in polyad quantum number effective resonant vibrational Hamiltonians using the symmetrized Hadamard product; the decomposition of the number of state generating function into regular and oscillatory contributions and its relation with Todd polynomials and topological characterization of energy bands; qualitative aspects of resonant oscillators and fractional monodromy as one of new generalizations of Hamiltonian monodromy.
Physics of Atomic Nuclei 01/2012; 75(1). · 0.54 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Hamiltonian monodromy is known to be the first obstruction to the existence of global action coordinates in integrable systems. Its manifestation in quantum systems can be seen as characteristic defects of the regular lattice formed by the joint eigenvalues of mutually commuting quantum operators. The relation between topology of singular fibers of classical integrable fibrations and patterns formed by joint spectrum of corresponding quantum systems is discussed. The notion of the sign of 'elementary monodromy defect' is introduced on the basis of 'cut and glue' construction of the lattice defects. Special attention is paid to non-elementary defects which generically appear in phyllotaxis patterns and can be associated with plant morphology.
Journal of Physics A Mathematical and Theoretical 10/2010; 43(43):434033. · 1.77 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Hamiltonian monodromy is known to be the simplest obstruction to the existence of global action-angle variables in integrable models of classical dynamics. Recently, the corresponding quantum monodromy concept is introduced and shown to be an important qualitative feature of many different realistic models and concrete physical quantum systems. Vibrational structure of simple molecules, electronic states of hydrogen atom in external fields, coupling of angular momenta is discussed as basic physical examples. Starting from these examples new qualitative features of molecular systems leading naturally to generalized monodromy notions is introduced. Going finally to really complex systems the tentative relation between phyllotaxis and monodromy is suggested.
[Show abstract][Hide abstract] ABSTRACT: A system exhibits monodromy if we take the system around a closed loop
in its parameter space, and we find that the system does not come back
to its original state. Many systems have this property, including
hydrogen in crossed fields, cylindrically symmetric barrier systems,
such as the ``mexican hat'' potential, the spherical pendulum, dipolar
molecules in fields, and near-linear molecules. Atoms in a trap can
display a newly discovered dynamical manifestation of monodromy. We show
the behavior in computations, and provide a theoretical explanation.
[Show abstract][Hide abstract] ABSTRACT: a b s t r a c t Monodromy is the simplest obstruction to the existence of global action–angle variables in integrable Hamiltonian dynamical sys-tems. We consider one of the simplest possible systems with monodromy: a particle in a circular box containing a cylindrically symmetric potential-energy barrier. Systems with monodromy have nontrivial smooth connections between their regular Liouville tori. We consider a dynamical connection produced by an appro-priate time-dependent perturbation of our system. This turns studying monodromy into studying a physical process. We explain what aspects of this process are to be looked upon in order to uncover the interesting and somewhat unexpected dynamical behavior resulting from the nontrivial properties of the connection. We compute and analyze this behavior.
Annals of Physics 04/2009; 324(9). · 3.32 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: A system exhibits monodromy if we take the system around a closed loop in its parameter space, and we find that the system does not come back to its original state. Many systems have this property, including quasi-linear molecules, atoms in a trap or a hydrogen atom in crossed fields. Using classical perturbation theory, Sadovskii and Cushman predicted the presence of monodromy in perpendicular fields. It shows up as a defect in the lattice of quantum states. When the fields are tilted from perpendicular, these lattice defects undergo a series of bifurcations. Atoms in a trap can display a newly discovered dynamical manifestation of monodromy. This phenomenon will also occur with oriented dipolar molecules in fields or with quasilinear molecules. (Supported by NSF and Region Nord--Pas-de-Calais)
[Show abstract][Hide abstract] ABSTRACT: Hamiltonian monodromy —a topological property of the bundle of regular tori of a static Hamiltonian system which obstructs the existence of global action-angle variables— occurs in a number of integrable dynamical systems. Using as an example a simple integrable system of a particle in a circular box with quadratic potential barrier, we describe a time-dependent process which shows that monodromy in the static system leads to interesting dynamical effects.
[Show abstract][Hide abstract] ABSTRACT: We present a method which enables one to calculate generating functions counting the number of linearly independent tensor operators of different degrees which should be included in phenomenological effective Hamiltonians constructed from boson creation and annihilation operators for several degrees of freedom in the presence of resonances and symmetry. The method is based on the application of the Molien generating function technique and the Hadamard product of rational functions. The latter leads to the representation of the answer in a form of a rational function. The technique is illustrated by the example of effective Hamiltonians for vibrational polyads in a methane-type molecule, which is a dynamical system with nine degrees of freedom formed by one non-degenerate, one doubly degenerate and two triply degenerate modes in resonance 2:1:1:2:2:2:1:1:1.
Journal of Physics A Mathematical and Theoretical 01/2008; 4115. · 1.77 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Presence of energy bands in quantum energy spectra of molecules reflects the existence of “slow” and “fast” motions in corresponding
classical problem. Generic qualitative modifications of energy bands under the variation of some strict or approximate integrals
or motion considered as control parameters are analyzed within purely quantum description, within semi-quantum one (slow dynamical
variables are classical; fast variables are quantum) and within purely classical one. In quantum approach the reorganization
of bands is seen from the redistribution of energy levels between bands. In semi-quantum approach the system of bands is represented
by a complex vector bundle with the base space being the classical phase space for slow variables. The topological invariants
(Chern classes) of the bundle are related to the number of states in bands through Fedosov deformation quantization. In purely
classical description the reorganization of energy bands is manifested through the presence of Hamiltonian monodromy.
Journal of Mathematical Chemistry 01/2008; 44(4):1009-1022. · 1.23 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We present a one-parameter family of systems with fractional monodromy, which arises from a 1:2 diagonal action of a dynamical symmetry SO(2). In a regime of adiabatic separation of slow and fast motions, we relate the presence of fractional monodromy to a redistribution of states both in the quantum and in the semi-quantum spectra.
Journal of Physics A Mathematical and Theoretical 10/2007; 40(43):13075. · 1.77 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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.
[Show abstract][Hide abstract] ABSTRACT: Qualitative changes in the rotational structure of a finite particle quantum system are studied. The crossover phenomenon is explained from the point of view of consecutive quantum bifurcations. The generic organization of bifurcations is related to the stratification of the space of dynamical variables imposed by the invariance group of the Hamiltonian.
[Show abstract][Hide abstract] ABSTRACT: We present a 1-parameter family of systems with fractional monodromy and adiabatic separation of motion. We relate the presence of monodromy to a redistribution of states both in the quantum and semi-quantum spectrum. We show how the fractional monodromy arises from the non diagonal action of the dynamical symmetry of the system and manifests itself as a generic property of an important subclass of adiabatically coupled systems.
[Show abstract][Hide abstract] ABSTRACT: Within the qualitative approach to the study of finite particle quantum systems different possible ways of the generalization of Hamiltonian monodromy are discussed. It is demonstrated how several simple integrable models like non-linearly coupled resonant oscillators, or coupled rotators, lead to physically natural generalizations of the monodromy concept. Fractional monodromy, bidromy, and the monodromy in the case of multi-valued energy-momentum maps are briefly reviewed.
[Show abstract][Hide abstract] ABSTRACT: We consider perturbations of the hydrogen atom by sufficiently small homogeneous static electric and magnetic fields of al possible mutual orientations. Normalizing with regard to the Keplerian symmetry, we uncover resonances and conjecture tha the parameter space of this family of dynamical systems is stratified into zones centred on the resonances. The 1 : 1 resonanc corresponds to the orthogonal field limit, studied earlier by Cushman & Sadovskií (Cushman & Sadovskií 2000 Physica 142, 166–196). We describe the structure of the 1 : 1 zone, where the system may have monodromy of different kinds, and conside briefly the 1 : 2 zone.
Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences 01/2007; 463(2083):1771-1790. · 2.38 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We consider a generalization of the 1:1:2 resonant swing–spring [see H. Dullin, A. Giacobbe, R.H. Cushman, Physica D 190 (2004) 15] which is suggested both by the symmetries of this system and by its physical and in particular molecular realizations [see R.H. Cushman, H.R. Dullin, A. Giacobbe, D.D. Holm, M. Joyeux, P. Lynch, D.A. Sadovskií, B.I. Zhilinskií, Phys. Rev. Lett. 93 (2004) 024302-1–024302-4]. Our generic integrable system is detuned off the exact Fermi resonance 1:2. The three-dimensional (3D) image of its energy–momentum map EM consists either of two or three qualitatively different non-intersecting 3D regions: a regular region at low vibrational excitation, a region with monodromy similar to that studied for the exact resonance, and in some cases—an intermediate region in which the 3D set of regular values of EM is partially self-overlapping while remaining connected. In the presence of this latter region, the system has an interesting property which we called bidromy. We analyze monodromy and bidromy for a concrete integrable classical Hamiltonian system of three coupled oscillators and for its quantum analog. We also show that the bifurcation involved in the transition from the regular region to the region with monodromy can be regarded as a special resonant equivariant analog of the Hamiltonian Hopf bifurcation.
[Show abstract][Hide abstract] ABSTRACT: We introduce fractional monodromy in order to characterize certain non-isolated critical values of the energy–momentum map
of integrable Hamiltonian dynamical systems represented by nonlinear resonant two-dimensional oscillators. We give the formal
mathematical definition of fractional monodromy, which is a generalization of the definition of monodromy used by other authors
before. We prove that the 1:(−2) resonant oscillator system has monodromy matrix with half-integer coefficients and discuss
manifestations of this monodromy in quantum systems.
Communicated by Eduard Zehnder
Annales Henri Poincare 09/2006; 7(6):1099-1211. · 1.53 Impact Factor