of Hamiltonian manifolds

Let be a connected, compact symplectic manifold equipped with a Hamiltonian action. We prove that, as fundamental groups of topological spaces, , where is the symplectic quotient at any value in the image of the moment map .

     

E ⊗ ϵ Jahn–Teller anharmonic coupling for an octahedral system

The coupling between doubly degenerate electronic states and doubly degenerate vibrations is analyzed for an octahedral system on the basis of the introduction of an anharmonic Morse potential for the vibronic part. The vibrations are described by anharmonic coherent states and their linear coupling with the electronic states is considered. The matrix elements of the vibronic interaction are built and the energy levels corresponding to the interaction Hamiltonian derived. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Potential energy surfaces in algebraic molecular models using coherent states

The potential energy surfaces provided by the coherent states formalism for the case of interacting one-dimensional oscillators is investigated. The case of two interacting oscillators modelled with Morse potentials are considered in detail. First the traditional treatment is presented in order to identify the need to establish a new transformation between the parameters and the classical variables that allow the full Hamiltonian to be recovered. To this end the connection between the su(2) algebraic model and the usual treatment in configuration space was taken into account. It is shown that the coherent state approach when applied to recover the potential energy surface (PES) from a polyad preserving Hamiltonian is viable only in systems with a local mode behaviour. For molecules with a normal mode behaviour the approach is still valid, but the polyad must be broken in order to recover the PES. Our approach is evaluated through the calculation of the force constants for H₂O and CO₂, representative examples of local and normal behaviours.     

An application of hypervirial perturbation theory to calculate energy eigenvalues of the Morse potential for various diatomic molecules

The Schrödinger equation with the potentialV(r)=D[exp(?2β(r?r _e))?2exp(?β(r?r _e))] is treated in the framework of the hypervirial-renormalization parameter scheme. The energy eigenvalues of various eigenstates for different molecules are calculated.     

Morse Theory and Central Configurations in the Spatial N-body Problem

In the spirit of Palmore and Pacella, Morse Theory is used to obtain a lower bound for the number of central configurations in the spatial N-body problem. The homology of the configuration ellipsoid with the collision and collinear manifolds removed and the SO(3) symmetry quotiented out is calculated. As intermediate steps, homology calculations are carried out for several additional manifolds naturally arising in the N-body problem.     

Approximating the Partition Function of Morse Oscillators

A simple method is proposed leading to reasonable approximations of the partition function of Morse oscillators in terms of elementary functions.     

VC‐dimension on manifolds: a first approach

The Vapnik–Chervonenkis‐dimension is an index of the capacity of a learning machine. It has been computed in several cases, but always in a Euclidean context. This paper extends the notion to classifiers acting in the more general environment of a manifold. General properties are proved, and some examples of simple classifiers on elementary manifolds are given. A large part of the research is directed toward a still open problem on product manifolds. Copyright © 2007 John Wiley & Sons, Ltd.     

Multiplicity of solutions of a zero mass nonlinear equation on a Riemannian manifold

The relation between the number of solutions of a nonlinear equation on a Riemannian manifold and the topology of the manifold itself is studied. The technique is based on Ljusternik-Schnirelmann category and Morse theory.     

A Uniform Exponential Spectrum for Linear Flows on Vector Bundles

For linear flows on vector bundles we define a uniform exponential spectrum. For a compact invariant set for the projected flow we obtain this spectrum by taking all accumulation points for the time tending to infinity of the union over the finite time exponential growth rates for all initial values in this set. Using direct arguments we show that for a connected compact invariant set this spectrum is a closed interval whose boundary points are Lyapunov exponents. For a compact invariant set on which the flow is chain transitive we show that this spectrum coincides with the Morse spectrum. In particular, this approach admits a straightforward analytic proof for the regularity and continuity properties of the Morse spectrum without using cohomology or ergodicity results.     

Bifurcation for strongly indefinite functional and applications to Hamiltonian system and noncooperative elliptic system

In this paper, we discuss the bifurcation problems for strongly indefinite functional via Morse theory. The generalized topological degree for a class of vector fields is defined. As applications, we study the bifurcation problems for Hamiltonian system and noncooperative elliptic system.