In this paper we present the results of a semiclassical investigation and a quantum mechanical study of the bound energy spectrum of the Henon–Heiles Hamiltonian (HHH) and of the Barbanis Hamiltonian (BH). We have derived a simple semiclassical formula for the energy levels E, and for their sensitivity dE/de with respect to the strength e of the nonlinear coupling for the HHH, and established general relations between E and its derivatives dnE/den (n⩾1). Numerical quantum mechanical computations of the energy levels were conducted for the HHH and for the BH. The nonlinear coupling constant was adjusted so that for the HHH there will be ∼150 states up to the classical critical energy Ec and ∼300 states up to the dissociation energy ED. The E values were obtained by direct diagonalization using a basis containing 760 states, while the values of dE/de were computed utilizing the Hellmann–Feynman theorem. Good agreement between the semiclassical and the quantum mechanical spectra was observerd well above Ec. ...

Quantum manifestations of classical stochasticity. I. Energetics of some nonlinear systems

A variety of correlation functions computed over a microcanonical ensemble for the Henon–Heiles system are investigated. We find the general trend is that of a gradual change to some form of decaying behavior as the motion becomes predominantly chaotic. The decay of a mode energy correlation function indicates a time scale for intramolecular energy redistribution.

/pdf/on-correlation-functions-and-the-onset-of-chaotic-motion-3d8jmdimw6.pdf

On correlation functions and the onset of chaotic motion

On the onset of chaotic motion in terms of generalized Langevin equations

Using MORI’S method [1], MO has studied several hamiltonian systems with a definite and small degree of freedom n.

MO Stochasticity Criterion

Recurrence relations (RR) play an important role in many domains of Science. Mathematics, Physics or Computer Science are obvious examples. It is rather intriguing to remark that few attempts to handle them as generic objects have been made in the field of Computer Algebra [24]. The only genuine effort is due to Verbaeten [1] who studied recurrence relations of hypergeometric functions. The same type of functions was considered by Lafferty [2], who was interested in reducing them to elementary or special functions by doing pattern matching with RR.

Synthesizing recurrence relations I: analysis of the problems

The conditions under which the phase space trajectories of a perturbed Hamiltonian system preserve conditional periodicity are investigated. It is shown that the sufficient and necessary condition for the stability of the system is the self-adjointness of the Liouville operator in the Hilbert space of functions absolute square integrable on the energy surface of the system in the phase space.

A. Lonke

Papers

Stability criterion for Hamiltonian systems