Boris Chirikov

Quantum chaos : between order and disorder

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Chirikov standard map

(1959) Chirikov criterion (1969) Classical map: ¯ p = p + K sin x ¯ x = x + ¯ p (1979) Quantum map (kicked rotator): ¯ ψ = e −î p 2 /2 e −iK / cosˆx ψ (1959-2008) Hamiltonian classical / quantum chaos : H (ˆ p , ˆ x) = ˆ p 2 / 2 + K cosˆx m δ (t − m) [ ˆ p , ˆ x ] = −i (2001) Quantum computations (2008) Ongoing experiments with cold atoms and Bose-Einstein condensates

AN EXAMPLE OF CHAOTIC EIGENSTATES IN A COMPLEX ATOM

Abstract Statistically processing a group of excited states with the total angular momentum and parity Jπ = 1+ in the cerium atom reveals that their eigenfunctions are random superpositions of some few basic states. A possible dynamical mechanism responsible for the formation of those chaotic states is briefly discussed.

Quantum ergodicity and localization in conservative systems: the Wigner band random matrix model

First theoretical and numerical results on the global structure of the energy shell, the Green function spectra and the eigenfunctions, both localized and ergodic, in a generic conservative quantum system are presented. In case of quantum localization the eigenfunctions are shown to be typically narrow and solid, with centers randomly scattered within the semicircle energy shell while the Green function spectral density (local spectral density of states) is extended over the whole shell, but sparse.First theoretical and numerical results on the global structure of the energy shell, the Green function spectra and the eigenfunctions, both localized and ergodic, are presented for the Wigner band random matrix ensemble, which is believed to provide a description for a broad class of conservative quantum systems which are strongly chaotic in the classical limit. In case of quantum localization the eigenfunctions are shown to be typically narrow and solid, with centers randomly scattered within the semicircle energy shell while the Green function spectral density (local spectral density of states) is extended over the whole shell, but sparse.

Marginal local instability of quasi-periodic motion☆

Abstract In this paper, we analytically prove a long suspected link between integrable hamiltonian systems and average linear growth with time of separation distance between initially close phase space states. Specifically, it is shown that almost all solutions to the linearized variational equations derived from bounded, integrable hamiltonian systems exhibit an average linear growth with time, becoming unbounded at t →∞. The orbits of bounded, integrable hamiltonian systems are thus always locally marginally unstable, forever lying on that sharp border which divides completely stable from completely unstable motion.

Some numerical studies or arnold diffusion in a simple model

Arnold diffusion is described in terms of a Hamiltonian representing two nonlinear oscillators with a weak linear coupling and a driving force acting on one of them. (AIP)

Quantum chaos: Pinball scattering

Classical and semiclassical periodic orbit expansions are applied to the dynamics of a point particle scattering elastically oo several disks in a plane. Fredholm determinants, zeta functions, and convergence of their cycle expansions are tested and applied to evaluation of classical escape rates and quantum resonances. The results demonstrate applicability of the Ruelle and Gutzwiller type periodic orbit expressions for chaotic systems.