Wave Functions in Classically Chaotic Systems

N. Raje, V B Sheorey

Jakub Zakrzewski, Roman Marcinek

The wave-function localization in a classically, strongly chaotic system---quadric oscillator model---is discussed in the adiabatic approach. A comparison is made between the coordinate-based traditional method and the periodic-orbit-based scheme of Eckhardt, Hose, and Pollak [Phys. Rev. A 39, 3776 (1989)] for channel localized wave functions. A second type of localization of the ring type, observed in the model, is accounted for by an adiabatic approach in polar coordinates.

M. S. Santhanam

I. Puha

Stefan Giller, Janiak, Jarosław

A way of construction of semiclassical wave function (SWF) based on the Maslov - Fedoriuk approach is proposed which appears to be appropriate also for systems with chaotic classical limits. Some classical constructions called skeletons are considered. The skeletons are generalizations of Arnolds' tori able to gather chaotic dynamics. SWF's are continued by caustic singularities in the configuration space rather then in the phase space using complex time method. The skeleton formulation provides us with a new algorithm for the semiclassical approximation method which is applied to construct SWF's as well as to calculate energy spectra for the circular and rectangular billiards as well as to construct the simplest SWF's and the respective spectrum for the Bunimovich stadium. The scar phenomena are considered and a possibility of their description by the skeleton method is discussed. PACS number(s): 03.65.-w, 03.65.Sq, 02.30.Jr, 02.30.Lt, 02.30.Mv Key Words: Schrödinger equation, semiclassical expansion, Lagrange manifolds, classical trajectories, chaotic dynamics, quantum chaos, scars

Kazuo Hida

The effect of dissipation on the recovery of the classical chaotic dynamics is discussed in the semiclassical regime of the quantum nonintegrable systems. In the classically chaotic Hamiltonian systems, the semiclassical approach breaks down within rather short time scale due to the interference of the wave function. It is shown that the dissipation suppresses the interference and confines the quantum fluctuation around the semiclassical trajectory which is described by the semiclassical Langevin equation. We also show that the semiclassical condition obtained by Schmid is equivalent to the condition for the suppression of interference. The discussion is valid regardless whether the classical motion is chaotic or not and therefore dissipation helps the recovery of the classical dynamics in the chaotic systems.