Quantum Surface of Section Method: Eigenstates and Unitary Quantum Poincare Evolution

Abstract

The unitary representation of exact quantum Poincare mapping is constructed. It is equivalent to the compact representation in a sense that it yields equivalent quantization condition with important advantage over the compact version: since it preserves the probability it can be literally interpreted as the quantum Poincare mapping which generates quantum time evolution at fixed energy between two successive crossings with surface of section (SOS). SOS coherent state representation (SOS Husimi distribution) of arbitrary (either stationary or evolving) quantum SOS state (vector from the Hilbert space over the configurational SOS) is introduced. Dynamical properties of SOS states can be quantitatively studied in terms of the so called localization areas which are defined through information entropies of their SOS coherent state representations. In the second part of the paper I report on results of extensive numerical application of quantum SOS method in a generic but simple 2-dim Hamiltonian system, namely semiseparable oscillator. I have calculated the stretch of 13500 consecutive eigenstates with the largest sequential quantum number around 18 million and obtained the following results: (i) the validity of the semiclassical Berry-Robnik formula for level spacing statistics was confirmed and using the concept of localization area the states were quantitatively classified as regular or chaotic, (ii) the classical and quantum Poincare evolution were performed and compared, and expected agreement was found, (iii) I studied few examples of wavefunctions and particularly, SOS coherent state representation of regular and chaotic eigenstates and analyzed statistical properties of their zeros which were shown on the chaotic component of 2-dim SOS to be uniformly distributed with the cubic repulsion between nearest neighbours.

27 pages, file in plain LaTeX, 15 figures upon request submitted to Physica D in March 1995

References in corpus (4)

• Spectral Statistics in the Quantized Cardioid Billiard
• General Quantum Surface of Section Method
• Quantization of Sinai's Billiard - A Scattering Approach
• Quantum Surface of Section Method: Demonstration of semiclassical Berry-Robnik energy level spacing distribution in a generic 2-dim Hamiltonian system

Cited by in corpus (23)

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• Random Analytic Chaotic Eigenstates
• Uni-directional transport properties of a serpent billiard
• Exact statistics of complex zeros for Gaussian random polynomials with real coefficients
• Comment on Energy Level Statistics in the Mixed Regime
• Quantum mechanics of chaotic billiards
• In-out decomposition of boundary integral equations
• Propagating wave correlations in complex systems
• Phenomenology of quantum eigenstates in mixed-type systems: lemon billiards with complex phase space structure
• Level spacing statistics of classically integrable systems -Investigation along the line of the Berry-Robnik approach-
• Wavefunction Statistics using Scar States
• Parametric statistics of zeros of Husimi representations of quantum chaotic eigenstates and random polynomials
• Random-Matrix Ensembles for Semi-Separable Systems
• Unified theory of bound and scattering molecular Rydberg states as quantum maps
• Multichannel Quantum Defect Theory: a Quantum Poincare Map
• Further results on the power-law decay of the fraction of the mixed eigenstates in kicked-top model with mixed-type classical phase space
• Short wave length approximation of a boundary integral operator for homogeneous and isotropic elastic bodies
• Statistical properties of spectral fluctuations for a quantum system with infinitely many components
• Numerical accuracy of Bogomolny's semiclassical quantization scheme in quantum billiards
• Deviations from Berry--Robnik Distribution Caused by Spectral Accumulation