Quantum surface of section method: eigenstates and unitary quantum Poincaré evolution

Abstract

The unitary representation of the exact quantum Poincaré mapping is constructed. It is equivalent to the compact representation in a sense that it yields an equivalent quantization condition with an important advantage over the compact representation in a sense that it yields an equivalent quantization condition with an important advantage over the compact version: since it preserves the probability it can be literally interpreted as the quantum Poincaré mapping which generates the quantum time evolution at fixed energy between two successive crossings with the surface of section (SOS). An SOS coherent state representation (SOS Husimi distribution) of an arbitrary (either stationary or evolving) quantum SOS state (vestor 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 an extensive numerical application of the quantum SOS method in a generic but simple 2-dim. Hamiltonian system, namely the semiseparable oscillator. I have calculated the stretch of 13 445 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 the level spacing statistics was confirmed and using the concept of the localization area the states were quantitatively classified as regular or chaotic, (ii) the classical and quantum Poincaré 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 the nearest neighbours.