Applications of Bogomolny's Semiclassical Quantization to Integrable and Nonintegrable Systems

Abstract

<p>Bogomolny's transfer operator (T-operator) method is used to calculate semiclassical energy eigenvalues and eigenfunctions for the quantum analogues of several Hamiltonian systems. The calculations are performed with a finite approximation to the T-operator in coordinate space. We demonstrate the success of this technique for the integrable systems, the circle alld the 45º wedge billards, as well as for nonintegrable systems, the 41º and 30º wedge billiards (both displaying mixed behaviour) and the 49º and 60° wedge billiards (both showing hard chaos). For the 49° wedge, an alternate partition involving the symbolic sequences is studied. We also focus on properties of the eigenvalues of the T-operator with the objective of finding a reliable characteristic to describe the manifestation of chaos in quantum systems. In particular, we discuss the special connection between the T-operator eigenvalues and quantum numbers of integrable system. In addition, we investigate the distributions of phase separations of the T-operator eigenvalues and show that they may reflect the dynamical properties of Hamiltonian systems.</p>