Classical and quantum chaos in the wedge billiard

Abstract

<p>We examine the relationship between the classical mechanics and the quantum eigenvalues of the wedge billiard. Special emphasis is placed on applications of the periodic orbit theory developed by Gutzwiller (1971) and recently modified by others. Evidence is presented suggesting that words formed from a two letter alphabet, consisting of T and V, uniquely label all periodic orbits. The periodic orbits are found (for several different wedge angles) using a two-dimensional Newton method and their actions, periods, stability exponents, and Maslov indices calculated. It is found that the periodic orbits are heavily pruned at all wedge angles. Families of non-isolated periodic orbits and their implications for the Gutzwiller trace formula are discussed. The scaling properties of the classical periodic orbits are also examined. A large matrix diagonalization was used to accurately calculate the first 200 and 300 quantum eigenvalues for the 60° and 49° wedges respectively. A theory is developed which allows one to use the quantum eigenvalues to obtain information about the classical periodic orbits. This theory is tested on the 49° and 60° wedges and shows that the quantum eigenvalues 'know' about the classical periodic orbits. Several quantization schemes derived from the Gutzwiller periodic orbit theory are presented and then tested for the 49° and 60° wedges. Eigenvalues computed using the classical periodic orbits as input are compared to the 'exact' eigenvalues from the matrix diagonalization. A functional relation which uses the dynamical zeta function expressed as a product gives the best results.</p>