The short periodic orbit (PO) approach was developed in order to understand the structure of stationary states of quantum autonomous Hamiltonian systems corresponding to a classical chaotic Hamiltonian. In this chapter, we will describe the method for the case of a two-dimensional chaotic billiard where the Schrödinger equation reduces to the Helmholtz equation; then, it can directly be applied to evaluate the acoustic eigenfunctions of a two-dimensional cavity. This method consists of the short-wavelength construction of a basis of wavefunctions related to unstable short POs of the billiard, and the evaluation of matrix elements of the Laplacian in order to specify the eigenfunctions.

Introduction

The theoretical study of wave phenomena in systems with irregular motion received a big impetus after the works by Gutzwiller (summarized in Gutzwiller 1990). He derived a semiclassical approach providing the energy spectrum of a classically chaotic Hamiltonian system as a function of its POs. This formalism is very efficient for the evaluation of mean properties of eigenvalues and eigenfunctions (Berry 1985, Bogomolny 1988), but it suffers from a very serious limitation when a description of individual eigenfunctions is required: the number of used POs proliferates exponentially with the complexity of the eigenfunction. In this way, the approach loses two of the common advantages of asymptotic techniques: simplicity in the calculation and, more important, simplicity in the interpretation of the results.

Based on numerical experiments in the Bunimovich stadium billiard (Vergini & Wisniacki 1998), we have derived a short PO approach (Vergini 2000), which was successfully verified in the stadium billiard (Vergini & Carlo 2000): the first 25 eigenfunctions were computed by using five periodic orbits.