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arXiv:nlin/0207034 · Journal of Physics A Mathematical and General

Behaviour of boundary functions for quantum billiards

A B cker, S F rstberger, R Schubert, Frank Steiner

Published Nov 19, 2002• 21 citations

Abstract

We study the behaviour of the normal derivative of eigenfunctions of the Helmholtz equation inside billiards with Dirichlet boundary condition. These boundary functions are of particular importance because they uniquely determine the eigenfunctions inside the billiard and also other physical quantities of interest. Therefore, they form a reduced representation of the quantum system, analogous to the Poincaré section of the classical system. For the normal derivatives we introduce an equivalent to the standard Green function and derive an integral equation on the boundary. Based on this integral equation we compute the first two terms of the mean asymptotic behaviour of the boundary functions for large energies. The first term is universal and independent of the shape of the billiard. The second one is proportional to the curvature of the boundary. The asymptotic behaviour is compared with numerical results for the stadium billiard, different limaçon billiards and the circle billiard, and good agreement is found. Furthermore, we derive an asymptotic completeness relation for the boundary functions.

DOI: 10.1088/0305-4470/35/48/306

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We study the behaviour of the normal derivative of eigenfunctions of the Helmholtz equation inside billiards with Dirichlet boundary condition. These boundary functions are of particular importance because they uniquely determine the eigenfunctions inside the billiard and also other physical quantities of interest. Therefore, they form a reduced representation of the quantum system, analogous to the Poincaré section of the classical system. For the normal derivatives we introduce an equivalent to the standard Green function and derive an integral equation on the boundary. Based on this integral equation we compute the first two terms of the mean asymptotic behaviour of the boundary functions for large energies. The first term is universal and independent of the shape of the billiard. The second one is proportional to the curvature of the boundary. The asymptotic behaviour is compared with numerical results for the stadium billiard, different limaçon billiards and the circle billiard, and good agreement is found. Furthermore, we derive an asymptotic completeness relation for the boundary functions.

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A simple expansion method for numerically calculating the energy levels and the corresponding wave functions of a quantum particle in a two-dimensional infinite potential well with arbitrary shape (quantum billiard) is presented. The method permits the study of quantum billiards in an introductory quantum mechanics course. According to the method, wave functions inside the billiard are expressed in terms of an expansion of a complete set of orthonormal functions defined in a surrounding rectangle for which the Dirichlet boundary conditions apply, while approximating the billiard boundary by a potential energy step of a sufficiently large size. Numerical implementations of the method are described and applied to determine the energies and wave functions for quarter-circle, circle, and triangle billiards. Finally, the expansion method is applied to investigate the quantum signatures of chaos in a classically chaotic generic-triangle billiard.

For the representation of eigenstates on a Poincare section at the boundary of a billiard different variants have been proposed. We compare these Poincare Husimi functions, discuss their properties, and based on this select one particularly suited definition. For the mean behavior of these Poincare Husimi functions an asymptotic expression is derived, including a uniform approximation. We establish the relation between the Poincare Husimi functions and the Husimi function in phase space from which a direct physical interpretation follows. Using this, a quantum ergodicity theorem for the Poincare Husimi functions in the case of ergodic systems is shown.

We introduce an additional method to solve Schr\"odinger's equation for a free particle in an infinite well of arbitrary shape (the Helmholtz equation with Dirichlet boundary conditions), a problem of interest in the area of quantum chaos. We expand the wave function in a basis of products of sine functions, then use the constraint operator to contain the wave function to a region within the domain of the basis functions. In this manner, a quantum billiard problem of arbitrary shape can be solved. Several methods exist to solve problems of this sort, but as recent work reviewing these methods has shown, all have shortcomings. Our work represents a different direction in the solution of these problems. Our method is different in that it provides a means of computing an eigenbasis. It is also interesting from a physical standpoint in that it can represent the Hamiltonian of a classically chaotic system in the basis of a classically regular system. \textcopyright{} 1996 The American Physical Society.

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