Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review Letters

Exact and Quasiclassical Fredholm Solutions of Quantum Billiards

Bertrand Georgeot, R. E. Prange

Published Apr 10, 1995• 53 citations
Publisher

Abstract

Quantum billiards are much studied as perhaps the simplest case which presents the central difficulty that the quasiclassical approximation is expressed as a divergent series. We find here, using the Fredholm method, an exact Green's function for billiards expressed as a ratio of absolutely convergent series. We make the quasiclassical approximation to this ratio. The method provides a convergence argument for previous results and an extension of results obtained for the spectrum to the full Green's function.

DOI
10.1103/physrevlett.74.2851

Related papers

Spectral Sum Rules for the Circular Aharonov-Bohm Quantum Billiard

Frank Steiner

This work presents the results of a quantitative investigation of the “sum rule method” recently proposed by the author for calculating low-lying energy levels. The system considered in detail is the circular Aharonov-Bohm quantum billiard recently introduced by Berry and Robnik. Exact, expressions are derived for the spectral zeta function at positiveinteger values as a function of the magnetic flux. Using the zeta function for fixed angular momentum, we observe a very fast convergence to the exact ground state energy (“precocious convergence”).

Fortschritte der Physik/Progress of Physics•Jan 1, 1987•23 citations

Wave packet construction in two-dimensional quantum billiards: Blueprints for the square, equilateral triangle, and circular cases

M. A. Doncheski, S. Heppelmann, R. W. Robinett +1 more

We present quasianalytical and numerical calculations of Gaussian wave packet solutions of the Schrödinger equation for two-dimensional infinite well and quantum billiard problems with equilateral triangle, square, and circular footprints. These cases correspond to N=3, N=4, and N→∞ regular polygonal billiards and infinite wells, respectively. In each case the energy eigenvalues and wave functions are given in terms of familiar special functions. For the first two systems, we obtain closed form expressions for the expansion coefficients for localized Gaussian wave packets in terms of the eigenstates of the particular geometry. For the circular case, we discuss numerical approaches. We use these results to discuss the short-time, quasiclassical evolution in these geometries and the structure of wave packet revivals. We also show how related half-well problems can be easily solved in each of the three cases.

arXiv:quant-ph/0307070•American Journal of Physics•May 12, 2003•33 citations

Quantum Aharonov-Bohm Billiard System

Der-San Chuu, De‐Hone Lin

The Green's functions of the two and three-dimensional relativistic Aharonov-Bohm (A-B) systems are given by the path integral approach. In addition the exact radial Green's functions of the spherical A-B quantum billiard system in two and three-dimensional are obtained via the perturbation techanique of $δ$-function.

arXiv:chao-dyn/9909022•Sep 15, 1999•0 citations

Boundary Integral Method for Stationary States of Two-Dimensional Quantum Systems

Ioan Kosztin, Klaus J. Schulten

The boundary integral method for calculating the stationary states of a quantum particle in nano-devices and quantum billiards is presented in detail at an elementary level. According to the method, wave functions inside the domain of the device or billiard are expressed in terms of line integrals of the wavefunction and its normal derivative along the domain's boundary; the respective energy eigenvalues are obtained as the roots of Fredholm determinants. Numerical implementations of the method are described and applied to determine the energy level statistics of billiards with circular and stadium shapes and demonstrate the quantum mechanical characteristics of chaotic motion. The treatment of other examples as well as the advantages and limitations of the boundary integral method are discussed.

arXiv:physics/9702022•International Journal of Modern Physics C•Apr 1, 1997•42 citations

Expansion method for stationary states of quantum billiards

David L. Kaufman, Ioan Kosztin, Klaus J. Schulten

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.

American Journal of Physics•Feb 1, 1999•56 citations

Uniform approximation for diffractive contributions to the trace formula in billiard systems

Martin Sieber, Nicolas Pavloff, C. Schmit

We derive contributions to the trace formula for the spectral density accounting for the role of diffractive orbits in two-dimensional billiard systems with corners. This is achieved by using the exact Sommerfeld solution for the Green function of a wedge. We obtain a uniformly valid formula for single-diffractive orbits which interpolates between formerly separate approaches (the geometrical theory of diffraction and Gutzwiller's trace formula). It yields excellent numerical agreement with exact quantum results, also in cases where other methods fail.

arXiv:chao-dyn/9610006•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Mar 1, 1997•35 citations
Quantum billiards are much studied as perhaps the simplest case which presents the central difficulty that the quasiclassical approximation is expressed as a divergent series. We find here, using the Fredholm method, an exact Green's function for billiards expressed as a ratio of absolutely convergent series. We make the quasiclassical approximation to this ratio. The method provides a convergence argument for previous results and an extension of results obtained for the spectrum to the full Green's function.
This work presents the results of a quantitative investigation of the “sum rule method” recently proposed by the author for calculating low-lying energy levels. The system considered in detail is the circular Aharonov-Bohm quantum billiard recently introduced by Berry and Robnik. Exact, expressions are derived for the spectral zeta function at positiveinteger values as a function of the magnetic flux. Using the zeta function for fixed angular momentum, we observe a very fast convergence to the exact ground state energy (“precocious convergence”).
We present quasianalytical and numerical calculations of Gaussian wave packet solutions of the Schrödinger equation for two-dimensional infinite well and quantum billiard problems with equilateral triangle, square, and circular footprints. These cases correspond to N=3, N=4, and N→∞ regular polygonal billiards and infinite wells, respectively. In each case the energy eigenvalues and wave functions are given in terms of familiar special functions. For the first two systems, we obtain closed form expressions for the expansion coefficients for localized Gaussian wave packets in terms of the eigenstates of the particular geometry. For the circular case, we discuss numerical approaches. We use these results to discuss the short-time, quasiclassical evolution in these geometries and the structure of wave packet revivals. We also show how related half-well problems can be easily solved in each of the three cases.
The Green's functions of the two and three-dimensional relativistic Aharonov-Bohm (A-B) systems are given by the path integral approach. In addition the exact radial Green's functions of the spherical A-B quantum billiard system in two and three-dimensional are obtained via the perturbation techanique of δδδ-function.
The boundary integral method for calculating the stationary states of a quantum particle in nano-devices and quantum billiards is presented in detail at an elementary level. According to the method, wave functions inside the domain of the device or billiard are expressed in terms of line integrals of the wavefunction and its normal derivative along the domain's boundary; the respective energy eigenvalues are obtained as the roots of Fredholm determinants. Numerical implementations of the method are described and applied to determine the energy level statistics of billiards with circular and stadium shapes and demonstrate the quantum mechanical characteristics of chaotic motion. The treatment of other examples as well as the advantages and limitations of the boundary integral method are discussed.
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.
We derive contributions to the trace formula for the spectral density accounting for the role of diffractive orbits in two-dimensional billiard systems with corners. This is achieved by using the exact Sommerfeld solution for the Green function of a wedge. We obtain a uniformly valid formula for single-diffractive orbits which interpolates between formerly separate approaches (the geometrical theory of diffraction and Gutzwiller's trace formula). It yields excellent numerical agreement with exact quantum results, also in cases where other methods fail.