An efficient Fredholm method for calculation of highly excited states of billiards
arXiv:nlin/0703030 · doi:10.1088/1751-8113/40/46/004
Abstract
A numerically efficient Fredholm formulation of the billiard problem is presented. The standard solution in the framework of the boundary integral method in terms of a search for roots of a secular determinant is reviewed first. We next reformulate the singularity condition in terms of a flow in the space of an auxiliary one-parameter family of eigenproblems and argue that the eigenvalues and eigenfunctions are analytic functions within a certain domain. Based on this analytic behavior we present a numerical algorithm to compute a range of billiard eigenvalues and associated eigenvectors by only two diagonalizations.
arXiv comment15 pages, 10 figures; included systematic study of accuracy with 2 new figures, movie to Fig. 4, http://www.quantumchaos.de/Media/0703030media.avi
References in corpus (15)
- High-Power Directional Emission from Microlasers with Chaotic Resonators
- Unidirectional light emission from high-Q modes in optical microcavities
- Semiclassical Description of Tunneling in Mixed Systems: The Case of the Annular Billiard
- Classical wave experiments on chaotic scattering
- Theory of the spatial structure of non-linear lasing modes
- Boundary Integral Method for Stationary States of Two-Dimensional Quantum Systems
- Expanded boundary integral method and chaotic time-reversal doublets in quantum billiards
- Direct Scattering Processes and Signatures of Chaos in Quantum Waveguides
- Quantum Necking in Stressed Metallic Nanowires
- Inside-Outside Duality for Planar Billiards -- A Numerical Study
- Vibrating soap films: An analog for quantum chaos on billiards
- Numerical aspects of eigenvalue and eigenfunction computations for chaotic quantum systems
- "Scars" in parametrically excited surface waves
- Fredholm methods for billiard eigenfunctions in the coherent state representation
- Quasi-orthogonality on the boundary for Euclidean Laplace eigenfunctions