An efficient Fredholm method for calculation of highly excited states of billiards

Details An efficient Fredholm method for calculation of highly excited states of billiards An efficient Fredholm method for calculation of highly excited states of billiards

2007-03-15

arxiv.org/abs/nlin/0703030v3

Nonlinear Sciences

Chaotic Dynamics

15 pages, 10 figures; included systematic study of accuracy with 2 new figures, movie to Fig. 4, http://www.quantumchaos.de/

Scientific paper

10.1088/1751-8113/40/46/004

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.

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