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

Tureci, Hakan E.; Schwefel, Harald G. L.

doi:10.1088/1751-8113/40/46/004

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

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Schwefel, Harald G. L.
Max Planck Research Group, Max Planck Institute for the Science of Light, Max Planck Society;

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Citation

Tureci, H. E., & Schwefel, H. G. L. (2007). An efficient Fredholm method for the calculation of highly excited states of billiards. JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, 40(46), 13869-13882. doi:10.1088/1751-8113/40/46/004.

Cite as: https://hdl.handle.net/11858/00-001M-0000-002D-6D02-A

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.