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Journal Article

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


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

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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
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.