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

Hakan E. Tureci·Harald G. L. Schwefel
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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.

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Hakan E. Tureci, Harald G. L. Schwefel. 2007-09-19. An efficient Fredholm method for calculation of highly excited states of billiards. https://doi.org/10.1088/1751-8113%2F40%2F46%2F004

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