Efficient Boundary Integral Method for Quantum Billiards

Calculating highly excited eigenvalues of the Laplace equation and their corresponding eigenfunctions are of great current interest in many areas. We present an efficient algorithm based on a novel Fredholm formulation of the Laplace eigenvalue problem, in the spirit of the scattering quantization method proposed by the authors in the context of the basis function expansion technique.\footnote{H.~E. Tureci, H.~G.~L. Schwefel, Ph. Jacquod, and A.~Douglas Stone. Modes of wave-chaotic dielectric resonators. {\em Progress In Optics}, 47, 2005.} We also point out the connection to the scaling eigenfunctions\footnote{A.~H.~Barnett. Quasi-orthogonality on the boundary for Euclidean Laplace eigenfunctions. {\em submitted, Comm.\ Pure Appl.\ Math.}, 2004} and show how this method can be generalized to dielectric cavities.