Computing Laplacian Eigenvalues using the Fredholm Determinant of a Boundary Integral Operator

Lin Zhao

Mathematics Dept, Dartmouth College

Numerical computation of the eigenvalues of the Laplacian to high accuracy is known to be difficult. The boundary integral method has been used to tackle this problem, turning it into a nonlinear eigenvalue problem. We present a method of computing eigenvalues of the planar Dirichlet Laplacian by first constructing the Fredholm determinant of a boundary integral operator using the Nystrom method with product quadrature to handle weakly singular kernels, then solving for zeros through Chebyshev expansion and polynomial root finding. Singular values are also used to obtain high accuracy at close eigenfrequencies. An exterior resonance can occur for concave domains which results in spurious eigenvalues. We overcome this by using a combined layer potential. Our analysis gives exponential convergence rate for analytic domains.

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