Isospectral Shapes with Neumann and Alternating Boundary Conditions
Driscoll, Tobin A.
Department of Mathematical Sciences
The best-known negative answer to Mark Kac's question, "Can one hear the shape of a drum?" is a pair of octagons discovered by Gordon, Webb, and Wolpert. Their nonconstructive proof of Dirichlet isospectrality has since been supplemented by experiment and high-accuracy computation of many of the eigenvalues of these drums. In this paper we compute the Neumann modes of these regions, also known to be identical, to high precision. Additionally, we carry out the computations for two cases in which the boundary conditions alternate between Dirichlet and Neumann around the sides. There is overwhelming numerical evidence that the regions remain isospectral, though to our knowledge there has been no analytic demonstration of this fact.
Eigen values , Laplacian , chaotic billiards , isospectrality