Eigenmodes of Isospectral Drums

Abstract

Recently it was proved that there exist nonisomeric planar regions that have identical Laplace spectra. That is, one cannot "hear the shape of a drum." All known examples of such regions are bounded by polygons with reentrant corners. While the isospectrality can be proven mathematically, analytical techniques are unable to produce eigenvalues themselves. Furthermore, standard numerical methods for computing the eigenvalues, such as adaptive finite elements, are highly inefficient. Physical experiments have been performed to measure the spectra, but the accuracy and flexibility of this method are limited. We describe an algorithm due to Descloux and Tolley that blends finite elements with domain decomposition, and show that, with a modification that doubles its accuracy, this algorithm can be used to compute efficiently the eigenvalues for polygonal regions. We present results accurate to twelve digits for the most famous pair of isospectral drums, as well as results for another pair.