Resolving isospectral ‘drums’ by counting nodal domains

Several types of systems have been put forward during the past few decades to show that there exist isospectral systems which are metrically different. One important class consists of Laplace–Beltrami operators for pairs of flat tori in R with n 4. We propose that the spectral ambiguity can be resolved by comparing the nodal sequences (the numbers of nodal domains of eigenfunctions, arranged by increasing eigenvalues). In the case of isospectral flat tori in four dimensions—where a four-parameter family of isospectral pairs is known—we provide heuristic arguments supported by numerical simulations to support the conjecture that the isospectrality is resolved by the nodal count. Thus one can count the shape of a drum (if it is designed as a flat torus in four dimensions). PACS numbers: 02.40.Vh, 05.45.Mt Mathematics Subject Classification: 58J50, 58J53