Cardinality of isospectral planar drums

Ascertain the possible cardinalities and structure of isospectral equivalence classes of bounded planar domains with Dirichlet boundary conditions; in particular, determine whether there exist uniform upper bounds or infinite families of pairwise non-congruent planar domains sharing the same Dirichlet Laplace spectrum.

Background

Multiple constructions show that distinct domains (and manifolds) can be isospectral, demonstrating that the spectrum does not determine shape in full generality. However, the size and structure of isospectral classes among planar “drums” remain poorly understood.

The authors explicitly raise the question of “how many drums can sound the same,” calling for classification or bounds on the number of non-congruent domains that share a spectrum.