Almost 50 years ago, Kac posed the now-famous question Can one hear the shape of a drum?, that is, if two planar domains are isospectral with respect to the Dirichlet (or Neumann) Laplacian, must they necessarily be congruent? This question was answered in the negative about 20 years ago with the construction of pairs of polygonal domains with special group-theoretically motivated symmetries, which are simultaneously Dirichlet and Neumann isospectral.
We wish to revisit these examples from an analytical perspective, recasting the arguments in terms of elliptic forms and intertwining operators. This allows us to prove in particular that the isospectrality property holds for a far more general class of elliptic operators than the Laplacian, as it depends purely on what the intertwining operator does to the form domains. We also show that the same type of intertwining operator cannot intertwine the Robin Laplacian on such domains.
This is joint work with Wolfgang Arendt (Ulm) and Tom ter Elst (Auckland).