One cannot hear the density of a drum (and further aspects of isospectrality)

It is well known that certain pairs of planar domains have the same spectra of the Laplacian operator. We prove that these domains are still isospectral for a wider class of physical problems, including the cases of heterogeneous drums and quantum billiards in an external field. In particular we show that the isospectrality is preserved when the density or the potential is symmetric under reflections along the folding lines of the domain. These results are also confirmed numerically using the finite-difference method: We find that the pairs of numerical matrices obtained in the discretization are exactly isospectral up to machine precision.