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

TL;DR

This paper demonstrates that isospectrality of planar domains extends beyond the Laplacian to include physical problems like heterogeneous drums and quantum billiards, with numerical confirmation of the spectral equivalence.

Contribution

It proves that isospectrality persists for a broader class of physical problems, including heterogeneous densities and external potentials, under certain symmetry conditions.

Findings

Isospectrality extends to heterogeneous drums and quantum billiards.

Numerical methods confirm spectral equivalence up to machine precision.

Symmetry under reflection preserves isospectrality.

Abstract

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 of quantum billiards in an external field. In particular we show that the isospectrality is preserved when the density or the potential are 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.