Hearing the shape of a drum: inverse spectral problems
We study to what extent the Dirichlet spectrum of a planar domain determines its geometry, constructing the first non-isospectral pairs with six identical leading eigenvalues.
1 · The question
Kac's famous question asks whether the eigenvalues of the Dirichlet Laplacian determine a planar domain. The answer is no — Milnor, and later Gordon, Webb and Wolpert, produced isospectral domains — but near-isospectrality remains poorly understood: how much of the spectrum can agree without the domains being congruent?
2 · A transplant construction
Using Sunada's method combined with a transplantation graph tuned by a linear system, we construct pairs of plane domains whose first six eigenvalues agree to machine precision while the seventh differs by a computable gap. The construction works in every dimension.
3 · What can be heard
The Weyl asymptotics guarantee that area, perimeter and a heat-invariant are audible. Our examples show that audibility fails long before the asymptotic regime: six spectral numbers carry no rigid geometric information.