Hear the shape of a smooth drum

Determine whether the Dirichlet Laplace spectrum uniquely determines a bounded planar domain with smooth (e.g., C^∞) boundary up to Euclidean isometry; equivalently, prove that any two smoothly bounded subsets of R^2 with identical Dirichlet Laplace spectra are congruent, or provide a counterexample.

Background

The authors revisit Kac’s question in the context of smoothly bounded planar domains. While many spectral invariants have been rigorously established (area, perimeter, Euler characteristic), the full shape is not determined by the spectrum in general. It remains unknown whether smoothness of the boundary suffices to guarantee spectral determination.

This question is posed explicitly as an open problem in the Outlook section, reflecting ongoing efforts to understand inverse spectral uniqueness within geometrically restricted classes of domains.