Complete list of spectral invariants that can be heard

Identify and characterize the full set of geometric and topological quantities of bounded planar domains in R^2 that are determined by the Dirichlet Laplace spectrum (i.e., the complete collection of spectral invariants), extending beyond known invariants such as area, perimeter, Euler characteristic, and corner angles.

Background

The survey documents numerous spectral invariants discovered over the past century, including Weyl’s law (dimension and volume), boundary length for planar domains, Euler characteristic for smoothly bounded domains, and corner angles for polygonal domains.

Nevertheless, a comprehensive characterization of all quantities that are spectrally determined remains an open question. The authors explicitly pose “What all can one hear?” as an open question, seeking a complete description of spectral invariants for planar drums.

References

Can one hear the shape of a convex drum? Can one hear the shape of a smooth drum? How many drums can sound the same? What all can one hear? These are just a few of numerous open questions in the rich field of spectral geometry.

— 112 years of listening to Riemannian manifolds (2406.18369 - Mårdby et al., 26 Jun 2024) in Section 7 (We’re still listening…), sec:outlook