Definitions

# Hearing the shape of a drum

To hear the shape of a drum is to infer information about the shape of the drumhead from the sound it makes, i.e., from the list of basic harmonics, via the use of mathematical theory. "Can One Hear the Shape of a Drum?" was the witty title of an article by Mark Kac in the American Mathematical Monthly 1966 (see References below), but these questions can be traced back all the way to Hermann Weyl.

The frequencies at which a drumhead can vibrate depend on its shape. Known mathematical formulas tell us the frequencies if we know the shape. A central question is: can they tell us the shape if we know the frequencies? No other shape than a square vibrates at the same frequencies as a square. Is it possible for two different shapes to yield the same set of frequencies? Kac did not know the answer to that question.

## In the language of mathematicians

Somewhat more formally, we are given a domain D, typically in the plane but sometimes in higher dimension, and the eigenvalues of a Dirichlet problem for the Laplacian, which we will denote by λn. The question is: what can be inferred on D if one knows only the values of λn? Two domains are said to be isospectral (or homophonic) if they have the same eigenvalues. Another way to pose the question is: are there two distinct domains that are isospectral?

Almost immediately, Milnor produced a pair of 16-dimensional tori that have the same eigenvalues but different shapes. However, the problem in two dimensions remained open until 1992, when Gordon, Webb, and Wolpert constructed, based on the Sunada method, a pair of regions in the plane that have different shapes but identical eigenvalues. The regions are non-convex polygons (see picture). The proof that both regions have the same eigenvalues is rather elementary and uses the symmetries of the Laplacian. This idea has been generalized by Buser et al., who constructed numerous similar examples. So, the answer to Kac' question is: for many shapes, one cannot hear the shape of the drum completely. However, some information can be inferred.

On the other hand, Zelditch proved that the answer to Kac' question is positive if one imposes restrictions to certain convex planar regions with analytic boundary. It is not known whether two non-convex analytic domains can have the same eigenvalues.

## Weyl's formula

Weyl's formula states that one can infer the area V of the drum by counting how many of the λns are quite small. We define N(R) to be the number of eigenvalues smaller than R and we get

$V=\left(2pi\right)^d lim_\left\{Rtoinfty\right\}frac\left\{N\left(R\right)\right\}\left\{R^\left\{d/2\right\}\right\},$

where d is the dimension. Weyl also conjectured that the next term in the approximation below would give the perimeter of D. In other words, if A denotes the length of the perimeter (or the surface area in higher dimension), then one should have

$,N\left(R\right)=\left(2pi\right)^\left\{-d\right\}VR^\left\{d/2\right\}+cAR^\left\{\left(d-1\right)/2\right\}+o\left(R^\left\{\left(d-1\right)/2\right\}\right).,$

where c is some constant that depends only on the dimension. For smooth boundary, this was proved by V. Ja. Ivrii in 1980.

## The Weyl-Berry conjecture

For non-smooth boundaries, Michael Berry conjectured in 1979 that the correction should be of the order of

$R^\left\{D/2\right\}$

where D is the Hausdorff dimension of the boundary. This was disproved by J. Brossard and R. A. Carmona, who then suggested one should replace the Hausdorff dimension with the upper box dimension. In the plane, this was proved if the boundary has dimension 1 (1993), but mostly disproved for higher dimensions (1996). Both results are by Lapidus and Pomerance.

## References

• (In Russian).
• . (Revised and enlarged second edition to appear in 2005.)