# Hearing shapes of drums — mathematical and physical aspects of isospectrality

@article{Giraud2010HearingSO, title={Hearing shapes of drums — mathematical and physical aspects of isospectrality}, author={Olivier Giraud and Koen Thas}, journal={Reviews of Modern Physics}, year={2010}, volume={82}, pages={2213-2255} }

In a celebrated paper ''Can one hear the shape of a drum?'' M. Kac [Am. Math. Monthly 73, 1 (1966)] asked his famous question about the existence of nonisometric billiards having the same spectrum of the Laplacian. This question was eventually answered positively in 1992 by the construction of noncongruent planar isospectral pairs. This review highlights mathematical and physical aspects of isospectrality.

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#### 49 Citations

We can't hear the shape of drum: revisited in 3D case

- Mathematics, Physics
- 2017

Can one hear the shape of a drum? was proposed by Kac in 1966. The simple answer is NO as shown through the construction of iso-spectral domains. There already exists 17 families of planar domains… Expand

On Inaudible Properties of Broken Drums - Isospectral Domains with Mixed Boundary Conditions

- Mathematics
- 2011

Since Kac raised the question "Can one hear the shape of a drum?", various families of non-smooth counterexamples have been constructed using the transplantation method, which we consider for domains… Expand

Hearing the shape of a triangle

- Mathematics
- 2012

In 1966 Mark Kac asked the famous question 'Can one hear the shape of a drum?'. While this was later shown to be false in general, it was proved by C. Durso that one can hear the shape of a triangle.… Expand

Disjointness-preserving operators and isospectral Laplacians

- Mathematics
- Journal of Spectral Theory
- 2021

All the known counterexamples to Kac' famous question "can one hear the shape of a drum", i.e., does isospectrality of two Laplacians on domains imply that the domains are congruent, consist of pairs… Expand

The geometry of drums

- Mathematics, Physics
- 2017

We introduce the new concept of D-geometry (or "drum geometry"), which has been recently discovered by the author in \cite{KT-DRUMS} when constructing and classifying isospectral and length… Expand

Isospectral drums and simple groups

- Mathematics
- 2015

Virtually every known pair of isospectral but nonisometric manifolds - with as most famous members isospectral bounded $\mathbb{R}$-planar domains which makes one "not hear the shape of a drum" [13]… Expand

On inaudible properties of broken drums - Isospectrality with mixed Dirichlet-Neumann boundary conditions

- Mathematics
- 2011

We study isospectrality for manifolds with mixed Dirichlet-Neumann boundary conditions and express the well-known transplantation method in graph- and representation-theoretic terms. This leads to a… Expand

The 2-Transitive Transplantable Isospectral Drums

- Mathematics, Physics
- 2011

In this paper we investigate pairs of Euclidean TI-domains which are isospectral but not congruent. For Riemannian manifolds there are several examples which are isospectral but not isometric, see… Expand

Kac's isospectrality question revisited in neutrino billiards.

- Physics, Medicine
- Physical review. E
- 2020

It is demonstrated that the transplantation method fails and thus isospectrality is lost when changing from the nonrelativistic to the relativistic case. Expand

Spectral statistics of 'cellular' billiards

- Physics, Mathematics
- 2011

For a bounded domain whose boundary contains a number of flat pieces Γi, i = 1, ..., l we consider a family of non-symmetric billiards Ω constructed by patching several copies of Ω0 along Γis. It is… Expand

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