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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References

SHOWING 1-10 OF 128 REFERENCES
Can one hear the shape of a Lie-type geometry?
In a celebrated paper 'Can one hear the shape of a drum?' (Kac 1966 Am. Math. Mon. 73 1–23) Kac asked his famous question on the consequences of isospectrality, which was eventually answeredExpand
LETTER TO THE EDITOR: 'Can one hear the shape of a drum?': revisited
A famous inverse problem posed by M Kac 'Can one hear the shape of a drum?' is concerned with isospectrality of drums or planer billiards, and the first counter example was constructed by Gordon,Expand
Isospectral drums in , involution graphs and Euclidean TI-domains
The widely investigated question 'Can one hear the shape of a drum?' which Kac posed in his published lecture (Kac 1966 Am. Math. Mon. 73 1–23) was eventually answered negatively in Gordon et alExpand
Can one hear the shape of a drum? revisited
In a landmark paper, Mark Kac in 1966 [Amer. Math. Monthly, 73, pp. 1–23] showed that geometric properties of regions in $R^2 $ can be obtained by studying the asymptotic properties of the spectrumExpand
LETTER TO THE EDITOR: Kac's question, planar isospectral pairs and involutions in projective space
In a paper published in Am. Math. Mon. (1966 73 1?23), Kac asked his famous question 'Can one hear the shape of a drum?'. Gordon et al answered this question negatively by constructing planarExpand
One cannot hear the shape of a drum
We use an extension of Sunada's theorem to construct a nonisometric pair of isospectral simply connected domains in the Euclidean plane, thus answering negatively Kac's question, can one hear theExpand
On "hearing the shape of drums": An experimental study using vibrating smectic films
Sunada's theorem provides a method for building pairs of drums of different shapes which possess the same spectrum of eigenfrequencies. Our study concerns the isospectrality of a family of pairs ofExpand
Resolving isospectral 'drums' by counting nodal domains
Several types of systems have been put forward during the past few decades to show that there exist isospectral systems which are metrically different. One important class consists ofExpand
Linear Representations and Isospectrality with Boundary Conditions
We present a method for constructing families of isospectral systems, using linear representations of finite groups. We focus on quantum graphs, for which we give a complete treatment. However, theExpand
Coincidence of length spectra does not imply isospectrality
Penrose–Lifshits mushrooms are planar domains coming in nonisometric pairs with the same geodesic length spectrum. Recently Zelditch raised the question whether such billiards also have the sameExpand
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