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arXiv · 1701.05984

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

Xiao Hui Liu·Jia Chang Sun·Jian Wen Cao
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Abstract

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 which are non-isometric but display the same spectra of frequencies. These frequencies, deduced from the eigenvalues of the Laplacian, are determined by solving the wave equation in a domain, which is subject to Dirichlet boundary conditions. This paper revisits the serials of reflection rule inherent in the 17 families of iso-spectral domains. In accordance with the reflection rule visualized by red-blue-black, we construct real 3D isospectral models successfully. What is more, accompanying with the proof of transplantation method, we also use the numerical method to verify the isospectrality of the 3D models.

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Xiao Hui Liu, Jia Chang Sun, Jian Wen Cao. 2017-01-21. We can't hear the shape of drum: revisited in 3D case. https://arxiv.org/abs/1701.05984

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