Concerning the Laplace operator with homogeneous Dirichlet boundary conditions, the classical notion of isospectrality assumes that two domains are related when they give rise to the same spectrum. In two dimensions, non isometric, isospectral domains exist. It is not known however if all the eigenvalues relative to a specific domain can be preserved under suitable continuous deformation of its geometry. We show that this is possible when the 2D Laplacian is replaced by a finite dimensional version and the geometry is modified by respecting certain constraints. The analysis is carried out in a very small finite dimensional space, but it can be extended to more accurate finite-dimensional representations of the 2D Laplacian, with an increase of computational complexity. The aim of this paper is to introduce the preliminary steps in view of more serious generalizations.

Isospectral Domains for Discrete Elliptic Operators / Fatone, Lorella; Funaro, Daniele. - In: JOURNAL OF SCIENTIFIC COMPUTING. - ISSN 0885-7474. - 75:1(2018), pp. 405-426. [10.1007/s10915-017-0541-5]

Isospectral Domains for Discrete Elliptic Operators

Funaro, Daniele
2018

Abstract

Concerning the Laplace operator with homogeneous Dirichlet boundary conditions, the classical notion of isospectrality assumes that two domains are related when they give rise to the same spectrum. In two dimensions, non isometric, isospectral domains exist. It is not known however if all the eigenvalues relative to a specific domain can be preserved under suitable continuous deformation of its geometry. We show that this is possible when the 2D Laplacian is replaced by a finite dimensional version and the geometry is modified by respecting certain constraints. The analysis is carried out in a very small finite dimensional space, but it can be extended to more accurate finite-dimensional representations of the 2D Laplacian, with an increase of computational complexity. The aim of this paper is to introduce the preliminary steps in view of more serious generalizations.
2018
31-ago-2017
75
1
405
426
Isospectral Domains for Discrete Elliptic Operators / Fatone, Lorella; Funaro, Daniele. - In: JOURNAL OF SCIENTIFIC COMPUTING. - ISSN 0885-7474. - 75:1(2018), pp. 405-426. [10.1007/s10915-017-0541-5]
Fatone, Lorella; Funaro, Daniele
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11380/1150224
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