Isospectral domains for discrete elliptic operators
This is an incremental theoretical result for mathematicians studying spectral geometry, but it is limited to a toy model and does not address the original open problem.
The paper shows that isospectrality (preserving all eigenvalues) under continuous geometric deformation is possible for a finite-dimensional version of the 2D Laplacian, but only in a very small finite-dimensional space. The result is preliminary and does not extend to the actual Laplacian.
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.