Isospectral domains for discrete elliptic operators

TL;DR

This paper explores how the spectrum of a finite-dimensional approximation of the 2D Laplacian can be preserved under geometric deformations, providing foundational steps for future generalizations.

Contribution

It introduces a finite-dimensional model that allows eigenvalue preservation under domain deformation, a step towards understanding isospectrality in more complex settings.

Findings

Eigenvalues can be preserved under domain deformation in finite-dimensional models.

The approach can be extended to more accurate finite-dimensional representations.

The analysis provides a basis for future research in isospectral domain deformation.

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.