Equispectrality and Transplantation

TL;DR

This paper introduces a novel numerical method that uses discretized volumes, symmetry groups, and graph theory to identify equispectral domains for boundary value problems, enhancing spectral analysis techniques.

Contribution

It develops a new approach combining group theory and graph analysis to determine equispectral discretized volumes, a novel contribution in numerical spectral methods.

Findings

Similarity of auxiliary matrices is necessary and sufficient for equispectrality.

The method can identify equispectral volumes using symmetry and graph structures.

A simple example demonstrates the method's feasibility.

Abstract

We present a technique novel in numerical methods. It compiles the domain of the numerical methods as a discretized volume. Congruent elements are glued together to compile the domain over which the solution of a boundary value problem of a linear operator is sought. We associate a group and a graph to that volume. When the group is symmetry of the boundary value problem under investigation, one can specify the structure of the solution, and find out if there are equispectral volumes of a given type. We show that similarity of the so called auxiliary matrices is sufficient and necessary for two discretized volumes to be equispectral. A simple example demonstrates the feasibility of the suggested method.