Linear Representations and Isospectrality with Boundary Conditions

TL;DR

This paper introduces a new method for constructing isospectral systems using linear group representations, applicable to quantum graphs, drums, and manifolds, unifying and extending existing techniques like Sunada's method.

Contribution

The paper presents a general framework for creating isospectral systems via linear representations, encompassing and extending Sunada's method to various geometrical settings.

Findings

Reproduced known isospectral drums using the new method

Generated new examples of isospectral systems

Unified existing methods under a common framework

Abstract

We present a method for constructing families of isospectral systems, using linear representations of finite groups. We focus on quantum graphs, for which we give a complete treatment. However, the method presented can be applied to other systems such as manifolds and two-dimensional drums. This is demonstrated by reproducing some known isospectral drums, and new examples are obtained as well. In particular, Sunada's method is a special case of the one presented.