The Isospectral Fruits of Representation Theory: Quantum Graphs and Drums

TL;DR

This paper introduces a novel method based on representation theory for constructing isospectral objects like quantum graphs and drums, providing new examples and insights into their theoretical underpinnings.

Contribution

It develops a general framework for creating isospectral objects using representation theory and assembly techniques, unifying and extending previous methods including Sunada's theorem.

Findings

New isospectral quantum graphs and drums are constructed.

Sunada's theorem is shown as a special case of the new method.

The paper provides a gallery of novel isospectral examples.

Abstract

We present a method which enables one to construct isospectral objects, such as quantum graphs and drums. One aspect of the method is based on representation theory arguments which are shown and proved. The complementary part concerns techniques of assembly which are both stated generally and demonstrated. For that purpose, quantum graphs are grist to the mill. We develop the intuition that stands behind the construction as well as the practical skills of producing isospectral objects. We discuss the theoretical implications which include Sunada's theorem of isospectrality arising as a particular case of this method. A gallery of new isospectral examples is presented and some known examples are shown to result from our theory.