Families of isospectral and isoscattering quantum graphs

TL;DR

This paper introduces a new method using germ graphs and M-function formalism to construct large families of isospectral and isoscattering quantum graphs, differing from traditional approaches and validated through microwave network experiments.

Contribution

The paper presents a novel approach to constructing isospectral and isoscattering quantum graphs without embedding into larger symmetric graphs, extending to dissipative cases.

Findings

Successful construction of isospectral and isoscattering graph pairs

Experimental validation using microwave networks

Extension of formalism to dissipative quantum graphs

Abstract

A concept of germ graphs and the M-function formalism are employed to construct large families of isospectral and isoscattering graphs. This approach represents a complete departure from the original approach pioneered by Sunada, where isospectral graphs are obtained as quotients of a certain large symmetric graph. Using the M-function formalism and the symmetries of the graph itself we construct isospectral and isoscattering pairs. In our novel approach isospectral pairs do not need to be embedded into a larger symmetric graph as in Sunada's approach. We demonstrate that the introduced formalism can also be extended to graphs with dissipation. The theoretical predictions are validated experimentally using microwave networks emulating open quantum graphs with dissipation.