Inverse problems in quantum graphs and accidental degeneracy

TL;DR

This paper develops a comprehensive algebraic and combinatorial framework for solving inverse spectral problems in quantum graphs, revealing the role of superalgebras and symmetry in degeneracy phenomena.

Contribution

It introduces a new algebraic approach to inverse spectral problems in quantum graphs and demonstrates how combinatorial and algebraic structures explain degeneracies.

Findings

Inverse spectral problems can be formulated with algebraic equations related to graph topology.

Superalgebra structures explain accidental degeneracies in certain quantum graphs.

Spectral data can be used to reconstruct graph topology unambiguously in specific cases.

Abstract

A general treatment of the spectral problem of quantum graphs and tight-binding models in finite Hilbert spaces is given. The direct spectral problem and the inverse spectral problem are written in terms of simple algebraic equations containing information on the topology of a quantum graph. The inverse problem is shown to be combinatorial, and some low dimensional examples are explicitly solved. For a {\it window\ }graph, a commutator and anticommutator algebra (superalgebra) is identified as the culprit behind accidental degeneracy in the form of triplets, where configurational symmetry {\it alone\ }fails to explain the result. For a M\"obius cycloacene graph, it is found that the accidental triplet cannot be explained with a superalgebra, but that the graph can be built unambiguously from the spectrum using combinatorial methods. These examples are compared with a more symmetric but less degenerate system, i.e. a {\it car wheel\ } graph which possesses neither triplets, nor superalgebra.