# Zero Measure and Singular Continuous Spectra for Quantum Graphs

## TL;DR

This paper constructs a class of quantum graphs with spectra that are purely singular continuous or of zero Lebesgue measure, using advanced spectral theory and a new uniqueness result to analyze their spectral properties.

## Contribution

It introduces a novel class of quantum graphs with zero measure and singular continuous spectra, employing a new local Borg--Marchenko result and Kotani theory for aperiodic subshifts.

## Key findings

- Quantum graphs with zero Lebesgue measure spectrum
- Presence of nontrivial singular continuous spectrum
- Application of new uniqueness results in spectral analysis

## Abstract

We introduce a dynamically defined class of unbounded, connected, equilateral metric graphs on which the Kirchhoff Laplacian has zero Lebesgue measure spectrum and a nontrivial singular continuous part. A new local Borg--Marchenko uniqueness result is obtained in order to utilize Kotani theory for aperiodic subshifts satisfying Boshernitzan's condition.