Distinguishing co-spectral quantum graphs by scattering

TL;DR

This paper introduces a scattering-based method to distinguish co-spectral quantum graphs by attaching leads and comparing their scattering matrices, effectively resolving co-spectrality in various graph classes.

Contribution

The authors develop a novel approach using scattering data to differentiate co-spectral Schrödinger operators on metric graphs, extending the ability to distinguish graphs beyond eigenvalues alone.

Findings

Eigenvalues and scattering data together distinguish co-spectral graphs in several cases.

The method applies to graphs with up to 6 vertices, trees with up to 9 vertices, and fuzzy balls.

Scattering approach enhances graph isomorphism detection in quantum graph analysis.

Abstract

We propose a simple method for resolution of co-spectrality of Schr\"odinger operators on metric graphs. Our approach consists of attaching a lead to them and comparing the $S$-functions of the corresponding scattering problems on these (non-compact) graphs. We show that in several cases -- including general graphs on at most 6 vertices, general trees on at most 9 vertices, and general fuzzy balls -- eigenvalues and scattering data are together sufficient to distinguish co-spectral metric graphs.