Spectrum of Schr\"odinger operators on subcovering graphs

TL;DR

This paper explores the spectral properties of Schr"odinger operators on periodic graphs and their subcoverings, providing criteria for isospectrality and analyzing band edge asymptotics as graphs are rolled into nanotube-like structures.

Contribution

It introduces a criterion for when subcovering graphs are isospectral to original graphs and analyzes spectral band edge asymptotics for large chiral vectors.

Findings

Subcovering graphs can be isospectral to the original graph under certain conditions.

Spectral band edges have specific asymptotic behaviors as chiral vectors grow long.

Connections between spectra of original and subcovering graphs are characterized.

Abstract

We consider discrete Schr\"odinger operators with real periodic potentials on periodic graphs. The spectra of the operators consist of a finite number of bands. By "rolling up" a periodic graph along some appropriate directions we obtain periodic graphs of smaller dimensions called subcovering graphs. For example, rolling up a planar hexagonal lattice along different directions will lead to nanotubes with various chiralities. We describe connections between spectra of the Schr\"odinger operators on a periodic graph and its subcoverings. In particular, we provide a simple criterion for the subcovering graph to be isospectral to the original periodic graph. By isospectrality of periodic graphs we mean that the spectra of the Schr\"odinger operators on the graphs consist of the same number of bands and the corresponding bands coincide as sets. We also obtain asymptotics of the band edges of the Schr\"odinger operator on the subcovering graph as the "chiral" (roll up) vectors are long enough.