Periodic spectrum of $n$-cubic quantun graphs

TL;DR

This paper analyzes the spectra of periodic Schrödinger operators on infinite n-cubic quantum graphs using Floquet-Bloch theory, deriving dispersion relations for 2D rectangles and n-cubes to facilitate spectral analysis.

Contribution

It provides an analytic derivation of dispersion relations for periodic quantum graphs generated by rectangles and n-cubes, advancing spectral analysis methods.

Findings

Derived dispersion relations for 2D rectangular and n-cubic quantum graphs.

Facilitated spectral analysis of periodic Schrödinger operators on these graphs.

Provided a framework for further spectral studies of quantum graphs.

Abstract

We study the spectrum of some periodic differential operators, in particular the periodic Schr\"{o}dinger operator acting on infinite $n$-cubic graphs. Using Floquet-Bloch theory, we derive and analyze on the dispersion relations of the periodic quantum graph generated by 2-dimensional rectangles, and also $n$-cubes. Our proof is analytic. These dispersion relations define the spectra of the associated periodic operator, thus facilitating further analysis of the spectra.