Dispersion relations of periodic quantum graphs associated with Archimedean tiling (I)

TL;DR

This paper derives and analyzes the dispersion relations of periodic quantum graphs linked to various Archimedean tilings using Floquet-Bloch theory, revealing simple, symmetric spectra with point and continuous components.

Contribution

It provides explicit dispersion relations for quantum graphs associated with multiple Archimedean tilings, expanding understanding of their spectral properties.

Findings

Spectra consist of point and absolutely continuous parts.

Dispersion relations are simple and symmetric.

Analysis of the structure of the spectra.

Abstract

There are totally 11 kinds of Archimedean tiling for the plane. Applying the Floquet-Bloch theory, we derive the dispersion relations of the periodic quantum graphs associated with a number of Archimedean tiling, namely the triangular tiling {$(3^6)$}, the elongated triangular tiling {$(3^3,4^2)$}, the trihexagonal tiling {$(3,6,3,6)$} and the truncated square tiling {$(4,8^2)$}. The derivation makes use of characteristic functions, with the help of the symbolic software Mathematica. The resulting dispersion relations are surprisingly simple and symmetric. They show that in each case the spectrum is composed of point spectrum and an absolutely continuous spectrum. We further analyzed on the structure of the absolutely continuous spectra. Our work is motivated by the studies on the periodic quantum graphs associated with hexagonal tiling in \cite{KP} and \cite{KL}.