Large gaps in point-coupled periodic systems of manifolds

TL;DR

This paper investigates the spectral properties of quantum systems on periodically structured manifolds, revealing that such systems exhibit infinitely many spectral gaps, especially prominent at high energies.

Contribution

It introduces a general theoretical framework for analyzing quantum motion on complex periodic manifolds and demonstrates the prevalence of spectral gaps through various examples.

Findings

Spectra have infinitely many gaps.

Spectral gaps dominate at high energies.

Examples include spherical chains and square-lattice carpets.

Abstract

We study a free quantum motion on periodically structured manifolds composed of elementary two-dimensional "cells" connected either by linear segments or through points where the two cells touch. The general theory is illustrated with numerous examples in which the elementary components are spherical surfaces arranged into chains in a straight or zigzag way, or two-dimensional square-lattice "carpets". We show that the spectra of such systems have an infinite number of gaps and that the latter dominate the spectrum at high energies.