Schr\"odinger operators on armchair nanotubes. II

TL;DR

This paper analyzes the spectral properties of Schr"odinger operators with periodic potentials on armchair nanotubes, detailing the structure of the spectrum, including gaps, eigenvalues, and resonances, with asymptotic behavior at high energies.

Contribution

It provides a detailed description of the absolutely continuous spectrum, including multiplicity, gap endpoints, and resonance gaps, for Schr"odinger operators on armchair nanotubes.

Findings

Spectrum consists of absolutely continuous parts and eigenvalues with infinite multiplicity.

Endpoints of spectral gaps are characterized by eigenvalues or resonances.

Asymptotic behavior of spectral gaps at high energy is determined.

Abstract

We consider the Schr\"odinger operator with a periodic potential on quasi-1D models of armchair single-wall nanotubes. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite number of eigenvalues with infinite multiplicity. We describe the absolutely continuous spectrum of the Schr\"odinger operator: 1) the multiplicity, 2) endpoints of the gaps, they are given by periodic or antiperiodic eigenvalues or resonances (branch points of the Lyapunov function), 3) resonance gaps, where the Lyapunov function is non-real. We determine the asymptotics of the gaps at high energy.