Magnetic layers with periodic point perturbations

TL;DR

This paper analyzes the spectral properties of a quantum particle in a magnetic field within a layered structure with periodic point impurities, providing explicit spectral band descriptions based on the model parameters.

Contribution

It introduces a method to explicitly describe spectral bands of a quantum system with periodic point perturbations and magnetic fields using Landau-Zak transformation and Krein's formula.

Findings

Spectral bands can be absolutely continuous or degenerate.

Explicit spectral descriptions depend on model parameters.

The approach applies to systems with rational flux and finite impurities per cell.

Abstract

We study spectral properties of a spinless quantum particle confined to an infinite planar layer with hard walls which interacts with a periodic lattice of point perturbations and a homogeneous magnetic field perpendicular to the layer. It is supposed that the lattice cell contains a finite number of impurities and the flux through the cell is rational. Using the Landau-Zak transformation, we convert the problem into investigation of the corresponding fiber operators which is performed by means of Krein's formula. This yields an explicit description of the spectral bands which may be absolutely continuous or degenerate, depending on the parameters of the model.