On semiclassical dispersion relations of Harper-like operators

TL;DR

This paper analyzes the semiclassical spectral properties of Harper-like operators, focusing on dispersion relations, band widths, and the influence of geometric features and symmetries in the quantum Hamiltonian.

Contribution

It provides asymptotic formulas for dispersion relations and explores how geometric characteristics affect energy band structures in Harper-like operators.

Findings

Derived asymptotic formulas for dispersion relations

Identified how geometric features influence energy bands

Analyzed the impact of symmetries on spectral properties

Abstract

We describe some semiclassical spectral properties of Harper-like operators, i.e. of one-dimensional quantum Hamiltonians periodic in both momentum and position. The spectral region corresponding to the separatrices of the classical Hamiltonian is studied for the case of integer flux. We derive asymptotic formula for the dispersion relations, the width of bands and gaps, and show how geometric characteristics and the absence of symmetries of the Hamiltonian influence the form of the energy bands.