Band functions of Iwatsuka models : power-like and flat magnetic fields

TL;DR

This paper studies the spectral properties of Iwatsuka models with magnetic fields that are either power-like or flat at infinity, providing detailed asymptotics of band functions and implications for quantum state currents.

Contribution

It extends previous asymptotic analysis of band functions to magnetic fields that are constant at infinity, showing exponential convergence to Landau levels.

Findings

Band functions tend to Landau levels at large frequencies.

Asymptotics are exponential for flat magnetic fields.

Control of quantum state currents near spectral thresholds.

Abstract

In this note we consider the Iwatsuka model with a postive increasing magnetic field having finite limits. The associated magnetic Laplacian is fibred through partial Fourier transform, and, for large frequencies, the band functions tend to the Landau levels, which are thresholds in the spectrum. The asymptotics of the band functions is already known when the magnetic field converge polynomially to its limits. We complete this analysis by giving the asymptotics for a regular magnetic field which is constant at infinity, showing that the band functions converge now exponentially fast toward the thresholds. As an application, we give a control on the current of quantum states localized in energy near a threshold.