Edge currents and eigenvalue estimates for magnetic barrier Schr\"odinger operators

TL;DR

This paper analyzes magnetic Schr"odinger operators with a magnetic barrier, studying their spectral properties, eigenvalue asymptotics, and edge currents, revealing insights into their band functions and stability under perturbations.

Contribution

It provides a detailed spectral analysis of magnetic barrier Schr"odinger operators, including eigenvalue asymptotics, effective mass calculations, and bounds on edge currents, with new results on their stability and spectral decomposition.

Findings

Band functions have specific asymptotic behaviors for large wave numbers.

Eigenvalues of related operators describe band function asymptotics.

Lower bounds on magnetic edge currents are established.

Abstract

We study two-dimensional magnetic Schr\"odinger operators with a magnetic field that is equal to b>0 for x > 0 and (-b) for x < 0. This magnetic Schr\"odinger operator exhibits a magnetic barrier at x=0. The unperturbed system is invariant with respect to translations in the y-direction. As a result, the Schr\"odinger operator admits a direct integral decomposition. We analyze the band functions of the fiber operators as functions of the wave number and establish their asymptotic behavior. Because the fiber operators are reflection symmetric, the band functions may be classified as odd or even. The odd band functions have a unique absolute minimum. We calculate the effective mass at the minimum and prove that it is positive. The even band functions are monotone decreasing. We prove that the eigenvalues of an Airy operator, respectively, harmonic oscillator operator, describe the asymptotic behavior of the band functions for large negative, respectively positive, wave numbers. We prove a Mourre estimate for perturbations of the magnetic Schr\"odinger operator and establish the existence of absolutely continuous spectrum in certain energy intervals. We prove lower bounds on magnetic edge currents for states with energies in the same intervals. We also prove that these lower bounds imply stable lower bounds for the asymptotic currents. We study the perturbation by slowly decaying negative potentials and establish the asymptotic behavior of the eigenvalue counting function for the infinitely-many eigenvalues below the bottom of the essential spectrum.