Spectral Properties of Finite Quantum Hall Systems

TL;DR

This paper reviews the spectral properties of finite quantum Hall systems, analyzing eigenvalues and currents in different energy intervals related to Landau levels and spectral gaps.

Contribution

It provides a classification of eigenvalues and their associated currents in finite quantum Hall systems with confining potentials, extending understanding of spectral behavior.

Findings

Eigenvalues in the first Landau band show different current magnitudes.

Eigenvalues in the spectral gap have quantum currents of order one.

Spectral properties depend on the energy interval and boundary conditions.

Abstract

In this note we review spectral properties of magnetic random Schroedinger operators H_omega=H_0+V_omega + U_l + U_r defined on L^2(R x [-L/2,L/2],dx dy) with periodic boundary conditions along y. U_l and U_r are two confining potentials for x<-L/2 and x>L/2 respectively and vanish for -L/2<x<L/2. We describe the spectrum in two energy intervals and we classify it according to the quantum mechanical current of eigenstates along the periodic direction. The first interval lies in the first Landau band of the bulk Hamiltonian, and contains intermixed eigenvalues with a quantum mechanical current of O(1) and O(e^{-gamma B(log L)^2}) respectively. The second interval lies in the first spectral gap of the bulk Hamiltonian, and contains only eigenvalues with a quantum mechanical current of O(1).