Spectral Properties of Finite Quantum Hall Systems

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).