Spectral properties of finite quantum hall systems

In this note we review spectral properties of magnetic random Schrodinger operators H-omega = H-0 + V-omega + U-l + U-T defined on L-2 (R x [- L/2, L/2], dx dy) with periodic boundary conditions along y. U-l and U-T are two confining potentials for x less than or equal to - L/2 and x greater than or equal to L/2 respectively and vanish for -L/2 less than or equal to x less than or equal to 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(-gammaB(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).