Discrete spectrum of quantum Hall effect Hamiltonians II: Periodic edge potentials

Abstract

We consider the unperturbed operator H-0 := (-i del-A)(2) + W, self-adjoint in L-2(R-2). Here A is a magnetic potential which generates a constant magnetic field b > 0, and the edge potential W = (W) over bar is a T-periodic non-constant bounded function depending only on the first coordinate x is an element of R of (x, y) is an element of R-2. Then the spectrum sigma(H-0) of H-0 has a band structure, the band functions are bT-periodic, and generically there are infinitely many open gaps in sigma(H-0). We establish explicit sufficient conditions which guarantee that a given band of sigma(H-0) has a positive length, and all the extremal points of the corresponding band function are non-degenerate. Under these assumptions we consider the perturbed operators H-+/- = H-0 +/- V where the electric potential V is an element of L-infinity(R-2) is non-negative and decays at infinity. We investigate the asymptotic distribution of the discrete spectrum of H-+/- in the spectral gaps of H-0. We introduce an effective Hamiltonian which governs the main asymptotic term; this Hamiltonian could be interpreted as a 1D Schrodinger operator with infinite-matrix-valued potential. Further, we restrict our attention on perturbations V of compact support. We find that there are infinitely many discrete eigenvalues in any open gap in the spectrum sigma(H-0), and the convergence of these eigenvalues to the corresponding spectral edge is asymptotically Gaussian.