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

摘要

We consider the unperturbed operator H_0 := (-i ▽ - 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 is a T-periodic non-constant bounded function depending only on the first coordinate x ∈ R of (x, y) ∈ R~2 . Then the spectrum σ(H_0) of H_0 has a band structure, the band functions are bT-periodic, and generically there are infinitely many open gaps in σ(H_0). We establish explicit sufficient conditions which guarantee that a given band of σ(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 ∈ L∞(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 Hq. We introduce an effective Hamiltonian which governs the main asymptotic term; this Hamiltonian could be interpreted as a ID 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 σ(H_0), and the convergence of these eigenvalues to the corresponding spectral edge is asymptotically Gaussian.