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

Abstract

We consider the unperturbed operator H₀:=(−i∇−A)²+W, self-adjoint in L²(R²). 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)∈R². Then the spectrum σ(H₀) of H₀ has a band structure, the band functions are b𝒯-periodic, and generically there are infinitely many open gaps in σ(H₀). We establish explicit sufficient conditions which guarantee that a given band of σ(H₀) 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_±=H₀±V where the electric potential V∈L^∞(R²) is non-negative and decays at infinity. We investigate the asymptotic distribution of the discrete spectrum of H_± in the spectral gaps of H₀. We introduce an effective Hamiltonian which governs the main asymptotic term; this Hamiltonian could be interpreted as a 1D Schrödinger 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(H₀), and the convergence of these eigenvalues to the corresponding spectral edge is asymptotically Gaussian.