Discrete Spectrum of Quantum Hall Effect Hamiltonians II. Periodic Edge Potentials

Abstract

We consider the unperturbed operator $H_0: = (-i \nabla - {\bf A})^2 + W$, self-adjoint in $L^2({\mathbb R}^2)$. Here ${\bf A}$ is a magnetic potential which generates a constant magnetic field $b>0$, and the edge potential $W = \bar{W}$ is a ${\mathcal T}$-periodic non constant bounded function depending only on the first coordinate $x \in {\mathbb R}$ of $(x,y) \in {\mathbb R}^2$. Then the spectrum $σ(H_0)$ of $H_0$ has a band structure, the band functions are $b {\mathcal T}$-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 $H_{\pm} = H_0 \pm V$ where the electric potential $V \in L^{\infty}({\mathbb R}^2)$ is non-negative and decays at infinity. We investigate the asymptotic distribution of the discrete spectrum of $H_\pm$ 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 Schroedinger 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 of $σ(H_0)$, and the convergence of these eigenvalues to the corresponding spectral edge is asymptotically Gaussian.