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

TL;DR

This paper analyzes the spectral properties of quantum Hamiltonians with periodic edge potentials under magnetic fields, establishing conditions for spectral gaps and describing the asymptotic distribution of eigenvalues introduced by perturbations.

Contribution

It provides explicit conditions for spectral band positivity, non-degeneracy of extremal points, and characterizes the asymptotic distribution of eigenvalues in spectral gaps under localized perturbations.

Findings

Spectral bands have positive length under certain conditions.

Infinitely many eigenvalues appear in spectral gaps due to perturbations.

Eigenvalues converge to spectral edges with Gaussian asymptotics.

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 $\sigma(H_0)$ of $H_0$ has a band structure, the band functions are $b {\mathcal T}$-periodic, and generically there are infinitely many open gaps in $\sigma(H_0)$. 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 $H_{\pm} = H_0 \pm V$ 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 $\sigma(H_0)$, and the convergence of these eigenvalues to the corresponding spectral edge is asymptotically Gaussian.