Discrete Spectrum of Quantum Hall Effect Hamiltonians I. Monotone Edge Potential

TL;DR

This paper analyzes the spectral properties of quantum Hall Hamiltonians with monotone edge potentials, focusing on the asymptotic distribution of eigenvalues in spectral gaps under perturbations, and introduces an effective Hamiltonian for this analysis.

Contribution

It introduces an effective Hamiltonian involving a pseudo-differential operator to describe eigenvalue asymptotics in spectral gaps of quantum Hall Hamiltonians with monotone edge potentials.

Findings

Spectral gaps can contain infinitely many eigenvalues under certain conditions.

A geometric condition on the support of the perturbation determines the finiteness of eigenvalues in gaps.

Eigenvalues converge to spectral edges with Gaussian rate when the geometric condition is violated.

Abstract

We consider the unperturbed operator $H_0 : = (-i \nabla - A)^2 + W$, 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 $W$ is a non-decreasing non constant bounded function depending only on the first coordinate $x \in \R$ of $(x,y) \in \R^2$. Then the spectrum of $H_0$ has a band structure and is absolutely continuous; moreover, the assumption $\lim_{x \to \infty}(W(x) - W(-x)) < 2b$ implies the existence of infinitely many spectral gaps for $H_0$. We consider the perturbed operators $H_{\pm} = H_0 \pm V$ where the electric potential $V \in L^{\infty}(\R^2)$ 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 involves a pseudo-differential operator with generalized anti-Wick symbol equal to $V$. Further, we restrict our attention on perturbations $V$ of compact support and constant sign. We establish a geometric condition on the support of $V$ which guarantees the finiteness of the eigenvalues of $H_{\pm}$ in any spectral gap of $H_0$. In the case where this condition is violated, we show that, generically, the convergence of the infinite series of eigenvalues of $H_+$ (resp. $H_-$) to the left (resp. right) edge of a given spectral gap, is Gaussian.