# Discrete spectrum of quantum Hall effect Hamiltonians I. Monotone edge potentials

### Vincent Bruneau

Université Bordeaux 1, Talence, France### Pablo L. Miranda

Universidad de Chile, Santiago, Chile### Georgi Raikov

Pontificia Universidad Católica de Chile, Santiago de Chile, Chile

## Abstract

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 $W$ is a non-decreasing non constant bounded function depending only on the first coordinate $x∈R$ of $(x,y)∈R_{2}$. Then the spectrum of $H_{0}$ has a band structure and is absolutely continuous; moreover, the assumption $lim_{x→∞}(W(x)−W(−x))<2b$ implies the existence of infinitely many spectral gaps for $H_{0}$. 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 $H_{0}$. 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 $H_{±}$ 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.

## Cite this article

Vincent Bruneau, Pablo L. Miranda, Georgi Raikov, Discrete spectrum of quantum Hall effect Hamiltonians I. Monotone edge potentials. J. Spectr. Theory 1 (2011), no. 3, pp. 237–272

DOI 10.4171/JST/11