A Trace Formula for Long-Range Perturbations of the Landau Hamiltonian

TL;DR

This paper analyzes how the eigenvalues of a Landau Hamiltonian with a long-range electric potential cluster and accumulate near Landau levels, providing estimates and asymptotic distributions of these eigenvalues.

Contribution

It introduces a new estimate for the rate at which eigenvalue clusters shrink to Landau levels and describes their asymptotic distribution using the mean-value transform of the potential.

Findings

Eigenvalue clusters shrink at a quantifiable rate as the cluster index increases.

The asymptotic density of eigenvalues within a cluster is explicitly characterized.

The distribution depends on the mean-value transform of the long-range part of the potential.

Abstract

We consider the Landau Hamiltonian perturbed by a long-range electric potential $V$. The spectrum of the perturbed operator consists of eigenvalue clusters which accumulate to the Landau levels. First, we obtain an estimate of the rate of the shrinking of these clusters to the Landau levels as the number of the cluster $q$ tends to infinity. Further, we assume that there exists an appropriate $\V$, homogeneous of order $-\rho$ with $\rho \in (0,1)$, such that $V(x) = \V(x) + O(|x|^{-\rho - \epsilon})$, $\epsilon > 0$, as $|x| \to \infty$, and investigate the asymptotic distribution of the eigenvalues within a given cluster, as $q \to \infty$. We obtain an explicit description of the asymptotic density of the eigenvalues in terms of the mean-value transform of $\V$.