Discrete spectrum distribution of the Landau Operator Perturbed by an Expanding Electric Potential

TL;DR

This paper investigates how an expanding electric potential affects the discrete spectrum of the Landau operator, deriving a quasi-classical formula for the eigenvalue count as the potential expands infinitely.

Contribution

It introduces a novel analysis of the Landau operator perturbed by an expanding electric potential, providing a new quasi-classical spectral counting formula.

Findings

Derived a quasi-classical formula for eigenvalue counting

Analyzed spectral distribution under expanding potentials

Extended understanding of Landau operator perturbations

Abstract

Under a perturbation by a decaying electric potential, the Landau Hamiltonian acquires some discrete eigenvalues between the Landau levels. We study the perturbation by an "expanding" electric potential $V(t^{-1}x)$, $t>0$, and derive a quasi-classical formula for the counting function of the discrete spectrum as $t\to \infty$.