The Fate of the Landau Levels under Perturbations of Constant Sign

TL;DR

This paper investigates how Landau levels in a 2D Schrödinger operator with a constant magnetic field are affected by bounded electric potentials, showing they disappear under fixed sign perturbations and can reappear with arbitrary multiplicity otherwise.

Contribution

It provides a detailed analysis of the spectral stability of Landau levels under sign-restricted perturbations, revealing conditions for their persistence or disappearance.

Findings

Landau levels cease to be eigenvalues with fixed sign perturbations.

Any Landau level can become an eigenvalue with arbitrary multiplicity under non-fixed sign perturbations.

The spectral behavior depends critically on the sign of the electric potential perturbation.

Abstract

We show that the Landau levels cease to be eigenvalues if we perturb the 2D Schr\"odinger operator with constant magnetic field, by bounded electric potentials of fixed sign. We also show that, if the perturbation is not of fixed sign, then any Landau level may be an eigenvalue of arbitrary large, finite or infinite, multiplicity of the perturbed problem.