On a quantum Hamiltonian in a unitary magnetic field with axisymmetric potential

TL;DR

This paper analyzes a magnetic Schr{"o}dinger Hamiltonian with an axisymmetric potential, revealing the spectral structure, asymptotic behavior of band functions, and properties of quantum states in a magnetic field.

Contribution

It provides a detailed spectral analysis of a magnetic Hamiltonian with axisymmetric potential, including asymptotics of band functions and implications for quantum state localization.

Findings

Band functions have finite limits at Landau levels.

Infinite band functions intersect each energy level.

Quantum states have a bulk component even away from thresholds.

Abstract

We study a magnetic Schr{\"o}dinger Hamiltonian, with axisymmetric potential in any dimension. The associated magnetic field is unitary and non constant. The problem reduces to a 1D family of singular Sturm-Liouville operators on the half-line indexed by a quantum number. We study the associated band functions. They have finite limits that are the Landau levels. These limits play the role of thresholds in the spectrum of the Hamiltonian. We provide an asymptotic expansion of the band functions at infinity. Each Landau level concerns an infinity of band functions and each energy level is intersected by an infinity of band functions. We show that among the band functions that intersect a fixed energy level, the derivative can be arbitrary small. We apply this result to prove that even if they are localized in energy away from the thresholds, quantum states possess a bulk component. A similar result is also true in classical mechanics.