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

Abstract

We study a magnetic Schrödinger 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.