  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.
