This study is about a magnetic Hamiltonian with axisymmetric potential in R3. The associated magnetic field is planar, unitary and non-constant. The problem reduces to a 1D family of singular Sturm-Liouville operators on the half-line. We study the associated band functions, in particular their behavior at infinity and we describe the quantum state localized in energy near the Landau levels that play the role of threshold in the spectrum. We compare our Hamiltonian to the "de Gennes" operators arising in the study of a 2D Hamiltonian with monodimensional, odd and discontinuous magnetic field. We show in particular that the ground state energy is higher in dimension 3.
