On the lowest energy of a 3D magnetic hamiltonian with axisymmetric potential

We study the bottom of the spectrum of a magnetic hamiltonian with axisymmetrical 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 to associated band functions and we compare it 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 lowest energy is higher in dimension 3.