Splitting of Landau levels in the presence of external potentials

We study the quantum mechanical problem of a charged particle moving in a constant magnetic field and an azimuthilly symmetric external potential. We show that the lowest Landau level splits in a definite way if the external potential is a monotone function of the radial distance in the plane orthogonal to the magnetic field. For two-dimensional systems we give sufficient conditions such that all Landau levels are completely ordered. For the lowest Landau level this result holds for nonhomogeneous magnetic fields as long as they are cylindrically symmetric.