Quantization and spectral geometry of a rigid body in a magnetic monopole field

A rigid body with a fixed point in an effective homogeneous magnetic field, which can be thought of as a magnetic monopole, is studied in the framework of geometric quantization. It turns out that there are two cases. The non-degenerate case, with SO(3) as configuration space, presents no obstruction to quantization, but has two non-equivalent quantizations. The degenerate case, which reduces to the study of a single particle in a constant magnetic field on S2, is quantizable if it satisfies the Dirac quantization condition. In the non-degenerate case, using techniques from harmonic analysis, the Schr?dinger equation for the two quantizations is explicitly solved for a spherical top, and for a symmetric top, in the case that the direction defined by the total magnetic term is along the symmetry axis. The same techniques plus Riemannian submersion theory are applied in the degenerate case to solve the Schr?dinger equation