Title screen

[References: 16 tit.]

We construct the quantum oscillator interacting with a constant magnetic field on complex projective spaces CPN, as well as on their noncompact counterparts, i.e., the N-dimensional Lobachewski spaces LN. We find the spectrum of this system and the complete basis of wave functions. Surprisingly, the inclusion of a magnetic field does not yield any qualitative change in the energy spectrum. For N>1 the magnetic field does not break the superintegrability of the system, whereas for N=1 it preserves the exact solvability of the system. We extend these results to the cones constructed over CPN and LN, and perform the Kustaanheimo-Stiefel transformation of these systems to the three dimensional Coulomb-like systems.