The quantum oscillator on complex projective space (Lobachewski space) in a constant magnetic field and the issue of generic boundary conditions

We perform a one-parameter family of self-adjoint extensions characterized by the parameter ω0. This allows us to get generic boundary conditions for the quantum oscillator on N-dimensional complex projective space (CPN) and on its non-compact version, i.e., Lobachewski space (LN) in the presence of a constant magnetic field. As a result, we get a family of energy spectra for the oscillator. In our formulation the already known result of this oscillator also belongs to the family. We have also obtained an energy spectrum which preserves all the symmetries (full-hidden symmetry and rotational symmetry) of the oscillator. The method of self-adjoint extensions has also been discussed for a conic oscillator in the presence of the constant magnetic field