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We perform a 1-parameter family of self-adjoint extensions characterized bythe parameter $\omega_0$. This allows us to get generic boundary conditions forthe quantum oscillator on $N$ dimensional complex projectivespace($\mathbb{C}P^N$) and on its non-compact version i.e., Lobachewskispace($\mathcal L_N$) in presence of constant magnetic field. As a result, weget a family of energy spectrums for the oscillator. In our formulation thealready known result of this oscillator is also belong to the family. We havealso obtained energy spectrum which preserve all the symmetry (full hiddensymmetry and rotational symmetry) of the oscillator. The method of self-adjointextensions have been discussed for conic oscillator in presence of constantmagnetic field also.Comment: Accepted in Journal of Physics

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