  We perform a 1-parameter family of self-adjoint extensions characterized by
the parameter $\omega_0$. This allows us to get generic boundary conditions for
the quantum oscillator on $N$ dimensional complex projective
space($\mathbb{C}P^N$) and on its non-compact version i.e., Lobachewski
space($\mathcal L_N$) in presence of constant magnetic field. As a result, we
get a family of energy spectrums for the oscillator. In our formulation the
already known result of this oscillator is also belong to the family. We have
also obtained energy spectrum which preserve all the symmetry (full hidden
symmetry and rotational symmetry) of the oscillator. The method of self-adjoint
extensions have been discussed for conic oscillator in presence of constant
magnetic field also.
