Quantization of AdS x S particle in static gauge

We quantize the particle dynamics in AdS_{N+1} x S^M spacetime in static gauge, which leads to the coordinate representation with wave functions depending only on spatial coordinates. The energy square operator is quadratic in canonical momenta and contains a scalar curvature term. We analyze the self-adjointness of this operator and calculate its spectrum. We then construct unitary representations of the isometry group SO(2,N) x SO(M+1) and calculate the quantum relation between the Casimir numbers.