Phases in noncommutative quantum mechanics on (pseudo)sphere

We compare the noncommutative quantum mechanics (NCQM) on sphere and the discrete part of the spectrum of NCQM on pseudosphere (Lobachevsky plane, or AdS2) in the presence of a constant magnetic field B with planar NCQM. We show, that (pseudo)spherical NCQM has a “critical point”, where the system becomes effectively one-dimensional, and two different “phases”, which the phases of the planar system originate from, specified by the sign of the parameter κ=1−Bθ. The “critical point” of (pseudo)spherical NCQM corresponds to the κ→∞ point of conventional planar NCQM, and to the “critical point” κ=0 of the so-called “exotic” planar NCQM, with a symplectic coupling of the (commutative) magnetic field.