Phases in noncommutative quantum mechanics on (pseudo)sphere
  We compare the non-commutative quantum mechanics (NCQM) on sphere and the discrete part of the spectrum of NCQM on pseudosphere (Lobachevsky plane, or $AdS_2$) 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 $\kappa=1-B\theta$. The ``critical point'' of (pseudo)spherical NCQM corresponds to the $\kappa\to\infty $ point of conventional planar NCQM, and to the ``critical point'' $\kappa=0$ of the so-called ``exotic'' planar NCQM, with a symplectic coupling of the (commutative) magnetic field. 
Stefano Bellucci and Armen Nersessian