We consider quantum mechanics on the noncommutative plane in the presence of magnetic field $B$. We show, that the model has two essentially different phases separated by the point $B\theta=c\hbar^2/e$, where $\theta$ is a parameter of noncommutativity. In this point the system reduces to exactly-solvable one-dimensional system. When $\kappa=1-eB\theta/c\hbar^2<0$ there is a finite number of states corresponding to the given value of the angular momentum. In another phase, i.e. when $\kappa>0$ the number of states is infinite. The perturbative spectrum near the critical point $\kappa=0$ is computed.Comment: 4 pages, presentation still improved, mistakes corrected, new sections adde

We split the generic conformal mechanical system into a "radial" and an "angular" part, where the latter is defined as the Hamiltonian system on the orbit of the conformal group, with the Casimir function in the role of the Hamiltonian. We reduce the analysis of the constants of motion of the full system to the study of certain differential equations on this orbit. For integrable mechanical systems, the conformal invariance renders them superintegrable, yielding an additional series of conserved quantities originally found by Wojciechowski in the rational Calogero model. Finally, we show that, starting from any N=4 supersymmetric "angular" Hamiltonian system one may construct a new system with full N=4 superconformal D(1,2;\alpha) symmetry.Comment: 9 pages revte

We define the ''maximally integrable'' isotropic oscillator on CP N and discuss its various properties, in particular, the behavior of the system with respect to a constant magnetic field. We show that the properties of the oscillator on CP N qualitatively differ in the NϾ1 and Nϭ1 cases. In the former case we construct the ''axially symmetric'' system which is locally equivalent to the oscillator. We perform the Kustaanheimo-Stiefel transformation of the oscillator on CP 2 and construct some generalized MIC-Kepler problem. We also define a Nϭ2 superextension of the oscillator on CP N and show that for NϾ1 the inclusion of a constant magnetic field preserves the supersymmetry of the system.

It is shown, that oscillators on the sphere and the pseudosphere are related, by the so-called Bohlin transformation, with the Coulomb systems on the pseudosphere. The even states of an oscillator yield the conventional Coulomb system on the pseudosphere, while the odd states yield the Coulomb system on the pseudosphere in the presence of magnetic flux tube generating spin 1/2. A similar relation is established for the oscillator on the ͑pseudo͒sphere specified by the presence of constant uniform magnetic field B 0 and the Coulomb-like system on pseudosphere specified by the presence of the magnetic field (B/2r 0 )(͉x 3 /x͉Ϫ⑀). The correspondence between the oscillator and the Coulomb systems the higher dimensions is also discussed.

We provide a systematic account of integrability of the spherical mechanics associated with the near horizon extremal Myers-Perry black hole in arbitrary dimension for the special case that all rotation parameters are equal. The integrability is established both in the original coordinates and in action-angle variables. It is demonstrated that the spherical mechanics associated with the black hole in d = 2n + 1 is maximally superintegrable, while its counterpart related to the black hole in d = 2n lacks for only one integral of motion to be maximally superintegrable.

We propose the general scheme of incorporation of the Dirac monopoles into mechanical systems on the three-dimensional conformal flat space. We found that any system (without monopoles) admitting the separation of variables in the elliptic or parabolic coordinates can be extended to the integrable system with the Dirac monopoles located at the foci of the corresponding coordinate systems. Particular cases of this class of system are the two-center MICZ-Kepler system in the Euclidean space, the limiting case when one of the background dyons is located at the infinity as well as the model of particle in parabolic quantum dot in the presence of parallel constant uniform electric and magnetic fields.

We define the "maximally integrable" isotropic oscillator on I CP N and discuss its various properties, in particular, the behaviour of the system with respect to a constant magnetic field. We show that the properties of the oscillator on I CP N qualitatively differ in the N > 1 and N = 1 cases. In the former case we construct the "axially symmetric" system which is locally equivalent to the oscillator. We perform the Kustaanheimo-Stiefel transformation of the oscillator on I CP 2 and construct some generalized MIC-Kepler problem. We also define a N = 2 superextension of the oscillator on I CP N and show that for N > 1 the inclusion of a constant magnetic field preserves the supersymmetry of the system.

An invariant definition of the operator Δ of the Batalin-Vilkovisky formalism is proposed. It is defined as the divergence of a Hamiltonian vector field with an odd Poisson bracket (anti-bracket). Its main properties, which follow from this definition, as well as an example of realization of Kählerian supermanifolds, are considered. The geometrical meaning of the Batalin-Vilkovisky formalism is discussed.