A general noncommutative quantum mechanical system in a central potential $V=V\left(r\right)\mathrm{}$ in two dimensions is considered. The spectrum is bounded from below and, for large values of the anticommutative parameter $\theta ,$ we find an explicit expression for the eigenvalues. In fact, any quantum mechanical system with these characteristics is equivalent to a commutative one in such a way that the interaction $V\left(r\right)\mathrm{}$ is replaced by $V=V(H_{\mathrm{HO}},L_z),$ where $H_{\mathrm{HO}}$ is the Hamiltonian of the two-dimensional harmonic oscillator and $L_z$ is the z component of the angular momentum. For other finite values of $\theta$ the model can be solved by using perturbation theory.

A discussion of the modification of the phase-space commutators in a quantum mechanical relativistic theory with an invariant length scale (DSR) is presented. Two examples are discussed where a classical behavior is approached in one case when the energy approaches the inverse of the invariant length which appears as a cutoff in the energy and in the second case when the mass is much larger than the inverse of the invariant length.

The possibility of detecting noncommutative space relics is analyzed using the Aharonov-Bohm effect. We show that, if space is noncommutative, the holonomy receives nontrivial kinematical corrections that will produce a diffraction pattern even when the magnetic flux is quantized. The scattering problem is also formulated, and the differential cross section is calculated. Our results can be extrapolated to high energy physics and the bound $\theta \sim [10\hspace{0.5em}\mathrm{TeV}{]}^{-2}$ is found. If this bound holds, then noncommutative effects could be explored in scattering experiments measuring differential cross sections for small angles. The bound state Aharonov-Bohm effect is also discussed.

The motion of relativistic particles around three dimensional black holes following the Hamilton-Jacobi formalism is studied. It follows that the Hamilton-Jacobi equation can be separated and reduced to quadratures in analogy with the four dimensional case. It is shown that: a) particles are trapped by the black hole independently of their energy and angular momentum, b) matter alway falls to the centre of the black hole and cannot understake a motion with stables orbits as in four dimensions. For the extreme values of the angular momentum of the black hole, we were able to find exact solutions of the equations of motion and trajectories of a test particle.