  We propose a Wigner quasiprobability distribution function for Hamiltonian
systems in spaces of constant curvature --in this paper on hyperboloids--,
which returns the correct marginals and has the covariance of the Shapiro
functions under SO(D,1) transformations. To the free systems obeying the
Laplace-Beltrami equation on the hyperboloid, we add a conic-oscillator
potential in the hyperbolic coordinate. As an example, we analyze the
1-dimensional case on a hyperbola branch, where this conic-oscillator is the
Poschl-Teller potential. We present the analytical solutions and plot the
computed results. The standard theory of quantum oscillators is regained in the
contraction limit to the space of zero curvature.

Quantum Physics