SO(2,2) representations in polar coordinates and Pöschl-Teller potentials

This work is devoted to show the interest of polar coordinates in the description of some unitary irreducible representations (or uir’s) of the S O ( 2 , 2 ) group where the support space are functions on the three dimensional pseudosphere H R 2 , 2 . We will show that the differential equations associated to such uir’s can be interpreted as quantum systems including centrifugal terms; in our case these equations lead to one-dimensional Pöschl-Teller systems. The solutions to these equations are computed and the uir’s are characterized in terms of polar coordinates. We will also discuss briefly the more standard pseudospherical coordinates on H R 2 , 2 in order to appreciate some of the differences. We will consider as well the (maximally superintegrable) free classical systems defined on the real pseudosphere H R 2 , 2 symmetric under S O ( 2 , 2 ) . The constants of motion are found and they are applied to find some bounded (therefore periodic) and unbounded orbits also in terms of polar coordinates.