Relativistic scattering amplitude in the P¨oschl-Teller double ring-shaped Coulomb potential

The scattering problems, in the presence of an external potential field, have become highly interesting topics in relativistic and non-relativistic quantum mechanics. It is well known that the scattering of a relativistic particle in the field of a potential can be treated exactly by finding the continuum solutions of the Dirac equation. In this research, we obtain the exact solution to the Dirac equation with the Pöschl-Teller double ring-shaped Coulomb (PTDRSC) potential for any spin-orbit quantum number k. The relativistic scattering amplitude for spin 1/2 particles in the field of this potential has been studied. The wave functions are being expressed in terms of the hyper-geometric series of the continuous states on the k/2π scale. In addition, a formula for the phase shifts has also been found. In the nonrelativistic limits, our solution to the Dirac particle converges to that of the Schrödinger one. At the high temperature, the partition function is being calculated in order to study the behavior of some thermodynamic properties.