A new class of Pöschl-Teller potentials with inverse square singularity and their spectra using the asymptotic iteration method

The aim of this work is to introduce a new family of potentials with inverse square singularity which we called the Pöschl-Teller family of potentials. We enforced the matrix representation of the wave operator to be symmetric and (2k+1) band-diagonal with respect to a square integrable basis set. This, in principle, is only satisfied for specific potential functions within the used basis set. The basis functions we used here are written in terms of Jacobi polynomials, which is the same basis used in the Tridiagonal Representation Approach (TRA). This yield a more general form of Pöschl-Teller potential that can have many terms which could be beneficial for modeling different physical systems where this potential applies. As an illustration, we have studied a specific new five-parameter potential that belongs to this new family and calculated the bound states for both s-wave and l-wave cases using the Asymptotic Iteration Method (AIM). Along the way, we have introduced new approximation schemes to deal with the l-wave centrifugal potential within the AIM at different approximation orders.