Eigenstates for Pöschl-Teller Potentials

It has been long known that the Schrödinger equation for a class of potentials of the form , usually referred to as Pöschl–Teller potentials, is exactly solvable. The eigenvalue problem

(in units with has physically significant solutions for , for both bound and continuum states. For , we find the solution , , which follows simply from the derivative relation . More generally, the Schrödinger equation has the bound state solutions

, , , ,

where the are associated Legendre polynomials.

The Schrödinger equation has, in addition, continuum positive-energy eigenstates with . The trivial case gives a free particle . The first two nontrivial solutions are and . These represent waves traveling left to right. A remarkable property of Pöschl-Teller potentials is that they are "reflectionless", meaning that waves are 100% transmitted through the barrier with no reflected waves.