Solvable potentials associated with su(1,1) algebras: a systematic study

Abstract

We consider a specific differential realization of the su(1,1) algebra and use it to explore such algebraic structures associated with shape-invariant potentials. Our approach combines elements of various methods of solving the Schrodinger equation, such as supersymmetric quantum mechanics (or the factorization method), algebraic techniques and special-function theory. In fact, it amounts to reformulating transformations mapping the Schrodinger equation into the differential equation of orthogonal polynomials in group-theoretical terms. Our systematic study recovers a number of earlier results in a natural unified way and also leads to new findings. The procedure presented here implicitly contains a similar treatment of the compact su(2) algebra as well. Possible generalizations of this approach (involving different realizations of the su(1,1) algebra, other algebraic structures and larger classes of potentials) are also outlined.