The authors give explicit point canonical transformations which map twelve types of shape invariant potentials (which are known to be exactly solvable) into two potential classes. The eigenfunctions in these two classes are given by hypergeometric and confluent hypergeometric functions respectively.

https://iopscience.iop.org/article/10.1088/0305-4470/25/13/013/pdf

Mapping of shape invariant potentials under point canonical transformations

Quantum mechanical potentials satisfying the property of shape invariance are well known to be algebraically solvable. Using a scaling ansatz for the change of parameters, the authors obtain a large class of new shape-invariant potentials which are reflectionless and possess an infinite number of bound states. They can be viewed as q-deformations of the single soliton solution corresponding to the Rosen-Morse potential. Explicit expressions for energy eigenvalues, eigenfunctions and transmission coefficients are given. Included in the potentials as a special case is the self-similar potential recently discussed by Shabat (1992) and Spiridonov (1992).

https://iopscience.iop.org/article/10.1088/0305-4470/26/18/003/pdf

New shape invariant potentials in supersymmetric quantum mechanics

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.

https://iopscience.iop.org/article/10.1088/0305-4470/27/11/031/pdf

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

Shape-invariant potentials in the sense of Gendenshtein (1983 JETP Lett. 38 356) which depend on more than two parameters are not known to date. Cooper et al (1987 Phys. Rev. D 36 2458) posed the problem of finding a class of shape-invariant potentials which depend on n parameters transformed by translation, but it was not solved. We analyse the problem using some properties of the Riccati equation and find the general solution.

Shape Invariant potentials depending on n parameters transformed by translation

It is demonstrated that seven exactly solvable potentials derived from supersymmetric quantum mechanics and presented as new ones, can, in fact, be obtained from one or another of the 12 known shape invariant potentials by a simple choice of the potential parameters. The remarkable result that the energy spectrum of three potentials is independent of certain parameters is also interpreted in a straightforward way.

http://iopscience.iop.org/article/10.1088/0305-4470/25/9/003/pdf

C.J. Maxwell

Papers

Shape invariance and the SWKB series