Shape invariance symmetries for quantum states of the superpotentials Atanh ωy+ B/ A and − Acot ωθ+ Bcsc ωθ

Abstract

Using shape invariance symmetries with respect to two different parameters n and m, we derive the general forms of the partner Hamiltonians for the superpotentials Atanh ωy+ B/ A and − Acot ωθ+ Bcsc ωθ, respectively. For the first model in a special case and the second model in the general case, the parameters m and n play the role of the quantization numbers for the spectrum and quantum states as finite and infinite ones, respectively. We also get a type of the shape invariance symmetry which is realized by the operators shifting only m and only n, respectively, for two models. Furthermore, for the models, other shape invariance symmetries based on the operators shifting the indices n and m of quantum states simultaneously and inversely as well as simultaneously and agreeably are derived.