Within the framework of supersymmetric quantum mechanics, we obtain a class of solvable potentials of shape invariance in two steps, where the parameters a 1 and a 2 of partner potentials are related to each other by translation a 2 = a 1 + α. It is found that discontinuity at some x-points is a characteristic of the two-step superpotentials, therefore giving rise to Dirac delta-function singularities to the corresponding potentials.

The gauge equivalent formulation of the Faddeev-Skyrme model is used for the study of the quantum theory. The rotational quantum excitations around the soliton solution of Hopf number unity are investigated by the method of collective coordinates. The quantum Hamiltonian of the system is found to coincide with the Hamiltonian of a symmetrical top rotating in SU (2). Thus, the irreducible representations of physical observables can be constructed.

The simplified algebraic structure of the shape invariance condition in two steps is developed by imposing an extra relation on the two superpotentials of the corresponding two-step potentials. This simplified version of potential algebra is found to be similar to that of the shape invariance condition in one step. The solvable potentials of shape invariance in two steps, with a translation change of parameters a1 = a0 + δ, are shown to possess such a simplified version of potential algebra. The condition under which the potential algebra of shape invariance in more than two steps becomes simplified is also discussed.

We revisit parasupersymmetric quantum mechanics of arbitrary order and present a set of nontrivial relations, which characterizes the most general multilinear part of the associated parasupersymmetric algebra. We then show that the formulation of multilinear relations leads immediately to a polynomial of parasupersymmetric Hamiltonian in terms of the corresponding parasupercharges. The deduction of higher derivative supersymmetric quantum mechanics directly via this parasupersymmetric formulation is discussed. The complete degenerate structure of the energy spectrum for parasupersymmetric quantum mechanics of order p is systematically analyzed. Finally, the notion of cyclic symmetry is introduced and the algebra of cyclic charge operators of arbitrary order is developed, based on the parasupersymmetric formalism.

Shape invariance condition in the framework of second-order supersymmetric quantum mechanics is studied. Two classes of solvable shape invariant potentials are consequently constructed, in which the parameters a 0 and a 1 of partner potentials are related to each other by translation a 1 = a 0 + α. In each class, general properties of the obtained shape invariant potentials are systematically investigated. The energy eigenvalues are algebraically determined and the corresponding eigenfunctions are expressed in terms of generalized associated Laguerre polynomials. It is found that these shape invariant potentials are inherently singular, characterized by the 1/x 2 singularity at the origin.