Algebraic shape invariant potentials in two steps

Abstract

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.

Shape Invariant Potentials in Second-Order Supersymmetric Quantum Mechanics

“…Additionally, the shape invariant potentials constructed in the framework of 2-SUSY QM are similar to those shape invariant potentials in two steps discussed in the context of standard SUSY QM. That is, there must be an analogous potential algebra underlying the 2-SUSY shape invariant potentials, thus providing an alternative way of getting the energy eigenvalues by algebraic methods [34,46,69,71].…”

Algebraic shape invariant potentials as the generalized deformed oscillator

“…Nevertheless, all the deformed oscillators can be accommodated within the framework as given in Eq. (22). To realize the algebra of the generalized deformed oscillator, it is natural to introduce the Fock space of eigenstates of the number operator N, which have the property: N |n = n |n and n|m = δ n,m (for n, m = 0, 1, 2, · · ·).…”

“…For the special value k = 1, it is readily known that the algebra defined in Eqs. (26) and (27) reduces to the generalized deformed oscillator algebra (22). In addition, we note that in defining the Z k -graded deformed oscillator algebra above, the creation operator a † is designated to increase the eigenvalue of the number operator N by 1 k unit, not by unity as in the conventional Z k -extended deformed oscillator.…”

“…Nevertheless, all the deformed oscillators can be accommodated within the framework as given in Eq. (22).…”

Two-step shape invariance in the framework of $$\mathcal{N}$$ -fold supersymmetry

“…For a quantum mechanical system with position-dependent mass the same approach was used to handle broken SUSY problem [22]. Recently a class of solvable potentials of translational SI in two steps were obtained [23,24]. It was found that discontinuity at some points was a characteristic of the two-step superpotentials, therefore giving rise to Dirac delta-function singularities in the corresponding potentials if they are considered in the whole line x ∈ (−∞, +∞).…”