An algebraic approach to shape invariance with reflections

We consider a realization of supersymmetric shape-invariant quantum systems where generalized ladder operators are constructed through the association of the supersymmetric partner operators (A) over cap (x, a(1)), (A) over cap (dagger)(x, a(1)), the parameter potential translation operator (T) over cap (a1) and the reflection operator (R) over cap. The basic algebraic relations as well as the system eigenstates and energy eigenvalues are obtained. As an illustration, we apply the generalized formalism for two important shape-invariant potential systems: the Scarf hyperbolic and the Morse potentials.