ALGEBRAIC PROPERTIES OF TRANSLATIONAL SHAPE INVARIANT POTENTIALS IN ARBITRARY STEPS

Within the framework of supersymmetric quantum mechanics, we investigate the general algebraic properties of translational shape invariant potentials in arbitrary k steps, in which the k remainders R-s (a(m)) are analytic functions of the parameter a(m) that is related to others by translation: a(m) = a(m-1) + delta. The present study is based on the fact that the simplified potential algebra of shape invariance condition in k steps is equivalent to that of generalized deformed oscillators with a built-in Z(k)-grading structure. We shall show that, despite the complexity in the study, the general algebraic properties still can be systematically determined.