2ND-ORDER DERIVATIVE SUPERSYMMETRY, Q-DEFORMATIONS AND THE SCATTERINGPROBLEM

In a search for pairs of quantum systems linked by dynamical symmetrie s, we give a systematic analysis of novel extensions of standard one-d imensional supersymmetric quantum mechanics. The most general supercha rges involving higher order derivatives are introduced, leading to an algebra which incorporates a higher order polynomial of the Hamiltonia n. We investigate the condition for irreducibility of such a higher or der generator to a product of standard first derivative Darboux transf ormations. As a new example of application of this approach we study t he quantum-mechanical radial problem including the scattering amplitud es. We also investigate the links between this higher derivative SUSY and a q-deformed supersymmetric quantum mechanics and introduce the no tion of self-similarity in momentum space. An explicit model for the s cattering amplitude is constructed in terms of a hypergeometric functi on which corresponds to a reflectionless potential with infinitely man y bound states.