Second Order Derivative Supersymmetry and Scattering Problem

Extensions of standard one-dimensional supersymmetric quantum mechanics are discussed. Supercharges involving higher order derivatives are introduced leading to an algebra which incorporates a higher order polynomial in the Hamiltonian. We study scattering amplitudes for that problem. We also study the role of a dilatation of the spatial coordinate leading to a q-deformed supersymmetric algebra. An explicit model for the scattering amplitude is constructed in terms of a hypergeometric function which corresponds to a reflectionless potential with infinitely many bound states. Department of Theoretical Physics, University of Sankt-Petersburg,198904 Sankt-Petersburg, Russia. E-mail: [email protected]; [email protected] Dipartimento di Fisica and INFN, Via Irnerio 46, 40126 Bologna, Italy. E-mail: [email protected] Laboratoire de Physique Nucléaire, Université Paris 7, 2 place Jussieu, F 75251 Paris Cedex 05, France and Division de Physique Théorique, Institut de Physique Nucléaire, F 91406 Orsay,France. E-mail: [email protected]