Deformed supersymmetry, $q$-oscillator algebra and related scattering problems in quantum mechanics

We describe extensions of the supersymmetric quantum mechanics (SSQM) (in one dimension) which are characterized by deformed algebras. The supercharges involving higher-order derivatives are introduced leading to a deformed algebra which incorpotates a higher-order polynomial of the hamiltonian. When supplementing them with dilatations one finds the class of $q$-deformed SUSY systems. For a special choice of $q$-selfsimilar potentials the energy spectrum is (partially) generated by the $q$-oscillator algebra. In contrast to the standard harmonic oscillators these systems exhibit a continuous spectrum. We investigate the scattering problem in the $q$-deformed SSQM and introduce the notion of self-similarity in momentum space for scattering data. An explicit model for the scattering amplitude of a $q$-oscillator is constructed in terms of a hypergeometric function which corresponds to a reflectionless potential with infinitely many bound states. The general scheme of realization of the $q$-oscillator algebra on the space of wave functions for a one-dimensional Schrödinger hamiltonian is developed. It shows the existence of non-Fock irreducible representations associated to the continuous part of the spectrum and directly related to the deformation.