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

A. A. Andrianov ; F. Cannata ; J. P. Dedonder ; M. V. Ioffe

Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 14, Tome 245 (1997), pp. 22-48

Résumé

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.