Supersymmetric quantum mechanics with reflections

We consider a realization of supersymmetric quantum mechanics where supercharges are differential-difference operators with reflections. A supersymmetric system with an extended Scarf I potential is presented and analyzed. Its eigenfunctions are given in terms of little -1 Jacobi polynomials which obey an eigenvalue equation of Dunkl type and arise as a q → -1 limit of the little q-Jacobi polynomials. Intertwining operators connecting the wavefunctions of extended Scarf I potentials with different parameters are presented. (paper)