Sarah Post (Centre de Recherches Mathématiques, Université de Montréal, Canada)

Quantum Hamiltonian Systems with Reflections

Abstract:
   In this talk, we discuss some recent results on quantum Hamiltonian systems with reflection. First, a novel realization of supersymmetric quantum mechanics is obtained by using as supercharge, differential-difference operators with reflections. As an example, we will present and analyze a supersymmetric system with an extended Scarf I potential. 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. Further, we will present a new infinite class of superintegrable quantum systems in the plane built from the same differential-difference operators. We give explicitly the wave function and construct higher-order integrals using the recurrence relations for the wave functions written in terms of Laguerre and little -1 Jacobi polynomials.
   This is joint work with L. Vinet and A. Zhedanov.