Interaction potential matrix elements for single-particle scattering off a simple harmonic oscillator

Summary

The matrix elements of the interaction potential for spinless two-particle scattering, where one of the particles is confined about the origin by a harmonic-oscillator potential, are explicitly evaluated. this problem is important in the quantum-mechanical theory of ultracold particle sticking to clean surfaces, and is directly relevant to studies of molecular adsoroption/desorption. A new result concerning the Hermite polynomials is obtained, as well as the discovery of a novel family of polynomials that are intimately connected with the diagonal matrix elements of the interaction potential. Recursion relations and a generating function are obtained for these polynomials. The zeros of these polynomials are also discussed.