Interaction potential matrix elements for single-particle scattering off a simple harmonic oscillator
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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.
PACS03.65.Nk Nonrelativistic scattering theory
PACS03.80 General theory of scattering
PACS34.10 General theories and models (including statistical theories, transition state, stochastic and trajectory models, etc.)
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