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arXiv:math-ph/0101013 · Journal of Physics A Mathematical and General

Integrable multi-boson systems and orthogonal polynomials

Anatol Odzijewicz, Maciej Horowski, Agnieszka Tereszkiewicz

Published May 9, 2001• 34 citations
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Abstract

The strict relation between a certain class of multi-boson Hamiltonian systems and the corresponding class of orthogonal polynomials is established. The correspondence is effectively used to integrate the systems. As an explicit example we integrate the class of multi-boson systems corresponding to q -Hahn class polynomials.
DOI
10.1088/0305-4470/34/20/308

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We present a formulation of quantum mechanics based on orthogonal polynomials. The wavefunction is expanded over a complete set of square integrable basis in configuration space where the expansion coefficients are orthogonal polynomials in the energy. Information about the corresponding physical systems (both structural and dynamical) are derived from the properties of these polynomials. We demonstrate that an advantage of this formulation is that the class of exactly solvable non-relativistic quantum mechanical problems becomes larger than in the conventional formulation (see, for example, Table 1 in the text).
An alternative and combinatorial proof is given for a connection between a system of Hahn polynomials and identities for symmetric elements in the Heisenberg algebra, which was first observed by Bender, Mead, and Pinsky [Phys. Rev. Lett. 56 (1986), J. Math. Phys. 28, 509 (1987)] and proved by Koornwinder [J. Phys. Phys. 30(4), 1989]. In the same vein two results announced by Bender and Dunne [J. Math. Phys. 29 (8), 1988] connecting a special one-parameter class of Hermitian operator orderings and the continuous Hahn polynomials are also proved.