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Representation of the quantum mechanical wavefunction by orthogonal polynomials in the energy and physical parameters. (English) Zbl 1452.81094

Summary: We present a formulation of quantum mechanics based on the theory of orthogonal polynomials. The wavefunction is expanded over a complete set of square integrable basis where the expansion coefficients are orthogonal polynomials in the energy and physical parameters. 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 quantum mechanical problems becomes larger than in the conventional formulation (see, for example, table 3 in the text). We limit our investigation in this work to the Askey classification scheme of hypergeometric orthogonal polynomials and focus on the Wilson polynomial and two of its limiting cases (the Meixner-Pollaczek and continuous dual Hahn polynomials). Nonetheless, the formulation is amenable to other classes of orthogonal polynomials.

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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