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Special Orthogonal Polynomials in Quantum Mechanics
Version 1
: Received: 27 January 2019 / Approved: 29 January 2019 / Online: 29 January 2019 (04:37:49 CET)
How to cite: Alhaidari, A.D. Special Orthogonal Polynomials in Quantum Mechanics. Preprints 2019, 2019010284. https://doi.org/10.20944/preprints201901.0284.v1 Alhaidari, A.D. Special Orthogonal Polynomials in Quantum Mechanics. Preprints 2019, 2019010284. https://doi.org/10.20944/preprints201901.0284.v1
Abstract
Using an algebraic method for solving the wave equation in quantum mechanics, we encountered a new class of orthogonal polynomials on the real line. One of these is a four-parameter polynomial with a discrete spectrum. Another that appeared while solving a Heun-type equation has a mix of continuous and discrete spectra. Based on these results and on our recent study of the solution space of an ordinary differential equation of the second kind with four singular points, we introduce a modification of the hypergeometric polynomials in the Askey scheme. Up to now, all of these polynomials are defined only by their three-term recursion relations and initial values. However, their other properties like the weight function, generating function, orthogonality, Rodrigues-type formula, etc. are yet to be derived analytically. This is an open problem in orthogonal polynomials.
Keywords
tridiagonal representation; orthogonal polynomials; potential functions; asymptotics; recursion relation; spectrum formula
Subject
Computer Science and Mathematics, Applied Mathematics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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