Preprint Communication Version 1 This version is not peer-reviewed

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 (doi: 10.20944/preprints201901.0284.v1). Alhaidari, A.D. Special Orthogonal Polynomials in Quantum Mechanics. Preprints 2019, 2019010284 (doi: 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.

Subject Areas

tridiagonal representation; orthogonal polynomials; potential functions; asymptotics; recursion relation; spectrum formula

Readers' Comments and Ratings (1)

Comment 1
Received: 16 February 2019
The commenter has declared there is no conflict of interests.
Comment: Table is missing. See published version in the special issue "Symmetry in Special Functions and Orthogonal Polynomials" of the Journal "Symmetry," Volume 11 (2019)
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