On an Energy-Dependent Quantum System with Solutions in Terms of a Class of Hypergeometric Para-Orthogonal Polynomials on the Unit Circle

Jorge A. Borrego-Morell, Cleonice F. Bracciali and Alagacone Sri Ranga
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Jorge A. Borrego-Morell: Departamento de Matemática, Campus Santa Cruz da Serra, UFRJ—Universidade Federal do Rio de Janeiro, Duque de Caxias 25255-030, Brazil
Cleonice F. Bracciali: Departamento de Matemática, Campus São José do Rio Preto, UNESP—Universidade Estadual Paulista, São Jose do Rio Preto 15054-000, Brazil
Alagacone Sri Ranga: Departamento de Matemática, Campus São José do Rio Preto, UNESP—Universidade Estadual Paulista, São Jose do Rio Preto 15054-000, Brazil

Mathematics, 2020, vol. 8, issue 7, 1-18

Abstract: We study an energy-dependent potential related to the Rosen–Morse potential. We give in closed-form the expression of a system of eigenfunctions of the Schrödinger operator in terms of a class of functions associated to a family of hypergeometric para-orthogonal polynomials on the unit circle. We also present modified relations of orthogonality and an asymptotic formula. Consequently, bound state solutions can be obtained for some values of the parameters that define the model. As a particular case, we obtain the symmetric trigonometric Rosen–Morse potential for which there exists an orthogonal basis of eigenstates in a Hilbert space. By comparing the existent solutions for the symmetric trigonometric Rosen–Morse potential, an identity involving Gegenbauer polynomials is obtained.

Keywords: orthogonal polynomials; Schrödinger equation; ordinary differential equations; energy-dependent potential; hypergeometric functions; asymptotic expansions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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