Solution of an Open Problem about Two Families of Orthogonal Polynomials

Assche, Walter Van

doi:10.3842/sigma.2019.005

Cited by 9 publications

(9 citation statements)

References 14 publications

(13 reference statements)

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“…This polynomial is defined by its three term recursion relation and some of its properties were addressed by W. Van Assche. 18 However, we do not know its important analytical properties such as its weight function, generating function, asymptotics, orthogonality relations, zeros, . .…”

Section: Table Imentioning

confidence: 99%

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Deformed Morse-like potential

Assi,

Alhaidari,

Bahlouli

2021

Preprint

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We introduce an exactly solvable one-dimensional potential that supports both bound and/or resonance states. This potential is a generalization of the well-known 1D Morse potential where we introduced a deformation that preserves the finite spectrum property. On the other hand, in the limit of zero deformation, the potential reduces to the exponentially confining potential well introduced recently by A. D. Alhaidari. The latter potential supports infinite spectrum which means that the zero deformation limit is a critical point where our system will transition from the finite spectrum limit to the infinite spectrum limit. We solve the corresponding Schrodinger equation and obtain the energy spectrum and the eigenstates using the tridiagonal representation approach.

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Section: Table Imentioning

confidence: 99%

“…This is also in agreement with the fact that the polynomial Hµ,ν (z −1 ; , θ) has discrete infinite spectrum. 18 Consequently, the wavefunction associated with the potential (1) is…”

Section: Table Imentioning

confidence: 99%

Deformed Morse-like potential

Assi,

Alhaidari,

Bahlouli

2021

Preprint

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“…are not yet known. This is still an open problem in orthogonal polynomials [21,23,24]. Nonetheless, it is conjectured that ( , ) ( ; , )…”

Section: 12mentioning

confidence: 99%

“…  is a pure discrete polynomial with an infinite spectrum   k z [21,23]. Consequently, we resort to numerical means for calculating this spectrum, which is needed for choosing the proper set of values for the coupling parameter A for a given energy and ratio ( ) B A (i.e., the PPS).…”

Section: 12mentioning

confidence: 99%

Bound states and the potential parameter spectrum

Alhaidari

Bahlouli

2020

Journal of Mathematical Physics

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In this article, we answer the following question: If the wave equation possesses bound states but it is exactly solvable for only a single non-zero energy, can we find all bound state solutions (energy spectrum and associated wavefunctions)? To answer this question, we use the "tridiagonal representation approach" to solve the wave equation at the given energy by expanding the wavefunction in a series of energy-dependent square integrable basis functions in configuration space. The expansion coefficients satisfy a three-term recursion relation, which is solved in terms of orthogonal polynomials. Depending on the selected energy we show that one of the potential parameters must assume a value from within a discrete set called the "potential parameter spectrum" (PPS). This discrete set is obtained from the spectrum of the above polynomials and can be either a finite or an infinite discrete set. Inverting the relation between the energy and the PPS gives the bound state energy spectrum. Therefore, the answer to the above question is affirmative.

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“…This is an open problem in orthogonal polynomials along with other similar ones. For discussions about these open problems, one may consult References [20,22,23]. Nonetheless, these polynomials can be written explicitly for any degree (albeit not in closed form) using the recursion (B1) and starting from the initial seed able to extract all physical information of the corresponding system (e.g., energy spectrum, scattering phase shift, density of states, etc.).…”

mentioning

confidence: 99%

Series solutions of Laguerre- and Jacobi-type differential equations in terms of orthogonal polynomials and physical applications

Alhaidari

2018

Journal of Mathematical Physics

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We introduce two ordinary second-order linear differential equations of the Laguerre-and Jacobi-type. Solutions are written as infinite series of square integrable functions in terms of the Laguerre and Jacobi polynomials, respectively. The expansion coefficients of the series satisfy threeterm recursion relations, which are solved in terms of orthogonal polynomials with continuous and/or discrete spectra. Most of these are well-known polynomials whereas few are not. We present physical applications of these differential equations in quantum mechanics.MSC: 34-xx, 81Qxx, 33C45, 33D45

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Solution of an Open Problem about Two Families of Orthogonal Polynomials

Cited by 9 publications

References 14 publications

Deformed Morse-like potential

Deformed Morse-like potential

Bound states and the potential parameter spectrum

Series solutions of Laguerre- and Jacobi-type differential equations in terms of orthogonal polynomials and physical applications

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