Geodesic
Energy spectrum design and potential function engineering
Teoretičeskaâ i matematičeskaâ fizika, Tome 216 (2023) no. 1, pp. 133-147

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Starting with an orthogonal polynomial sequence $\{p_n(s)\}_{n=0}^{\infty}$ that has a discrete spectrum, we design an energy spectrum formula $E_k=f(s_k)$, where $\{s_k\}$ is the finite or infinite discrete spectrum of the polynomial. Using a recent approach to quantum mechanics based not on potential functions but on orthogonal energy polynomials, we give a local numerical realization of the potential function associated with the chosen energy spectrum. We select the three-parameter continuous dual Hahn polynomial as an example. Exact analytic expressions are given for the corresponding bound-state energy spectrum, the phase shift of scattering states, and the wavefunctions. However, the potential function is obtained only numerically for a given set of physical parameters.
Keywords: energy spectrum design, potential function engineering, recursion relation, continuous dual Hahn polynomial, scattering phase shift, wavefunction.
Mots-clés : orthogonal polynomials 
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     author = {A. D. Alhaidari and T. J. Taiwo},
     title = {Energy spectrum design and potential function engineering},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {133--147},
     publisher = {mathdoc},
     volume = {216},
     number = {1},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2023_216_1_a8/}
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A. D. Alhaidari; T. J. Taiwo. Energy spectrum design and potential function engineering. Teoretičeskaâ i matematičeskaâ fizika, Tome 216 (2023) no. 1, pp. 133-147. http://geodesic.mathdoc.fr/item/TMF_2023_216_1_a8/