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 SIGMA, 2012, Volume 8, 080, 19 pages (Mi sigma757)

Novel enlarged shape invariance property and exactly solvable rational extensions of the Rosen–Morse II and Eckart potentials

Christiane Quesne

Physique Nucléaire Théorique et Physique Mathématique, Université Libre de Bruxelles, Campus de la Plaine CP229, Boulevard du Triomphe, B-1050 Brussels, Belgium

Abstract: The existence of a novel enlarged shape invariance property valid for some rational extensions of shape-invariant conventional potentials, first pointed out in the case of the Morse potential, is confirmed by deriving all rational extensions of the Rosen–Morse II and Eckart potentials that can be obtained in first-order supersymmetric quantum mechanics. Such extensions are shown to belong to three different types, the first two strictly isospectral to some starting conventional potential with different parameters and the third with an extra bound state below the spectrum of the latter. In the isospectral cases, the partner of the rational extensions resulting from the deletion of their ground state can be obtained by translating both the potential parameter $A$ (as in the conventional case) and the degree $m$ of the polynomial arising in the denominator. It therefore belongs to the same family of extensions, which turns out to be closed.

Keywords: quantum mechanics; supersymmetry; shape invariance.

DOI: https://doi.org/10.3842/SIGMA.2012.080

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ArXiv: 1208.6165
MSC: 81Q05; 81Q60
Received: August 30, 2012; in final form October 15, 2012; Published online October 26, 2012
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Citation: Christiane Quesne, “Novel enlarged shape invariance property and exactly solvable rational extensions of the Rosen–Morse II and Eckart potentials”, SIGMA, 8 (2012), 080, 19 pp.

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Marquette I. Quesne C., “Two-Step Rational Extensions of the Harmonic Oscillator: Exceptional Orthogonal Polynomials and Ladder Operators”, J. Phys. A-Math. Theor., 46:15 (2013), 155201
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13. Yves Grandati, Christiane Quesne, “Confluent Chains of DBT: Enlarged Shape Invariance and New Orthogonal Polynomials”, SIGMA, 11 (2015), 061, 26 pp.
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19. Kumari N., Yadav R.K., Khare A., Bagchi B., Mandal B.P., Ann. Phys., 373 (2016), 163–177
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25. R. K. Yadav, N. Kumari, A. Khare, B. P. Mandal, “Rationally extended shape invariant potentials in arbitrary $\mathrm{D}$-dimensions associated with exceptional $X_m$ polynomials”, Acta Polytech., 57:6 (2017), 477–487
26. A. Ramos, B. Bagchi, A. Khare, N. Kumari, B. P. Mandal, R. K. Yadav, “A short note on “Group theoretic approach to rationally extended shape invariant potentials” [Ann. Phys. 359 (2015) 46-54]”, Ann. Phys., 382 (2017), 143–149
27. S. Odake, “New determinant expressions of multi-indexed orthogonal polynomials in discrete quantum mechanics”, Prog. Theor. Exp. Phys., 2017, no. 5, 053A01
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