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

This article is cited in 30 scientific papers (total in 30 papers)

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.


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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

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.

Citation in format AMSBIB
\by Christiane Quesne
\paper Novel enlarged shape invariance property and exactly solvable rational extensions of the Rosen--Morse II and Eckart potentials
\jour SIGMA
\yr 2012
\vol 8
\papernumber 080
\totalpages 19

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    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  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    2. Odake S. Sasaki R., “Non-Polynomial Extensions of Solvable Potentials a La Abraham-Moses”, J. Math. Phys., 54:10 (2013), 102106  crossref  mathscinet  zmath  adsnasa  isi  scopus
    3. Odake S., “Recurrence Relations of the Multi-Indexed Orthogonal Polynomials”, J. Math. Phys., 54:8 (2013), 083506  crossref  mathscinet  zmath  adsnasa  isi  scopus
    4. Odake S. Sasaki R., “Extensions of Solvable Potentials with Finitely Many Discrete Eigenstates”, J. Phys. A-Math. Theor., 46:23 (2013), 235205  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    5. Quesne C., “Extending Romanovski Polynomials in Quantum Mechanics”, J. Math. Phys., 54:12 (2013), 122103  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    6. Odake S., Sasaki R., “Krein-Adler Transformations for Shape-Invariant Potentials and Pseudo Virtual States”, J. Phys. A-Math. Theor., 46:24 (2013), 245201  crossref  mathscinet  zmath  adsnasa  isi  scopus
    7. Ho C.-L. Lee J.-C. Sasaki R., “Scattering Amplitudes for Multi-Indexed Extensions of Solvable Potentials”, Ann. Phys., 343 (2014), 115–131  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    8. Schulze-Halberg A. Roy B., “Darboux Partners of Pseudoscalar Dirac Potentials Associated with Exceptional Orthogonal Polynomials”, Ann. Phys., 349 (2014), 159–170  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    9. Gomez-Ullate D. Grandati Y. Milson R., “Extended Krein-Adler Theorem for the Translationally Shape Invariant Potentials”, J. Math. Phys., 55:4 (2014), 043510  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    10. Schulze-Halberg A. Roy B., “Time Dependent Potentials Associated With Exceptional Orthogonal Polynomials”, J. Math. Phys., 55:12 (2014), 123506  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    11. Marquette I. Quesne Ch., “Combined State-Adding and State-Deleting Approaches To Type III Multi-Step Rationally Extended Potentials: Applications To Ladder Operators and Superintegrability”, J. Math. Phys., 55:11 (2014), 112103  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    12. Yadav R.K. Khare A. Mandal B.P., “the Scattering Amplitude For Rationally Extended Shape Invariant Eckart Potentials”, Phys. Lett. A, 379:3 (2015), 67–70  crossref  zmath  adsnasa  isi  scopus
    13. Yves Grandati, Christiane Quesne, “Confluent Chains of DBT: Enlarged Shape Invariance and New Orthogonal Polynomials”, SIGMA, 11 (2015), 061, 26 pp.  mathnet  crossref  mathscinet
    14. Bougie J., Gangopadhyaya A., Mallow J.V., Rasinariu C., “Generation of a Novel Exactly Solvable Potential”, Phys. Lett. A, 379:37 (2015), 2180–2183  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    15. Odake S., “Recurrence Relations of the Multi-Indexed Orthogonal Polynomials. II”, J. Math. Phys., 56:5 (2015), 053506  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    16. Quesne C., “Ladder Operators For Solvable Potentials Connected With Exceptional Orthogonal Polynomials”, Xxxth international colloquium on group theoretical methods in physics (icgtmp) (group30), Journal of Physics Conference Series, 597, IOP Publishing Ltd, 2015, 012064  crossref  isi  scopus
    17. Odake S., “Recurrence Relations of the Multi-Indexed Orthogonal Polynomials. III”, J. Math. Phys., 57:2 (2016), 023514  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    18. Odake S., “Recurrence relations of the multi-indexed orthogonal polynomials. IV. Closure relations and creation/annihilation operators”, J. Math. Phys., 57:11 (2016), 113503  crossref  mathscinet  zmath  isi  elib  scopus
    19. Kumari N., Yadav R.K., Khare A., Bagchi B., Mandal B.P., Ann. Phys., 373 (2016), 163–177  crossref  mathscinet  zmath  isi  elib  scopus
    20. Quesne C., “Quantum oscillator and Kepler–Coulomb problems in curved spaces: Deformed shape invariance, point canonical transformations, and rational extensions”, J. Math. Phys., 57:10 (2016), 102101  crossref  mathscinet  zmath  isi  elib  scopus
    21. Schulze-Halberg A. Roy B., “Generalized quantum nonlinear oscillators: Exact solutions and rational extensions”, J. Math. Phys., 57:10 (2016), 102103  crossref  mathscinet  zmath  isi  elib  scopus
    22. Mahdi K., Kasri Y., Grandati Y., Berard A., “SWKB and proper quantization conditions for translationally shape-invariant potentials”, Eur. Phys. J. Plus, 131:8 (2016), 259  crossref  isi  scopus
    23. A. Schulze-Halberg, P. Roy, “Quantum models with energy-dependent potentials solvable in terms of exceptional orthogonal polynomials”, Ann. Phys., 378 (2017), 234–252  crossref  mathscinet  zmath  isi  scopus
    24. S. Odake, “Casoratian identities for the discrete orthogonal polynomials in discrete quantum mechanics with real shifts”, Prog. Theor. Exp. Phys., 2017, no. 12, 123A02  crossref  mathscinet  isi
    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  crossref  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    27. S. Odake, “New determinant expressions of multi-indexed orthogonal polynomials in discrete quantum mechanics”, Prog. Theor. Exp. Phys., 2017, no. 5, 053A01  crossref  mathscinet  isi  scopus
    28. J. Bougie, A. Gangopadhyaya, C. Rasinariu, “The supersymmetric wkb formalism is not exact for all additive shape invariant potentials”, J. Phys. A-Math. Theor., 51:37 (2018), 375202  crossref  isi  scopus
    29. N. Kumari, R. K. Yadav, A. Khare, B. P. Mandal, “A class of exactly solvable rationally extended non-central potentials in two and three dimensions”, J. Math. Phys., 59:6 (2018), 062103  crossref  mathscinet  zmath  isi  scopus
    30. Morrison C.L., Shizgal B., “Pseudospectral Solution of the Schrodinger Equation For the Rosen-Morse and Eckart Potentials”, J. Math. Chem., 57:4 (2019), 1035–1052  crossref  mathscinet  zmath  isi  scopus
  • Symmetry, Integrability and Geometry: Methods and Applications
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