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Generalized Heisenberg Algebras, SUSYQM and Degeneracies: Infinite Well and Morse Potential
Symmetry, integrability and geometry: methods and applications, Tome 7 (2011) Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider classical and quantum one and two-dimensional systems with ladder operators that satisfy generalized Heisenberg algebras. In the classical case, this construction is related to the existence of closed trajectories. In particular, we apply these results to the infinite well and Morse potentials. We discuss how the degeneracies of the permutation symmetry of quantum two-dimensional systems can be explained using products of ladder operators. These products satisfy interesting commutation relations. The two-dimensional Morse quantum system is also related to a generalized two-dimensional Morse supersymmetric model. Arithmetical or accidental degeneracies of such system are shown to be associated to additional supersymmetry.
Keywords: generalized Heisenberg algebras; degeneracies; Morse potential; infinite well potential; supersymmetric quantum mechanics. 
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     title = {Generalized {Heisenberg} {Algebras,} {SUSYQM} and {Degeneracies:} {Infinite} {Well} and {Morse} {Potential}},
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}
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Véronique Hussin; Ian Marquette. Generalized Heisenberg Algebras, SUSYQM and Degeneracies: Infinite Well and Morse Potential. Symmetry, integrability and geometry: methods and applications, Tome 7 (2011). http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a23/

[1] Marquette I., “Superintegrability and higher order polynomial algebras”, J. Phys. A: Math. Theor., 43 (2010), 135203, 15 pp., arXiv: 0908.4399 | DOI | MR | Zbl

[2] Marquette I., “Construction of classical superintegrable systems with higher order integrals of motion from ladder operators”, J. Math. Phys., 51 (2010), 072903, 9 pp., arXiv: 1002.3118 | DOI | MR

[3] Fernández D. J., Hussin V., “Higher-order SUSY, linearized nonlinear Heisenberg algebras and coherent states”, J. Phys. A: Math. Gen., 32 (1999), 3603–3619 | DOI | MR | Zbl

[4] Carbello J. M., Fernández D. J., Negro J., Nieto L. M., “Polynomial Heisenberg algebras”, J. Phys. A: Math. Gen., 37 (2004), 10349–10362 | DOI | MR

[5] Marquette I., “Supersymmetry as a method of obtaining new superintegrable systems with higher order integrals of motion”, J. Math. Phys., 50 (2009), 122102, 10 pp., arXiv: 0908.1246 | DOI | MR | Zbl

[6] Junker G., Supersymmetric methods in quantum and statistical physics, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1996 | MR

[7] Marquette I., Winternitz P., “Superintegrable systems with third-order integrals of motion”, J. Phys. A: Math. Theor., 41 (2008), 304031, 10 pp., arXiv: 0711.4783 | DOI | MR | Zbl

[8] De Lange O. L., Raab R. E., Operator methods in quantum mechanics, The Clarendon Press, New York; Oxford University Press, 1991 | MR

[9] Delbecq C., Quesne C., “Nonlinear deformations of su(2) and su(1,1) generalizing Witten's algebra”, J. Phys. A: Math. Gen., 26 (1993), L127–L134 | DOI | MR | Zbl

[10] Eleonsky V. M., Korolev V. G., “On the nonlinear generalization of the Fock method”, J. Phys. A: Math. Gen., 28 (1995), 4973–4985 | DOI | MR | Zbl

[11] Eleonsky V. M., Korolev V. G., “On the nonlinear Fock description of quantum systems with quadratic spectra”, J. Phys. A: Math. Gen., 29 (1996), L241–L248 | DOI | MR | Zbl

[12] Ghosh A., Mitra P., Kundu A., “Multidimensional isotropic and anisotropic $q$-oscillator models”, J. Phys. A: Math. Gen., 29 (1996), 115–124, arXiv: hep-th/9511084 | DOI | MR | Zbl

[13] Quesne C., “Comment: J.-L. Chen, Y. Liu and M.-L. Ge Application of nonlinear deformation algebra to a physical system with Pöschl–Teller potential" [J. Phys. A: Math. Gen., 31, (1998), 6473–6481]”, J. Phys. A: Math. Gen., 32 (1999), 6705–6710, arXiv: math-ph/9911004 | DOI | MR | Zbl

[14] Chen J.-L., Zhang H.-B., Wang X.-H., Jing H., Zhao X.-G., “Raising and lowering operators for a two-dimensional hydrogen atom by an ansatz method”, Int. J. Theor. Phys., 39 (2000), 2043–2050 | DOI | Zbl

[15] Curado E. M. F., Rego-Monteiro M. A., “Multi-parametric deformed Heisenberg algebras: a route to complexity”, J. Phys. A: Math. Gen., 34 (2001), 3253–3264, arXiv: hep-th/0011126 | DOI | MR | Zbl

[16] Dong S.-H., Ma Z.-Q., “The hidden symmetry for a quantum system with an infinitely deep square-well potential”, Amer. J. Phys., 70 (2002), 520–521 | DOI | MR | Zbl

[17] Daoud M., Kibler M. R., “Fractional supersymmetry and hierarchy of shape invariant potentials”, J. Math. Phys., 47 (2006), 122108, 11 pp., arXiv: quant-ph/0609017 | DOI | MR | Zbl

[18] Dong S.-H., Factorization method in quantum mechanics, Fundamental Theories of Physics, 150, Springer, Dordrecht, 2007 | MR

[19] Curado E. M. F., Hassouni Y., Rego-Monteiro M. A., Rodrigues L. M. C. S., “Generalized Heisenberg algebra and algebraic method: the example of an infinite square-well potential”, Phys. Lett. A, 372 (2008), 3350–3355 | DOI | MR | Zbl

[20] Wang H.-B., Liu Y.-B., “Realizing the underlying quantum dynamical algebra $SU(2)$ in Morse potential,”, Chinese Phys. Lett., 27 (2010), 020301, 4 pp. | DOI

[21] Ioffe M. V., Nishnianidze D. N., “Exact solvability of two-dimensional real singular Morse potential”, Phys. Rev. A, 76 (2007), 052114, 5 pp., arXiv: 0709.2960 | DOI | MR

[22] Kuru S., Negro J., “Factorizations of one dimensional classical systems”, Ann. Physics, 323 (2008), 413–431, arXiv: 0709.4649 | DOI | MR | Zbl

[23] Cruz y Cruz S., Kuru S., Negro J., “Classical motion and coherent states for Pöschl–Teller potentials”, Phys. Lett. A, 372 (2008), 1391–1405 | DOI | MR | Zbl

[24] Dello Sbarba L., Hussin V., “Degenerate discrete energy spectra and associated coherent states”, J. Math. Phys., 48 (2007), 012110, 15 pp. | DOI | MR | Zbl

[25] Angelova M., Hussin V., “Generalized and Gaussian coherent states for the Morse potential”, J. Phys. A: Math. Theor., 41 (2008), 30416, 13 pp. | DOI | MR

[26] Angelova M., Hussin V., Squeezed coherent states and the Morse quantum system, arXiv: 1010.3277

[27] Dong S. H., Lemus R., Frank A., “Ladder operators for the Morse potential”, Int. J. Quant. Chem., 86 (2002), 433–439 | DOI

[28] Bagchi B., Mallik S., Quesne C., Infinite square well and periodic trajectories in classical mechanics, arXiv: physics/0207096

[29] Slater N. B., “Classical motion under a Morse potential”, Nature, 180 (1957), 1352–1353 | DOI | Zbl

[30] Itzykson C., Luck J. M., “Arithmetical degeneracies in simple quantum systems”, J. Phys. A: Math. Gen., 19 (1986), 211–239 | DOI | MR | Zbl

[31] Marquette I., “An infinite family of superintegrable systems from higher order ladder operators and supersymmetry”, Group 28: Physical and Mathematical Aspects of Symmetry: Proceedings of the 28th International Colloquium on Group-Theoretical Methods in Physics, J. Phys. Conf. Ser. (to appear) , arXiv: 1008.3073

[32] Post S., Winternitz P., A nonseparable quantum superintegrable system in 2D real Euclidean space, arXiv: 1101.5405 | MR