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Harmonic oscillator SUSY partners and evolution loops
Symmetry, integrability and geometry: methods and applications, Tome 8 (2012) Cet article a éte moissonné depuis la source Math-Net.Ru

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Supersymmetric quantum mechanics is a powerful tool for generating exactly solvable potentials departing from a given initial one. If applied to the harmonic oscillator, a family of Hamiltonians ruled by polynomial Heisenberg algebras is obtained. In this paper it will be shown that the SUSY partner Hamiltonians of the harmonic oscillator can produce evolution loops. The corresponding geometric phases will be as well studied.
Keywords: supersymmetric quantum mechanics, quantum harmonic oscillator, polynomial Heisenberg algebra, geometric phase. 
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     author = {David J. Fern\'andez},
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}
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David J. Fernández. Harmonic oscillator SUSY partners and evolution loops. Symmetry, integrability and geometry: methods and applications, Tome 8 (2012). http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a40/

[1] Adler V.E., “Nonlinear chains and {P}ainlevé equations”, Phys. D, 73 (1994), 335–351 | DOI | MR | Zbl

[2] Aharonov Y., Anandan J., “Phase change during a cyclic quantum evolution”, Phys. Rev. Lett., 58 (1987), 1593–1596 | DOI | MR

[3] Aizawa N., Sato H.T., “Isospectral Hamiltonians and $W_{1+\infty}$ algebra”, Progr. Theoret. Phys., 98 (1997), 707–718 ; arXiv: quant-ph/9601026 | DOI | MR

[4] Anandan J., Christian J., Wanelik K., “Resource letter GPP-1: geometric phases in physics”, Amer. J. Phys., 65 (1997), 180–185 ; arXiv: quant-ph/9702011 | DOI

[5] Andrianov A.A., Cannata F., “Nonlinear supersymmetry for spectral design in quantum mechanics”, J. Phys. A: Math. Gen., 37 (2004), 10297–10321 ; arXiv: hep-th/0407077 | DOI | MR | Zbl

[6] Andrianov A.A., Cannata F., Ioffe M.V., Nishnianidze D.N., “Systems with higher-order shape invariance: spectral and algebraic properties”, Phys. Lett. A, 266 (2000), 341–349 ; arXiv: quant-ph/9902057 | DOI | MR | Zbl

[7] Andrianov A.A., Ioffe M.V., Cannata F., Dedonder J.P., “Second order derivative supersymmetry, $q$-deformations and the scattering problem”, Internat. J. Modern Phys. A, 10 (1995), 2683–2702 ; arXiv: hep-th/9404061 | DOI | MR | Zbl

[8] Andrianov A.A., Ioffe M.V., Spiridonov V.P., “Higher-derivative supersymmetry and the Witten index”, Phys. Lett. A, 174 (1993), 273–279 ; arXiv: hep-th/9303005 | DOI | MR

[9] Ar{\i}k M., Atakishiyev N.M., Wolf K.B., “Quantum algebraic structures compatible with the harmonic oscillator Newton equation”, J. Phys. A: Math. Gen., 32 (1999), L371–L376 | DOI | MR | Zbl

[10] Bagchi B.K., Supersymmetry in quantum and classical mechanics, Chapman Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 116, Chapman Hall/CRC, Boca Raton, FL, 2001 | MR | Zbl

[11] Bagrov V.G., Samsonov B.F., “Darboux transformation of the Schrödinger equation”, Phys. Part. Nuclei, 28 (1997), 374–397 | DOI | MR

[12] Baye D., Sparenberg J.M., “Inverse scattering with supersymmetric quantum mechanics”, J. Phys. A: Math. Gen., 37 (2004), 10223–10249 | DOI | MR | Zbl

[13] Bermúdez D., Fernández D.J., “Non-Hermitian Hamiltonians and the Painlevé IV equation with real parameters”, Phys. Lett. A, 375 (2011), 2974–2978 ; arXiv: 1104.3599 | DOI | MR

[14] Bermúdez D., Fernández D.J., “Supersymmetric quantum mechanics and Painlevé IV equation”, SIGMA, 7 (2011), 025, 14 pp. ; arXiv: 1012.0290 | DOI | MR | Zbl

[15] Berry M.V., “Quantal phase factors accompanying adiabatic changes”, Proc. Roy. Soc. London Ser. A, 392 (1984), 45–57 | DOI | MR | Zbl

[16] Bohm A., Boya L.J., Kendrick B., “Derivation of the geometrical phase”, Phys. Rev. A, 43 (1991), 1206–1210 | DOI | MR

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

[18] Cariñena J.F., Perelomov A.M., Rañada M.F., “Isochronous classical systems and quantum systems with equally spaced spectra”, J. Phys. Conf. Ser., 87 (2007), 012007, 14 pp. | DOI

[19] Chruściński D., Jamiołkowski A., Geometric phases in classical and quantum mechanics, Progress in Mathematical Physics, 36, Birkhäuser Boston Inc., Boston, MA, 2004 | DOI | MR | Zbl

[20] Contreras-Astorga A., Fernández D.J., “Supersymmetric partners of the trigonometric Pöschl–Teller potentials”, J. Phys. A: Math. Theor., 41 (2008), 475303, 18 pp. ; arXiv: 0809.2760 | DOI | MR | Zbl

[21] Cooper F., Khare A., Sukhatme U., Supersymmetry in quantum mechanics, World Scientific Publishing Co. Inc., River Edge, NJ, 2001 | DOI | MR

[22] Correa F., Jakubský V., Nieto L.M., Plyushchay M.S., “Self-isospectrality, special supersymmetry, and their effect on the band structure”, Phys. Rev. Lett., 101 (2008), 030403, 4 pp. ; arXiv: 0801.1671 | DOI | MR | Zbl

[23] Daoud M., Kibler M.R., “Bosonic and $k$-fermionic coherent states for a class of polynomial Weyl–Heisenberg algebras”, J. Phys. A: Math. Theor., 45 (2012), 244036, 22 pp. ; arXiv: 1110.4799 | DOI | Zbl

[24] 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

[25] Daoud M., Kibler M.R., “Phase operators, temporally stable phase states, mutually unbiased bases and exactly solvable quantum systems”, J. Phys. A: Math. Theor., 43 (2010), 115303, 18 pp. ; arXiv: 1002.0955 | DOI | MR | Zbl

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

[27] Dubov S.Y., Eleonsky V.M., Kulagin N.E., “Equidistant spectra of anharmonic oscillators”, Sov. Phys. JETP, 75 (1992), 446–451 | MR

[28] Emmanouilidou A., Zhao X.G., Ao P., Niu Q., “Steering an eigenstate to a destination”, Phys. Rev. Lett., 85 (2000), 1626–1629 | DOI

[29] Fernández D.J., “Bogdan Mielnik: contributions to quantum control”, Geometric Methods in Physics (XXX Workshop, Bialowieza, Poland, June 26 – July 2, 2011), eds. P. Kielanowski, S.T. Ali, A. Odzijewicz, M. Schlichenmaier, T. Voronov (to appear)

[30] Fernández D.J., “Geometric phases and Mielnik's evolution loops”, Int. J. Theor. Phys., 33 (1994), 2037–2047 ; arXiv: hep-th/9410213 | DOI | Zbl

[31] Fernández D.J., “Supersymmetric quantum mechanics”, AIP Conf. Proc., 1287 (2010), 3–36 ; arXiv: 0910.0192 | DOI

[32] Fernández D.J., “SUSUSY quantum mechanics”, Internat. J. Modern Phys. A, 12 (1997), 171–176 ; arXiv: quant-ph/9609009 | DOI | MR | Zbl

[33] Fernández D.J., “Transformations of a wave packet in a Penning trap”, Nuovo Cimento B, 107 (1992), 885–893 | DOI

[34] Fernández D.J., Fernández-García N., “Higher-order supersymmetric quantum mechanics”, AIP Conf. Proc., 744 (2005), 236–273 ; arXiv: quant-ph/0502098 | DOI | MR

[35] 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

[36] Fernández D.J., Hussin V., Mielnik B., “A simple generation of exactly solvable anharmonic oscillators”, Phys. Lett. A, 244 (1998), 309–316 | DOI | MR | Zbl

[37] Fernández D.J., Mielnik B., “Controlling quantum motion”, J. Math. Phys., 35 (1994), 2083–2104 | DOI | MR

[38] Fernández D.J., Negro J., Nieto L.M., “Elementary systems with partial finite ladder spectra”, Phys. Lett. A, 324 (2004), 139–144 | DOI | Zbl

[39] Fernández D.J., Nieto L.M., del Olmo M.A., Santander M., “Aharonov–Anandan geometric phase for spin-$\frac12$ periodic Hamiltonians”, J. Phys. A: Math. Gen., 25 (1992), 5151–5163 | DOI | MR | Zbl

[40] Fernández D.J., Rosas-Ortiz O., “Inverse techniques and evolution of spin-$1/2$ systems”, Phys. Lett. A, 236 (1997), 275–279 | DOI | MR | Zbl

[41] Gangopadhyaya A., Mallow J.V., Rasinariu C., Supersymmetric quantum mechanics, World Scientific, Singapore, 2011 | Zbl

[42] Horváthy P.A., Plyushchay M.S., Valenzuela M., “Bosons, fermions and anyons in the plane, and supersymmetry”, Ann. Physics, 325 (2010), 1931–1975 ; arXiv: 1001.0274 | DOI | MR | Zbl

[43] Ioffe M.V., Nishnianidze D.N., “SUSY intertwining relations of third order in derivatives”, Phys. Lett. A, 327 (2004), 425–432 ; arXiv: hep-th/0404078 | DOI | MR | Zbl

[44] Lin Q.G., “Time evolution, cyclic solutions and geometric phases for general spin in an arbitrarily varying magnetic field”, J. Phys. A: Math. Gen., 36 (2003), 6799–6806 ; arXiv: quant-ph/0307099 | DOI | MR | Zbl

[45] Lin Q.G., “Time evolution, cyclic solutions and geometric phases for the generalized time-dependent harmonic oscillator”, J. Phys. A: Math. Gen., 37 (2004), 1345–1371 ; arXiv: quant-ph/0402159 | DOI | MR | Zbl

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

[47] Mateo J., Negro J., “Third-order differential ladder operators and supersymmetric quantum mechanics”, J. Phys. A: Math. Theor., 41 (2008), 045204, 28 pp. | DOI | MR | Zbl

[48] Mielnik B., “Evolution loops”, J. Math. Phys., 27 (1986), 2290–2306 | DOI | MR | Zbl

[49] Mielnik B., “Factorization method and new potentials with the oscillator spectrum”, J. Math. Phys., 25 (1984), 3387–3389 | DOI | MR | Zbl

[50] Mielnik B., “Global mobility of Schrödinger's particle”, Rep. Math. Phys., 12 (1977), 331–339 | DOI | MR

[51] Mielnik B., “Space echo”, Lett. Math. Phys., 12 (1986), 49–56 | DOI | MR

[52] Mielnik B., Nieto L.M., Rosas-Ortiz O., “The finite difference algorithm for higher order supersymmetry”, Phys. Lett. A, 269 (2000), 70–78 ; arXiv: quant-ph/0004024 | DOI | MR | Zbl

[53] Mielnik B., Ramírez A., “Ion traps: some semiclassical observations”, Phys. Scr., 82 (2010), 055002, 13 pp. | DOI | Zbl

[54] Mielnik B., Ramírez A., “Magnetic operations: a little fuzzy mechanics?”, Phys. Scr., 84 (2011), 045008, 17 pp. ; arXiv: 1006.1944 | DOI

[55] Mielnik B., Rosas-Ortiz O., “Factorization: little or great algorithm?”, J. Phys. A: Math. Gen., 37 (2004), 10007–10035 | DOI | MR | Zbl

[56] Mostafazadeh A., Dinamical invariants, adiabatic approximation and the geometric phase, Nova Science Publishers Inc., New York, 1992

[57] Plyushchay M.S., “Deformed Heisenberg algebra with reflection”, Nuclear Phys. B, 491 (1997), 619–634 ; arXiv: hep-th/9701091 | DOI | MR | Zbl

[58] Shapere A., Wilczek F. (Editors), Geometric phases in physics, Advanced Series in Mathematical Physics, 5, World Scientific Publishing Co. Inc., Teaneck, NJ, 1989 | MR | Zbl

[59] Sukumar C.V., “Supersymmetric quantum mechanics and its applications”, AIP Conf. Proc., 744 (2005), 166–235 | DOI | MR

[60] Veselov A.P., Shabat A.B., “Dressing chains and the spectral theory of the Schrödinger operator”, Func. Anal. Appl., 27 (1993), 81–96 | DOI | MR | Zbl