1887
banner image
No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
f
New ladder operators for a rational extension of the harmonic oscillator and superintegrability of some two-dimensional systems
Rent:
Rent this article for
Access full text Article
/content/aip/journal/jmp/54/10/10.1063/1.4823771
1.
1. C. V. Sukumar, “Supersymmetric quantum mechanics of one-dimensional systems,” J. Phys. A 18, 2917 (1985).
http://dx.doi.org/10.1088/0305-4470/18/15/020
2.
2. G. Junker and P. Roy, “Conditionally exactly solvable potentials: a supersymmetric construction method,” Ann. Phys. (N.Y.) 270, 155 (1998).
http://dx.doi.org/10.1006/aphy.1998.5856
3.
3. D. J. Fernández C and V. Hussin, “Higher-order SUSY, linearized nonlinear Heisenberg algebras and coherent states,” J. Phys. A 32, 3603 (1999).
http://dx.doi.org/10.1088/0305-4470/32/19/311
4.
4. D. J. Fernández C. and N. Fernández-García, “Higher-order supersymmetric quantum mechanics,” AIP Conf. Proc. 744, 236 (2004).
http://dx.doi.org/10.1063/1.1853203
5.
5. D. Bermúdez and D. J. Fernández C, “Supersymmetric quantum mechanics and Painlevé IV equation,” SIGMA 7, 025 (2011).
http://dx.doi.org/10.3842/SIGMA.2011.025
6.
6. S. Y. Dubov, V. M. Eleonskii, and N. E. Kulagin, “Equidistant spectra of anharmonic oscillators,” Sov. Phys. JETP 75, 446 (1992).
7.
7. S. Y. Dubov, V. M. Eleonskii, and N. E. Kulagin, “Equidistant spectra of anharmonic oscillators,” Chaos 4, 47 (1994).
http://dx.doi.org/10.1063/1.166056
8.
8. G. Junker and P. Roy, “Conditionally exactly solvable problems and non-linear algebras,” Phys. Lett. A 232, 155 (1997).
http://dx.doi.org/10.1016/S0375-9601(97)00422-2
9.
9. D. Gómez-Ullate, N. Kamran, and R. Milson, “The Darboux transformation and algebraic deformations of shape-invariant potentials,” J. Phys. A 37, 1789 (2004).
http://dx.doi.org/10.1088/0305-4470/37/5/022
10.
10. J. F. Cariñena, A. M. Perelomov, M. F. Rañada, and M. Santander, “A quantum exactly solvable nonlinear oscillator related to the isotonic oscillator,” J. Phys. A: Math. Theor. 41, 085301 (2008).
http://dx.doi.org/10.1088/1751-8113/41/8/085301
11.
11. J. M. Fellows and R. A. Smith, “Factorization solution of a family of quantum nonlinear oscillators,” J. Phys. A: Math. Theor. 42, 335303 (2009).
http://dx.doi.org/10.1088/1751-8113/42/33/335303
12.
12. D. Gómez-Ullate, N. Kamran, and R. Milson, “An extended class of orthogonal polynomials defined by a Sturm-Liouville problem,” J. Math. Anal. Appl. 359, 352 (2009).
http://dx.doi.org/10.1016/j.jmaa.2009.05.052
13.
13. C. Quesne, “Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry,” J. Phys. A: Math. Theor. 41, 392001 (2008).
http://dx.doi.org/10.1088/1751-8113/41/39/392001
14.
14. C. Quesne, “Solvable rational potentials and exceptional orthogonal polynomials in supersymmetric quantum mechanics,” SIGMA 5, 084 (2009).
http://dx.doi.org/10.3842/SIGMA.2009.084
15.
15. S. Odake and R. Sasaki, “Infinitely many shape invariant potentials and new orthogonal polynomials,” Phys. Lett. B 679, 414 (2009).
http://dx.doi.org/10.1016/j.physletb.2009.08.004
16.
16. Y. Grandati, “Solvable rational extensions of the isotonic oscillator,” Ann. Phys. (N.Y.) 326, 2074 (2011).
http://dx.doi.org/10.1016/j.aop.2011.03.001
17.
17. C.-L. Ho, “Prepotential approach to solvable rational potentials and exceptional orthogonal polynomials,” Prog. Theor. Phys. 126, 185 (2011).
http://dx.doi.org/10.1143/PTP.126.185
18.
18. D. Gómez-Ullate, N. Kamran, and R. Milson, “Two-step Darboux transformations and exceptional Laguerre polynomials,” J. Math. Anal. Appl. 387, 410 (2012).
http://dx.doi.org/10.1016/j.jmaa.2011.09.014
19.
19. S. Odake and R. Sasaki, “Exactly solvable quantum mechanics and infinite families of multi-indexed orthogonal polynomials,” Phys. Lett. B 702, 164 (2011).
http://dx.doi.org/10.1016/j.physletb.2011.06.075
20.
20. C. Quesne, “Rationally-extended radial oscillators and Laguerre exceptional orthogonal polynomials in kth-order SUSYQM,” Int. J. Mod. Phys. A 26, 5337 (2011).
http://dx.doi.org/10.1142/S0217751X11054942
21.
21. Y. Grandati, “Multistep DBT and regular rational extensions of the isotonic oscillator,” Ann. Phys. (N.Y.) 327, 2411 (2012).
http://dx.doi.org/10.1016/j.aop.2012.07.004
22.
22. Y. Grandati, “Solvable rational extensions of the Morse and Kepler-Coulomb potentials,” J. Math. Phys. 52, 103505 (2011).
http://dx.doi.org/10.1063/1.3651222
23.
23. C.-L. Ho, “Prepotential approach to solvable rational extensions of harmonic oscillator and Morse potentials,” J. Math. Phys. 52, 122107 (2011).
http://dx.doi.org/10.1063/1.3671966
24.
24. J. M. Carballo, D. J. Fernández C, J. Negro, and L. M. Nieto, “Polynomial Heisenberg algebras,” J. Phys. A 37, 10349 (2004).
http://dx.doi.org/10.1088/0305-4470/37/43/022
25.
25. J. Mateo and J. Negro, “Third-order differential ladder operators and supersymmetric quantum mechanics,” J. Phys. A: Math. Theor. 41, 045204 (2008).
http://dx.doi.org/10.1088/1751-8113/41/4/045204
26.
26. P. Winternitz, Ya. A. Smorodinsky, M. Uhlir, and I. Fris, “Symmetry groups in classical and quantum mechanics,” Sov. J. Nucl. Phys. 4, 444 (1967).
27.
27. E. G. Kalnins, J. M. Kress, W. Miller, Jr., and P. Winternitz, “Superintegrable systems in Darboux spaces,” J. Math. Phys. 44, 5811 (2003).
http://dx.doi.org/10.1063/1.1619580
28.
28. E. G. Kalnins, J. M. Kress, and W. Miller, Jr., “Second-order superintegrable systems in conformally flat spaces. V. Two- and three-dimensional quantum systems,” J. Math. Phys. 47, 093501 (2006).
http://dx.doi.org/10.1063/1.2337849
29.
29. C. Daskaloyannis and K. Ypsilantis, “Unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two dimensional manifold,” J. Math. Phys. 47, 042904 (2006).
http://dx.doi.org/10.1063/1.2192967
30.
30. Á. Ballesteros, A. Enciso, F. J. Herranz, and O. Ragnisco, “Superintegrability on N-dimensional curved spaces: Central potentials, centrifugal terms and monopoles,” Ann. Phys. (N.Y.) 324, 1219 (2009).
http://dx.doi.org/10.1016/j.aop.2009.03.001
31.
31. S. Gravel and P. Winternitz, “Superintegrability with third-order invariants in quantum and classical mechanics,” J. Math. Phys. 43, 5902 (2002).
http://dx.doi.org/10.1063/1.1514385
32.
32. S. Gravel, “Hamiltonians separable in Cartesian coordinates and third-order integrals of motion,” J. Math. Phys. 45, 1003 (2004).
http://dx.doi.org/10.1063/1.1633352
33.
33. I. Marquette, “Superintegrability with third order integrals of motion, cubic algebras, and supersymmetric quantum mechanics. I. Rational function potentials,” J. Math. Phys. 50, 012101 (2009).
http://dx.doi.org/10.1063/1.3013804
34.
34. I. Marquette, “Superintegrability with third order integrals of motion, cubic algebras, and supersymmetric quantum mechanics. II. Painlevé transcendent potentials,” J. Math. Phys. 50, 095202 (2009).
http://dx.doi.org/10.1063/1.3096708
35.
35. I. Marquette, “Superintegrability and higher order polynomial algebras,” J. Phys. A: Math. Theor. 43, 135203 (2010).
http://dx.doi.org/10.1088/1751-8113/43/13/135203
36.
36. E. G. Kalnins, J. M. Kress, and W. Miller, Jr., “A recurrence relation approach to higher order quantum superintegrability,” SIGMA 7, 031 (2011).
http://dx.doi.org/10.3842/SIGMA.2011.031
37.
37. B. Demircioǧlu, Ş. Kuru, M. Önder, and A. Verçin, “Two families of superintegrable and isospectral potentials in two dimensions,” J. Math. Phys. 43, 2133 (2002).
http://dx.doi.org/10.1063/1.1463217
38.
38. I. Marquette, “Supersymmetry as a method of obtaining new superintegrable systems with higher order integrals of motion,” J. Math. Phys. 50, 122102 (2009).
http://dx.doi.org/10.1063/1.3272003
39.
39. S. Post, S. Tsujimoto, and L. Vinet, “Families of superintegrable Hamiltonians constructed from exceptional polynomials,” J. Phys. A: Math. Theor. 45, 405202 (2012).
http://dx.doi.org/10.1088/1751-8113/45/40/405202
40.
40. I. Marquette and C. Quesne, “New families of superintegrable systems from Hermite and Laguerre exceptional orthogonal polynomials,” J. Math. Phys. 54, 042102 (2013).
http://dx.doi.org/10.1063/1.4798807
41.
41. F. Cooper, A. Khare, and U. Sukhatme, Supersymmetry in Quantum Mechanics (World Scientific, Singapore, 2000).
42.
42. I. Marquette and C. Quesne, “Two-step rational extensions of the harmonic oscillator: exceptional orthogonal polynomials and ladder operators,” J. Phys. A: Math. Theor. 46, 155201 (2013).
http://dx.doi.org/10.1088/1751-8113/46/15/155201
43.
43. A. P. Veselov and A. B. Shabat, “Dressing chains and the spectral theory of the Schrödinger operator,” Funct. Anal. Appl. 27, 81 (1993).
http://dx.doi.org/10.1007/BF01085979
44.
44. D. Bonatsos and C. Daskaloyannis, “Quantum groups and their applications in nuclear physics,” Prog. Part. Nucl. Phys. 43, 537 (1999).
http://dx.doi.org/10.1016/S0146-6410(99)00100-3
45.
45. S. Odake and R. Sasaki, “Krein-Adler transformations for shape-invariant potentials and pseudo virtual states,” J. Phys. A: Math. Theor. 46, 245201 (2013).
http://dx.doi.org/10.1088/1751-8113/46/24/245201
46.
46. S. Odake and R. Sasaki, “Extensions of solvable potentials with finitely many discrete eigenstates,” J. Phys. A: Math. Theor. 46, 235205 (2013).
http://dx.doi.org/10.1088/1751-8113/46/23/235205
http://aip.metastore.ingenta.com/content/aip/journal/jmp/54/10/10.1063/1.4823771
Loading
/content/aip/journal/jmp/54/10/10.1063/1.4823771
Loading

Data & Media loading...

Loading

Article metrics loading...

/content/aip/journal/jmp/54/10/10.1063/1.4823771
2013-10-08
2014-07-25

Abstract

New ladder operators are constructed for a rational extension of the harmonic oscillator associated with type III Hermite exceptional orthogonal polynomials and characterized by an even integer . The eigenstates of the Hamiltonian separate into + 1 infinite-dimensional unitary irreducible representations of the corresponding polynomial Heisenberg algebra. These ladder operators are used to construct a higher-order integral of motion for two superintegrable two-dimensional systems separable in cartesian coordinates. The polynomial algebras of such systems provide for the first time an algebraic derivation of the whole spectrum through their finite-dimensional unitary irreducible representations.

Loading

Full text loading...

/deliver/fulltext/aip/journal/jmp/54/10/1.4823771.html;jsessionid=1cf2k8fjh0yzo.x-aip-live-03?itemId=/content/aip/journal/jmp/54/10/10.1063/1.4823771&mimeType=html&fmt=ahah&containerItemId=content/aip/journal/jmp
true
true
This is a required field
Please enter a valid email address
This feature is disabled while Scitation upgrades its access control system.
This feature is disabled while Scitation upgrades its access control system.
752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: New ladder operators for a rational extension of the harmonic oscillator and superintegrability of some two-dimensional systems
http://aip.metastore.ingenta.com/content/aip/journal/jmp/54/10/10.1063/1.4823771
10.1063/1.4823771
SEARCH_EXPAND_ITEM