Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 10 (2015), 1 -- 21

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

QUANTAM LIÉNARD II EQUATION AND JACOBI'S LAST MULTIPLIER

A. Ghose Choudhury and Partha Guha

Abstract. In this survey the role of Jacobi's last multiplier in mechanical systems with a position dependent mass is unveiled. In particular, we map the Liénard II equation x" + f(x)x'2 + g(x) = 0 to a position dependent mass system. The quantization of the Liénard II equation is then carried out using the point canonical transformation method together with the von Roos ordering technique. Finally we show how their eigenfunctions and eigenspectrum can be obtained in terms of associated Laguerre and exceptional Laguerre functions. By employing the exceptional Jacobi polynomials we construct three exactly solvable potentials giving rise to bound-state solutions of the Schrödinger equation.

2010 Mathematics Subject Classification: 58A15; 58A30.
Keywords: Liénard II equation; position-dependent mass; Jacobi last multiplier; Schrödinger equation; exceptional Laguerre equation; exceptional Jacobi polynomial.

Full text

References

  1. F. Arias de Saavedra, J. Boronat, A. Polls and A. Fabrocini Effective mass of one 4He atom in liquid 3He , Phys. Rev B 50 (1994) 4248.

  2. B. Bagchi, S. Das, S. Ghosh and S. Poria, Nonlinear dynamics of a position-dependent mass driven Duffing-type oscillator, (2013) J. Phys. A Math: Theor. 46 032001. MR3007509. Zbl 1273.81102.

  3. B. Bagchi, Position-dependent mass models and their nonlinear characterization J. Phys. A: Math. Theor. 40 (2007) F1041--F1045 (arXiv:0706.0607 [quant-ph]). MR2441860(2009k:81070). Zbl 1127.81007.

  4. B. Bagchi, P. Gorain, C. Quesne and R. Roychoudhury, New approach to (quasi)-exactly solvable Schrödinger equations with a position-dependent effective mass Mod. Phys. Lett. A 19 (2004) 2765. MR2217356(2006k:81399).

  5. D. J. Ben Daniel and C. B. Duke, Space-charge effects on electron tunneling, Phys. Rev B 152(2) (1966) 683-692.

  6. A. Bhattacharjie and E. C. G. Sudarshan, A class of solvable potentials, Nuovo Cimento 25 (1962) 864. MR0152289(27 #2269). Zbl 0361.46061.

  7. S. Bochner, Über Sturm-Liouvillsche Polynomsysteme, Math. Z. 29 (1929) 730-736. MR1545034. JFM 55.0260.01.

  8. J. F. Cariñena, M. F. Rañada and M. Santander, A quantum exactly solvable non-linear oscillator with quasi-harmonic behaviour, Ann. Phys. 322 (2007) 434-59. (arXiv:math-ph/0604008) MR2289586(2008a:81244).

  9. 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 (2008) 085301. MR2451306(2009f:81060). Zbl 1138.81380.

  10. S. Cruz, Y. Cruz and O. Rosas-Ortiz, Position-dependent mass oscillators and coherent states, J. Phys. A: Math. Theor. 42, 185205 (2009). MR2591199(2011a:81127). Zbl 1162.81388.

  11. R. De, R. Dutt and U. Sukhatme Mapping of shape invariant potentials under point canonical transformations, J. Phys. A: Math. Gen. 25 (1992) L843-L850. MR1172067(93k:81038).

  12. A. de Souza Dutra and C. A. S. Almeida, Exact solvability of potentials with spatially dependent effective masses, Phys Lett. A 275 (2000) 25. MR1810988(2001k:81354). .

  13. M. R. Geller and W. Kohn, Quantum mechanics of electrons in crystals with graded composition, Phys. Rev. Lett. 70 (1993) 3103.

  14. A. Ghose Choudhury and Partha Guha, Quantization of the Liénard II equation and Jacobi's last multiplier, J. Phys. A: Math. Theor. 46 165202 (2013). Zbl 1276.81078.

  15. A. Ghose Choudhury, P. Guha and B. Khanra, On the Jacobi last multiplier, integrating factors and the Lagrangian formulation of differential equations of the Painlevé-Gambier classification, J. Math. Anal. Appl. 360 (2009), no. 2, 651--664. MR2561261(2011c:34224). Zbl 1183.34138.

  16. I. I. Goldman and V. D. Krivchenkov, Problems in quantum mechanics Pergamon, London 1961. MR0129305(23 #B2341). Zbl 0094.23407.

  17. D. Gómez-Ullate, N. Kamran and R. Milson, Quasi-exact solvability in a general polynomial setting Inverse Problems 23 (2007) 1915--1942, (arXiv:nlin/0610065 [nlin.SI]) MR2353321(2008g:81107). Zbl 1129.34056.

  18. D. Gómez-Ullate, N. Kamran and R. Milson, An extension of Bochner's problem: exceptional invariant subspaces J. Approx. Theory 162 (2010) 987-1006, (arXiv:0805.3376 [math-ph]). MR2610341(2011d:33016). Zbl 1214.34079.

  19. 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 (2009) 352-67, (arXiv:0807.3939 [math-ph]). MR2542180(2010j:34064). Zbl 1183.34033.

  20. B. Gonul, B. Gonul, D. Tutcu and O. Ozer, Supersymmetric approach to exactly solvable systems with position-dependent masses, Mod.Phys.Lett. A 17 (2002) 2057. MR1940547(2003i:81083). Zbl 1083.81525.

  21. T. Gora and F. Williams, Theory of electronic states and transport in graded mixed semiconductors, Phys. Rev, 177(3)(1969) 1179-1182.

  22. C. G. J. Jacobi, Sul principio dell'ultimo moltiplicatore, e suo uso come nuovo principio generale di meccanica, Giornale Arcadico di Scienze, Lettere ed Arti 99 (1844), 129-146.

  23. C. G. J. Jacobi, Theoria novi multiplicatoris systemati aequationum differentialium vulgarium applicandi,J. Reine Angew. Math 27 (1844), 199-268, Ibid 29 (1845), 213-279 and 333-376. MR1578395. Zbl 027.0793cj.

  24. N. Kamran and P. J. Olver, Lie algebras of differential operators and Lie-algebraic potentials, J. Math. Anal. Appl. 145 (1990) 342-356. MR1038161(91b:34145). Zbl 0693.34021.

  25. G. Lévai and O. Özer, An exactly solvable Schrödinger equation with finite positive position-dependent effective mass, J. Math. Phys. 51 (2010) 092103. MR2742794(2011j:81087). Zbl 1309.81079.

  26. G. Lévai, A search for shape-invariant solvable potentials J. Phys. A: Math. Gen. 22 (1989) 689. MR0986845(90k:81260). Zbl 0687.35081.

  27. T. L. Li and K. J. Kuhn, Band-offset ratio dependence on the effective-mass Hamiltonian based on a modified profile of the GaAs-AlxGa1-xAs quantum well, Phys. Rev B 47 (1993) 12760.

  28. B. Midya and B. Roy, A generalized quantum nonlinear oscillator, J. Phys. A: Math. Theor. 42 (2009) 285301. MR2519734(2011d:81111). Zbl 1167.81372.

  29. B. Midya and B. Roy, Exceptional orthogonal polynomials and exactly solvable potentials in position-dependent-mass Schrödinger Hamiltonians, Phys. Lett. A 373 (2009) 4117-22. MR2552404(2010j:81301). Zbl 1234.81072.

  30. A. Mostafazadeh, \mathcal PT-symmetric cubic anharmonic oscillator as a physical model, J. Phys. A: Math. Gen. 38 (2005) 6557-70 (arXiv:quant-ph/0411137); MR2167044(2006e:81060a). Zbl 1072.81020. J. Phys. A: Math. Gen. 38 8185 (erratum). MR2169337(2006e:81060b).

  31. R. L. Norton Cam design and Manufacturing Handbook, Second Edition, Industrial Press, Inc. 2009.

  32. M. C. Nucci and P. G. L. Leach, Jacobi's last multiplier and symmetries for the Kepler problem plus a lineal story, J. Phys. A: Math. Gen. 37, (2004), 7743-7753. MR2090011(2005g:70012). Zbl 1065.70007.

  33. S. Odake and R. Sasaki, Infinitely many shape invariant potentials and new orthogonal polynomials, Phys. Letts. B 679, (2009) 414. MR2569488(2011b:81102).

  34. S. Odake and R. Sasaki, Infinitely many shape-invariant potentials and cubic identities of the Laguerre and Jacobi polynomials J. Math.Phys. 51, (2010) 053513. MR2666991(2011k:81105). Zbl 1310.81059.

  35. A. Puente, L. Serra and M. Casas, Dipole excitation of Na clusters with a non-local energy density functional, Z Phys. D 31 (1994) 283.

  36. C. Quesne, J. Phys. A: Math. Theor. 41, 392001 (2008). MR0470692. Zbl 0361.46061.

  37. C. Quesne, Solvable Rational Potentials and Exceptional Orthogonal Polynomials in Supersymmetric Quantum Mechanics, SIGMA 5 (2009), 084, 24pp. MR2559677 (2011c:81119). Zbl 1188.81071.

  38. C. Quesne, Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry, J. Phys. A: Math. Theor. 41 (2008) 392001. MR2439200(2010f:42054). Zbl 1192.81174.

  39. V. Chithiika Ruby, M. Senthilvelan and M. Lakshmanan, Exact quantization of a PT-symmetric (reversible) Liénard-type nonlinear oscillator, J. Phys. A: Math. Theor. 45 (2012) 382002. MR2971248. Zbl 1252.81055.

  40. M. Sabatini, On the period function of x''+f(x)x\sp \prime 2+g(x)=0. J. Differential Equations 196 (2004), no. 1, 151-168. MR2025190(2004i:34069). Zbl 1048.34068.

  41. A. G. M. Schmidt, Time evolution for harmonic oscillators with position-dependent mass Phys. Scr. 75 (2007) 480-483. MR2299232(2007m:81074).

  42. L. Serra and E. Lipparini, Spin response of unpolarized quantum dots, Europhys. Lett. 40 (1997) 667.

  43. O. Von Roos, Position-dependent effective masses in semiconductor theory, Phys. Rev. B 27 (1983) 7547-7552.

  44. Y. Weissman and J. Jortner, The isotonic oscillator, Phys. Letts. A70 (1979) 177-179. MR0587593(81i:81021).

  45. E. Witten, Dynamical breaking of supersymmetry Nucl. Phys. B 188 (1981) 513-54. Zbl 1258.81046.

  46. E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge University Press, Cambridge, 1988. MR0992404(90a:01112).

  47. Q. G. Zhu and H. Kroemer, Interface connection rules for effective-mass wave-function at an abrupt heterojunction between two different semiconductors, Phys. Rev B 27 (1983) 3519.




A. Ghose Choudhury Partha Guha
E-mail: aghosechoudhury@gmail.com E-mail: partha@bose.res.in
Department of Physics, S. N. Bose National Centre for Basic Sciences,
Surendranath College, JD Block, Sector III, Salt Lake,
Mahatma Gandhi Road, Kolkata - 700098, India.
Calcutta-700009, India.



http://www.utgjiu.ro/math/sma