Skip to main content
SpringerLink
Log in

Quantum Hamilton–Jacobi route to exceptional Laguerre polynomials and the corresponding rational potentials

  • Published:
Pramana Aims and scope Submit manuscript

Abstract

A method to construct the multi-indexed exceptional Laguerre polynomials using the isospectral deformation technique and quantum Hamilton–Jacobi (QHJ) formalism is presented. For a given potential, the singularity structure of the quantum momentum function, defined within the QHJ formalism, allows us to find its solutions. We show that this singularity structure can be exploited to construct the generalised superpotentials, which lead to rational potentials with exceptional polynomials as solutions. We explicitly construct such rational extensions of the radial oscillator and their solutions, which involve exceptional Laguerre orthogonal polynomials having two indices. The weight functions of these polynomials are also presented. We also discuss the possibility of constructing more rational potentials with interesting solutions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D Gómez-Ullate, N Kamran and R Milson, J. Math. Anal. Appl. 359, 352 (2009), arXiv:0807.3939

    Article  MathSciNet  Google Scholar 

  2. C Quesne, J. Phys. A 41, 392001 (2008), arXiv:0807.4087

    Article  MathSciNet  Google Scholar 

  3. A M Garcia-Ferrero, D Gómez-Ullate and R Milson, J. Math. Anal. Appl. 472, 584 (2019), arXiv:1603.04358

    Article  MathSciNet  Google Scholar 

  4. C Liaw, L L Littlejohn, R Milson and J Stewart, J. Approx. Theory 202, 5 (2016), arXiv:1407.4145

    Article  MathSciNet  Google Scholar 

  5. C-L Ho, S Odake and R Sasaki, SIGMA 7, 107 (2011), arXiv:0912.5447

    ADS  Google Scholar 

  6. D Gómez-Ullate, F Marcellán and R Milson, J. Math. Anal. Appl. 399, 480 (2013), arXiv:1204.2282

    Article  MathSciNet  Google Scholar 

  7. D Gómez-Ullate, Y Grandati and R Milson, J. Phys. A 51, 345201 (2018)

    Article  MathSciNet  Google Scholar 

  8. N Bonneux and A B J Kuijlaars, arXiv:1708.03106

  9. A B J Kuijlaars and R Milson, J. Approx. Theory 200, 28 (2015), arXiv:1412.6364

    Article  MathSciNet  Google Scholar 

  10. D Gomez-Ullate, N Kamran and R Milson, J. Phys. A 37, 1780 (2004), arXiv:quant-ph/0308062

    ADS  Google Scholar 

  11. S Odake and R Sasaki, Phys. Lett. B 679, 414 (2009), arXiv:0906.0142

    Article  ADS  MathSciNet  Google Scholar 

  12. D Gomez-Ullate, N Kamran and R Milson, J. Phys. A 43, 434016 (2010), arXiv:1002.2666

    Article  ADS  MathSciNet  Google Scholar 

  13. R Sasaki, S Tsujimoto and A Zhedanov, J. Phys. A 43, 315204 (2010), arXiv:1004.4711

    Article  ADS  MathSciNet  Google Scholar 

  14. A Ramos, J. Phys. A 44, 342001 (2011)

    Article  MathSciNet  Google Scholar 

  15. B Bagchi, Y Grandati and C Quesne, J. Math. Phys. 56, 062103 (2015), arXiv:math-ph/1411.7857

    Article  ADS  MathSciNet  Google Scholar 

  16. Y Grandati, Phys. Lett. A 376, 2866 (2012), arXiv:1203.4149

    Article  ADS  MathSciNet  Google Scholar 

  17. D Gömez-Ullate, Y Grandati and R Milson, J. Phys. A 47, 015203 (2014), arXiv:1306.5143

    Article  ADS  MathSciNet  Google Scholar 

  18. D Dutta and P Roy, J. Math. Phys. 51, 042101 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  19. R K Yadav, N Kumari, A Khare and B P Mandal, Ann. Phys. 359, 46 (2015), arXiv:1502.07455 and the references there in.

    Article  ADS  Google Scholar 

  20. K V S Chaitanya, S Sree Ranjani, P K Panigrahi, R Radhakrishnan and V Srinivasan, Pramana – J. Phys. 85, 55 (2015), arXiv:1110.3738

    Google Scholar 

  21. S Sree Ranjani, R Sandhya and A K Kapoor, Int. J. Mod. Phys. A 30, 1550146 (2015), arXiv:1503.01394

    Article  Google Scholar 

  22. J F Cari\({\tilde{{\rm n}}}\)ena, A M Perelomov, M F Ra\({\tilde{{\rm n}}}\)ada and M Santander, J. Phys. A 41, 085301 (2008), arXiv:0711.4899

  23. J M Fellows and R A Smith, J. Phys. A 42, 335303 (2009)

    Article  MathSciNet  Google Scholar 

  24. I Marquette and C Quesne, J. Math. Phys. 54, 042102 (2013), arXiv:1211.2957

    Article  ADS  MathSciNet  Google Scholar 

  25. D Dutta and P Roy, J. Math. Phys. 52, 032104 (2011), arXiv:1103.1246

    Article  ADS  MathSciNet  Google Scholar 

  26. S Odake and R Sasaki, J. Phys. A 44, 353001 (2011), arXiv:1104.0473

    Article  Google Scholar 

  27. B Midya and B Roy, Phys. Lett. A 373, 4117 (2009), arXiv:0910.1209

    Article  ADS  MathSciNet  Google Scholar 

  28. A G Choudhury and P Guha, Surveys. Math. Appl. 10, 1 (2015)

    MathSciNet  Google Scholar 

  29. C Quesne, SIGMA 5, 46 (2009), arXiv:0807.4650

    Google Scholar 

  30. R Sasaki, Universe 2, 2 (2014)

    Google Scholar 

  31. S Odake and R Sasaki, Phys. Lett. B 702, 164 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  32. C Quesne, Mod. Phys. Lett. A 26, 1843 (2011), arXiv:1106.1990

    Article  ADS  Google Scholar 

  33. D Gomez-Ullate, N Kamran and R Milson, J. Math. Anal. Appl. 387, 410 (2012)

    Article  MathSciNet  Google Scholar 

  34. S Odake and R Sasaki, SIGMA 13, 20 (2017)

    Google Scholar 

  35. Y Grandati and C Quesne, J. Math. Phys. 54, 073512 (2013), arXiv: 1211.5308v2

    Article  ADS  MathSciNet  Google Scholar 

  36. S Bochner, Math. Z. 29, 730 (1929)

    Article  MathSciNet  Google Scholar 

  37. E L Ince, Ordinary differential equations (Dover Publications, New York, 1956)

    Google Scholar 

  38. A Erdélyi, W Magnus, F Oberhettinger and F G Tricomi, Higher transcendental functions (McGraw-Hill, New York, 1953)

    MATH  Google Scholar 

  39. F Cooper, A Khare and U P Sukhatme, Supersymmetric quantum mechanics (World Scientific, Singapore, 2001)

    Book  Google Scholar 

  40. L Gendenshtein, JETP Lett. 38, 356 (1983)

    ADS  Google Scholar 

  41. S Sree Ranjani, K G Geojo, A K Kapoor and P K Panigrahi, Mod. Phys. Lett. A 19, 1457 (2004), arXiv:quant-ph/0211168

    Article  ADS  Google Scholar 

  42. S Sree Ranjani, P K Panigrahi, A K Kapoor, A Khare and A Gangopadhyay, J. Phys. A 45, 055210 (2012), arXiv:1009.1944

    Article  ADS  MathSciNet  Google Scholar 

  43. R Sandhya, S Sree Ranjani and A K Kapoor, Ann. Phys. 359, 125 (2015), arXiv:1412.4244

    Article  Google Scholar 

Download references

Acknowledgements

The author acknowledges the financial support of the Science and Engineering Research Board (SERB) under the extramural project scheme, Project Number EMR / 2016 / 005002. The author thanks A K Kapoor, S Rau and T Shreecharan for useful discussions and the researchers who make their manuscripts available on the arXiv and the people who maintain this archive.

Author information

Authors and Affiliations

  1. Faculty of Science and Technology, ICFAI Foundation for Higher Education, Dontanapally, Hyderabad, 501 203, India

    S Sree Ranjani

Authors
  1. S Sree Ranjani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S Sree Ranjani.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sree Ranjani, S. Quantum Hamilton–Jacobi route to exceptional Laguerre polynomials and the corresponding rational potentials. Pramana - J Phys 93, 29 (2019). https://doi.org/10.1007/s12043-019-1787-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12043-019-1787-2

Keywords

PACS Nos

Search

Navigation