Skip to main content
SpringerLink
Log in

Nonlocal Darboux transformation of the two-dimensional stationary Schrödinger equation and its relation to the Moutard transformation

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

We consider a nonlocal Darboux transformation of the two-dimensional stationary Schrödinger equation and establish its relation to the Moutard transformation. We show that the Moutard transformation is a special case of the nonlocal Darboux transformation and obtain new examples of solvable two-dimensional stationary Schrödinger operators with smooth potentials as an application of the nonlocal Darboux transformation.

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. L. D. Landau and E. M. Lifshits, Course of Theoretical Physics [in Russian], Vol. 3, Quantum Mechanics: Non-Relativistic Theory, Nauka, Moscow (1974); English transl. prev. ed., Pergamon, London (1958).

    Google Scholar 

  2. P. M. Morse and K. U. Ingard, Theoretical Acoustics, McGraw-Hill, New York (1968).

    Google Scholar 

  3. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (Classics Appl. Math., Vol. 33), SIAM, Philadelphia, Pa. (2001).

    Book  MATH  Google Scholar 

  4. A. P. Veselov and S. P. Novikov, Soviet Math. Dokl., 30, 705–708 (1984).

    Google Scholar 

  5. P. G. Grinevich, A. E. Mironov, and S. P. Novikov, Russian Math. Surv., 65, 580–582 (2010).

    Article  ADS  MathSciNet  Google Scholar 

  6. P. G. Grinevich, Russian Math. Surv., 55, 1015–1083 (2000).

    Article  ADS  MathSciNet  Google Scholar 

  7. H. Risken, The Fokker–Planck Equation (Springer Ser. Synergetics, Vol. 18), Springer, Berlin (1989).

    Book  MATH  Google Scholar 

  8. A. G. Kudryavtsev, Phys. Lett. A, 377, 2477–2480 (2013); arXiv:1211.5685v1 [math-ph] (2012).

    Article  ADS  MathSciNet  Google Scholar 

  9. T. Moutard, J. Ècole Polyt., 45, 1–11 (1878).

    Google Scholar 

  10. I. A. Taimanov and S. P. Tsarev, Theor. Math. Phys., 157, 1525–1541 (2008); arXiv:0801.3225v2 [math-ph] (2008).

    Article  MathSciNet  Google Scholar 

  11. A. G. Kudryavtsev, “The nonlocal Darboux transformation of the 2D stationary Schrödinger equation and its relation to the Moutard transformation,” arXiv:1303.5858v3 [math-ph] (2013).

    Google Scholar 

  12. V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin (1991).

    Book  MATH  Google Scholar 

  13. A. A. Andrianov, M. V. Ioffe, and D. N. Nishnianidze, Phys. Lett. A, 201, 103–110 (1995); arXiv:hep-th/9404120v1 (1994).

    Article  ADS  MathSciNet  Google Scholar 

  14. M. V. Ioffe, J. Phys. A: Math. Gen., 37, 10363–10374 (2004); arXiv:hep-th/0405241v1 (2004).

    Article  ADS  MathSciNet  Google Scholar 

  15. A. A. Andrianov and M. V. Ioffe, J. Phys. A: Math. Theor., 45, 503001 (2012).

    Article  MathSciNet  Google Scholar 

  16. M. S. Bardavelidze, M. V. Ioffe, and D. N. Nishnianidze, Phys. Lett. A, 377, 195–199 (2013); arXiv:1211.1461v1 [hep-th] (2012).

    Article  ADS  MathSciNet  Google Scholar 

  17. I. Sh. Akhatov, R. K. Gazizov, and N. Kh. Ibragimov, J. Soviet Math., 55, 1401–1450 (1991).

    Article  Google Scholar 

  18. G. W. Bluman, A. F. Cheviakov, and S. C. Anco, Applications of Symmetry Methods to Partial Differential Equations (Appl. Math. Sci., Vol. 168), Springer, New York (2010).

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mechanics, RAS, Moscow, Russia

    A. G. Kudryavtsev

Authors
  1. A. G. Kudryavtsev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. G. Kudryavtsev.

Additional information

Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 187, No. 1, pp. 12–20, April, 2016.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kudryavtsev, A.G. Nonlocal Darboux transformation of the two-dimensional stationary Schrödinger equation and its relation to the Moutard transformation. Theor Math Phys 187, 455–462 (2016). https://doi.org/10.1134/S0040577916040024

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0040577916040024

Keywords

Search

Navigation