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Some properties of higher-order Darboux transformations

  • Physics Of Elementary Particles And Field Theory
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Abstract

We introduce the operator of the Darboux transformation of order n as an nth-order differential operator transforming the solution of the one-dimensional Schrödinger equation with a given potential into the solution corresponding to a different potential. For n=1 we obtain the usual Darboux transformation. It is shown that when n≠1 this transformation can always be represented as a product of n first-order Darboux transformations. We discuss the relationship between higher-order Darboux transformations and the supersymmetric approach to quantum mechanics and also the algebra of the dynamical symmetry of the system.

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References

  1. L. M. Berkovich, Factorization and Transformation of Ordinary Differential Equations [in Russian], Saratov State Univ. (1989).

  2. G. A. Natanzon, Vest. LGU No. 10, 22–28 (1971); Teor. Mat. Fiz.,38, No. 2, 219 (1979).

  3. A. M. Goncharenko, V. A. Karpenko, and V. N. Mogilevich, Teor. Mat. Fiz.,80, No. 1, 59 (1991).

    Google Scholar 

  4. N. Kubota, M. Yamagushi, and C. Iwata, Natur. Sci. Rept. Ochanamosu Univ.,39, No. 2, 20 (1969).

    Google Scholar 

  5. J. M. Cinochio, Ann. Phys.,152, No. 1, 203 (1984).

    Google Scholar 

  6. G. Levai, J. Phys. A,24, No. 1, 131 (1991).

    Google Scholar 

  7. A. S. Vshivtsev and V. N. Sorokin, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 1, 95 (1994).

  8. Yu. S. Dubov, V. M. Eleonskii, and N. E. Kulagin, Zh. Éksp. Teor. Fiz.,102, No. 3(9), 814 (1992).

    Google Scholar 

  9. G. Darboux, Compt. Rend. Acad. Sci.,94, 1456 (1882).

    Google Scholar 

  10. G. Levai, J. Phys. A,24, 131 (1991).

    Google Scholar 

  11. Zhu Dongpei, J. Phys. A,20, No. 13, 4331 (1987).

    Google Scholar 

  12. W. M. Zheng, J. Math. Phys.,25, No. 1,, 88 (1984).

    Google Scholar 

  13. J. Delsart, J. Math. Pure Appl.,17, 213 (1938).

    Google Scholar 

  14. B. M. Levitan, Inverse Strum—Liouville Problems [in Russian], Nauka, Moscow (1984).

    Google Scholar 

  15. A. P. Veselov and A. B. Shabat, Funkts. Anal. Prilozh.,27, No. 2, 1 (1993).

    Google Scholar 

  16. M. G. Krein, Dokl. Akad. Nauk SSSR,113, No. 5, 970 (1957).

    Google Scholar 

  17. É. R. Ains, Ordinary Differential Equations [in Russian], GONTI, Moscow (1939).

    Google Scholar 

  18. T. Fulton and R. C. Newton, Nuovo Cim.,3, 677 (1956).

    Google Scholar 

  19. W. R. Theis, Z. Naturforsch.,11a, 889(1956).

    Google Scholar 

  20. L. D. Faddeev, Usp. Mat. Nauk,105,, 4No 4(88), 57 (1959).

    Google Scholar 

  21. V. G. Bagrov and A. S. Vshivtsev, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 7, 19 (1988).

  22. Cao Xuan Chuan, J. Phys. A,24, No. 19, L1155 (1991).

  23. R. W. Haymaker and A. R. R. Rau, Am. Phys.,54, No. 10, 928 (1986).

    Google Scholar 

  24. I. A. Malkin, Preprint Inst. Teor. Phys. Acad. Sci. Ukr. SSR, No. 75, Kiev (1979).

  25. V. G. Bagrov, A. V. Shapovalov, and I. V. Shirokov, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 9, 19 (1991).

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Authors
  1. B. F. Samsonov
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  2. I. N. Ovcharov
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Additional information

Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 3–8, July, 1995.

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Samsonov, B.F., Ovcharov, I.N. Some properties of higher-order Darboux transformations. Russ Phys J 38, 651–656 (1995). https://doi.org/10.1007/BF00560263

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  • DOI: https://doi.org/10.1007/BF00560263

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