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Theoretical and Mathematical Physics

, Volume 104, Issue 2, pp 1051–1060 | Cite as

Darboux transformation, factorization, and supersymmetry in one-dimensional quantum mechanics

  • V. G. Bagrov
  • B. F. Samsonov
Article

Abstract

We introduce an N-order Darboux transformation operator as a particular case of general transformation operators. It is shown that this operator can always be represented as a product of N first-order Darboux transformation operators. The relationship between this transformation and the factorization method is investigated. Supercharge operators are introduced. They are differential operators of order N. It is shown that these operators and super-Hamiltonian form a superalgebra of order N. For N=2, we have a quadratic superalgebra analogous to the Sklyanin quadratic algebras. The relationship between the transformation introduced and the inverse scattering problem in quantum mechanics is established. An elementary N-parametric potential that has exactly N predetermined discrete spectrum levels is constructed. The paper concludes with some examples of new exactly soluble potentials.

Keywords

Quantum Mechanic Differential Operator Discrete Spectrum Factorization Method Inverse Scattering 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    G. Darboux,Lecons sur la theorie generale des surfaces et les application geometriques du calcul infinitesimal. Deuxiem partie, Gauthier-Villars et fils, Paris (1889).Google Scholar
  2. 2.
    T.E. Infeld and H. Hull,Rev. Mod. Phys.,53, 21 (1951).Google Scholar
  3. 3.
    H. Hull and T.E. Infeld,Phys. Rev.,74, 905 (1948).Google Scholar
  4. 4.
    B. Melnik,J. Math. Phys.,25, No. 12, 3387–3389 (1984).Google Scholar
  5. 5.
    N.A. Alves and E.D. Filho,J. Phys. A: Math. Gen.,21, No. 15, 3215–3225 (1988).Google Scholar
  6. 6.
    L.D. Faddeev,Usp. Mat. Nauk,105, No. 4(88), 57–119 (1959).Google Scholar
  7. 7.
    B.M. Levitan,Inverse Sturm-Liouville Problems [In Russian] Nauka, Moscow (1984).Google Scholar
  8. 8.
    Z.S. Agranovich and V.A. Marchenko,Inverse Scattering Problem [In Russian], Kharkov Univ., Kharkov (1969).Google Scholar
  9. 9.
    P.B. Abraham and H.E. Moses,Phys. Rev. A.,22, No. 4, 1333–1340 (1980).Google Scholar
  10. 10.
    D.L. Pursey,Phys. Rev. D.,33, No. 4, 1048–1055 (1986).Google Scholar
  11. 11.
    M. Luban and D.L. Pursey,Phys. Rev. D.,33, No. 2, 431–436 (1986).Google Scholar
  12. 12.
    V.G. Bagrov, A.V. Shapovalov, and I.V. Shirokov,Teor. Mat. Fiz.,87, No. 3, 426–433 (1992).Google Scholar
  13. 13.
    V.G. Bagrov, A.V. Shapovalov, and I.V. Shirokov,Phys. Lett. A.,147, No. 7, 348–350 (1990).Google Scholar
  14. 14.
    E. Witten,Nucl. Phys.,B185, 513 (1981).Google Scholar
  15. 15.
    V.G. Bagrov and A.S. Vshivtsev,Izv. Vyssh. Uchebn. Zaved., Fiz., No. 7, 19 (1988).Google Scholar
  16. 16.
    J. Delsart,J. Math. Pures Appl.,17, 213–230 (1938).Google Scholar
  17. 17.
    A.P. Veselov and A.B. Shabat,Funkts. Anal. Prilozh.,27, No. 2, 1–21 (1993).Google Scholar
  18. 18.
    E.K. Sklyanin,Funkts. Anal. Prilozh.,16, 27 (1982);17, 34 (1983).Google Scholar
  19. 19.
    Ya.I. Granovskii, A.S. Zhedanov, and I.M. Lutsenko,Zh. Exp. Teor. Fiz.,99, 369 (1991).Google Scholar
  20. 20.
    M.G. Krein,Dokl. Akad. Nauk SSSR,113, No. 5, 970–973 (1957).Google Scholar
  21. 21.
    V.P. Berezovskii and A.I. Pashnev,Teor. Mat. Fiz.,70, 146 (1987).Google Scholar
  22. 22.
    Yu.S. Dubov, V.M. Eleonskii, and N.E. Kulagin,Zh. Exp. Teor. Fiz.,102, 814 (1992).Google Scholar
  23. 23.
    G.A. Natanzon,Vestn. Leningr. Gos. Univ., No. 10, 22 (1971);Teor. Mat. Fiz.,38, 219 (1979).Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • V. G. Bagrov
    • 1
    • 2
  • B. F. Samsonov
    • 1
    • 2
  1. 1.Tomsk State UniversityUSSR
  2. 2.Institute of Strong-Current ElectronicsSiberian Division of the Russian Academy of SciencesUSSR

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