Mathematical Physics, Analysis and Geometry

, Volume 16, Issue 2, pp 179–193 | Cite as

Darboux Transformations for Energy-Dependent Potentials and the Klein–Gordon Equation

  • Axel Schulze-Halberg


We construct explicit Darboux transformations for a generalized Schrödinger-type equation with energy-dependent potential, a special case of which is the stationary Klein–Gordon equation. Our results complement and generalize former findings (Lin et al., Phys Lett A 362:212–214, 2007).


Generalized Schrödinger equation Energy-dependent potential Darboux transformation Klein–Gordon equation 

Mathematics Subject Classifications (2010)

81Q05 34A05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aktosun, T., van der Mee, C.: A unified approach to Darboux transformations. Inverse Probl. 25, 105003 (2009)ADSCrossRefGoogle Scholar
  2. 2.
    Andrianov, A.A., Borisov, N.V., Ioffe, M.V.: Quantum systems with identical energy spectra. JETP Lett. 39, 93–97 (1984)ADSGoogle Scholar
  3. 3.
    Andrianov, A.A., Borisov, N.V., Ioffe, M.V.: Factorization method and Darboux transformation for multidimensional Hamiltonians. Theor. Math. Phys. 61, 1078–1089 (1984)CrossRefGoogle Scholar
  4. 4.
    Andrianov, A.A., Borisov, N.V., Ioffe, M.V., Eides, M.I.: Supersymmetric mechanics: a new look at the equivalence of quantum systems. Theor. Math. Phys. 61, 965–972 (1984)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Andrianov, A.A., Ioffe, M.V., Nishnianidze, D.N.: Higher order SUSY in quantum mechanics and integrability of two-dimensional Hamiltonians. Zapiski Nauch. Semin. POMI 224, 68 (1995)Google Scholar
  6. 6.
    Belyavskii, V.I., Goldfarb, M.V., Kopaev, Y.V.: Binding energy of Coulomb acceptors in quantum-well systems. Fiz. Teh. Poluprovodn. 31, 1095–1099 (2007)Google Scholar
  7. 7.
    Cooper, F., Khare, A., Sukhatme, U.: Supersymmetry and quantum mechanics. Phys. Rep. 251, 267–388 (1995)MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    Crater, H.W., van Alstine, P.: Two-body Dirac equations for particles interacting through world scalar and vector potentials. Phys. Rev. D 36, 3007–3036 (1987)MathSciNetADSCrossRefGoogle Scholar
  9. 9.
    Darboux, M.G.: Sur une proposition relative aux équations linéaires. C. R. Acad. Sci. Paris 94, 1456–1459 (1882)MATHGoogle Scholar
  10. 10.
    Debergh, N., Pecheritsin, A.A., Samsonov, B.F., van den Bossche, B.: Darboux transformations of the one-dimensional stationary Dirac equation. J. Phys. A 35, 3279–3288 (2002)MathSciNetADSMATHCrossRefGoogle Scholar
  11. 11.
    De Sanctis, M., Quintero, P.: A new energy-dependent quark interaction from a Tamm–Dancoff reduction of an effective field theory quark model. Eur. Phys. J. A 39, 1434–6001 (2009)CrossRefGoogle Scholar
  12. 12.
    Formanek, J., Lombard, R.J., Mares, J.: Wave equations with energy-dependent potentials. Czechoslov. J. Phys. 54, 289–315 (2004)MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    Gonzalez-Lopez, A., Kamran, N.: The multidimensional Darboux transformation. J. Geom. Phys. 26, 202–226 (1998)MathSciNetADSMATHCrossRefGoogle Scholar
  14. 14.
    Gu, C., Hu, H., Zhou, Z.: Darboux transformations in integrable systems. In: Mathematical Physics Studies, vol. 26. Springer, Dordrecht (2005)Google Scholar
  15. 15.
    Günther, U., Samsonov, B.F., Stefani, F.: A globally diagonalizable α2-dynamo operator, SUSY QM and the Dirac equation. J. Phys. A 40, F169–F176 (2007)ADSMATHCrossRefGoogle Scholar
  16. 16.
    Ioffe, M.V.: Supersymmetrical separation of variables in two-dimensional Quantum Mechanics. SIGMA 6, 75–85 (2010)MathSciNetGoogle Scholar
  17. 17.
    Ioffe, M.V., Nishnianidze, D.N., Valinevich, P.A.: New exactly solvable two-dimensional quantum model not amenable to separation of variables. J. Phys. A 43, 485303 (2010)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Landsberg, G.T.: Solid State Theory: Methods and Applications. Wiley-Interscience, London (1969)Google Scholar
  19. 19.
    Li, Y.: Some water wave equations and integrability. J. Nonlin. Math. Phys. 12, 466–481 (2005)CrossRefGoogle Scholar
  20. 20.
    Lin, J., Li, Y.-S., Qian, X.-M.: The Darboux transformation of the Schrödinger equation with an energy-dependent potential. Phys. Lett. A 362, 212–214 (2007)MathSciNetADSMATHCrossRefGoogle Scholar
  21. 21.
    Ma, W.X.: Darboux transformations for a Lax integrable system in 2n dimensions. Lett. Math. Phys. 39, 33–49 (1997)MathSciNetADSMATHCrossRefGoogle Scholar
  22. 22.
    Matveev, V.B., Salle, M.A.: Darboux Transformations and Solitons. Springer, Berlin (1991)MATHCrossRefGoogle Scholar
  23. 23.
    Milanovic, V., Ikonic, Z.: On the optimization of resonant intersubband nonlinear optical susceptibilities in semiconductor quantum wells. IEEE J. Quantum Electron. 32, 1316–1323 (1996)ADSCrossRefGoogle Scholar
  24. 24.
    Mourad, J., Sazdjian, H.: The two-fermion relativistic wave equations of constraint theory in the Pauli–Schrödinger form. J. Math. Phys. 35, 6379–6406 (1994)MathSciNetADSMATHCrossRefGoogle Scholar
  25. 25.
    Pavlov, B.S., Strepetov, A.V.: Exactly solvable model of electron scattering by an inhomogeneity in a thin conductor. Theor. Math. Phys. 90, 152–156 (1992)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Pozdeeva, E., Schulze-Halberg, A.: Darboux transformations for a generalized Dirac equation in two dimensions. J. Math. Phys. 51, 113501 (15 pp.) (2010)MathSciNetADSCrossRefGoogle Scholar
  27. 27.
    Schulze-Halberg, A.: Darboux transformations for effective mass Schrödinger equations with energy-dependent potentials. Int. J. Mod. Phys. A 23, 537–546 (2008)MathSciNetADSMATHCrossRefGoogle Scholar
  28. 28.
    Schulze-Halberg, A.: Modified Darboux transformations with foreign auxiliary equations. Phys. Lett. A 375, 2513–2518 (2011)MathSciNetADSMATHCrossRefGoogle Scholar
  29. 29.
    Suzko, A.A., Schulze-Halberg, A.: Darboux transformations and supersymmetry for the generalized Schrödinger equations in (1 + 1) dimensions. J. Phys. A 42, 295203 (14 pp.) (2009)CrossRefGoogle Scholar
  30. 30.
    Tomic, S., Milanovic, V., Ikonic, Z.: Optimization of intersubband resonant second-order susceptibility in asymmetric graded AlxGa1-xAs quantum wells using supersymmetric quantum mechanics. Phys. Rev. B 56, 1033–1036 (1997)ADSCrossRefGoogle Scholar
  31. 31.
    Yurov, A.V.: Darboux transformation for Dirac equations with (1 + 1) potentials. Phys. Lett. A 225, 51–59 (1997)MathSciNetADSMATHCrossRefGoogle Scholar
  32. 32.
    Zhang, J.E., Li, Y.: Bidirectional solitons on water. Phys. Rev. E 67, 016306 (8 pp.) (2003)ADSGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of Mathematics and Actuarial ScienceIndiana University NorthwestGaryUSA

Personalised recommendations