Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 043, 26 pages      arXiv:1111.0120
Contribution to the Special Issue “Geometrical Methods in Mathematical Physics”

Darboux Integrals for Schrödinger Planar Vector Fields via Darboux Transformations

Primitivo B. Acosta-Humánez a and Chara Pantazi b
a) Departamento de Matemáticas y Estadística Universidad del Norte, Km. 5 via Puerto Colombia, Barranquilla, Colombia
b) Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, (EPSEB), Av. Doctor Marañón, 44-50, 08028 Barcelona, Spain

Received March 05, 2012, in final form July 06, 2012; Published online July 14, 2012

In this paper we study the Darboux transformations of planar vector fields of Schrödinger type. Using the isogaloisian property of Darboux transformation we prove the ''invariance'' of the objects of the ''Darboux theory of integrability''. In particular, we also show how the shape invariance property of the potential is important in order to preserve the structure of the transformed vector field. Finally, as illustration of these results, some examples of planar vector fields coming from supersymmetric quantum mechanics are studied.

Key words: Darboux theory of integrability; Darboux transformations; differential Galois theory; Schrödinger equation; supersymmetric quantum mechanics.

pdf (451 kb)   tex (30 kb)


  1. Acosta-Humánez P.B., Galoisian approach to supersymmetric quantum mechanics. The integrability analysis of the Schrödinger equation by means of differential Galois theory, VDM Verlag, Dr Müller, Berlin, 2010.
  2. Acosta-Humánez P.B., Lázaro-Ochoa J.T., Morales-Ruiz J.J., Pantazi Ch., On the integrability of polynomial fields in the plane by means of Picard-Vessiot theory, arXiv:1012.4796.
  3. Acosta-Humánez P.B., Morales-Ruiz J.J., Weil J.A., Galoisian approach to integrability of Schrödinger equation, Rep. Math. Phys. 67 (2011), 305-374, arXiv:1008.3445.
  4. Berkovich L.M., Evlakhov S.A., The Euler-Imshenetski-Darboux transformation of second-order linear equations, Program. Comput. Software 32 (2006), 154-165.
  5. Blázquez-Sanz D., Pantazi Ch., A note on the Darboux theory of integrability of non autonomous polynomial differential systems, Preprint, 2011.
  6. Blecua P., Boya L.J., Segui A., New solvable quantum-mechanical potentials by iteration of the free V=0 potential, Nuovo Cimento B 118 (2003), 535-546, quant-ph/0311139.
  7. Carnicer M.M., The Poincaré problem in the nondicritical case, Ann. of Math. (2) 140 (1994), 289-294.
  8. Cerveau D., Lins Neto A., Holomorphic foliations in CP(2) having an invariant algebraic curve, Ann. Inst. Fourier (Grenoble) 41 (1991), 883-903.
  9. Christopher C., Llibre J., Algebraic aspects of integrability for polynomial systems, Qual. Theory Dyn. Syst. 1 (1999), 71-95.
  10. Christopher C., Llibre J., Pantazi Ch., Walcher S., Inverse problems for invariant algebraic curves: explicit computations, Proc. Roy. Soc. Edinburgh Sect. A 139 (2009), 287-302.
  11. Christopher C., Llibre J., Pantazi Ch., Walcher S., Inverse problems for multiple invariant curves, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), 1197-1226.
  12. Christopher C., Llibre J., Pantazi Ch., Zhang X., Darboux integrability and invariant algebraic curves for planar polynomial systems, J. Phys. A: Math. Gen. 35 (2002), 2457-2476.
  13. Christopher C., Llibre J., Pereira J.V., Multiplicity of invariant algebraic curves in polynomial vector fields, Pacific J. Math. 229 (2007), 63-117.
  14. Cooper F., Khare A., Sukhatme U., Supersymmetry in quantum mechanics, World Scientific Publishing Co. Inc., River Edge, NJ, 2001.
  15. Darboux G., Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré, Bull. Sci. Math. (2) 2 (1878), 60-96, 123-144, 151-200.
  16. Darboux G., Sur une proposition relative aux équations linéaires, Comptes Rendus Acad. Sci. 94 (1882), 1456-1459.
  17. Darboux G., Théorie des Surfaces, II, Gauthier-Villars, Paris, 1889.
  18. García I.A., Giacomini H., Giné J., Generalized nonlinear superposition principles for polynomial planar vector fields, J. Lie Theory 15 (2005), 89-104.
  19. García I.A., Giné J., Generalized cofactors and nonlinear superposition principles, Appl. Math. Lett. 16 (2003), 1137-1141.
  20. Gendenshteïn L.E., Derivation of exact spectra of the Schrödinger equation by means of supersymmetry, JETP Lett. 38 (1983), 356-359.
  21. Giné J., Llibre J., A family of isochronous foci with Darboux first integral, Pacific J. Math. 218 (2005), 343-355.
  22. Ince E.L., Ordinary differential equations, Dover Publications, New York, 1944.
  23. Jouanolou J.P., Équations de Pfaff algébriques, Lecture Notes in Mathematics, Vol. 708, Springer, Berlin, 1979.
  24. Kalnins E.G., Kress J.M., Miller W., Families of classical subgroup separable superintegrable systems, J. Phys. A: Math. Theor. 43 (2010), 092001, 8 pages, arXiv:0912.3158.
  25. Kalnins E.G., Kress J.M., Miller W., Superintegrability and higher order integrals for quantum systems, J. Phys. A: Math. Theor. 43 (2010), 265205, 21 pages, arXiv:1002.2665.
  26. Kaplansky I., An introduction to differential algebra, Hermann, Paris, 1957.
  27. Kolchin E.R., Differential algebra and algebraic groups, Pure and Applied Mathematics, Vol. 54, Academic Press, New York, 1973.
  28. Kovacic J.J., An algorithm for solving second order linear homogeneous differential equations, J. Symbolic Comput. 2 (1986), 3-43.
  29. Llibre J., On the integrability of the differential systems in dimension two and of the polynomial differential systems in arbitrary dimension, J. Appl. Anal. Comput. 1 (2011), 33-52.
  30. Llibre J., Pantazi Ch., Darboux theory of integrability for a class of nonautonomous vector fields, J. Math. Phys. 50 (2009), 102705, 19 pages.
  31. Llibre J., Rodríguez G., Configurations of limit cycles and planar polynomial vector fields, J. Differential Equations 198 (2004), 374-380.
  32. Llibre J., Zhang X., Rational first integrals in the Darboux theory of integrability in Cn, Bull. Sci. Math. 134 (2010), 189-195.
  33. Maciejewski A.J., Przybylska M., Yoshida H., Necessary conditions for classical super-integrability of a certain family of potentials in constant curvature spaces, J. Phys. A: Math. Theor. 43 (2010), 382001, 15 pages, arXiv:1004.3854.
  34. Nikiforov A.F., Uvarov V.B., Special functions of mathematical physics. A unified introduction with applications, Birkhäuser Verlag, Basel, 1988.
  35. Pantazi Ch., Inverse problems of the Darboux theory of integrability for planar polynomial differential systems, Ph.D. thesis, Universitat Autonoma de Barcelona, 2004.
  36. Prelle M.J., Singer M.F., Elementary first integrals of differential equations, Trans. Amer. Math. Soc. 279 (1983), 215-229.
  37. Ramis J.P., Martinet J., Théorie de Galois différentielle et resommation, in Computer Algebra and Differential Equations, Comput. Math. Appl., Academic Press, London, 1990, 117-214.
  38. Schlomiuk D., Algebraic particular integrals, integrability and the problem of the center, Trans. Amer. Math. Soc. 338 (1993), 799-841.
  39. Singer M.F., Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc. 333 (1992), 673-688.
  40. Spiridonov V., Universal superpositions of coherent states and self-similar potentials, Phys. Rev. A 52 (1995), 1909-1935, quant-ph/9601030.
  41. Teschl G., Mathematical methods in quantum mechanics. With applications to Schrödinger operators, Graduate Studies in Mathematics, Vol. 99, American Mathematical Society, Providence, RI, 2009.
  42. Tremblay F., Turbiner A.V., Winternitz P., An infinite family of solvable and integrable quantum systems on a plane, J. Phys. A: Math. Theor. 42 (2009), 242001, 10 pages, arXiv:0904.0738.
  43. van der Put M., Singer M.F., Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften, Vol. 328, Springer-Verlag, Berlin, 2003.
  44. Weil J.A., Introduction to differential algebra and differential Galois theory, CIMPA-UNESCO Lectures, Hanoi, 2001.
  45. Witten E., Dynamical breaking of supersymmetry, Nuclear Phys. B 188 (1981), 513-554.
  46. Zoladek H., Polynomial Riccati equations with algebraic solutions, in Differential Galois Theory (Bedlewo, 2001), Banach Center Publ., Vol. 58, Polish Acad. Sci., Warsaw, 2002, 219-231.

Previous article  Next article   Contents of Volume 8 (2012)