Skip to main content
SpringerLink
Log in

Oscillation Theorems for the Wronskian of an Arbitrary Sequence of Eigenfunctions of Schrödinger’s Equation

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

The work of Adler provides necessary and sufficient conditions for the Wronskian of a given sequence of eigenfunctions of Schrödinger’s equation to have constant sign in its domain of definition. We extend this result by giving explicit formulas for the number of real zeros of the Wronskian of an arbitrary sequence of eigenfunctions. Our results apply in particular to Wronskians of classical orthogonal polynomials, thus generalizing classical results by Karlin and Szegő. Our formulas hold under very mild conditions that are believed to hold for generic values of the parameters. In the Hermite case, our results allow to prove some conjectures recently formulated by Felder et al.

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. Simon B., Teschl G., Gesztesy F.: Zeros of the Wronskian and renormalized oscillation theory. Am. J. Math 118, 571–594 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  2. Krüger, H., Teschl, G.: Relative oscillation theory, weighted zeros of the Wronskian, and the spectral shift function. Commun. Math. Phys. 287(2), 613–640 (2009)

  3. Freeman, N.C., Nimmo, J.J.C.: Soliton solutions of the Korteweg de Vries and the Kadomtsev–Petviashvili equations: the Wronskian technique. Proc. R. Soc. Lond. A 389, 319–329 (1983)

  4. Matveev, V.B., Salle, M. A.: Darboux transformations and solitons. Springer-Verlag, Berlin (1991)

  5. Infeld, L., Hull, T.E.: The factorization method. Rev. Mod. Phys. 23, 21 (1951)

  6. Crum, M.M.: Associated Sturm–Liouville systems. Q. J. Math. Oxford Ser. (2) 6, 121–127 (1955)

  7. Adler V.E.: A modification of Crum’s method. Theor. Math. Phys. 101, 1381–1386 (1994)

    Article  MATH  Google Scholar 

  8. Krein M.G.: A continual analogue of a Christoffel formula from the theory of orthogonal polynomials. Dokl. Akad. Nauk. SSSR 113, 970–973 (1957)

    MATH  MathSciNet  Google Scholar 

  9. Gesztesy F., Simon B., Teschl G.: Spectral deformations of one-dimensional Schrödinger operators. J. Anal. Math. 70, 267–324 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  10. Deift P.A.: Applications of a commutation formula. Duke Math. J. 45, 267–310 (1987)

    Article  MathSciNet  Google Scholar 

  11. Gomez-Ullate, D., Grandati, Y., Milson, R.: Extended Krein–Adler theorem for the translationally shape invariant potentials. J. Math. Phys. 55, 043510 (2014)

  12. Karlin S., Szegő G.: On certain determinants whose elements are orthogonal polynomials. J. Anal. Math. 8, 1–157 (1960)

    Article  Google Scholar 

  13. Gomez-Ullate D., Kamran N., Milson R.: An extended class of orthogonal polynomials defined by a Sturm–Liouville problem. J. Math. Anal. Appl. 359, 352–367 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  14. Gomez-Ullate D., Kamran N., Milson R.: An extension of Bochner’s problem: exceptional invariant subspaces. J. Approx. Theory 162, 987–1006 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Odake, S., Sasaki, R.: Infinitely many shape invariant potentials and new orthogonal polynomials. Phys. Lett. B 679, 414 (2009)

  16. Gómez-Ullate D., Kamran N., Milson R.: Exceptional orthogonal polynomials and the Darboux transformation. J. Phys. A 43, 434016–434032 (2010)

    Article  MathSciNet  Google Scholar 

  17. Gómez-Ullate D., Kamran N., Milson R.: Two-step Darboux transformations and exceptional Laguerre polynomials. J. Math. Anal. Appl. 387, 410–418 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  18. Gomez-Ullate D., Kamran N., Milson R.: A conjecture on exceptional orthogonal polynomials. Found. Comput. Math. 13, 615–666 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  19. Durán A.: Exceptional Charlier and Hermite orthogonal polynomials. J. Approx. Theory 182, 29–58 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  20. Durán A.: Exceptional Meixner and Laguerre orthogonal polynomials. J. Approx. Theory 184, 176–208 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  21. Gómez-Ullate, D., Grandati, Y., Milson, R.: Rational extensions of the quantum harmonic oscillator and exceptional hermite polynomials. J. Phys. A 47, 015203 (2014)

  22. Gomez-Ullate D., Marcellan F., Milson R.: Asymptotic and interlacing properties of zeros of exceptional Jacobi and Laguerre polynomials. J. Math. Anal. Appl. 399, 480–495 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  23. Clarkson P.A.: The fourth Painlevé equation and associated special polynomials. J. Math. Phys. 44, 5350–5374 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  24. Clarkson P.A.: On rational solutions of the fourth Painlevé equation and its Hamiltonian. CRM Proc. Lect. Notes 39, 103–118 (2005)

    MathSciNet  Google Scholar 

  25. Filipuk G.V., Clarkson P.A.: The symmetric fourth Painlevé hierarchy and associated special polynomials. Stud. Appl. Math. 121, 157–188 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  26. Felder G., Hemery A.D., Veselov A.P.: Zeros of Wronskians of Hermite polynomials and Young diagrams. Phys. D 241, 2131–2137 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  27. Zhang, L., Filipuk, G.: On certain Wronskians of multiple orthogonal polynomials (2014). arXiv:1402.1569 [math.CA]

  28. Szegő, G.: Orthogonal polynomials. Am. Math. Soc. Colloq. Publ. 23 (1975) (American Mathematical Society, Providence, 1975, 4th edition)

  29. Dimitrov D.K., Rafaeli R.F.: Monotonicity of zeros of Laguerre polynomials. J. Comput. Appl. Math. 233, 699–702 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  30. Dorwart H.L.: Irreducibility of polynomials. Am. Math. Monthly 42, 369–381 (1935)

    Article  MathSciNet  Google Scholar 

  31. Schur, I.: Einige Sätze über Primzahlen mit Anwendungen auf Irreduzibilitätsfragen, II. Sitzungsber. Preuss. Akad. Wiss. Berlin Phys. Math. Kl. 14, 370–391 (1929)

  32. Durán A.J., Pérez M., Varona L.J.: Some conjectures on Wronskian and Casorati determinants of orthogonal polynomials. Exp. Math. 24(1), 123–132 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Física Teórica II, Universidad Complutense de Madrid, 28040, Madrid, Spain

    MªÁngeles García-Ferrero & David Gómez-Ullate

  2. Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), C/ Nicolas Cabrera 15, 28049, Madrid, Spain

    MªÁngeles García-Ferrero & David Gómez-Ullate

Authors
  1. MªÁngeles García-Ferrero
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David Gómez-Ullate
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Gómez-Ullate.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

García-Ferrero, M., Gómez-Ullate, D. Oscillation Theorems for the Wronskian of an Arbitrary Sequence of Eigenfunctions of Schrödinger’s Equation. Lett Math Phys 105, 551–573 (2015). https://doi.org/10.1007/s11005-015-0751-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-015-0751-4

Mathematics Subject Classification

Keywords

Search

Navigation