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Letters in Mathematical Physics

, Volume 105, Issue 4, pp 551–573 | Cite as

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

  • MªÁngeles García-Ferrero
  • David Gómez-UllateEmail author
Article

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.

Mathematics Subject Classification

34C10 33C45 81Q80 

Keywords

Sturm–Liouville problem Wronskian determinant zeros oscillation classical orthogonal polynomials exceptional polynomials 

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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • MªÁngeles García-Ferrero
    • 1
    • 2
  • David Gómez-Ullate
    • 1
    • 2
    Email author
  1. 1.Departamento de Física Teórica IIUniversidad Complutense de MadridMadridSpain
  2. 2.Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM)MadridSpain

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