Oscillation theorems for the Wronskian of an arbitrary sequence of eigenfunctions of Schrödinger's equation

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.

arXiv comment16 pages, typed in AMS-LaTeX

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• Exceptional Jacobi polynomials
• Exceptional Laguerre polynomials
• Moments of random matrices and hypergeometric orthogonal polynomials
• Shape invariance and equivalence relations for pseudowronskians of Laguerre and Jacobi polynomials
• Coefficients of Wronskian Hermite polynomials
• Complex exceptional orthogonal polynomials and quasi-invariance
• Third-order ladder operators, generalized Okamoto and exceptional orthogonal polynomials
• On Integrable Ermakov-Painlevé IV Systems
• On the general family of third-order shape-invariant Hamiltonians related to generalized Hermite polynomials