Oscillation theorems for the Wronskian of an arbitrary sequence of eigenfunctions of Schr\"odinger's equation

TL;DR

This paper extends Adler's criteria by providing explicit formulas for counting real zeros of Wronskians of eigenfunctions of Schrödinger's equation, with applications to orthogonal polynomials and validation of recent conjectures.

Contribution

It introduces explicit formulas for zeros of Wronskians of eigenfunctions, generalizing classical results and applying to orthogonal polynomials, including Hermite cases.

Findings

Formulas for the number of real zeros of Wronskians.

Application to classical orthogonal polynomials.

Validation of recent conjectures in Hermite case.

Abstract

The work of Adler provides necessary and sufficient conditions for the Wronskian of a given sequence of eigenfunctions of Schr\"odinger'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 Szeg\H{o}. 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.