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arXiv · 1211.5308

Disconjugacy, regularity of multi-indexed rationally-extended potentials, and Laguerre exceptional polynomials

Yves Grandati·Christiane Quesne
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Abstract

The power of the disconjugacy properties of second-order differential equations of Schrödinger type to check the regularity of rationally-extended quantum potentials connected with exceptional orthogonal polynomials is illustrated by re-examining the extensions of the isotonic oscillator (or radial oscillator) potential derived in kth-order supersymmetric quantum mechanics or multistep Darboux-Bäcklund transformation method. The function arising in the potential denominator is proved to be a polynomial with a nonvanishing constant term, whose value is calculated by induction over k. The sign of this term being the same as that of the already known highest-degree term, the potential denominator has the same sign at both extremities of the definition interval, a property that is shared by the seed eigenfunction used in the potential construction. By virtue of disconjugacy, such a property implies the nodeless character of both the eigenfunction and the resulting potential.

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Yves Grandati, Christiane Quesne. 2012-12-09. Disconjugacy, regularity of multi-indexed rationally-extended potentials, and Laguerre exceptional polynomials. https://arxiv.org/abs/1211.5308

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