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

Yves Grandati (1) , Christiane Quesne (2)

1 FCN - Fédération de Chimie de Nancy
2 Physique Nucléaire Théorique et Physique Mathématique

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.

Mots clés: Quantum mechanics; supersymmetry; orthogonal polynomials

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]; Physique Quantique [quant-ph]

Yves Grandati, Christiane Quesne. Disconjugacy, regularity of multi-indexed rationally-extended potentials, and Laguerre exceptional polynomials. 2012. ⟨hal-00755966v2⟩