New rational extensions of solvable potentials with finite bound state spectrum

Yves Grandati (1)

1 FCN - Fédération de Chimie de Nancy

Using the disconjugacy properties of the Schrödinger equation, it is possible to develop a new type of generalized SUSY QM partnership which allows to generate new solvable rational extensions for translationally shape invariant potentials having a finite bound state spectrum. For this we prolong the dispersion relation relating the energy to the quantum number out of the physical domain until a disconjugacy sector. The prolonged excited states Riccati-Schrödinger (RS) functions are used to build Darboux-Bäcklund transforms which give regular isospectral extensions of the initial potential. We give the spectra of these extensions in terms of new orthogonal polynomials and study their shape invariance properties.

Mots clés: supersymmetric quantum mechanics; solvable systems; disconjugacy; shape invariance; rational extensions; orthogonal polynomials; Darboux transformations

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

Yves Grandati. New rational extensions of solvable potentials with finite bound state spectrum. 2012. ⟨hal-00680522⟩