Solvable rational extensions of the isotonic oscillator

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Abstract

Combining recent results on rational solutions of the Riccati-Schr\"odinger equations for shape invariant potentials to the finite difference B\"acklund algorithm and specific symmetries of the isotonic potential, we show that it is possible to generate the three infinite sets (L1, L2 and L3 families) of regular rational solvable extensions of this potential in a very direct and transparent way.

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Infinitely many shape invariant potentials and new orthogonal polynomials

Satoru Odake, Ryu Sasaki (2009)

Three sets of exactly solvable one-dimensional quantum mechanical potentials are presented. These are shape invariant potentials obtained by deforming the radial oscillator and the trigonometric/hyperbolic P\"oschl-Teller potentials in terms of their degree \ell polynomial eigenfunctions. We present the entire eigenfunctions for these Hamiltonians (\ell=1,2,...) in terms of new orthogonal polynomials. Two recently reported shape invariant potentials of Quesne and G\'omez-Ullate et al's are the first members of these infinitely many potentials.

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Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry

(2008)

We construct two new exactly solvable potentials giving rise to bound-state solutions to the Schr\"odinger equation, which can be written in terms of the recently introduced Laguerre- or Jacobi-type \(X_1\) exceptional orthogonal polynomials. These potentials, extending either the radial oscillator or the Scarf I potential by the addition of some rational terms, turn out to be translationally shape invariant as their standard counterparts and isospectral to them.