Solvable rational extensions of the isotonic oscillator

Abstract

Combining recent results on rational solutions of the Riccati-Schrödinger equations for shape invariant potentials to the finite difference Bäcklund 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.

References in corpus (24)

• An extended class of orthogonal polynomials defined by a Sturm-Liouville problem
• Infinitely many shape invariant potentials and new orthogonal polynomials
• An Extension of Bochner's Problem: Exceptional Invariant Subspaces
• Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry
• Another set of infinitely many exceptional (X_{\ell}) Laguerre polynomials
• Conditionally exactly solvable potentials: A supersymmetric construction method
• Solvable Rational Potentials and Exceptional Orthogonal Polynomials in Supersymmetric Quantum Mechanics
• Isospectrality of conventional and new extended potentials, second-order supersymmetry and role of PT symmetry
• The finite difference algorithm for higher order supersymmetry
• A quantum exactly solvable non-linear oscillator related with the isotonic oscillator
• Exceptional orthogonal polynomials and the Darboux transformation
• New possibilities for supersymmetry breakdown in quantum mechanics and second order irreducible Darboux transformations
• Group theoretical approach to the intertwined Hamiltonians
• Properties of the Exceptional ($X_{\ell }$) Laguerre and Jacobi Polynomials
• Infinitely many shape invariant potentials and cubic identities of the Laguerre and Jacobi polynomials
• The Darboux transformation and algebraic deformations of shape-invariant potentials
• Supersymmetry and algebraic deformations
• Integrability of Riccati equation from a group theoretical viewpoint
• An update on PT-symmetric complexified Scarf II potential, spectral singularities and some remarks on the rationally-extended supersymmetric partners
• Exotic supersymmetry of the kink-antikink crystal, and the infinite period limit
• Self-isospectrality, mirror symmetry, and exotic nonlinear supersymmetry
• Rational solutions for the Riccati-Schrödinger equations associated to translationally shape invariant potentials
• Supersymmetric quantum mechanics based on higher excited states
• Supersymmetric Method for Constructing Quasi-Exactly and Conditionally-Exactly Solvable Potentials

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• A class of exactly solvable rationally extended Calogero-Wolfes type 3-body problems
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• Shape Invariant Rational Extensions And Potentials Related to Exceptional Polynomials
• New 1-step extension of the Swanson oscillator and superintegrability of its two-dimensional generalization
• Ladder operators for solvable potentials connected with exceptional orthogonal polynomials
• A short note on "Group theoretic approach to rationally extended shape invariant potentials" [Annals of Physics 359, 46 (2015)]
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• Symmetries and the compatibility condition for the new translational shape invariant potentials
• Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators
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• Gauss-Seed Nets of Sturm-Liouville Problems With Energy-Independent Characteristic Exponents and Related Sequences of Exceptional Orthogonal Polynomials I. Canonical Darboux Transformations Using AEH Functions
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• Single-Source Nets of Algebraically-Quantized Reflective Liouville Potentials on the Line I. Almost-Everywhere Holomorphic Solutions of Rational Canonical Sturm-Liouville Equations with Second-Order Poles
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• Disconjugacy, regularity of multi-indexed rationally-extended potentials, and Laguerre exceptional polynomials
• Explicit solutions of Schrödinger and KdV equations in terms of square roots of the generalised matrix eigenvalues
• Lectures on exceptional orthogonal polynomials and rational solutions to Painlevé equations
• Singular Isotonic Oscillator, Supersymmetry and Superintegrability
• Generalized Baecklund-Darboux transformation: conservation laws, rational extensions and bispectrality
• Deformed oscillator algebra approach of some quantum superintegrable Lissajous systems on the sphere and of their rational extensions
• Infinite families of shape invariant potentials with n parameters subject to translation
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