Novel Enlarged Shape Invariance Property and Exactly Solvable Rational Extensions of the Rosen-Morse II and Eckart Potentials

Abstract

The existence of a novel enlarged shape invariance property valid for some rational extensions of shape-invariant conventional potentials, first pointed out in the case of the Morse potential, is confirmed by deriving all rational extensions of the Rosen-Morse II and Eckart potentials that can be obtained in first-order supersymmetric quantum mechanics. Such extensions are shown to belong to three different types, the first two strictly isospectral to some starting conventional potential with different parameters and the third with an extra bound state below the spectrum of the latter. In the isospectral cases, the partner of the rational extensions resulting from the deletion of their ground state can be obtained by translating both the potential parameter $A$ (as in the conventional case) and the degree $m$ of the polynomial arising in the denominator. It therefore belongs to the same family of extensions, which turns out to be closed.

References in corpus (23)

• Supersymmetry and Quantum Mechanics
• 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
• Exactly Solvable Quantum Mechanics and Infinite Families of Multi-indexed Orthogonal Polynomials
• Another set of infinitely many exceptional (X_{\ell}) Laguerre polynomials
• Two-step Darboux transformations and exceptional Laguerre polynomials
• Solvable Rational Potentials and Exceptional Orthogonal Polynomials in Supersymmetric Quantum Mechanics
• Polynomial SUSY in Quantum Mechanics and Second Derivative Darboux Transformation
• Higher-order supersymmetric quantum mechanics
• Isospectrality of conventional and new extended potentials, second-order supersymmetry and role of PT symmetry
• Solvable rational extensions of the isotonic oscillator
• N-fold Supersymmetry in Quantum Mechanics - General Formalism -
• A quantum exactly solvable non-linear oscillator related with the isotonic oscillator
• Higher-order SUSY, exactly solvable potentials, and exceptional orthogonal polynomials
• Properties of the Exceptional ($X_{\ell }$) Laguerre and Jacobi Polynomials
• Generation of a Complete Set of Supersymmetric Shape Invariant Potentials from an Euler Equation
• The Darboux transformation and algebraic deformations of shape-invariant potentials
• Supersymmetry and algebraic deformations
• New rational extensions of solvable potentials with finite bound state spectrum
• Isospectral Potentials from Modified Factorization
• Method for Generating Additive Shape Invariant Potentials from an Euler Equation
• Global Solutions of Certain Second-Order Differential Equations with a High Degree of Apparent Singularity

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