Solvable Rational Potentials and Exceptional Orthogonal Polynomials in Supersymmetric Quantum Mechanics

Abstract

New exactly solvable rationally-extended radial oscillator and Scarf I potentials are generated by using a constructive supersymmetric quantum mechanical method based on a reparametrization of the corresponding conventional superpotential and on the addition of an extra rational contribution expressed in terms of some polynomial $g$. The cases where $g$ is linear or quadratic are considered. In the former, the extended potentials are strictly isospectral to the conventional ones with reparametrized couplings and are shape invariant. In the latter, there appears a variety of extended potentials, some with the same characteristics as the previous ones and others with an extra bound state below the conventional potential spectrum. Furthermore, the wavefunctions of the extended potentials are constructed. In the linear case, they contain $(\nu +1)$th-degree polynomials with $\nu =0,1,2,...$, which are shown to be $X_1$-Laguerre or $X_1$-Jacobi exceptional orthogonal polynomials. In the quadratic case, several extensions of these polynomials appear. Among them, two different kinds of $(\nu +2)$th-degree Laguerre-type polynomials and a single one of $(\nu +2)$th-degree Jacobi-type polynomials with $\nu =0,1,2,...$ are identified. They are candidates for the still unknown $X_2$-Laguerre and $X_2$-Jacobi exceptional orthogonal polynomials, respectively.

v2: additions in secs. 1 and 4, 5 new references; v3: published version

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Cited by in corpus (64)

• Infinitely many shape invariant potentials and new orthogonal polynomials
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• Solvable rational extensions of the isotonic oscillator
• Infinitely many shape invariant discrete quantum mechanical systems and new exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials
• Asymptotic behaviour of zeros of exceptional Jacobi and Laguerre polynomials
• Properties of the Exceptional ($X_{\ell }$) Laguerre and Jacobi Polynomials
• Extended Krein-Adler theorem for the translationally shape invariant potentials
• Novel Enlarged Shape Invariance Property and Exactly Solvable Rational Extensions of the Rosen-Morse II and Eckart Potentials
• Exceptional Laguerre and Jacobi polynomials and the corresponding potentials through Darboux-Crum Transformations
• Modification of Crum's Theorem for `Discrete' Quantum Mechanics
• Solvable rational extensions of the Morse and Kepler-Coulomb potentials
• New ladder operators for a rational extension of the harmonic oscillator and superintegrability of some two-dimensional systems
• Combined state-adding and state-deleting approaches to type III multi-step rationally-extended potentials: applications to ladder operators and superintegrability
• Exceptional Askey-Wilson type polynomials through Darboux-Crum transformations
• Supersymmetric Quantum Mechanics with Reflections
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials
• Parametric symmetries in exactly solvable real and P T symmetric complex potentials
• Rationally-extended Dunkl oscillator on the line
• Ground-state isolation and discrete flows in a rationally extended quantum harmonic oscillator
• Extending Romanovski polynomials in quantum mechanics
• Inter-relations between additive shape invariant superpotentials
• The confluent supersymmetry algorithm for Dirac equations with pseudoscalar potentials
• A Bochner type classification theorem for exceptional orthogonal polynomials
• Scattering Amplitudes for Multi-indexed Extensions of Solvable Potentials
• ABC of ladder operators for rationally extended quantum harmonic oscillator systems
• Confluent Chains of DBT: Enlarged Shape Invariance and New Orthogonal Polynomials
• A class of exactly solvable rationally extended non-central potentials in Two and Three Dimensions
• Generation of a Novel Exactly Solvable Potential
• Equivalences of the Multi-Indexed Orthogonal Polynomials
• Shape invariance and equivalence relations for pseudowronskians of Laguerre and Jacobi polynomials
• Rational extension and Jacobi-type {\boldmath{$X_m$}} solutions of a quantum nonlinear oscillator
• Exactness of SWKB for Shape Invariant Potentials
• Exceptional Legendre Polynomials and Confluent Darboux Transformations
• Spectral analysis for the exceptional $X_m$-Jacobi equation
• Ladder operators for solvable potentials connected with exceptional orthogonal polynomials
• Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators
• The Supersymmetric WKB Formalism is Not Exact for All Additive Shape Invariant Potentials
• Exactly solvable potentials with finitely many discrete eigenvalues of arbitrary choice
• 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
• Higher order superintegrability, Painlevé transcendents and representations of polynomial algebras
• Moment Representations of Exceptional $X_1$ Orthogonal Polynomials
• Heun-Polynomial Representation of Regular-at-Infinity Solutions for the Basic SUSY Ladder of Hyperbolic Pöschl-Teller Potentials Starting from the Reflectionless Symmetric Potential Well
• 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
• Dutt-Khare-Varshni Potential and Its Quantized-by-Heun-Polynomials SUSY Partners as Nontrivial Examples of Solvable Potentials Explicitly Expressible in Terms of Elementary Functions
• Non-polynomial extensions of solvable potentials a la Abraham-Moses
• A new way to classify 2D higher order quantum superintegrable systems
• Generalized Langer Correction and the Exactness of WKB for all Conventional Potentials
• Exactly Solvable Quantum Mechanics
• Time and band limiting for exceptional polynomials
• Disconjugacy, regularity of multi-indexed rationally-extended potentials, and Laguerre exceptional polynomials
• A generic classification of exceptional orthogonal X1-polynomials based on Pearson distributions family
• Rational extensions of the trigonometric Darboux-Pöschl-Teller potential based on para-Jacobi polynomials
• Cubic algebras, induced representations and general solution of the exceptional Laguerre equation $X_1$
• Ladder operators and coherent states for the Rosen-Morse system and its rational extensions
• Revisiting an isospectral extension of the Morse potential
• The bispectral problem, the Darboux process, monodromy and the Hermite operator
• Quantum Hamilton-Jacobi Quantization and Shape Invariance
• Composition of Two Potentials
• Deformed oscillator algebra approach of some quantum superintegrable Lissajous systems on the sphere and of their rational extensions
• Deformed Shape Invariant Superpotentials in Quantum Mechanics and Expansions in Powers of $\hslash$
• Infinite families of shape invariant potentials with n parameters subject to translation
• Determinantal Formulas for Exceptional Orthogonal Polynomials