Infinitely many shape invariant discrete quantum mechanical systems and new exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials

Abstract

Two sets of infinitely many exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials are presented. They are derived as the eigenfunctions of shape invariant and thus exactly solvable quantum mechanical Hamiltonians, which are deformations of those for the Wilson and Askey-Wilson polynomials in terms of a degree \ell (\ell=1,2,...) eigenpolynomial. These polynomials are exceptional in the sense that they start from degree \ell\ge1 and thus not constrained by any generalisation of Bochner's theorem.

7 pages; one reference added, published in Phys. Lett. B682 (2009) 130-136

References in corpus (9)

• Infinitely many shape invariant potentials and new orthogonal polynomials
• Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry
• Solvable Rational Potentials and Exceptional Orthogonal Polynomials in Supersymmetric Quantum Mechanics
• Orthogonal Polynomials from Hermitian Matrices
• Exactly solvable `discrete' quantum mechanics; shape invariance, Heisenberg solutions, annihilation-creation operators and coherent states
• Exact solution in the Heisenberg picture and annihilation-creation operators
• Exactly Solvable Birth and Death Processes
• Orthogonal Polynomials from Hermitian Matrices II
• q-oscillator from the q-Hermite Polynomial

Cited by in corpus (44)

• Exactly Solvable Quantum Mechanics and Infinite Families of Multi-indexed Orthogonal Polynomials
• Another set of infinitely many exceptional (X_{\ell}) Laguerre polynomials
• Properties of the Exceptional ($X_{\ell }$) Laguerre and Jacobi Polynomials
• Infinitely many shape invariant potentials and cubic identities of the Laguerre and Jacobi polynomials
• Modification of Crum's Theorem for `Discrete' Quantum Mechanics
• An update on PT-symmetric complexified Scarf II potential, spectral singularities and some remarks on the rationally-extended supersymmetric partners
• Discrete Quantum Mechanics
• Multi-indexed (q-)Racah Polynomials
• Trends in supersymmetric quantum mechanics
• Extensions of solvable potentials with finitely many discrete eigenstates
• Exceptional Askey-Wilson type polynomials through Darboux-Crum transformations
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials
• Dual Christoffel transformations
• Information Entropy of conditionally exactly solvable potentials
• Prepotential approach to solvable rational extensions of Harmonic Oscillator and Morse potentials
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials : III
• Inter-relations between additive shape invariant superpotentials
• Generation of a Novel Exactly Solvable Potential
• Equivalences of the Multi-Indexed Orthogonal Polynomials
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials : II
• A new family of shape invariantly deformed Darboux-Pöschl-Teller potentials with continuous \ell
• Exceptional orthogonal polynomials and new exactly solvable potentials in quantum mechanics
• Global Solutions of Certain Second-Order Differential Equations with a High Degree of Apparent Singularity
• Exactness of SWKB for Shape Invariant Potentials
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials IV : closure relations and creation/annihilation operators
• Shape Invariant Rational Extensions And Potentials Related to Exceptional Polynomials
• The Supersymmetric WKB Formalism is Not Exact for All Additive Shape Invariant Potentials
• Exactly Solvable Discrete Quantum Mechanical Systems and Multi-indexed Orthogonal Polynomials of the Continuous Hahn and Meixner-Pollaczek Types
• Exactness of Semiclassical Quantization Rule for Broken Supersymmetry
• Generalized Rayleigh and Jacobi processes and exceptional orthogonal polynomials
• Solvable Discrete Quantum Mechanics: q-Orthogonal Polynomials with |q|=1 and Quantum Dilogarithm
• Quasi-Hermitian Hamiltonians associated with exceptional orthogonal polynomials
• New Determinant Expressions of the Multi-indexed Orthogonal Polynomials in Discrete Quantum Mechanics
• Exactly and quasi-exactly solvable `discrete' quantum mechanics
• Exceptional Meixner and Laguerre orthogonal polynomials
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials V : Racah and $q$-Racah types
• Dual Polynomials of the Multi-Indexed ($q$-)Racah Orthogonal Polynomials
• Type II Multi-indexed Little $q$-Jacobi and Little $q$-Laguerre Polynomials
• Hypercomplex Fock States for Discrete Electromagnetic Schrödinger Operators: A Bayesian Probability Perspective
• Casoratian Identities for the Discrete Orthogonal Polynomials in Discrete Quantum Mechanics with Real Shifts
• Generalized Langer Correction and the Exactness of WKB for all Conventional Potentials
• New Finite Type Multi-Indexed Orthogonal Polynomials Obtained From State-Adding Darboux Transformations
• Integrable models for quantum media excited by laser radiation: a method, physical interpretation, and examples
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials VI : Meixner-Pollaczek and continuous Hahn types