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Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry

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C Quesne

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We construct two new exactly solvable potentials giving rise to bound-state solutions to the Schrödinger equation, which can be written in terms of the recently introduced Laguerre- or Jacobi-type X1 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.


PACS

03.65.Ge Solutions of wave equations: bound states

02.10.De Algebraic structures and number theory

02.10.Ud Linear algebra

03.65.Ta Foundations of quantum mechanics; measurement theory

03.65.Fd Algebraic methods

MSC

15A18 Eigenvalues, singular values, and eigenvectors

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other quantum-mechanical equations

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) (See also 42C05 for general orthogonal polynomials and functions)

81U15 Exactly and quasi-solvable systems

81Q60 Supersymmetric quantum mechanics

Subjects

Mathematical physics

Quantum information and quantum mechanics

Dates

Issue 39 (3 October 2008)

Received 8 July 2008, in final form 8 August 2008

Published 29 August 2008



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