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      Solvable Rational Potentials and Exceptional Orthogonal Polynomials in Supersymmetric Quantum Mechanics

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          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.

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          The Factorization Method

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            Supersymmetry and Quantum Mechanics

            , , (2010)
            In the past ten years, the ideas of supersymmetry have been profitably applied to many nonrelativistic quantum mechanical problems. In particular, there is now a much deeper understanding of why certain potentials are analytically solvable and an array of powerful new approximation methods for handling potentials which are not exactly solvable. In this report, we review the theoretical formulation of supersymmetric quantum mechanics and discuss many applications. Exactly solvable potentials can be understood in terms of a few basic ideas which include supersymmetric partner potentials, shape invariance and operator transformations. Familiar solvable potentials all have the property of shape invariance. We describe new exactly solvable shape invariant potentials which include the recently discovered self-similar potentials as a special case. The connection between inverse scattering, isospectral potentials and supersymmetric quantum mechanics is discussed and multi-soliton solutions of the KdV equation are constructed. Approximation methods are also discussed within the framework of supersymmetric quantum mechanics and in particular it is shown that a supersymmetry inspired WKB approximation is exact for a class of shape invariant potentials. Supersymmetry ideas give particularly nice results for the tunneling rate in a double well potential and for improving large \(N\) expansions. We also discuss the problem of a charged Dirac particle in an external magnetic field and other potentials in terms of supersymmetric quantum mechanics. Finally, we discuss structures more general than supersymmetric quantum mechanics such as parasupersymmetric quantum mechanics in which there is a symmetry between a boson and a para-fermion of order \(p\).
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              On the Connection between Phase Shifts and Scattering Potential

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                Author and article information

                Journal
                12 June 2009
                2009-08-21
                Article
                10.3842/SIGMA.2009.084
                0906.2331
                d4d09659-7347-40cb-abbb-7ff9e532ed23

                http://creativecommons.org/licenses/by-nc-sa/3.0/

                History
                Custom metadata
                ULB/229/CQ/09/2
                SIGMA 5 (2009), 084, 24 pages
                v2: additions in secs. 1 and 4, 5 new references; v3: published version
                math-ph math.MP quant-ph

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