Rationally-extended radial oscillators and Laguerre exceptional
  orthogonal polynomials in kth-order SUSYQM

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Abstract

A previous study of exactly solvable rationally-extended radial oscillator potentials and corresponding Laguerre exceptional orthogonal polynomials carried out in second-order supersymmetric quantum mechanics is extended to \(k\)th-order one. The polynomial appearing in the potential denominator and its degree are determined. The first-order differential relations allowing one to obtain the associated exceptional orthogonal polynomials from those arising in a (\(k-1\))th-order analysis are established. Some nontrivial identities connecting products of Laguerre polynomials are derived from shape invariance.

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Is Open Access

Generation of a Complete Set of Supersymmetric Shape Invariant Potentials from an Euler Equation

Jeffry V. Mallow, Asim Gangopadhyaya, Jonathan Bougie (2010)

In supersymmetric quantum mechanics, shape invariance is a sufficient condition for solvability. We show that all conventional additive shape invariant superpotentials that are independent of \(\hbar\) obey two partial differential equations. One of these is equivalent to the one-dimensional Euler equation expressing momentum conservation for inviscid fluid flow, and it is closed by the other. We solve these equations, generate the set of all conventional shape invariant superpotentials, and show that there are no others in this category. We then develop an algorithm for generating all additive shape invariant superpotentials including those that depend on \(\hbar\) explicitly.