New ladder operators for a rational extension of the harmonic oscillator
  and superintegrability of some two-dimensional systems

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Abstract

New ladder operators are constructed for a rational extension of the harmonic oscillator associated with type III Hermite exceptional orthogonal polynomials and characterized by an even integer \(m\). The eigenstates of the Hamiltonian separate into \(m+1\) infinite-dimensional unitary irreducible representations of the corresponding polynomial Heisenberg algebra. These ladder operators are used to construct a higher-order integral of motion for two superintegrable two-dimensional systems separable in cartesian coordinates. The polynomial algebras of such systems provide for the first time an algebraic derivation of the whole spectrum through their finite-dimensional unitary irreducible representations.

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Dressing chains and the spectral theory of the Schr�dinger operator

A. P. Veselov, A. B. Shabat (1993)

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Solvable Rational Potentials and Exceptional Orthogonal Polynomials in Supersymmetric Quantum Mechanics

Christiane Quesne (2009)

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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Superintegrable Systems in Darboux spaces

W. Miller, J M Kress, E. G. Kalnins … (2003)

Almost all research on superintegrable potentials concerns spaces of constant curvature. In this paper we find by exhaustive calculation, all superintegrable potentials in the four Darboux spaces of revolution that have at least two integrals of motion quadratic in the momenta, in addition to the Hamiltonian. These are two-dimensional spaces of nonconstant curvature. It turns out that all of these potentials are equivalent to superintegrable potentials in complex Euclidean 2-space or on the complex 2-sphere, via "coupling constant metamorphosis" (or equivalently, via Staeckel multiplier transformations). We present tables of the results.