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

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.

22 pages, published version

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• ABC of ladder operators for rationally extended quantum harmonic oscillator systems
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• Algebraic calculations for spectrum of superintegrable system from exceptional orthogonal polynomials
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• Nonlinear supersymmetry as a hidden symmetry
• Higher order superintegrability, Painlevé transcendents and representations of polynomial algebras
• New families of superintegrable systems from k-step rational extensions, polynomial algebras and degeneracies
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• Recurrence approach and higher rank polynomial algebras for superintegrable monopole systems
• Recurrence relations and general solution of the exceptional Hermite equation
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• Lectures on exceptional orthogonal polynomials and rational solutions to Painlevé equations
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