New families of superintegrable systems from Hermite and Laguerre exceptional orthogonal polynomials

Abstract

In recent years, many exceptional orthogonal polynomials (EOP) were introduced and used to construct new families of 1D exactly solvable quantum potentials, some of which are shape invariant. In this paper, we construct from Hermite and Laguerre EOP and their related quantum systems new 2D superintegrable Hamiltonians with higher-order integrals of motion and the polynomial algebras generated by their integrals of motion. We obtain the finite-dimensional unitary representations of the polynomial algebras and the corresponding energy spectrum. We also point out a new type of degeneracies of the energy levels of these systems that is associated with holes in sequences of EOP.

24 pages

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Cited by in corpus (24)

• Classical and Quantum Superintegrability with Applications
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• Trends in supersymmetric quantum mechanics
• Exceptional Laguerre polynomials
• Combined state-adding and state-deleting approaches to type III multi-step rationally-extended potentials: applications to ladder operators and superintegrability
• Connection between quantum systems involving the fourth Painleve transcendent and $k$-step rational extensions of the harmonic oscillator related to Hermite EOP
• On realizations of polynomial algebras with three generators via deformed oscillator algebras
• Multi-indexed Jacobi polynomials and Maya diagrams
• DARBOUX partners of pseudoscalar Dirac potentials associated with exceptional orthogonal polynomials
• New quasi-exactly solvable class of generalized isotonic oscillators
• Shape invariance and equivalence relations for pseudowronskians of Laguerre and Jacobi polynomials
• Ladder operators for solvable potentials connected with exceptional orthogonal polynomials
• Higher Order Quantum Superintegrability: a new "Painlevé conjecture"
• Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators
• Algebraic calculations for spectrum of superintegrable system from exceptional orthogonal polynomials
• Quantum, classical symmetries and action-angle variables by factorization of superintegrable systems
• New families of superintegrable systems from k-step rational extensions, polynomial algebras and degeneracies
• Relativistic Potentials with Rational Extensions
• Nonstationary deformed singular oscillator: quantum invariants and the factorization method
• A family of fourth-order superintegable systems with rational potentials related to Painlevé VI
• Recurrence relations and general solution of the exceptional Hermite equation
• SUSY partners and $S$-matrix poles of the one dimensional Rosen-Morse II Hamiltonian
• Cubic algebras, induced representations and general solution of the exceptional Laguerre equation $X_1$