Connection between quantum systems involving the fourth Painleve transcendent and $k$-step rational extensions of the harmonic oscillator related to Hermite EOP

Abstract

The purpose of this communication is to point out the connection between a 1D quantum Hamiltonian involving the fourth Painleve transcendent P${}_{\mathrm{IV}}$, obtained in the context of second-order supersymmetric quantum mechanics and third-order ladder operators, with a hierarchy of families of quantum systems called $k$-step rational extensions of the harmonic oscillator and related with multi-indexed $X_{m_1,m_2,...,m_k}$ Hermite exceptionnal orthogonal polynomials of type III. The connection between these exactly solvable models is established at the level of the equivalence of the Hamiltonians using rational solutions of the fourth Painleve equation in terms of generalized Hermite and Okamoto polynomials. We also relate the different ladder operators obtained by various combinations of supersymmetric constructions involving Darboux-Crum and Krein-Adler supercharges, their zero modes and the corresponding energies. These results will demonstrate and clarify the relation observed for a particular case in previous papers.

27 pages

References in corpus (7)

• Hamiltonians separable in cartesian coordinates and third-order integrals of motion
• Superintegrability with third order integrals of motion, cubic algebras and supersymmetric quantum mechanics I:Rational function potentials
• Combined state-adding and state-deleting approaches to type III multi-step rationally-extended potentials: applications to ladder operators and superintegrability
• An infinite family of superintegrable systems from higher order ladder operators and supersymmetry
• Non-hermitian Hamiltonians and Painlevé IV equation with real parameters
• Complex oscillator and Painlevé IV equation
• Painlevé IV Coherent States

Cited by in corpus (17)

• Trends in supersymmetric quantum mechanics
• Poles of Painlevé IV Rationals and their Distribution
• Higher Order Quantum Superintegrability: a new "Painlevé conjecture"
• Fifth-order superintergrable quantum system separating in Cartesian coordinates. Doubly exotic potentials
• Third-order ladder operators, generalized Okamoto and exceptional orthogonal polynomials
• A fourth-order superintegrable system with a rational potential related to Painleve VI
• On Integrable Ermakov-Painlevé IV Systems
• Fourth Painlevé and Ermakov equations: quantum invariants and new exactly-solvable time-dependent Hamiltonians
• Higher order superintegrability, Painlevé transcendents and representations of polynomial algebras
• A family of fourth-order superintegable systems with rational potentials related to Painlevé VI
• An affine Weyl group characterization of polynomial Heisenberg algebras
• Recurrence relations and general solution of the exceptional Hermite equation
• On the general family of third-order shape-invariant Hamiltonians related to generalized Hermite polynomials
• Polynomially Superintegrable Hamiltonians Separating in Cartesian Coordinates
• Lectures on exceptional orthogonal polynomials and rational solutions to Painlevé equations
• Complete classification of rational solutions of $A_{2n}$-Painlevé systems
• Representations of quadratic Heisenberg-Weyl algebras and polynomials in the fourth Painlevé transcendent