Group Approach to the Factorization of the Radial Oscillator Equation

Fernández, J C David; Negro, J.; Olmo, M. A. del

doi:10.1006/aphy.1996.0138

Cited by 63 publications

(53 citation statements)

References 15 publications

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“…If b = 2 these eigenvalues reduce to those of [38]. Compare also with the eigenfunctions and eigenvalues given in [15, pp.…”

Section: Radial Oscillator-like Potentialssupporting

confidence: 52%

Group Theoretical Approach to the Intertwined Hamiltonians

Cariñena¹,

Ramos²,

C.³

2001

Annals of Physics

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We show that the finite difference Bäcklund formula for the Schrödinger Hamiltonians is a particular element of the transformation group on the set of Riccati equations considered by two of us in a previous paper. Then, we give a group theoretical explanation to the problem of Hamiltonians related by a first order differential operator. A generalization of the finite difference algorithm relating eigenfunctions of three different Hamiltonians is found, and some illustrative examples of the theory are analyzed, finding new potentials for which one eigenfunction and its corresponding eigenvalue is exactly known.

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“…If b = 2 these eigenvalues reduce to those of [38]. Compare also with the eigenfunctions and eigenvalues given in [15, pp.…”

Section: Radial Oscillator-like Potentialssupporting

confidence: 52%

Group Theoretical Approach to the Intertwined Hamiltonians

Cariñena¹,

Ramos²,

C.³

2001

Annals of Physics

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“…To perform the higher-order SUSY transformations onto this potential we will explore once again the factorization method [8,31,32]. The radial oscillator Hamiltonian can be factorized directly in four different ways.…”

Section: Example 2 the Radial Oscillatormentioning

confidence: 99%

Supersymmetric quantum mechanics and Painlevé equations

Bermúdez

Fernández

2014

AIP Conference Proceedings

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In these lecture notes we shall study first the supersymmetric quantum mechanics (SUSY QM), specially when applied to the harmonic and radial oscillators. In addition, we will define the polynomial Heisenberg algebras (PHA), and we will study the general systems ruled by them: for zero and first order we obtain the harmonic and radial oscillators, respectively; for second and third order PHA the potential is determined by solutions to Painlevé IV (PIV) and Painlevé V (PV) equations. Taking advantage of this connection, later on we will find solutions to PIV and PV equations expressed in terms of confluent hypergeometric functions. Furthermore, we will classify them into several solution hierarchies, according to the specific special functions they are connected with.

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“…It has been shown in previous references [27] that this radial Hamiltonian can be factorized in two ways by means of two sets of differential operators…”

Section: Quantum Algebraic Treatmentmentioning

confidence: 99%

Superintegrability of the Fock–Darwin system

et al. 2017

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The Fock-Darwin system is analysed from the point of view of its symmetry properties in the quantum and classical frameworks. The quantum Fock-Darwin system is known to have two sets of ladder operators, a fact which guarantees its solvability. We show that for rational values of the quotient of two relevant frequencies, this system is superintegrable, the quantum symmetries being responsible for the degeneracy of the energy levels. These symmetries are of higher order and close a polynomial algebra. In the classical case, the ladder operators are replaced by ladder functions and the symmetries by constants of motion. We also prove that the rational classical system is superintegrable and its trajectories are closed. The constants of motion are also generators of symmetry transformations in the phase space that have been integrated for some special cases. These transformations connect different trajectories with the same energy. The coherent states of the quantum superintegrable system are found and they reproduce the closed trajectories of the classical one.

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Group Approach to the Factorization of the Radial Oscillator Equation

Cited by 63 publications

References 15 publications

Group Theoretical Approach to the Intertwined Hamiltonians

Group Theoretical Approach to the Intertwined Hamiltonians

Supersymmetric quantum mechanics and Painlevé equations

Superintegrability of the Fock–Darwin system

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