A simple generation of exactly solvable anharmonic oscillators

Hussin, Véronique; Mielnik, Bogdan

doi:10.1016/s0375-9601(98)00298-9

Cited by 85 publications

(97 citation statements)

References 39 publications

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“…However, the previous remark will be useful in the interpretation of equations (3) and (4). We can rewrite them as…”

Section: Dilation Symmetry and Reduction Of A Linear Second-order Difmentioning

confidence: 99%

“…Theorem 4.1 (Finite difference Bäcklund algorithm [4,28,34,35]). Let w k (x), w l (x) be two solutions of the Riccati equations…”

Section: Transformation Group On the Set Of Riccati Equationsmentioning

confidence: 99%

“…We have already said that the finite difference algorithm, based on the Theorem 4.1, appeared in [4] when the authors wanted to iterate the standard first order intertwining technique. This idea has been kept also in subsequent works [5,19], and in all of these articles the algorithm has been shown to be of use for obtaining new exactly solvable Hamiltonians.…”

Section: Finite Difference Algorithm and Intertwined Hamiltonians Fromentioning

confidence: 99%

“…It is closely related with the existence of intertwining operators [4,5], Supersymmetric Quantum Mechanics [6] and Darboux transformations [7], among other questions. Indeed, most of the exactly solvable potentials can be obtained by making use of an appropriate intertwining operator transformation.…”

Section: Introductionmentioning

confidence: 99%

“…The key to this technique is to intertwine two different Hamiltonians by a differential operator A, usually of first order [3,4,5,8,9,10] or higher [11,12,13,14,15,16,17,18], provided one of them is solvable. Concerning iterations of the first order intertwining technique, one of the authors (DJF), Hussin and Mielnik [4] have recently used a finite difference algorithm which provides algebraically the solution of the key Riccati equation at a given iteration step in terms of two solutions of the corresponding Riccati equation at the previous step associated to two different 'factorization energies'. This procedure has been successfully applied in order to obtain new exactly solvable Hamiltonians departing from the harmonic oscillator and Coulomb potentials [4,5,19].…”

Section: Introductionmentioning

confidence: 99%

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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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“…However, the previous remark will be useful in the interpretation of equations (3) and (4). We can rewrite them as…”

Section: Dilation Symmetry and Reduction Of A Linear Second-order Difmentioning

confidence: 99%

“…Theorem 4.1 (Finite difference Bäcklund algorithm [4,28,34,35]). Let w k (x), w l (x) be two solutions of the Riccati equations…”

Section: Transformation Group On the Set Of Riccati Equationsmentioning

confidence: 99%

Section: Finite Difference Algorithm and Intertwined Hamiltonians Fromentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Group Theoretical Approach to the Intertwined Hamiltonians

Cariñena¹,

Ramos²,

C.³

2001

Annals of Physics

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Isospectral orthogonal polynomials from the Darboux transforms

Peña

Morales

Zamora-Gallardo

et al. 2004

Int J of Quantum Chemistry

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ABSTRACT:Orthogonal polynomials (OP) are used in many branches of the mathematical and physical sciences; in particular they are part of the eigenfunctions of quantum chemical (QC) potential models. Recently, in the search for new solvable potentials to be useful in QC applications, the use of supersymmetry (SUSY) and Hamiltonian intertwining methods have shown the existence of isospectral potential partners. Also, it has been proved that SUSY is equivalent to the standard Darboux transform (DT) applied to Sturm-Liouville-type problems. Consequently, because of the similarity between the differential equations of OP and of SL, we consider the application of the standard and generalized DT to find the isospectral OP that are partners of the standard one. To attain this purpose, we use a pair of transformations, a point canonical and a gauge, to convert the SL differential equation into a Schrö dingerlike equation allowing OP solutions. As a useful application of our proposal, we consider some of the most important OP by obtaining the generalized isospectral OP partners. Also, we shown the associated isospectral QC potentials for each of the orthogonal polynomials considered in this work. The proposed procedure can be used to obtain new orthogonal polynomials as well as to find new solvable QC potential models.

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Exactly solvable quantum potentials with special functions solutions

Peña

Ovando

Morales

et al. 2008

Int J of Quantum Chemistry

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ABSTRACT:In this work, a canonical point and a gauge transformation applied to the g(x) times differential equations (DE) for special functions (SF) is used to convert it into a Schrödinger-like equation. This method leads to exactly solvable potentials allowing special functions solutions. For specific applications, keeping in mind the general DE for SF, the choice of the coefficients that identify the SF under study allows to find the equivalent of the Witten superpotential and wavefunctions for each g (x). That is, the choice of a particular g(x) leads straightforwardly to those exactly solvable potentials having as wavefunctions the SF under consideration. As a useful application of the proposed method, we considered explicitly the potentials associated to the standard Hermite, Laguerre, Legendre, and Bessel SF in the case of some particular forms of g (x). However, the proposed approach can be directly applied to any other standard SF with the aim to obtain exactly solvable potentials that can be used in the modelling of quantum chemical applications.

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A simple generation of exactly solvable anharmonic oscillators

Cited by 85 publications

References 39 publications

Group Theoretical Approach to the Intertwined Hamiltonians

Group Theoretical Approach to the Intertwined Hamiltonians

Isospectral orthogonal polynomials from the Darboux transforms

Exactly solvable quantum potentials with special functions solutions

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