SUSY partners of the truncated oscillator, Painlevé transcendents and Bäcklund transformations

C., David J. Fernández; Morales-Salgado, V S

doi:10.1088/1751-8113/49/19/195202

Cited by 12 publications

(4 citation statements)

References 46 publications

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“…Now let us turn to the equations (25)- (26). From the equation (25), C r = 0, we find the function G 4 (r, θ):…”

Section: B the Linear Compatibility Conditionmentioning

confidence: 99%

“…At this point, all eight coefficients A rrr , A rrθ , A rθθ , A θθθ , B rr , B rθ , B θθ , C r in (12) vanish. In fact, the main equation to be solved is C θ = 0, presented in (26).…”

Section: B the Linear Compatibility Conditionmentioning

confidence: 99%

“…Substituting G 1 , G 2 , G 3 , G 4 into the determining equation C θ = 0, (26), and collecting powers of r we get three equations that must be satisfied simultaneously in order for Y in (5) to be an integral of motion:…”

Section: B the Linear Compatibility Conditionmentioning

confidence: 99%

“…Pioneering work is due to Drach 22,23 . For more recent work see 6,[11][12][13]26,33,34,36,37,[50][51][52]58,[60][61][62][63]65 .…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Fourth order superintegrable systems separating in polar coordinates. I. Exotic potentials

Escobar-Ruiz¹,

Vieyra²,

Winternitz³

et al. 2017

J. Phys. A: Math. Theor.

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We present all real quantum mechanical potentials in a two-dimensional Euclidean space that have the following properties: 1. They allow separation of variables of the Schrödinger equation in polar coordinates, 2. They allow an independent fourth order integral of motion, 3. It turns out that their angular dependent part S(θ) does not satisfy any linear equation. In this case S(θ) satisfies a nonlinear ODE that has the Painlevé property and its solutions can be expressed in terms of the Painlevé transcendent P 6 . We also study the corresponding classical analogs of these potentials. The polynomial algebra of the integrals of motion is constructed in the classical case.

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“…Now let us turn to the equations (25)- (26). From the equation (25), C r = 0, we find the function G 4 (r, θ):…”

Section: B the Linear Compatibility Conditionmentioning

confidence: 99%

Section: B the Linear Compatibility Conditionmentioning

confidence: 99%

Section: B the Linear Compatibility Conditionmentioning

confidence: 99%

“…Pioneering work is due to Drach 22,23 . For more recent work see 6,[11][12][13]26,33,34,36,37,[50][51][52]58,[60][61][62][63]65 .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Fourth order superintegrable systems separating in polar coordinates. I. Exotic potentials

Escobar-Ruiz¹,

Vieyra²,

Winternitz³

et al. 2017

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

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Affine Weyl group symmetries of Frobenius Painlevé equations

Wang

2019

Math Methods in App Sciences

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In this paper, we introduce a Frobenius Painlevé IV equation and the corresponding Hamilton system, and we give the symmetric form of the Frobenius Painlevé IV equation. Then, we construct the Lax pair of the Frobenius Painlevé IV equation. Furthermore, we recall the Frobenius modified KP hierarchy and the Frobenius KP hierarchy by bilinear equations, then we show how to get Frobenius Painlevé IV equation from the Frobenius modified KP hierarchy. In order to study the different aspects of the Frobenius Painlevé IV equation, we give the similarity reduction and affine Weyl group symmetry of the equation. Similarly, we introduce a Frobenius Painlevé II equation and show the connection between the Frobenius modified KP hierarchy and the Frobenius Painlevé II equation.

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Trends in Supersymmetric Quantum Mechanics

David²

2019

Integrability, Supersymmetry and Coherent States

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Along the years, supersymmetric quantum mechanics (SUSY QM) has been used for studying solvable quantum potentials. It is the simplest method to build Hamiltonians with prescribed spectra in the spectral design. The key is to pair two Hamiltonians through a finite order differential operator. Some related subjects can be simply analyzed, as the algebras ruling both Hamiltonians and the associated coherent states. The technique has been applied also to periodic potentials, where the spectra consist of allowed and forbidden energy bands. In addition, a link with nonlinear second-order differential equations, and the possibility of generating some solutions, can be explored. Recent applications concern the study of Dirac electrons in graphene placed either in electric or magnetic fields, and the analysis of optical systems whose relevant equations are the same as those of SUSY QM. These issues will be reviewed briefly in this paper, trying to identify the most important subjects explored currently in the literature. KeywordsSupersymmetric quantum mechanics • Coherent states • Painlevé equations • Painlevé transcendents • Polynomial Heisenberg algebras • Factorization method • Exact solutions • Spectral design • Graphene Dedicated to my dear friend and colleague Véronique Hussin. D. J. Fernández C ( )

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SUSY partners of the truncated oscillator, Painlevé transcendents and Bäcklund transformations

Cited by 12 publications

References 46 publications

Fourth order superintegrable systems separating in polar coordinates. I. Exotic potentials

Fourth order superintegrable systems separating in polar coordinates. I. Exotic potentials

Affine Weyl group symmetries of Frobenius Painlevé equations

Trends in Supersymmetric Quantum Mechanics

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