Supersymmetric quantum mechanics and Painlevé equations

Bermúdez, David; Fernández, J C David

doi:10.1063/1.4861699

Cited by 35 publications

(37 citation statements)

References 61 publications

(74 reference statements)

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“…Let us note that some rational PIV solutions derived here coincide with several ones contained in tables 26.1 and 26.2 of [30]. A further study of the hierarchies of PIV solutions which can be generated by applying the SUSY techniques to this truncated harmonic oscillator is still required (see however [12,28,29,45]).…”

Section: Discussionmentioning

confidence: 74%

Supersymmetric partners of the harmonic oscillator with an infinite potential barrier

C.¹,

Morales-Salgado²

2013

J. Phys. A: Math. Theor.

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Supersymmetry transformations of first and second order are used to generate Hamiltonians with known spectra departing from the harmonic oscillator with an infinite potential barrier. It is studied also the way in which the eigenfunctions of the initial Hamiltonian are transformed. The first and certain second order supersymmetric partners of the initial Hamiltonian possess third-order differential ladder operators. Since systems with this kind of operators are linked with the Painlevé IV equation, several solutions of this non-linear second-order differential equation will be simply found.

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Section: Discussionmentioning

confidence: 74%

Supersymmetric partners of the harmonic oscillator with an infinite potential barrier

C.¹,

Morales-Salgado²

2013

J. Phys. A: Math. Theor.

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“…In the quantum mechanical context, it is related to the method of factorization first proposed by Schrodinger, and SQM has led to substantial advances in the field of integrable systems. 10 In nonlinear filtering theory (and numerous other applications such as time evolution of gene pool and the dynamical model of neural activity) one is led to consider the Fokker-Planck and Kolmogorov equations. For a certain class of Langevin equations, this is mathematically related to Euclidean quantum mechanics.…”

Section: Introductionmentioning

confidence: 99%

Supersymmetry and nonlinear filtering: operator perspective

Balaji

2014

SPIE Proceedings

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Supersymmetry (SUSY), or Bose-Fermi symmetry, is an attempt to provide a unified description of all of the fundamental interactions. Although originally introduced in the quantum field theory context, it was noted by Witten that the understanding of certain features of SUSY was simpler in the quantum mechanical setting. Some aspects of the vast subject of supersymmetric quantum mechanics, and the supersymmetric Fokker-Planck equation, of relavance to continuous-time nonlinear filtering theory are briefly reviewed. The applicability of certain remarkable results in supersymmetric quantum mechanics to nonlinear filtering is noted and illustrated with a few examples.

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“…Some results obtained in [8,9,27] are used and adapted to our context. As general references related to these subjects, we recommend [3,13,17,18,19,28,30].…”

Section: Introductionmentioning

confidence: 99%

Darboux transformations and second order difference equations

Dobrogowska,

2018

Preprint

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In this paper we implement the Darboux transformation, as well as an analogue of Crum's theorem, for a discrete version of Schrödinger equation. The technique is based on the use of first order operators intertwining two difference operators of second order. This method, which has been applied successfully for differential cases, leads also to interesting non trivial results in the discrete case. The technique allows us to construct the solutions for a wide class of difference Schrödinger equations. The exact solutions for some special potentials are also found explicitly.

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Supersymmetric quantum mechanics and Painlevé equations

Cited by 35 publications

References 61 publications

Supersymmetric partners of the harmonic oscillator with an infinite potential barrier

Supersymmetric partners of the harmonic oscillator with an infinite potential barrier

Supersymmetry and nonlinear filtering: operator perspective

Darboux transformations and second order difference equations

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