Higher-order supersymmetric quantum mechanics

Fernández, J C David; Fernández‐García, Nicolás

doi:10.1063/1.1853203

Cited by 111 publications

(149 citation statements)

References 72 publications

(191 reference statements)

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“…These equations immediately lead to the higher-order SUSY QM [16][17][18][19][20][21][22][23][24][25][26][27][28][29]. In this treatment, the standard SUSY algebra with two generators…”

Section: Higher-order Susy Qmmentioning

confidence: 99%

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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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“…These equations immediately lead to the higher-order SUSY QM [16][17][18][19][20][21][22][23][24][25][26][27][28][29]. In this treatment, the standard SUSY algebra with two generators…”

Section: Higher-order Susy Qmmentioning

confidence: 99%

“…Then, starting from H 0 we have generated a chain of factorized Hamiltonians in the way 27) where the final potential V k (x) can be determined using equations (2.23) and (2.24). In addition, since we are departing from k solutions of the initial Riccati equation, {α 1 (x, i ); i = 1, .…”

Section: Higher-order Susy Qmmentioning

confidence: 99%

Supersymmetric quantum mechanics and Painlevé equations

Bermúdez

Fernández

2014

AIP Conference Proceedings

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“…22,23,[39][40][41] The correspondence between the initial and final Hamiltonians H (1) and H (2) is determined by the intertwining relations…”

Section: Laguerre Eop and Reducible Kth Order Susyqmmentioning

confidence: 99%

Disconjugacy, regularity of multi-indexed rationally extended potentials, and Laguerre exceptional polynomials

Grandati

Quesne

2013

Journal of Mathematical Physics

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The power of the disconjugacy properties of second-order differential equations of Schrödinger type to check the regularity of rationally extended quantum potentials connected with exceptional orthogonal polynomials is illustrated by re-examining the extensions of the isotonic oscillator (or radial oscillator) potential derived in kth-order supersymmetric quantum mechanics or multistep Darboux-Bäcklund transformation method. The function arising in the potential denominator is proved to be a polynomial with a nonvanishing constant term, whose value is calculated by induction over k. The sign of this term being the same as that of the already known highest degree term, the potential denominator has the same sign at both extremities of the definition interval, a property that is shared by the seed eigenfunction used in the potential construction. By virtue of disconjugacy, such a property implies the nodeless character of both the eigenfunction and the resulting potential. C 2013 AIP Publishing LLC.

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“…In general, to obtain a SUSY partner from a given initial potential, one uses eigenfunctions of such Hamiltonian having specific properties [24][25][26][27]. In our case, we will use just the wave functions of the anti-bound states in order to obtain a hierarchy of SUSY partners of the hyperbolic step potential called Rosen Morse II potentials.…”

Section: Introductionmentioning

confidence: 99%

The hyperbolic step potential: Anti-bound states, SUSY partners and Wigner time delays

2017

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We study the scattering produced by a one dimensional hyperbolic step potential, which is exactly solvable and shows an unusual interest because of its asymmetric character. The analytic continuation of the scattering matrix in the momentum representation has a branch cut and an infinite number of simple poles on the negative imaginary axis which are related with the so called anti-bound states. This model does not show resonances. Using the wave functions of the antibound states, we obtain supersymmetric (SUSY) partners which are the series of Rosen Morse II potentials. We have computed the Wigner reflection and transmission time delays for the hyperbolic step and such SUSY partners. Our results show that the more bound states a partner Hamiltonian has the smaller is the time delay. We also have evaluated time delays for the hyperbolic step potential in the classical case and have obtained striking similitudes with the quantum case.

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Higher-order supersymmetric quantum mechanics

Cited by 111 publications

References 72 publications

Supersymmetric quantum mechanics and Painlevé equations

Supersymmetric quantum mechanics and Painlevé equations

Disconjugacy, regularity of multi-indexed rationally extended potentials, and Laguerre exceptional polynomials

The hyperbolic step potential: Anti-bound states, SUSY partners and Wigner time delays

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