Supersymmetric partners for the associated Lamé potentials

Fernandez, Clara; Ganguly, Asish

doi:10.1134/s1063778810020146

Cited by 2 publications

(2 citation statements)

References 21 publications

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“…Although in this case the spectrum consists of an infinite number of nondegenerate discrete energy levels, the method works as well for Hamiltonians with mixed spectrum (discrete and continuous) or even when there is just a continuous one (see, e.g., [173]). This is what happens for periodic potentials [102][103][104][105][106][107][108][109],…”

Section: Susy Qm and Exactly Solvable Potentialsmentioning

confidence: 91%

“…In addition, the analysis of the confluent algorithm, the degenerate case in which all the factorization energies tend to a single one, was also elaborated [74,[93][94][95][96][97][98][99][100][101]. The SUSY techniques for exactly solvable periodic potentials, as the Lamé and associated Lamé potentials, have been explored as well [102][103][104][105][106][107][108][109]. Some other groups have addressed the same subjects through different viewpoints, e.g., the N -fold supersymmetry by Tanaka and collaborators [110][111][112][113][114], the hidden non-linear supersymmetry by Plyushchay et al [115][116][117][118][119], among others.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Susy Qm and Exactly Solvable Potentialsmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

Trends in Supersymmetric Quantum Mechanics

David²

2019

Integrability, Supersymmetry and Coherent States

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Soliton defects in one-gap periodic system and exotic supersymmetry

et al. 2014

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By applying Darboux-Crum transformations to the quantum one-gap Lamé system, we introduce an arbitrary countable number of bound states into forbidden bands. The perturbed potentials are reflectionless and contain two types of soliton defects in the periodic background. The bound states with finite number of nodes are supported in the lower forbidden band by the periodicity defects of the potential well type, while the pulse type bound states in the gap have infinite number of nodes and are trapped by defects of the compression modulations nature. We investigate the exotic nonlinear N = 4 supersymmetric structure in such paired Schrödinger systems, which extends an ordinary N = 2 supersymmetry and involves two bosonic generators composed from Lax-Novikov integrals of the subsystems. One of the bosonic integrals has a nature of a central charge, and allows us to liaise the obtained systems with the stationary equations of the Korteweg-de Vries and modified Korteweg-de Vries hierarchies. This exotic supersymmetry opens the way for the construction of self-consistent condensates based on the Bogoliubov-de Gennes equations and associated with them new solutions to the Gross-Neveu model. They correspond to the kink or kink-antikink defects of the crystalline background in dependence on whether the exotic supersymmetry is unbroken or spontaneously broken.

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Supersymmetric partners for the associated Lamé potentials

Abstract: The general solution of the stationary Schrödinger equation for the associated Lamé potentials with an arbitrary real energy is found. The supersymmetric partners are generated by employing seeds solutions for factorization energies inside the gaps.

Cited by 2 publications

References 21 publications

Trends in Supersymmetric Quantum Mechanics

Trends in Supersymmetric Quantum Mechanics

Soliton defects in one-gap periodic system and exotic supersymmetry

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