Wronskian differential formula for confluent supersymmetric quantum mechanics

Bermúdez, David; Fernández, J C David; Fernández‐García, Nicolás

doi:10.1016/j.physleta.2011.12.020

Cited by 36 publications

(54 citation statements)

References 36 publications

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“…In this way we generalize our earlier results obtained for the restricted case of the AFF model with integer values of ν only [42,43], that were based on the Darboux transformations of the quantum harmonic oscillator. Coherently with the indicated above peculiarity of the half-integer values of the parameter ν from the point of view of the Klein four-group transformations, we will see how the Jordan states [44,45,46,47,48,41,49] enter the construction at ν = Z + 1/2 via the confluent Darboux transformations. We also trace out the structural changes in the spectra of the rationally deformed AFF systems which happen at half-integer ν under continuous variation of this parameter.…”

Section: Introductionmentioning

confidence: 78%

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Klein four-group and Darboux duality in conformal mechanics

Inzunza

Plyushchay

2019

Phys. Rev. D

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We study the Klein four-group (K 4 ) symmetry of the time-dependent Schrödinger equation for the conformal mechanics model of de Alfaro-Fubini-Furlan (AFF) with confining harmonic potential and coupling constant g = ν(ν + 1) ≥ −1/4. We show that it undergoes a complete or partial (at half-integer ν) breaking on eigenstates of the system, and is the automorphism of the osp(2, 2) superconformal symmetry in super-extensions of the model by inducing a transformation between the exact and spontaneously broken phases of N = 2 Poincaré supersymmetry. We exploit the K 4 symmetry and its relation with the conformal symmetry to construct the dual Darboux transformations which generate spectrally shifted pairs of the rationally deformed AFF models. Two distinct pairs of intertwining operators originated from Darboux duality allow us to construct complete sets of the spectrum generating ladder operators that identify specific finite-gap structure of a deformed system and generate three distinct related versions of nonlinearly deformed sl(2, R) algebra as its symmetry. We show that at half-integer ν, the Jordan states associated with confluent Darboux transformations enter the construction, and the spectrum of rationally deformed AFF systems undergoes structural changes.constant g = ν(ν + 1) ≥ −1/4 1 . Its non-relativistic conformal symmetry and supersymmetric extensions [5,6,7,8,9,10] find a variety of interesting applications including the particles dynamics in black hole backgrounds [11,12,13,14,15,16], cosmology [17,18,19], non-relativistic AdS/CFT correspondence [20,21,22,23,24], QCD confinement problem [25,26], and physics of Bose-Einstein condensates [27,28].On the other hand, the time-dependent Schrödinger equation for the AFF conformal mechanics model reveals a discrete Klein four-group symmetry generated by transformation of the parameter ν → −ν − 1, and by the spatial Wick rotation x → ix accompanied by the time reflection t → −t. In the picture of the stationary Schrödinger equation the time reflection transforms into the change of the eigenvalue's sign E → −E. The discrete symmetry K 4 , however, turns out to be completely broken at the level of the quantum states when ν is not a half-integer number : application of the group generators to physical eigenstates produces formal eigenstates which do not satisfy the necessary boundary conditions. In the case of half-integer values of the parameter ν the K 4 discrete symmetry breaks partially, and transformation ν → −ν − 1, as we shall see, turns out to be a true symmetry nontrivially realized on the spectrum of the system. In the physics of anyons, where the AFF model is used to generate the transmutation of statistics, halfinteger values of ν correspond to the two-particle system of identical fermions [29,30,31]. In the context of the problem we consider here, even though the new solutions with arbitrary value of ν generated by transformations of the discrete group are not acceptable from the physical point of view, the analogs of such non-physical states in other q...

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Section: Introductionmentioning

confidence: 78%

“…Following the analysis and ideas presented in refs. [44,45,46,47,49], the constructions can be generalized to the case of higher order Jordan states defined via relations (L − λ * )Ω (0) * = ψ * , (L − λ * )Ω (k) * = Ω (k−1) * , k = 1, . .…”

Section: Application: Examplementioning

confidence: 99%

Klein four-group and Darboux duality in conformal mechanics

Inzunza

Plyushchay

2019

Phys. Rev. D

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“…In the confluent SUSY algorithm, the transformation functions form a Jordan chain of the Hamiltonian that governs the initial quantum problem, that is, they are generalized eigenvectors of that Hamiltonian [2]. As such, the transformation functions admit both an integral and a differential representation [20] [7], the relationship between which was reported on recently [9] [19]. One of the principal open questions regarding the confluent SUSY algorithm concerns the regularity of the potential in the SUSY-transformed system.…”

Section: Introductionmentioning

confidence: 99%

Recursive representation of Wronskians in confluent supersymmetric quantum mechanics

Contreras-Astorga¹,

Schulze-Halberg²

2017

J. Phys. A: Math. Theor.

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A recursive form of arbitrary-order Wronskian associated with transformation functions in the confluent algorithm of supersymmetric quantum mechanics (SUSY) is constructed. With this recursive form regularity conditions for the generated potentials can be analyzed. Moreover, as byproducts we obtain new representations of solutions to Schrödinger equations that underwent a confluent SUSY-transformation.

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“…Recent works on the confluent algorithm include its application to the inverted oscillator [7] and to the Dirac equation for pseudoscalar potentials [8]. Further results concern its equivalence with the zero-mode supersymmetry scheme [9], a representation using a differential formula [10,11] and higher-order confluent SUSY transformations [12]. The purpose of the present note is to apply the confluent SUSY algorithm to models that are governed by the spheroidal equation [13], a particular case of the confluent Heun class [14].…”

Section: Introductionmentioning

confidence: 99%

“…By combining the point transformation (35) with the series (10) and (11), we obtain solutions Ψ j , j a nonnegative integer, of bound-state type to our boundary-value problem (36), (37) in the form…”

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confidence: 99%