Nonlinear Supersymmetry as a Hidden Symmetry

Plyushchay, Mikhail S.

doi:10.1007/978-3-030-20087-9_6

Cited by 4 publications

(3 citation statements)

References 141 publications

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“…We, however, did not compute commutators between ladder operators of different types, but with a quick inspection one can notice that new structures are generated. Though the resulting picture is expected to be rather complicated and requires a separate study, it should be similar to that appearing in the case of ν = 0, which was analyzed in detail in [43], as well as to that in the P T -regularized two-particle Calogero systems [69,70], and can be described as follows. Any extended system composed from a pair of the AFF systems characterized by the parameters ν and ν + m, m ∈ Z, are described, as we showed, by the superconformal osp(2, 2) symmetry in the case of m = 1, while a non-linear deformation of this superalgebra should appear when m > 0.…”

Section: Application: Examplementioning

confidence: 96%

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: Application: Examplementioning

confidence: 96%

Klein four-group and Darboux duality in conformal mechanics

Inzunza

Plyushchay

2019

Phys. Rev. D

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“…Similarly to the two-dimensional anisotropic oscillator [20] and related symmetry algebra, many papers were devoted to two-dimensional superintegrable systems [21,22,23,24] with deformations for which the wavefunctions were written in terms of exceptional orthogonal polynomials [25,26,27,28,29,30,31,32,33,34,35,36,37,38]. Since of their first appearance, exceptional orthogonal polynomials have found application in many different physical contexts [39,40,41,42,43,44,45]. One class of Hamiltonians with richer structures such as different "regimes"for degeneracies were discovered, leading to the path of different types of spectral design for the quantum systems.…”

Section: Introductionmentioning

confidence: 99%

Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators

Latini¹,

Marquette²,

Zhang³

2022

Preprint

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We introduce the general polynomial algebras characterizing a class of higher order superintegrable systems that separate in Cartesian coordinates. The construction relies on underlying polynomial Heisenberg algebras and their defining higher order ladder operators. One feature of these algebras is that they preserve by construction some aspects of the structure of the gl(n) Lie algebra. Among the classes of Hamiltonians arising in this framework are various deformations of harmonic oscillator and singular oscillator related to exceptional orthogonal polynomials and even Painlevé and higher order Painlevé analogs. As an explicit example, we investigate a new three-dimensional superintegrable system related to Hermite exceptional orthogonal polynomials of type III. Among the main results is the determination of the degeneracies of the model in terms of the finite-dimensional irreducible representations of the polynomial algebra.

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“…There is a large literature on the classification and analysis of superintegrable systems (see the review [21]) and they naturally occur in many applications in physics (additional integrals being referred to as 'hidden symmetries' [2,22]).…”

Section: Introductionmentioning

confidence: 99%

The role of commuting operators in quantum superintegrable systems

Fordy

2020

J. Phys. A: Math. Theor.

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We discuss the role of commuting operators for quantum superintegrable systems, showing how they are used to build eigenfunctions. These ideas are illustrated in the context of resonant harmonic oscillators, the Krall-Sheffer operators, with polynomial eigenfunctions, and the Calogero-Moser system with additional harmonic potential. The construction is purely algebraic, avoiding the use of separation of variables and differential equation theory.

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Nonlinear Supersymmetry as a Hidden Symmetry

Cited by 4 publications

References 141 publications

Klein four-group and Darboux duality in conformal mechanics

Klein four-group and Darboux duality in conformal mechanics

Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators

The role of commuting operators in quantum superintegrable systems

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