Equivalence between free quantum particles and those in harmonic potentials and its application to instantaneous changes

Steuernagel, Ole

doi:10.1140/epjp/i2014-14114-3

Cited by 17 publications

(17 citation statements)

References 34 publications

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“…where ξ ′ = dξ/dτ .The total derivative term is essential for getting the QHO's propagator from the free particle system's propagator [57,32]. In the context of (super)conformal symmetry we discuss, it is important that the action potential term A g = g dt x 2 is invariant under the indicated transformation, A g → g dτ ξ 2 .…”

Section: Osp(2|2) and Super-schrödinger Symmetriesmentioning

confidence: 99%

Hidden symmetries of rationally deformed superconformal mechanics

Inzunza

Plyushchay

2019

Phys. Rev. D

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We study the spectrum generating closed nonlinear superconformal algebra that describes N = 2 super-extensions of rationally deformed quantum harmonic oscillator and conformal mechanics models with coupling constant g = m(m + 1), m ∈ N. It has a nature of a nonlinear finite W superalgebra being generated by higher derivative integrals, and generally contains several different copies of either deformed superconformal osp(2|2) algebra in the case of superextended rationally deformed conformal mechanics models, or deformed super-Schrödinger algebra in the case of super-extension of rationally deformed harmonic oscillator systems.

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Section: Osp(2|2) and Super-schrödinger Symmetriesmentioning

confidence: 99%

Hidden symmetries of rationally deformed superconformal mechanics

Inzunza

Plyushchay

2019

Phys. Rev. D

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“…and the first two terms are zero, from Eq. (8). When there is some spatial symmetry in the spline, some of the exponential terms may combine to give sines and cosines.…”

Section: A Linear Splinesmentioning

confidence: 99%

The evolution of piecewise polynomial wave functions

Andrews

2017

Eur. Phys. J. Plus

157

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For a non-relativistic particle, we consider the evolution of wave functions that consist of polynomial segments, usually joined smoothly together. These spline wave functions are compact (that is, they are initially zero outside a finite region), but they immediately extend over all available space as they evolve. The simplest splines are the square and triangular wave functions in one dimension, but very complicated splines have been used in physics. In general the evolution of such spline wave functions can be expressed in terms of antiderivatives of the propagator; in the case of a free particle or an oscillator, all the evolutions are expressed exactly in terms of Fresnel integrals. Some extensions of these methods to two and three dimensions are discussed.

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“…A similar picture also is valid for the two-particle Calogero system without confining potential term and omitted center of mass degree of freedom, i.e. for the system (3.1), and the AFF model (3.7) [2,25,73,74]. The Calogero model and its deformations, in turn, are intimately related to the soliton solutions of the Korteweg-de Vries equation and higher equations of its hierarchy [75,76,77,41].…”

Section: Application: Examplementioning

confidence: 70%

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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Equivalence between free quantum particles and those in harmonic potentials and its application to instantaneous changes

Cited by 17 publications

References 34 publications

Hidden symmetries of rationally deformed superconformal mechanics

Hidden symmetries of rationally deformed superconformal mechanics

The evolution of piecewise polynomial wave functions

Klein four-group and Darboux duality in conformal mechanics

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