Abstract
We study the Klein four-group (${K}_{4}$) symmetry of the time-dependent Schr\"odinger equation for the conformal mechanics model of de Alfaro-Fubini-Furlan (AFF) with confining harmonic potential and coupling constant $g=\ensuremath{\nu}(\ensuremath{\nu}+1)\ensuremath{\ge}\ensuremath{-}1/4$. We show that it undergoes a complete or partial (at half-integer $\ensuremath{\nu}$) breaking on eigenstates of the system, and is the automorphism of the $\mathfrak{osp}(2,2)$ superconformal symmetry in super-extensions of the model by inducing a transformation between the exact and spontaneously broken phases of $\mathcal{N}=2$ Poincar\'e 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 $\mathfrak{sl}(2,\mathbb{R})$ algebra as its symmetry. We show that at half-integer $\ensuremath{\nu}$, the Jordan states associated with confluent Darboux transformations enter the construction, and the spectrum of rationally deformed AFF systems undergoes structural changes.
Highlights
In quantum mechanics, symmetries map the states of a system into its states
An interesting and important case from this point of view is presented by the conformal mechanics model of de Alfaro, Fubini and Furlan (AFF) [2] with confining harmonic potential and coupling constant g 1⁄4 νðν þ 1Þ ≥ −1=4.1 Its nonrelativistic 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], nonrelativistic AdS/CFT correspondence [20,21,22,23,24], QCD
We study in detail the action of transformations of the Klein four-group on the states of the AFF system, its relation to the conformal symmetry, and its nontrivial role in N 1⁄4 2 super-extensions of the AFF model and their ospð2; 2Þ superconformal symmetry
Summary
Symmetries map the states of a system into its states. If the ground state is invariant under the corresponding transformations, one says that the symmetry is unbroken, otherwise symmetry is (spontaneously) broken. We study in detail the action of transformations of the Klein four-group on the states of the AFF system, its relation to the conformal symmetry, and its nontrivial role in N 1⁄4 2 super-extensions of the AFF model and their ospð2; 2Þ superconformal symmetry This superconformal symmetry, as will be shown, is based essentially on the simplest case of the Darboux dual schemes which produce the same but spectrally shifted pairs of the quantum systems. Their distinct intertwining operators allow us to construct the complete sets of the spectrum generating ladder operators for them and identify the nonlinearly deformed versions of conformal slð; RÞ algebra which describe their symmetries 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. In three Appendixes some technical details necessary for the main text are presented
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