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The Free Particle--Oscillator--Inverted Oscillator Triangle: Conformal Bridges, Metaplectic Rotations and $\mathfrak{osp}(1|2)$ Structure
We study the free particle (FP), harmonic oscillator (HO) and inverted harmonic oscillator (IHO) as parabolic, elliptic and hyperbolic realizations of one conformal/metaplectic structure. Within this framework, we develop a common free-particle construction of HO bound and coherent states and IHO re
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