The Free Particle--Oscillator--Inverted Oscillator Triangle: Conformal Bridges, Metaplectic Rotations and $\mathfrak{osp}(1|2)$ Structure

Abstract

We study the free particle (FP), the harmonic oscillator (HO) and the inverted harmonic oscillator (IHO) as parabolic, elliptic and hyperbolic realizations of one conformal/metaplectic structure, naturally extended to the superconformal algebra $\mathfrak{osp}(1|2)$. Since the corresponding self-adjoint Hamiltonians have different spectra, the relations between them are not ordinary unitary equivalences. They are instead bridge transformations between different realizations of the same conformal module. We show that the zero-energy Jordan states of the FP are mapped to HO bound states and to the two IHO Gamow families, while FP plane waves are mapped to HO coherent states and, after light-cone Mellin decomposition, to the IHO scattering data. The direct FP--IHO bridge is a real metaplectic quarter-rotation, in contrast with the stationary FP--HO conformal bridge, which is nonunitary in the Schr\"odinger representation but becomes unitary as a change of polarization to the Fock--Bargmann representation. The IHO transmission and reflection amplitudes are obtained as Fourier--Mellin connection coefficients, equivalently as Weber/Stokes connection data. We also describe the hyperbolic Cayley--Niederer map for the time-dependent Schr\"odinger equation, the Wigner/separatrix picture, and the coherent-state and Bogoliubov-transformation aspects of the construction. Some physical applications of the hyperbolic sector are briefly discussed, including quantum Hall saddle scattering, Schwinger-type production, Rindler/Unruh and near-horizon Hawking settings, and Berry--Keating/inverse-square structures.