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

Abstract

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 resonance and scattering data. The construction extends the FP--HO conformal bridge to both hyperbolic orientations using real metaplectic rotations through $\pm \pi /4$. These rotations map $2\Omega D$ to $\mp H_{\mathrm{IHO}}$ and carry the FP threshold Jordan module into the two Gamow families. Applied to Fourier characters, they give light-cone generating states whose Taylor and Mellin decompositions recover resonance and real-energy connection data, respectively. The familiar IHO scattering amplitudes and their Gamma-function poles thereby enter the same construction. We identify the crossed roles of the elliptic and hyperbolic generators in the two stationary bridges and develop the hyperbolic Cayley--Niederer map as their time-dependent counterpart. The $\mathfrak{osp}(1|2)$ extension keeps track of the two parity sectors and the linear generating operators. A phase-space analysis reconciles exact Liouville transport with nonzero stationary transmission, distinguishing asymptotic quantum currents from classical trajectories across separatrices. We also briefly discuss applications of the common quadratic structure to barrier scattering, time-dependent oscillators and quantum fields in gravitational backgrounds.

40 pages. Revised and shortened presentation; physical interpretation of IHO scattering clarified; references updated. Accepted for publication in JHEP