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 Schrödinger 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 Schrödinger 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.

• The paper introduces explicit conformal bridges that map free particle, harmonic oscillator, and inverted oscillator systems to reveal distinct spectral and algebraic structures.
• It demonstrates how non-unitary transformations and metaplectic rotations within sl(2,R) and osp(1|2) frameworks link bound, scattering, and resonant states.
• The methodology unifies diverse quantum phenomena by providing analytic tools for understanding barrier-top scattering, coherent-state formation, and PT-symmetric quantum mechanics.

Parabolic–Elliptic–Hyperbolic Triangle in Conformal and Superconformal Quantum Mechanics

Algebraic Foundations and Representation Structure

The paper rigorously analyzes the free particle (FP), harmonic oscillator (HO), and inverted harmonic oscillator (IHO) as distinct spectral manifestations—parabolic, elliptic, and hyperbolic—of a unified conformal/metaplectic algebraic structure. These systems are realized as eigenvalue problems associated with different generators within $$\mathfrak{sl}(2, \mathbb{R}) \simeq \mathfrak{sp}(2, \mathbb{R}) \simeq \mathfrak{su}(1,1)$$; their Hamiltonians are not linked by ordinary unitary equivalence due to the disparity in their spectra. Instead, the connection is established through "bridge transformations"—non-unitary and metaplectic mappings that relate their respective modules.

A crucial insight is the extension to the superconformal algebra $\mathfrak{osp}(1|2)$, where the odd generators (linear in $q, p$) facilitate transitions between the even and odd metaplectic sectors ($D^+_{1/4} \oplus D^+_{3/4}$). This superalgebraic structure is physically realized in coherent-state generation, scattering constructs, and ladder-state transformations.

Stationary Bridges: FP–HO vs. FP–IHO Transformations

The FP–HO bridge is classically implemented via the Cayley transformation—an analytically continued hyperbolic flow in the Schrödinger representation—mapping the dilation generator $2iD$ to $H_{+}$. This is non-unitary in $L^2(\mathbb{R}, dq)$ but becomes unitary when recast as a polarization change to the Fock–Bargmann space.

In contrast, the FP–IHO bridge is a real, unitary metaplectic quarter-rotation generated by $H_+$. The transformation sends $2D$ to $\pm H_-$, establishing a direct algebraic connection between the dilation operator and the IHO Hamiltonian. This duality yields two Gamow families, $\mathfrak{osp}(1|2)$0, and systematically reorganizes the free particle's threshold monomials into resonant states.

Free Particle Data: Threshold Jordan States and Plane Waves

The parabolic sector (FP) serves as the universal seed. Its threshold module at $\mathfrak{osp}(1|2)$1 forms two conformally invariant Jordan chains—polynomials in $\mathfrak{osp}(1|2)$2 with weights matching the metaplectic sectors—which are mapped to HO bound states and IHO Gamow states by respective bridges. The FP plane waves, as true translation eigenstates, become HO coherent states under the FP–HO bridge and light-cone coherent states under FP–IHO, whose Mellin decomposition encodes scattering amplitudes.

The additive Fourier characters of FP plane waves translate under the quarter-rotation into multiplicative Mellin characters in the IHO light-cone variable, directly linking the FP spectrum to hyperbolic scattering data.

Light-Cone Variables, Mellin Expansion, and Gamow Structures

Formulating the IHO in light-cone variables $\mathfrak{osp}(1|2)$3 exposes its hyperbolic nature—Hamiltonian flow is dilation-like in these directions with direct correspondence to the eigenfunction structure. The real-energy eigenstates are Mellin-type distributions on half-lines, with branch points at the separatrix. The analytic continuation yields discrete Gamow families as pole data, providing a transparent mapping for both resonance and scattering phenomena.

The IHO ladder operators $\mathfrak{osp}(1|2)$4 generate imaginary energy shifts, paralleling oscillator ladder operator algebra but in the hyperbolic regime.

Scattering Matrix as Fourier–Mellin Connection

The two-channel scattering problem for the IHO is solved as a connection problem between Mellin bases via the Fourier kernel. Mellin integrals produce Gamma-function coefficients:

$\mathfrak{osp}(1|2)$5

The associated connection coefficients exhibit explicit analytic structure: the transmission and reflection probabilities smoothly interpolate the classical separatrix, with pole singularities corresponding to Gamow resonances.

The indirect route via analytic continuation in the oscillator frequency $\mathfrak{osp}(1|2)$6 gives Gamow pole positions but requires explicit connection formulae (e.g., Weber/Stokes data) for scattering amplitudes.

Time-Dependent Conformal Bridges: Cayley–Niederer Transformations

The time-dependent FP–IHO bridge, manifested via a hyperbolic projective time map (Cayley–Niederer transformation), connects FP and IHO Schrödinger equations through scaling, M\"obius reparametrization, and a metaplectic phase. These transformations preserve Hilbert space structure and propagate Gaussian wave packets—a classical center follows the saddle trajectory, and the corresponding Wigner ellipses undergo hyperbolic squeezing.

Phase-Space and Semiclassical Interpretation

The phase-space portrait demonstrates channel geometry: transmitted ($\mathfrak{osp}(1|2)$7) and reflected ($\mathfrak{osp}(1|2)$8) hyperbolae, separated by the null separatrices of the hyperbolic flow. Quantum mechanically, the Wigner distribution shows the impossibility of sharply assigning a wave packet to one classical channel, and the quantum crossover is quantified by $\mathfrak{osp}(1|2)$9 and $q, p$0.

Branch prescriptions in Mellin eigenfunctions ($q, p$1) encode the analytic continuation across the separatrix—mirroring the construction in WKB theory.

Coherent States, Perelomov Squeezed States, and Bogoliubov Transformations

The coherent-state structure is extended: plane waves map to HO Glauber coherent states via FP–HO, and to generalized light-cone coherent states (Mellin expansion) via FP–IHO. SU(1,1) Perelomov states are realized as squeezed Gaussian packets in the metaplectic sectors. In the hyperbolic regime, Bogoliubov transformations capture stretching/squeezing dynamics, and, in field-theoretic settings, relate to particle production via nontrivial vacuum mixing.

IHO scattering states are not ordinary coherent states; they are generalized eigenstates of the noncompact generator, the Mellin basis.

Physical Implications and Applications

Universal Local Normal Forms

The inverted harmonic oscillator appears as the universal approximation near potential maxima, underpinning local barrier-top scattering (parabolic barrier, quantum Hall saddle, Schwinger production, Rindler/Unruh/Hawking physics, and black-hole barrier regions). The common feature is hyperbolic flow, light-cone variables, Mellin-type modes, and analytically controlled connection coefficients.

Quantum Hall Saddle

Projected guiding-center dynamics in quantum Hall systems reduces to an IHO due to the magnetic field's freezing of cyclotron motion, directly connecting point-contact transmission to the universal IHO scattering coefficients.

Schwinger, Rindler, and Hawking Problems

Schwinger-type pair production and Rindler/Unruh/Hawking thermal phenomena are governed by the same analytic structure: field modes are linked by hyperbolic Bogoliubov transformations, and connection coefficients from parabolic-cylinder functions or Mellin integrals encode vacuum mismatch and pair production rates.

Berry–Keating, Inverse-Square, and Newton–Hooke Correspondences

The structural relation between the IHO, dilation operator ($q, p$2), and supercritical inverse-square potential is elucidated via real metaplectic quarter-rotations. While not a direct operator equivalence, the analytic links underpin universal smooth counting laws (e.g., for Riemann zeros, limit cycles in RG flow), discrete scale invariance, and contributions to spectral theory.

Universality versus Model Dependence

The hyperbolic sector described (local light-cone variables, Mellin eigenfunctions, Gamma-function coefficients, transmission/reflection probabilities, Gamow pole structure) is universal and insensitive to global boundary conditions. Detailed extensions—greybody factors, self-adjoint domains, global interpretations—are model dependent and require separate specification.

Conclusion

This work establishes a formal, systematic translation mechanism—via conformal bridges, metaplectic rotations, and $q, p$3 extensions—between parabolic, elliptic, and hyperbolic quantum systems. The free particle acts as an organizational boundary, providing elementary Jordan and Fourier data. These are mapped to oscillator bound/coherent states and to IHO resonant and scattering states by explicit algebraic and analytic bridges.

The approach clarifies universal features behind a broad class of physical problems—from quantum barrier-top scattering to quantum field theory in curved spacetime—while identifying structural correspondences with boundary symmetries in holography. The implications reach beyond elementary quantum systems, offering insight into representation-theoretic dictionaries, non-Hermitian and PT-symmetric quantum mechanics, and integrable structures in broader mathematical physics.

Future directions include systematic exploration of PT-symmetric formulations, supersymmetric factorizations, extensions to field-theoretic and higher-dimensional contexts, and the development of bridge-based translation mechanisms for more general systems—including linear potentials and singular boundary conditions.