Affine Extension of the Free-Particle--Oscillator--Inverted-Oscillator Triangle

We study the one-dimensional linear-potential system as an affine extension of the conformal triangle formed by the free particle, harmonic oscillator (HO) and inverted harmonic oscillator (IHO) This paper determines how the linear-potential (LP) system fits into this picture

Algebraic positioning within the conformal structure

The free particle, harmonic oscillator and inverted harmonic oscillator are organized by the standard quadratic conformal generators Their basic dynamical integrals are PF = p + F t, XF = q − t/m p − F 2/2m t squared, and they generate translations and Galilean boosts The linear-potential Hamiltonian generates q˙ = p/m and p˙ = −F This suggests that HLP is a quadratic free-particle element supplemented by a linear Heisenberg–Galilei generator, rather than another quadratic representative of sl(2, R) The natural algebraic setting is consequently the non-semisimple Schrödinger algebra, equivalently the one-dimensional Jacobi algebra

Extension via affine and singular limits

The relation to the oscillator sectors is summarized by Hκ,F = p 2/2m + mκ squared q 2/2 + F q = p 2/2m + mκ squared Q squared − F 2/4mκ, Q = q + F/mκ, κ ≠ 0 For F = 0, the choices κ = 0, ω2, −omega2 reproduce FP, HO and IHO, respectively The limit κ → 0 is regular in the original family Hκ,F, where it yields HLP However, its HO and IHO realizations require controlled singular displaced-oscillator limits

Spectral transformations and Airy states

The LP Airy eigenstates arise from a cubic-phase transform of FP momentum eigenstates, from spectral condensation of highly excited displaced-HO states, and from the limit of a subdominant parabolic-cylinder scattering branch of the inverted oscillator The same Airy eigenstate is therefore reached through three complementary constructions: a regular cubic-phase change of spectral representation from FP plane waves, a singular condensation limit of highly excited displaced-HO states, and, as shown in Appendix A, the limit of a subdominant IHO parabolic-cylinder scattering branch

Planar extension and guiding-center dynamics

A planar extension in homogeneous crossed electric and magnetic fields gives a further interpretation A uniformly accelerated and rotating frame produces the electric interaction, the Coriolis/Landau coupling and the centrifugal IHO term; a compensating harmonic term isolates the Landau sector In guiding-center variables the electric field acts linearly on a noncommutative Heisenberg plane, generates the Hall drift, and supplements the Landau CBT by a displacement, a uniform drift and a scalar phase The quantum counterpart of the classical Hall drift is vD = −eE/mωc = −cE/B

Conclusion on system classification

The LP system is an affine–Heisenberg companion of the FP–HO–IHO conformal triangle, not a fourth homogeneous quadratic vertex The crossed-field extension provides a planar realization of the same distinction The electric field acts linearly on the noncommuting guidingcenter plane, produces the Hall drift, and supplements the Landau CBT by a displacement, drift and scalar phase

The IHO–LP spectral limit

At the Hamiltonian, action and propagator levels, the IHO–LP limit follows from the HO construction by the analytic continuation ω 7→ σiomega, σ = ±1 Spectrally, normalizable Hermite states are replaced by generalized parabolic-cylinder scattering states and an appropriate Stokes branch must be selected The two signs select conjugate complex rotations and Stokes sectors This is an analytic continuation of the differential equation, not a unitary continuation of a normalized HO state

The IHO–LP spectral edge

The dominant barrier branch tends locally to Bi Unlike the HO–LP limit, no condensation of discrete levels is involved: the IHO spectrum is continuous for every omega > 0 The two dashed spectral links therefore share the same local Airy turning-point normal form but differ globally: the HO route is a discrete-to-continuous large-level limit, whereas the IHO route is a continuous-scattering limit with vanishing transmission The finite turning point becomes the LP turning point while the IHO forbidden interval tends to the LP forbidden half-line q > E/F

Relativistic and geometric extensions

Rindler kinematics provides a relativistic completion of the regular FP–LP bridge Newton–Hooke mechanics arises from dS/AdS contractions and null reductions of homogeneous pp waves Accelerated and rotating frames in (A)dS may therefore provide a relativistic parent of Hκ,F, separating curvature κ from affine acceleration F/m The ordinary Galilean crossed-field problem arises from the magnetic-dominated contraction, so no transition at E = B occurs within the nonrelativistic system considered here

Future research directions

The guiding-center formulation points naturally to the subcritical, supercritical and critical phases of the noncommutative Landau problem, related by CBT and weak–strong duality It remains to determine how the electric-field affine intertwiner and Hall drift are realized across these phases A related relativistic direction starts from the identity of classical evolution between a charge in a constant (2 + 1)-dimensional electromagnetic field and the higher-derivative particle with torsion Geometric quantization gives internal discrete-series representations D±α of the universal cover of SL(2, R), while the physical constraint selects a discrete massive mass–spin tower and generalized massless and tachyonic states Clarifying the relation of this relativistic threefold structure to the Landau/HO, LP/Airy and IHO sectors remains open The ordinary Galilean crossed-field problem arises from the magnetic-dominated contraction, so no transition at E = B occurs within the nonrelativistic system considered here

Acknowledgments

The work was partially supported by the FONDECYT Project 1242046 The IHO–LP spectral limit follows from the HO construction by the analytic continuation ω 7→ σiomega, σ = ±1 The two signs select conjugate complex rotations and Stokes sectors This is an analytic continuation of the differential equation, not a unitary continuation of a normalized HO state For F > 0, the potential Vomega(q) = F q − momega squared q 2/2 has turning points q−(E, omega) = E/F + O(omega 2), q+(E, omega) = 2F/mω squared − E/F + O(ω 2), so that q− → E/F, while q+ → +∞ The slope at the finite turning point is somega = V′omega(q−) = √F squared − 2momega 2E → F Consequently, on fixed turning-point intervals, ξomega → κF (q − E/F) and the parabolic-cylinder equation reduces to the Airy equation The LP eigenstate selected in (5.3) is obtained from the IHO branch that is subdominant in the forbidden region The dominant barrier branch tends locally to Bi Unlike the HO–LP limit, no condensation of discrete levels is involved: the IHO spectrum is continuous for every omega > 0 The two dashed spectral links therefore share the same local Airy turning-point normal form but differ globally: the HO route is a discrete-to-continuous large-level limit, whereas the IHO route is a continuous-scattering limit with vanishing transmission

References

[1] U. Niederer, The maximal kinematical invariance group of the free Schrödinger equation, Helv. Phys. Acta 45 (1972) 802–810 [2] C. R. Hagen, Scale and conformal transformations in Galilean-covariant field theory, Phys. Rev. D 5 (1972) 377–388 [3] U. Niederer, The maximal kinematical invariance group of the harmonic oscillator, Helv. Phys. Acta 46 (1973) 191–200 [4] L. Inzunza, M. S. Plyushchay and A. Wipf, Conformal bridge between asymptotic freedom and confinement, Phys. Rev. D 101 (2020) 105019, arXiv:1912.11752 [5] L. Inzunza and M. S. Plyushchay, Conformal bridge transformation and PT symmetry, J. Phys. Conf. Ser. 2038 (2021) 012014, arXiv:2104.08351 [6] A. Alcala and M. S. Plyushchay, Projective time, Cayley transformations and the Schwarzian geometry of the free particle–oscillator correspondence, Phys. Rev. D 113 (2026) 086007, arXiv:2602.06378 [7] A. Alcala and M. S. Plyushchay, The free particle–oscillator–inverted oscillator triangle: conformal bridges, metaplectic rotations and osp(12) structure, arXiv:2605.09947 [8] V. Bargmann, On unitary ray representations of continuous groups, Ann. Math. 59 (1954) 1–46 [9] J.-M. Lévy-Leblond, Galilei group and nonrelativistic quantum mechanics, J. Math. Phys. 4 (1963) 776–788 [10] M. V. Berry and N. L. Balazs, Nonspreading wave packets, Am. J. Phys. 47 (1979) 264–267 [11] R. Berndt and R. Schmidt, Elements of the Representation Theory of the Jacobi Group, Progress in Mathematics 163, Birkhäuser, 1998 [12] J. Daboul and K. B. Wolf, Noncommuting contractions of oscillators with constant force, J. Phys. A: Math. Gen. 39 (2006) 4173–4180 [13] J. Daboul, G. S. Pogosyan and K. B. Wolf, Noncommuting limits of oscillator wave functions, Phys. At. Nucl. 70 (2007) 513–519 [14] R. W. Robinett, Quantum mechanical time-development operator for the uniformly accelerated particle, Am. J. Phys. 64 (1996) 803–808 [15] F. W. J. Olver, Uniform asymptotic expansions for Weber parabolic cylinder functions of large orders, J. Res. Natl. Bur. Stand. 63B (1959) 131–169 [16] G. Szegő, Orthogonal Polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. 23 (American Mathematical Society, Providence, 1975) [17] L. Inzunza and M. S. Plyushchay, Dynamics, symmetries, anomaly and vortices in a rotating cosmic string background, JHEP 01 (2022) 179 [18] M. H. Johnson and B. A. Lippmann, Motion in a constant magnetic field, Phys. Rev. 76 (1949) 828–832 [19] A. Alcala and M. S. Plyushchay, Weak–strong duality of the non-commutative Landau problem induced by a two-vortex permutation, and conformal bridge transformation, JHEP 08 (2023) 141 [20] M. S. Plyushchay, Relativistic particle with torsion and charged particle in a constant electromagnetic field: identity of evolution, Mod. Phys. Lett. A 10 (1995) 1463–1469, arXiv:hep-th/9309147 [21] C. Duval and P. A. Horváthy, The exotic Galilei group and the “Peierls substitution”, Phys. Lett. B 479 (2000) 284–290, arXiv:hep-th/0002233 [22] R. Jackiw and V. P. Nair, Anyon spin and the exotic central extension of the planar Galilei group, Phys. Lett. B 480 (2000) 237–238, arXiv:hep-th/0003130 [23] P. A. Horváthy and M. S. Plyushchay, Non-relativistic anyons, exotic Galilean symmetry and noncommutative plane, JHEP 06 (2002) 033 [24] P. A. Horváthy and M. S. Plyushchay, Anyon wave equations and the noncommutative plane, Phys. Lett. B 595 (2004) 547–555, arXiv:hep-th/0404137 [25] P. A. Horváthy and M. S. Plyushchay, Nonrelativistic anyons in external electromagnetic field, Nucl. Phys. B 714 (2005) 269–291, arXiv:hep-th/0502040 [26] M. S. Plyushchay, Majorana equation and exotics: higher derivative models, anyons and noncommutative geometry, Electron. J. Theor. Phys. 3, No. 10 (2006) 17–31, arXiv:mathph/0604022 [27] M. S. Plyushchay, The model of the relativistic particle with torsion, Nucl. Phys. B 362 (1991) 54–72 [28] W. Rindler, Kruskal space and the uniformly accelerated frame, Am. J. Phys. 34 (1966) 1174–1178 [29] H. Bacry and J.-M. Lévy-Leblond, Possible kinematics, J. Math. Phys. 9 (1968) 1605–1614 [30] G. W. Gibbons and C. E. Patricot, Newton–Hooke spacetimes, Hpp-waves and the cosmological constant, Class. Quantum Grav. 20 (2003) 5225–5239 [31] C. Duval, G. W. Gibbons and P. A. Horváthy, Celestial mechanics, conformal structures, and gravitational waves, Phys. Rev. D 43 (1991) 3907–3922 [32] S. Deser and O. Levin, Accelerated detectors and temperature in (anti-)de Sitter spaces, Class. Quantum Grav. 14 (1997) L163–L168 [33] V. de Alfaro, S. Fubini and G. Furlan, Conformal invariance in quantum mechanics, Nuovo Cim. A 34 (1976) 569–612 [34] L. Inzunza, M. S. Plyushchay and A. Wipf, Hidden symmetry and (super)conformal mechanics in a monopole background, JHEP 04 (2020) 028 [35] H. R. Lewis and W. B. Riesenfeld, An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field, J. Math. Phys. 10 (1969) 1458–1473

Crossed-field affine extension

The LP system describes a charged particle in a homogeneous electric field (F = −eE in one dimension) Adding a perpendicular constant magnetic field gives a planar extension: the pure Landau problem belongs to the quadratic oscillator/CBT sector, while the electric field acts linearly on the magnetic Heisenberg variables The Coriolis term linear in omega reproduces the orbital Larmor coupling, while (6.6) shows that the exact rotating-frame Hamiltonian differs from the crossed-field Hamiltonian HE,B by the centrifugal O(omega 2) IHO potential Adding the compensating isotropic HO term momega squared r 2/2 therefore yields HE,B; when a = 0, equivalently E = 0, HE,B reduces to the pure Landau Hamiltonian The electric field therefore shifts the cyclotron momentum and acts as a linear potential on the noncommutative guiding-center plane

The role of affine intertwiners

The three factors describe a scalar phase, a transverse displacement and a uniform drift This is the planar crossed-field analogue of the affine FP–LP bridge If SB denotes the CBT from the appropriate two-dimensional free-particle data to the pure Landau system, then the crossed-field bridge is simply SE,B(t) = exp imv2 Dt/2!

1. Improved Quantum State Evolution for Accelerated Systems: The system can accurately model quantum evolution of particles in a constant force field by using a Weyl displacement with a Bargmann phase instead of a quadratic metaplectic kernel. This allows for the direct mapping from free particle dynamics to linear potential dynamics, providing an exact quantum affine bridge as described in equation (3.8).

2. Improved Spectral Analysis via Cubic Phase Transform: The AI system can determine Airy energy eigenstates by performing a cubic-phase transform of free-particle momentum eigenstates, which is derived from the LP spectral problem in equation (5.4). This provides a method to find stationary solutions for systems with linear potentials that are not standard harmonic oscillators.

3. Improved Classical Dynamics for Crossed Fields: The system can simulate planar crossed-field dynamics by using an affine moving frame construction that combines translational and rotational parts. This allows the simulation of the Hall drift, which is generated by the electric field acting linearly on noncommuting Heisenberg variables, as shown in equation (6.10).

4. Improved Oscillator Limit Modeling: The AI can accurately predict behavior at singular limits for harmonic oscillators by using displaced-oscillator limits to derive LP and IHO spectral edges. This allows the system to transition between discrete oscillator spectra and continuous LP spectra, ensuring consistency across different potential regimes.