Affine Extension of the Free-Particle--Oscillator--Inverted-Oscillator Triangle
Abstract
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). Unlike these homogeneous quadratic Hamiltonians in the sl(2,ℝ) sector, the linear-potential Hamiltonian involves the Heisenberg ideal of the Schrödinger algebra. Its direct relation to the free particle is a regular accelerated-frame transformation, developed here at the levels of the classical action, canonical transformation, wave-function intertwiner and propagator; its HO and IHO realizations instead arise through singular displaced-oscillator limits. The Airy energy eigenstates follow from a cubic-phase transform of free-particle momentum eigenstates, from condensation of highly excited harmonic-oscillator levels, and from the limit of a subdominant parabolic-cylinder scattering branch of the inverted oscillator. We also develop a planar extension in homogeneous crossed electric and magnetic fields, where uniform acceleration generates the electric interaction, while uniform rotation produces the Landau coupling and the centrifugal inverted-oscillator term; the guiding-center dynamics yields the Hall drift.
Versions
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BibTeX
@misc{alcala2026affine,
title = {{Affine Extension of the Free-Particle--Oscillator--Inverted-Oscillator Triangle}},
author = {Andrey Alcala and Mikhail S. Plyushchay},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.20711},
eprint = {2607.20711},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.20711},
}
RIS
TY - UNPB
TI - Affine Extension of the Free-Particle--Oscillator--Inverted-Oscillator Triangle
AU - Andrey Alcala
AU - Mikhail S. Plyushchay
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2607.20711
UR - https://arxiv.org/abs/2607.20711
AB - 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). Unlike these homogeneous quadratic Hamiltonians in the sl(2,ℝ) sector, the linear-potential Hamiltonian involves the Heisenberg ideal of the Schrödinger algebra. Its direct relation to the free particle is a regular accelerated-frame transformation, developed here at the levels of the classical action, canonical transformation, wave-function intertwiner and propagator; its HO and IHO realizations instead arise through singular displaced-oscillator limits. The Airy energy eigenstates follow from a cubic-phase transform of free-particle momentum eigenstates, from condensation of highly excited harmonic-oscillator levels, and from the limit of a subdominant parabolic-cylinder scattering branch of the inverted oscillator. We also develop a planar extension in homogeneous crossed electric and magnetic fields, where uniform acceleration generates the electric interaction, while uniform rotation produces the Landau coupling and the centrifugal inverted-oscillator term; the guiding-center dynamics yields the Hall drift.
ER -