自由粒子 - 振荡器 - 倒易振荡器三角形的仿射扩展
Affine Extension of the Free-Particle--Oscillator--Inverted-Oscillator Triangle
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中文总结 AI 辅助
研究一维线性势系统作为自由粒子、谐振子和倒谐振子构成的共形三角形的仿射扩展，通过正则加速框架变换等方法展开研究，得出艾里能量本征态相关结论，并开发了平面扩展，得到电相互作用等结果。
中文摘要 AI 辅助
我们研究一维线性势系统，将其作为由自由粒子、谐振子（HO）和倒谐振子（IHO）构成的共形三角形的仿射扩展。与\(sl(2,\mathbb R)\)扇区中的这些齐次二次哈密顿量不同，线性势哈密顿量涉及薛定谔代数的海森堡理想。它与自由粒子的直接关系是一种正则加速框架变换，并在经典作用、正则变换、波函数交织器和传播子层面展开。其HO和IHO实现通过奇异位移振荡器极限产生。艾里能量本征态源于自由粒子动量本征态的三次相位变换、高激发谐振子能级的凝聚以及倒谐振子的次主导抛物柱面散射分支的极限。我们还在均匀交叉电场和磁场中开发了平面扩展，其中均匀加速产生电相互作用，均匀旋转产生朗道耦合和离心倒谐振子项；导向中心动力学产生霍尔漂移。
英文摘要
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 $sl(2,\mathbb R)$ 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.
