In this work, we present the inverted harmonic oscillator (IHO) Hamiltonian as a paradigm to understand the quantum mechanics of scattering and time-decay in a diverse set of physical systems. As one of the generators of area preserving transformations, the IHO Hamiltonian can be studied as a dilatation generator, squeeze generator, a Lorentz boost generator, or a scattering potential. In establishing these different forms, we demonstrate the physics of the IHO that underlies phenomena as disparate as the Hawking-Unruh effect and scattering in the lowest Landau level (LLL) in quantum Hall systems. We derive the emergence of the IHO Hamiltonian in the LLL in a gauge invariant way and show its exact parallels with the Rindler Hamiltonian that describes quantum mechanics near event horizons. This approach of studying distinct physical systems with symmetries described by isomorphic Lie algebras through the emergent IHO Hamiltonian enables us to reinterpret geometric response in the lowest Landau level in terms of relativistic effects such as Wigner rotation. Further, the analytic scattering matrix of the IHO points to the existence of quasinormal modes (QNMs) in the spectrum, which have quantized time-decay rates. We present a way to access these QNMs through wave packet scattering, thus proposing a novel effect in quantum Hall point contact geometries that parallels those found in black holes.

在这项工作中，我们将倒谐波振荡器（IHO）哈密顿量作为范例，以了解各种物理系统中散射和时间衰减的量子力学。作为区域保留变换的生成器之一，可以将IHO哈密顿量研究为膨胀生成器，挤压生成器，洛伦兹升压生成器或散射势。在建立这些不同的形式时，我们演示了IHO的物理学原理，该现象以霍金-安鲁效应和霍乱效应以及量子霍尔系统中最低的朗道能级（LLL）的散射为基础。我们以规范不变的方式推论了LLL中IHO哈密顿量的出现，并显示了它与描述事件视界附近的量子力学的Rindler哈密顿量的确切相似之处。这种通过新兴的IHO哈密顿量研究同构李代数描述的具有对称性的独特物理系统的方法，使我们能够根据相对论效应（例如维格纳旋转）重新解释最低Landau级别的几何响应。此外，IHO的解析散射矩阵指向频谱中存在准正交模式（QNM），这些准正交模式具有量化的时间衰减率。我们提出了一种通过波包散射来访问这些QNM的方法，从而在量子霍尔点接触几何中提出了与黑洞中发现的平行的新颖效应。IHO的解析散射矩阵指向频谱中存在准正交模式（QNM），并已量化了时间衰减率。我们提出了一种通过波包散射来访问这些QNM的方法，从而在量子霍尔点接触几何中提出了与黑洞中发现的平行的新颖效应。IHO的解析散射矩阵指向频谱中存在准正交模式（QNM），并已量化了时间衰减率。我们提出了一种通过波包散射来访问这些QNM的方法，从而在量子霍尔点接触几何中提出了与黑洞中发现的平行的新颖效应。