Physics of the Inverted Harmonic Oscillator: From the lowest Landau level to event horizons

TL;DR

This paper explores the inverted harmonic oscillator (IHO) as a unifying model to understand quantum scattering, time-decay, and relativistic phenomena across systems like quantum Hall effects and black hole horizons.

Contribution

It establishes the IHO Hamiltonian's role in diverse physical systems, revealing its connection to phenomena such as Hawking-Unruh effect and quantum Hall physics through symmetry and geometric response analysis.

Findings

Derivation of IHO in the lowest Landau level in a gauge-invariant manner

Identification of IHO's parallels with the Rindler Hamiltonian near event horizons

Proposal of a method to observe quasinormal modes via wave packet scattering

Abstract

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.