  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.
