Persistence of phase-space flattening for continuous-spectrum environments

Determine whether the shear-induced and hyperbolic mechanisms persist when the discrete two-mode environment is replaced by a continuous spectrum of harmonic oscillators with a specified spectral density, and establish whether local subsystem flattening holds generically under continuum couplings.

Background

The paper analyzes a closed three-mode quadratic system consisting of one subsystem and two environmental modes. It identifies two mechanisms by which the reduced subsystem loses local phase-space dependence: polynomial phase-space shear in the free-particle model and hyperbolic expansion and contraction above a critical coupling in the harmonic-oscillator model.

The authors propose extending this exactly solvable finite-mode framework to a continuum of harmonic oscillators characterized by a spectral density. The unresolved issue is whether the two identified mechanisms survive in that continuous-spectrum setting and whether local flattening of the reduced subsystem is a generic consequence of continuum couplings, rather than a feature specific to the discrete three-mode model.