Resonances and scattering states for the polaritonic model

Characterize resonances and scattering states for the one-dimensional scalar polaritonic model with quadratic dispersion ((k)=k^2) coupled to a continuum of two-level atoms with spatial density ((x)), in addition to the bound states analyzed in the paper.

Background

The paper studies bound states of a one-dimensional scalar polaritonic field with quadratic dispersion ((k)=k^2) coupled to a continuum of identical two-level atoms. For constant atomic density, the spectrum consists of two continuous bands separated by gaps, while localized density perturbations can generate bound states within those gaps.

The analysis is restricted to bound states and reduces the spectral problem, away from the resonant frequency, to a one-dimensional Schrödinger equation with a frequency-dependent effective potential. The authors explicitly identify the treatment of resonances and scattering states as unresolved, leaving the spectral and dynamical behavior of these non-bound components to future investigation.