  As a sequel of our preceding work [1] we carry out a comprehensive
comparative study between the quasi parton distribution functions (PDFs),
distribution amplitudes (DAs) and their light-cone counterparts for various
flavor-neutral mesons, in the context of the 't Hooft model, that is, the
two-dimensional QCD in the large $N$ limit. In contrast to the original
derivation via diagrammatic techniques exemplified by Dyson-Schwinger and
Bethe-Salpeter equations, here we employ the Hamiltonian operator approach to
reconstruct the celebrated 't Hooft equation in light-front quantization, and
Bars-Green equations in equal-time quantization. The novelty of our derivation
is to employ the soft momentum cutoff as the IR regulator. As a virtue of this
operator approach, the functional form of the quasi distributions can be
transparently built out of the Bars-Green wave functions and the Bogoliubov
angle with the aid of bosonization technique. Equipped with various bound-state
wave functions numerically inferred in [1], we then investigate how rapidly the
quasi distributions approach their light-cone counterparts with the increasing
meson momentum. We observe that, light mesons' quasi distributions approach the
light-cone distributions in a slower pace than the heavy quarkonia. Curiously,
lattice simulations of quasi distributions in four-dimensional QCD also
discover this feature. Furthermore, we also compute the partonic light-cone PDF
and quasi-PDF to one-loop order in perturbation theory, again employing the
momentum cutoff as the IR regulator. We explicitly verify one of the backbones
underlying the large momentum effective field theory (LaMET), namely, both
quasi-PDFs and light-cone PDFs in ${\rm QCD}_2$ indeed possess the same IR
behavior at leading order in $1/P^z$
