  The 't Hooft model, i.e. the two-dimensional quantum chromodynamics in the
limit of infinite number of colors, is interpolated by an angle parameter
$\delta$ between $\delta=0$ for the instant form dynamics (IFD) and
$\delta=\pi/4$ for the light-front dynamics (LFD). With this parameter
$\delta$, we formulate the interpolating mass gap equation which takes into
account the non-trivial vacuum effect on the bare fermion mass to find the
dressed fermion mass. Our interpolating mass gap solutions link the IFD and LFD
results with the $\delta$ parameter. We find the interpolation angle
independent characteristic energy function which satisfies the energy-momentum
dispersion relation of the dressed fermion, identifying the renormalized
fermion mass function and the wave function renormalization factor. Using the
dressed fermion propagator interpolating between IFD and LFD, we derive the
corresponding quark-antiquark bound-state equation in the interpolating
formulation. The mass spectra of mesons bearing the feature of the Regge
trajectories are found independent of the $\delta$-parameter for the equal mass
quark and antiquark bound-states. The Gell-Mann - Oakes - Renner relation for
the pionic ground-state in the zero fermion mass limit is confirmed indicating
that the spontaneous breaking of the chiral symmetry occurs regardless of the
quantization for $0 \le \delta \le \pi/4$. We obtain the corresponding
bound-state wave functions and discuss their reference frame dependence.
Applying them for the computation of the so-called quasi parton distribution
functions, we note a possibility of utilizing not only the reference frame
dependence but also the interpolation angle dependence to get an alternative
effective approach to the LFD-like results.
