  In this study the metal-insulator transition in the square-lattice Hubbard
model at half-filling is revisited in relation to the DOS and spectral
functions by means of the ladder dual fermion approximation (LDFA). For this
purpose, a new expression of the two-body Green's function in the form of
resolvents is proposed, which provides tractable and efficient means to
calculate the local vertex function with the Lanczos exact diagonalization (ED)
method. This makes it possible to use the Lanczos ED method as a solver of the
effective impurity Anderson model for LDFA, opening up the way to access low
temperatures for these perturbative extensions of the dynamical mean field
theory and to obtain accurate DOS and spectral functions on the real frequency
axis by a new variant of the maximum entropy method. It is found that for $U\le
3.5t$, as temperature decreases, the pseudogap formation due to
antiferromagnetic correlations in the quasiparticle peak of the spectral
function occurs at the X point $[k=(\pi,0)]$, spreads through the Fermi surface
and ends at the M$_2$ point $[k=(\pi/2,\pi/2)]$. The almost simultaneous
creation of the pseudogap and the loss of the Fermi liquid feature is
consistent with that expected in the Slater regime. Although the pseudogap
still appears in the quasi-particle-like single peak for $U\ge 4.0t$, the
Fermi-liquid feature is partially lost on the Fermi surface already at higher
temperatures as expected in the Mott-Heisenberg regime, in which local spins
are preformed at high temperatures. A sharp crossover from a pseudogap phase to
a Mott insulator at finite $U^{*} \approx 4.7t$ is found to occur below the
temperature of the pseudogap formation similar to a previous study with the
non-linear $\sigma$ model approach.
