  I establish the criteria and obtained analytical results for the pseudogap at
the Van Hove (antinodal) point on the Fermi surface in two dimensions. The
original criterion $\xi >> \xi_{th\_db}=v_F/\pi T$ is not applicable in this
case since Fermi velocity $v_F=0$. It turns out that the characteristic length
for the pseudogap crossover at the Van Hove point $\xi_{th\_vh} \propto
1/T^{1/2}$, which is significantly shorter than the one at the regular Fermi
surface points $\xi_{th\_db} \propto 1/T$. In particular, $\xi_{th\_vh}$ is
between one and two lattice spacing in the intermediate interaction regime of
the Hubbard model. I have also identified the regime where there is still a
single maximum in the spectral function, but single particle properties are
abnormal. Specifically, the imaginary part of the self-energy has a minimum
instead of maximum at the Fermi level and the slope of the real part of the
self-energy is positive instead of negative. The important advantages of an
analytical approach is that it provides results in both the Matsubara frequency
representation and in real frequencies representation. I compare Matsubara
frequency results with the exact numerical results of the Monte Carlo methods
in the Hubbard model and then show what they correspond to in the real
frequency self-energy and spectral function.
