  We reconsider constraints on the eigenvalue density of the Dirac operator in
the chiral symmetric phase of 2 flavor QCD at finite temperature. To avoid
possible ultra-violet(UV) divergences, we work on a lattice, employing the
overlap Dirac operator, which ensures the exact "chiral" symmetry at finite
lattice spacings. Studying multi-point correlation functions in various
channels and taking their thermodynamical limit (and then taking the chiral
limit), we obtain stronger constraints than those found in the previous
studies: both the eigenvalue density at the origin and its first and second
derivatives vanish in the chiral limit of 2 flavor QCD. In addition we show
that the axial U(1) anomaly becomes invisible in susceptibilities of scalar and
pseudo scalar mesons, suggesting that the 2nd order chiral phase transition
with the O(4) scaling is not realized in 2 flavor QCD. Possible lattice
artifacts when non-chiral lattice Dirac operator is employed are briefly
discussed.
