  The properties of the momentum space quark propagator in Landau gauge and
Gribov copy free Laplacian gauge are studied for the overlap quark action in
quenched lattice QCD. Numerical calculations are done on two lattices with
different lattice spacing $a$ and the same physical volume. We have calculated
the nonperturbative wave function renormalization function $Z(q)$ and the
nonperturbative mass function $M(p)$ for a variety of bare quark masses and
perform a simple linear extrapolation to the chiral limit. We focus on the
comparison of the behavior of $Z(q)$ and $M(p)$ in the chiral limit in the two
gauge fixing schemes as well as the behavior on two lattices with different
lattice spacing $a$. We find that the mass functions $M(p)$ are very similar
for the two gauges while the wave-function renormalization function $Z(q)$ is
more strongly infrared suppressed in the Laplacian gauge than in the Landau
gauge on the finer lattice. For Laplacian gauge, it seems that the finite $a$
error is large on the coarse lattice which has a lattice spacing $a$ of about
0.124 fm.
