  We discuss various issues concerning the behaviors near the boundary
(\sigma=0,\pi) and the midpoint (\sigma=\pi/2) of the open string coordinate
X(\sigma) and its conjugate momentum P(\sigma)=-i\delta/\delta X(\sigma) acting
on the matter projectors of vacuum string field theory. Our original interest
is in the dynamical change of the boundary conditions of the open string
coordinate from the Neumann one in the translationally invariant backgrounds to
the Dirichlet one in the D-brane backgrounds. We find that the Dirichlet
boundary condition is realized on a lump solution only partially and only when
its parameter takes a special value. On the other hand, the string midpoint has
a mysterious property: it obeys the Neumann (Dirichlet) condition in the
translationally invariant (lump) background.
