Boundary and midpoint behaviors of lump solutions in vacuum string field theory

We discuss various issues concerning the behaviors near the boundary (σ=0,π) and the midpoint (σ=π/2) of the open string coordinate X(σ) and its conjugate momentum P(σ)=-iδ/δX(σ) 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