S matrix of collective field theory

By applying the Lehmann-Symanzik-Zimmermann (LSZ) reduction formalism, we study the S matrix of collective field theory in which fermi energy is larger than the height of potential. We consider the spatially symmetric and antisymmetric boundary conditions. The difference is that S matrices are proportional to momenta of external particles in antisymmetric boundary condition, while they are proportional to energies in symmetric boundary condition. To the order of $g_{st}^2$, we find simple formulas for the S matrix of general potential. As an application, we calculate the S matrix of a case which has been conjectured to describe a "naked singularity".