This chapter presents an analytic approach which reduces the evaluation of quantum corrections to the corresponding quantum mechanical scattering problem. It considers scalar fields in the background of external sources described by a scalar field and analyzes effective Lagrangians in the context of quantum electrodynamics. The chapter shows that the renormalized ground-state energy for a repulsive potential is positive and suggests that the sign of the ground-state energy, or the Casimir energy, is actually strongly influenced by the renormalization procedure. It briefly describes the applications that have been considered using the scattering phase formulation and analyzes one-loop corrections to the energies of classical field configurations.