Negative Energy Densities in Quantum Field Theory With a Background Potential

We present a general procedure for calculating one-loop ``Casimir'' energy densities for a scalar field coupled to a fixed potential in renormalized quantum field theory. We implement direct subtraction of counterterms computed precisely in dimensional regularization with a definite renormalization scheme. Our procedure allows us to test quantum field theory energy conditions in the presence of background potentials spherically symmetric in some dimensions and independent of others. We explicitly calculate the energy density for several examples. For a square barrier, we find that the energy is negative and divergent outside the barrier, but there is a compensating divergent positive contribution near the barrier on the inside. We also carry out calculations with exactly solvable $\sech^2$ potentials, which arise in the study of solitons and domain walls.