  Quantum inequalities are bounds on negative time-averages of the energy
density of a quantum field. They can be used to rule out exotic spacetimes in
general relativity. We study quantum inequalities for a scalar field with a
background potential (i.e., a mass that varies with spacetime position) in
Minkowski space. We treat the potential as a perturbation and explicitly
calculate the first-order correction to a quantum inequality with an arbitrary
sampling function, using general results of Fewster and Smith. For an arbitrary
potential, we give bounds on the correction in terms of the maximum values of
the potential and its first three derivatives. The techniques we develop here
will also be applicable to quantum inequalities in general spacetimes with
small curvature, which are necessary to rule out exotic phenomena.
