  We study the phenomenon of negative energy densities in quantum field
theories with self-interaction. Specifically, we consider a class of integrable
models (including the sinh-Gordon model) in which we investigate the
expectation value of the energy density in one-particle states. In this
situation, we classify the possible form of the stress-energy tensor from first
principles. We show that one-particle states with negative energy density
generically exist in non-free situations, and we establish lower bounds for the
energy density (quantum energy inequalities). Demanding that these inequalities
hold reduces the ambiguity in the stress-energy tensor, in some situations
fixing it uniquely. Numerical results for the lowest spectral value of the
energy density allow us to demonstrate how negative energy densities depend on
the coupling constant and on other model parameters.
