  Ordinary and partial differential operators with an indefinite weight
function can be viewed as bounded perturbations of non-negative operators in
Krein spaces. Under the assumption that 0 and $\infty$ are not singular
critical points of the unperturbed operator it is shown that a bounded additive
perturbation leads to an operator whose non-real spectrum is contained in a
compact set and with definite type real spectrum outside this set. The main
results are quantitative estimates for this set, which are applied to
Sturm-Liouville and second order elliptic partial differential operators with
indefinite weights on unbounded domains.
