  Determining the structure of spectral densities is important for
understanding the behaviour of any quantum field theory (QFT). However, the
exact calculation of these quantities often requires a full non-perturbative
description of the theory, which for physically realistic theories such as
quantum chromodynamics (QCD) is currently unknown. Nevertheless, it is possible
to infer indirect information about these quantities. In this paper we
demonstrate an approach for constraining the form of spectral densities
associated with QFT propagators, which involves matching the short distance
expansion of the spectral representation with the operator product expansion
(OPE) of the propagators. As an application of this procedure we analyse the
scalar propagator in $\phi^4$-theory and the quark propagator in QCD, and show
that constraints are obtained on the spectral densities and the OPE
condensates. In particular, it is demonstrated that the perturbative and
non-perturbative contributions to the quark condensate in QCD can be
decomposed, and that the non-perturbative contributions are related to the
structure of the continuum component of the scalar spectral density.
