  The odd-frequency superconducting state appears generally in hybrid
structures consisting of conventional superconductors and other materials, and
features electrons that form temporally non-local Cooper pairs. The
quasiclassical theory of superconductivity has been extensively used to model
such systems, finding in many cases excellent agreement with experimental
measurements. Therefore, it is of interest to study effective models of
odd-frequency superconductivity to predict new phenomena associated with this
form of unconventional pairing. We establish necessary criteria that the
quasiclassical Green functions in odd-frequency superconducting systems in the
dirty limit must satisfy in order to be physically reasonable, including
conservation of spectral weight. We show that it is possible to write down
effective models which satisfy all the abovementioned criteria, but which still
predict different behavior when it comes to the density of states and the
magnetic response of the superconductor. For instance, an effective model for
the odd-frequency anomalous Green function that gives a conserved spectral
weight can yield either a peaked or gapped density of states at the Fermi
energy, and exhibit conventional, zero, or unconventional Meissner response.
This finding demonstrates the importance of carefully considering the
properties of effective models describing odd-frequency superconductivity in
order to obtain physically reasonable results.
