  Time-local master equations are more generally applicable than is often
recognised, but at first sight it would seem that they can only safely be used
in time intervals where the time evolution is invertible. Using the
Jaynes-Cummings model, we here construct an explicit example where two
different Hamiltonians, corresponding to two different non-invertible and
non-Markovian time evolutions, will lead to arbitrarily similar time-local
master equations. This illustrates how the time-local master equation on its
own in this case does not uniquely determine the time evolution. The example is
nevertheless artificial in the sense that a rapid change in (at least) one of
the Hamiltonians is needed. The change must also occur at a very specific
instance in time. If a Hamiltonian is known not to have such very specific
behaviour, but is "physically well-behaved", then one may conjecture that a
time-local master equation also determines the time evolution when it is not
invertible.
