Investigating the generality of time-local master equations

Time-local master equations are more generally applicable than is often recognized, 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 noninvertible and non-Markovian time evolutions, 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 behavior 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.