  The Loschmidt echo is a popular quantity that allows making predictions about
the stability of quantum states under time evolution. In our work, we present
an approach that allows us to find a differential equation that can be used to
compute the Loschmidt echo. This approach, while in essence perturbative, has
the advantage that it converges at finite order. We demonstrate that the
approach for generically chosen matrix Hamiltonians often offers advantages
over Taylor and cumulant expansions even when we truncate at finite order. We
then apply the approach to two ordinary band Hamiltonians (multi-Weyl
semimetals and AB bilayer graphene) to obtain the Loschmidt echo after a quench
for an arbitrary starting state and find that the results readily generalize to
find transmission amplitudes and specific contributions to the partition
function, too. We then test our methods on many body spin and fermionic
Hamiltonians and find that while the approach still offers advantages, more
care has to be taken than in a generic case.
