  We consider the vector and the Lichnerowicz wave equations on the
Schwarzschild spacetime, which correspond to the Maxwell and linearized
Einstein equations in harmonic gauges (or, respectively, in Lorenz and de
Donder gauges). After a complete separation of variables, the radial mode
equations form complicated systems of coupled linear ODEs. We outline a precise
abstract strategy to decouple these systems into sparse triangular form, where
the diagonal blocks consist of spin-$s$ scalar Regge-Wheeler equations (for
spins $s=0,1,2$). Building on the example of the vector wave equation, which we
have treated previously, we complete a successful implementation of our
strategy for the Lichnerowicz wave equation. Our results go a step further than
previous more ad-hoc attempts in the literature by presenting a full and
maximally simplified final triangular form. These results have important
applications to the quantum field theory of and the classical stability
analysis of electromagnetic and gravitational perturbations of the
Schwarzschild black hole in harmonic gauges.
