Quasinormal modes describe the ringdown of compact objects deformed by small
perturbations. In generic theories of gravity that extend General Relativity,
the linearized dynamics of these perturbations is described by a system of
coupled linear differential equations of second order. We first show, under
general assumptions, that such a system can be brought to a Schr\”odinger-like
form. We then devise an analytic approximation scheme to compute the spectrum
of quasinormal modes. We validate our approach using a toy model with a
controllable mixing parameter $\varepsilon$ and showing that the analytic
approximation for the fundamental mode agrees with the numerical computation
when the approximation is justified. The accuracy of the analytic approximation
is at the (sub-) percent level for the real part and at the level of a few
percent for the imaginary part, even when $\varepsilon$ is of order one. Our
approximation scheme can be seen as an extension of the approach of Schutz and
Will to the case of coupled systems of equations, although our approach is not
phrased in terms of a WKB analysis, and offers a new viewpoint even in the case
of a single equation.
