  We propose a simple method to prove the linear mode stability of a black hole
when the perturbed field equations take the form of a system of coupled
Schr\"odinger equations. The linear mode stability of the spacetime is
guaranteed by the existence of an appropriate $S$-deformation. Such an
$S$-deformation is related to the Riccati transformation of a solution to the
Schr\"odinger system with zero energy. We apply this formalism to some examples
and numerically study their stability.
