Stability analysis of black holes by the $S$-deformation method for coupled systems

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.