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 Schrödinger 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 Schrödinger system with zero energy. We apply this formalism to some examples and numerically study their stability.