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.
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Kimura et al. (2019) studied this question.
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