Robustness of the $S$-deformation method for black hole stability analysis

Understand

The $S$-deformation method is a useful way to show the linear mode stability of a black hole when the perturbed field equation takes the form of the Schr\"odinger equation.

• While previous works where many explicit examples are studied suggest that this method works well, general discussion is not given yet explicitly.
• In this paper, we show the existence of a regular $S$-deformation when a black hole spacetime is stable under some reasonable assumptions.
• This $S$-deformation is constructed from a solution of a differential equation.