# A simple test for stability of black hole by $S$-deformation

## TL;DR

This paper introduces a numerical method to find a deformation function that proves black hole stability by transforming the potential in the Schrödinger-like perturbation equation, simplifying stability analysis especially near marginal stability.

## Contribution

The paper presents a simple, numerically driven approach to identify a deformation function for stability analysis, avoiding trial-and-error methods and applicable even in near-marginal cases.

## Key findings

- Numerical solution can find a regular deformation function when the potential has a small negative region.
- The method works effectively even for nearly marginally stable spacetimes.
- The stability condition aligns with the existence of a regular solution to the differential equation.

## Abstract

We study a sufficient condition to prove the stability of a black hole when the master equation for linear perturbation takes the form of the Schr\"odinger equation. If the potential contains a small negative region, usually, the $S$-deformation method was used to show the non-existence of unstable mode. However, in some cases, it is hard to find an appropriate deformation function analytically because the only way known so far to find it is a try-and-error approach. In this paper, we show that it is easy to find a regular deformation function by numerically solving the differential equation such that the deformed potential vanishes everywhere, when the spacetime is stable. Even if the spacetime is almost marginally stable, our method still works. We also discuss a simple toy model which can be solved analytically, and show the condition for the non-existence of a bound state is the same as that for the existence of a regular solution for the differential equation in our method. From these results, we conjecture that our criteria is also a necessary condition.