potential. Due to quasinormal modes (QNM) oscillations, we discuss the stability of black holes by using the S-deformation technique [@Ishibashi:2003ap]. Our result reduces to previous expressions derived in the literature, such as the Regge-Wheeler potential [@Regge:1957td] or more recently [@Anabalon:2014lea; @Cisterna:2015uya], in presence of a scalar field. We find that the stability analysis is reduced to an algebraic problem where three functions characterizing the black hole should be positive outside the horizon. We finally apply this formalism to various solutions, checking at the same time that our calculations are correct, by confronting them to cases previously discussed in the literature.

The model
=========

Following the same strategy as [@Kobayashi:2012kh], we consider the most general scalar-tensor theory in four dimensions having second-order field equations, both for the metric and the scalar field $$\begin{aligned}
S=\int{\rm d}^4x\sqrt{-g}\Bigl[L_2+L_3+L_4+L_5\Bigr] \label{action1}\,,\end{aligned}$$ where $$\begin{aligned}
\label{eq:action}
L_2&=K(\phi, X),\\
L_3&=-G_3(\phi, X)\Box\phi,\\
L_4&=G_4(\phi, X)R+G_{4X}\left[(\Box\phi)^2-(\nabla_\mu\nabla_\nu\phi)^2\right],\\
L_5&=G_5(\phi, X)G_{\mu\nu}\nabla^\mu\nabla^\nu\phi-\frac{1}{6}G_{5X}\bigl[(\Box\phi)^3\nonumber\\
&-3\Box\phi(\nabla_\mu\nabla_\nu\phi)^2+2(\nabla_\mu\nabla_\nu\phi)^3\bigr],\end{aligned}$$ where $K$ and $G_i$ are arbitrary functions of $\phi$ and $X\equiv-(\partial\phi)^2/2$, and $G_{iX}\equiv\partial_X G_i$

Background
==========

We consider a static spherically symmetric spacetime for which the metric can be written in the following form $$\begin{aligned}
{\rm d}\bar s^2&=\bar g_{\mu\nu}{\rm d} x^\mu{\rm d} x^\nu\,,\\
\label{eq:metricB}
&=-A(r){\rm d} t^2+\frac{{\rm d} r^2}{B(r)}+C(r)\left({\rm d}\theta^2+\sin^2\theta\,{\rm d} \varphi^2\right).\end{aligned}$$ We 