Operator $\mathrm{\Delta }-aS$ on warped product manifolds

In this work we studied the stability of the family of operators $L_a=\mathrm{\Delta }-aS$, $a\in \mathbb{R}$, in a warped product of an infinite interval or real line by one compact manifold, where $\mathrm{\Delta }$ is the Laplacian and $S$ is the scalar curvature of the resulting manifold.