  In this paper we deal with the so-called "spectral inequalities", which yield
a sharp quantification of the unique continuation for the spectral family
associated with the Schr\"odinger operator in $ \mathbb{R}^d$ \begin{equation*}
H_{g,V} = \Delta_g + V(x), \end{equation*} where $\Delta_g$ is the
Laplace-Beltrami operator with respect to an analytic metric $g$, which is a
perturbation of the Euclidean metric, and $V(x)$ a real valued analytic
potential vanishing at infinity.
