  Sinai's walk can be thought of as a random walk on $\mathbb {Z}$ with random
potential $V$, with $V$ weakly converging under diffusive rescaling to a
two-sided Brownian motion. We consider here the generator $\mathbb {L}_N$ of
Sinai's walk on $[-N,N]\cap \mathbb {Z}$ with Dirichlet conditions on $-N,N$.
By means of potential theory, for each $h>0$, we show the relation between the
spectral properties of $\mathbb {L}_N$ for eigenvalues of order
$o(\exp(-h\sqrt{N}))$ and the distribution of the $h$-extrema of the rescaled
potential $V_N(x)\equiv V(Nx)/\sqrt{N}$ defined on $[-1,1]$. Information about
the $h$-extrema of $V_N$ is derived from a result of Neveu and Pitman
concerning the statistics of $h$-extrema of Brownian motion. As first
application of our results, we give a proof of a refined version of Sinai's
localization theorem.
