  In this paper we consider a finite state time discrete Markov chain that
mimics the behaviour of solutions of the stochastic differential equation
$dX=-U'(X)dt+\epsilon dL$, where $U$ is a multi-well potential with $n\geq 2$
local minima and L is a symmetric \alpha-stable L\'evy process (L\'evy flights
process). We investigate the spectrum of the generator of this Markov chain in
the limit $\epsilon\to 0$ and localize the top n eigenvalues
$\lambda^\epsilon_1,\dots, \lambda^\epsilon_n$. These eigenvalues turn out to
be of the same algebraic order $O(\epsilon^\alpha)$ and are well separated from
the rest of the spectrum by a spectral gap. We also determine the limits
$\lim_{\epsilon\to 0}\epsilon^{-\alpha} \lambda^\epsilon_i$, $1\leq i\leq n$,
and show that the corresponding eigenvectors are approximately constant over
the domains which correspond to the potential wells of $U$.
