  This paper is devoted to the construction of an \emph{eigen-solution} for the
Fokker-Planck operator with heavy tail equilibrium. We propose an
\textit{alternative} method in dimension 1, which will be generalizable in
higher dimension. The later method is inspired by the work of H. Koch on
non-linear KdV equation \cite{Koch}. As a consequence of this construction, we
recover the result of G. Lebeau and M. Puel \cite{LebPu} on the fractional
diffusion limit for the Fokker-Planck equation.
