  The paper deals with the Neumann spectral problem for a singularly perturbed
second order elliptic operator with bounded lower order terms. The main goal is
to provide a refined description of the limit behaviour of the principal
eigenvalue and eigenfunction. Using the logarithmic transformation we reduce
the studied problem to additive eigenvalue problem for a singularly perturbed
Hamilton-Jacobi equation. Then assuming that the Aubry set of the Hamiltonian
consists of a finite number of points or limit cycles situated in the domain or
on its boundary, we find the limit of the eigenvalue and formulate the
selection criterium that allows us to choose a solution of the limit
Hamilton-Jacobi equation which gives the logarithmic asymptotics of the
principal eigenfunction.
