  The paper deals with a Dirichlet spectral problem for a singularly perturbed
second order elliptic operator with rapidly oscillating locally periodic
coefficients. We study the limit behaviour of the first eigenpair (ground
state) of this problem. The main tool in deriving the limit (effective) problem
is the viscosity solutions technique for Hamilton-Jacobi equations. The
effective problem need not have a unique solution. We study the non-uniqueness
issue in a particular case of zero potential and construct the higher order
term of the ground state asymptotics.
