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For a general class of difference operators H-epsilon = T-epsilon + V-epsilon on l(2) ((epsilon Z)(d)), where V- epsilon is a multi-well potential and a is a small parameter. we analyze the asymptotic behavior as epsilon -> 0 of the (low-lying) eigenvalues and eigenfunctions. We show that the first it eigenvalues of H converge to the first it eigenvalues of the direct suns of harmonic oscillators oil R-d located at the several wells. Our proof is microlocal.