Harmonic approximation of difference operators

For a general class of difference operators Hε=Tε+Vε on ℓ2(d(εZ)), where Vε is a multi-well potential and ε is a small parameter, we analyze the asymptotic behavior as ε→0 of the (low-lying) eigenvalues and eigenfunctions. We show that the first n eigenvalues of Hε converge to the first n eigenvalues of the direct sum of harmonic oscillators on Rd located at the several wells. Our proof is microlocal.