Harmonic Approximation of Difference Operators

For a general class of difference operators $H_\varepsilon = T_\varepsilon + V_\varepsilon$on$\ell^2(\varepsilon$\mathbb{Z}$^d)$,where$V_\varepsilon$ is a multi-well potential and $\epsilon$ is a small parameter, we analyze the asymptotic behavior as $\epsilon \to 0$ of the (low-lying) eigenvalues and eigenfunctions. We show that the first $n$ eigenvalues of $H_{\epsilon }$ converge to the first $n$ eigenvalues of the direct sum of harmonic oscillators on ${\mathbb{R}}^d$ located at the several wells. Our proof is microlocal.