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 $\varepsilon$ is a small parameter, we analyze the asymptotic behavior as $\varepsilon\to 0$ of the (low-lying) eigenvalues and eigenfunctions. We show that the first $n$ eigenvalues of $H_\varepsilon$ 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.