Harmonic Approximation of Difference Operators

Abstract

For a general class of difference operators $H_{\epsilon }=T_{\epsilon }+V_{\epsilon }$ on ${\ell }^2(\epsilon {\mathbb{Z}}^d)$, where $V_{\epsilon }$ 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.

30 pages

References in corpus (2)

• An asymptotic maximum principle for essentially linear evolution models
• Agmon-Type Estimates for a Class of Difference Operators