# Harmonic Approximation of Difference Operators

## TL;DR

This paper studies the asymptotic behavior of eigenvalues and eigenfunctions of a class of difference operators with multi-well potentials as the discretization parameter approaches zero, showing convergence to harmonic oscillators.

## Contribution

It provides a microlocal analysis demonstrating that low-lying eigenvalues of difference operators converge to those of harmonic oscillators at multiple wells as the discretization becomes finer.

## Key findings

- Eigenvalues converge to harmonic oscillator eigenvalues at wells
- Eigenfunctions localize near potential wells
- Asymptotic behavior characterized by microlocal analysis

## Abstract

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.