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arXiv · 1706.06357

Harmonic Approximation of Difference Operators

Markus Klein·Elke Rosenberger
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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.

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Markus Klein, Elke Rosenberger. 2017-06-20. Harmonic Approximation of Difference Operators. https://doi.org/10.1016/j.jfa.2009.09.004

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