{"paper":{"title":"Harmonic Approximation of Difference Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP","math.SP"],"primary_cat":"math-ph","authors_text":"Elke Rosenberger, Markus Klein","submitted_at":"2017-06-20T10:30:50Z","abstract_excerpt":"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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.06357","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}