{"paper":{"title":"A Multi-Scale Analysis Scheme on Abelian Groups with an Application to Operators Dual to Hill's Equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.SP","authors_text":"David Damanik (Rice University), Michael Goldstein (University of Toronto), Milivoje Lukic (University of Toronto, Rice University)","submitted_at":"2014-09-07T18:14:28Z","abstract_excerpt":"We present an abstract multiscale analysis scheme for matrix functions $(H_{\\varepsilon}(m,n))_{m,n\\in \\mathfrak{T}}$, where $\\mathfrak{T}$ is an Abelian group equipped with a distance $|\\cdot|$. This is an extension of the scheme developed by Damanik and Goldstein for the special case $\\mathfrak{T} = \\mathbb{Z}^\\nu$. Our main motivation for working out this extension comes from an application to matrix functions which are dual to certain Hill operators. These operators take the form $H_{\\tilde\\omega}=-\\frac{d^2}{dx^2} + \\varepsilon U(\\tilde\\omega x)$, where $U$ is a real smooth function on th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.2147","kind":"arxiv","version":2},"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"}