{"paper":{"title":"Defect Modes and Homogenization of Periodic Schr\\\"odinger Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MP"],"primary_cat":"math-ph","authors_text":"M. A. Hoefer, M. I. Weinstein","submitted_at":"2010-09-05T14:22:30Z","abstract_excerpt":"We consider the discrete eigenvalues of the operator $H_\\eps=-\\Delta+V(\\x)+\\eps^2Q(\\eps\\x)$, where $V(\\x)$ is periodic and $Q(\\y)$ is localized on $\\R^d,\\ \\ d\\ge1$. For $\\eps>0$ and sufficiently small, discrete eigenvalues may bifurcate (emerge) from spectral band edges of the periodic Schr\\\"odinger operator, $H_0 = -\\Delta_\\x+V(\\x)$, into spectral gaps. The nature of the bifurcation depends on the homogenized Schr\\\"odinger operator $L_{A,Q}=-\\nabla_\\y\\cdot A \\nabla_\\y +\\ Q(\\y)$. Here, $A$ denotes the inverse effective mass matrix, associated with the spectral band edge, which is the site of t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.0922","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"}