{"paper":{"title":"Spectral Properties of Grain Boundaries at Small Angles of Rotation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP","math.SP"],"primary_cat":"math-ph","authors_text":"Martin Kohlmann, Rainer Hempel","submitted_at":"2010-09-21T10:12:34Z","abstract_excerpt":"We study some spectral properties of a simple two-dimensional model for small angle defects in crystals and alloys. Starting from a periodic potential $V \\colon \\R^2 \\to \\R$, we let $V_\\theta(x,y) = V(x,y)$ in the right half-plane $\\{x \\ge 0\\}$ and $V_\\theta = V \\circ M_{-\\theta}$ in the left half-plane $\\{x < 0\\}$, where $M_\\theta \\in \\R^{2 \\times 2}$ is the usual matrix describing rotation of the coordinates in $\\R^2$ by an angle $\\theta$. As a main result, it is shown that spectral gaps of the periodic Schr\\\"odinger operator $H_0 = -\\Delta + V$ fill with spectrum of $R_\\theta = -\\Delta + V_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.4039","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"}