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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_"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1009.4039","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2010-09-21T10:12:34Z","cross_cats_sorted":["math.MP","math.SP"],"title_canon_sha256":"213c09fd4f2c64e61d9932ad4552633e6f4aa45ca097aa3a3256ab870156a299","abstract_canon_sha256":"53fd85a8d7510da59915575ee3c4e52bb7f2f8b2befc46e82bbf4e990968645e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:15:04.360361Z","signature_b64":"9jalhsUx7bWKKPRrYZe6/ueBJy8l4lgAmkmrrIdrfj7UoChUGx1SUbFDHHocM7vUWhfw6pr3qcAW82Cqavr8Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"86879fe39a9b30cc1d2a2ec47393fbd6ab0b63175a6abc04625aaab45a9f622f","last_reissued_at":"2026-05-18T04:15:04.359745Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:15:04.359745Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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