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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"},"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":"1009.0922","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2010-09-05T14:22:30Z","cross_cats_sorted":["math.AP","math.MP"],"title_canon_sha256":"7b4adc27200145f42c747734bc732fb9c7f35285d31cbbf00d3a679708f0c767","abstract_canon_sha256":"3af0611c8b622e40cfda2ddc965b5a466e745d3f5f43dd39b4a3664a3e7fd4a2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:08:44.799922Z","signature_b64":"h3thX47TXUA+9c+gt7Ln6iB4k0NdW+UYbPJVEVAi7rbTXUzK+daYsC04fzSgZfrPRXt9cw7AP9Jh6AT+OYpSBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b0e2339f2fea8df31faff9a14cb3b49e01bfc4cbfd2a78818de543332f240be5","last_reissued_at":"2026-05-18T04:08:44.799428Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:08:44.799428Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $L_{A,Q}=-\\nabla_\\y\\cdot A \\nabla_\\y +\\ Q(\\y)$. 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