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Here ${\\bf A}$ is a magnetic potential which generates a constant magnetic field $b>0$, and the edge potential $W = \\bar{W}$ is a ${\\mathcal T}$-periodic non constant bounded function depending only on the first coordinate $x \\in {\\mathbb R}$ of $(x,y) \\in {\\mathbb R}^2$. Then the spectrum $\\sigma(H_0)$ of $H_0$ has a band structure, the band functions are $b {\\mathcal T}$-periodic, and generically there are infinitely many open gaps in $\\sigma(H_0)$. We establish explicit sufficien"},"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":"1101.1079","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-01-05T20:57:04Z","cross_cats_sorted":["math.AP","math.MP","math.SP"],"title_canon_sha256":"165e27b193e508fc8861135d12c47e9da506ee4b2859baaf0f44f6182d3d33b4","abstract_canon_sha256":"9ed332572ca84e7f79a5274719568f2ca4e145ecec459dbe562a4ec57992739d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:21:10.586369Z","signature_b64":"kOdKhPtT+AnapQv68lwTp15+Wunms7Jwbw6li12ZtMU/rbj0dwD7Qvz8MXVHSMCX6nl2T+gJ8MCpXSqRRL3GAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e6066c85404560edf508f522483735f70dc82631efc2f8b6aae6de7bc440364a","last_reissued_at":"2026-05-18T04:21:10.585894Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:21:10.585894Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Discrete Spectrum of Quantum Hall Effect Hamiltonians II. 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