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Under these assumptions, we prove that the spectrum of $H$ contains a half-line. This, in particular implies the Bethe-Sommerfeld Conjecture for the Schr\\\"odinger operator with a periodic magnetic potential in all dimensions."},"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":"0907.0887","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2009-07-05T20:27:09Z","cross_cats_sorted":[],"title_canon_sha256":"3d5459c896eb7812f47d21b29f8ee755fb43e7dd60d172adb24d55141a8fda6b","abstract_canon_sha256":"273bbc79b0a0b475932c2f39c325cd18a35ed385e407f2dd0340f79622d98ed0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:13:05.179395Z","signature_b64":"71Aqm71fDnqqbUkI4ZfA87voICE4eFsXk9SoI7dT/fT4WRF3UVH7z3SjGvt1dA25Pz+PTdWDDRoVst21TrBIDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dd6ee205ee5a342b2b010797161879143149f8061a26364169eb3566a36b2b75","last_reissued_at":"2026-05-18T02:13:05.178803Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:13:05.178803Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bethe-Sommerfeld conjecture for periodic operators with strong perturbations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"A.V. 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