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We denote by $\\mathbf{L}_{\\mathrm{per}}$ the set of such operators. The main result of this work is as follows: for an arbitrary $L>0$ and for arbitrary pairwise disjoint intervals $(\\alpha_j,\\beta_j)\\subset[0,L]$, $j=1,...,m$ ($m\\in\\mathbb{N}$) we construct the family of operators $\\{\\mathbf{A}^\\varepsilon\\in \\mathbf{L}_{\\mathrm{pe"},"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":"1201.3729","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2012-01-18T09:24:49Z","cross_cats_sorted":["math-ph","math.AP","math.MP"],"title_canon_sha256":"5e6bd706fb421c6a671185760a4c8c7f7cca7b765bcb141ecf802f798f5c264e","abstract_canon_sha256":"18087797690bd1e07137084568d21e2b21c023aa07cfed882e42d96356661507"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:03:08.726308Z","signature_b64":"kZlLSgQWGvc3Igoc3DAlXjNKbS/zPvllGgFM5a0dw2A2SDPR9o1uVaR7Yh9R1tgbw7V6j2pKSUUI4Qe/8FHXDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ce3a3cf339ce157e32514ca6dde7c0124ff91902f0ffa2753b4d623feb5a1b74","last_reissued_at":"2026-05-18T04:03:08.725444Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:03:08.725444Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Periodic elliptic operators with asymptotically preassigned spectrum","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.AP","math.MP"],"primary_cat":"math.SP","authors_text":"Andrii Khrabustovskyi","submitted_at":"2012-01-18T09:24:49Z","abstract_excerpt":"We deal with operators in $\\mathbb{R}^n$ of the form $$\\mathbf{A}=-{1\\over \\mathbf{b}(x)}\\sum\\limits_{k=1}^n\\ds{\\partial\\over\\partial x_k}(\\mathbf{a}(x){\\partial \\over\\partial x_k})$$ where $\\mathbf{a}(x),\\mathbf{b}(x)$ are positive, bounded and periodic functions. We denote by $\\mathbf{L}_{\\mathrm{per}}$ the set of such operators. The main result of this work is as follows: for an arbitrary $L>0$ and for arbitrary pairwise disjoint intervals $(\\alpha_j,\\beta_j)\\subset[0,L]$, $j=1,...,m$ ($m\\in\\mathbb{N}$) we construct the family of operators $\\{\\mathbf{A}^\\varepsilon\\in \\mathbf{L}_{\\mathrm{pe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.3729","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1201.3729","created_at":"2026-05-18T04:03:08.725587+00:00"},{"alias_kind":"arxiv_version","alias_value":"1201.3729v2","created_at":"2026-05-18T04:03:08.725587+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.3729","created_at":"2026-05-18T04:03:08.725587+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZY5DZ4ZZZYKX","created_at":"2026-05-18T12:27:30.460161+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZY5DZ4ZZZYKX4MSR","created_at":"2026-05-18T12:27:30.460161+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZY5DZ4ZZ","created_at":"2026-05-18T12:27:30.460161+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ZY5DZ4ZZZYKX4MSRJSTN3Z6ACJ","json":"https://pith.science/pith/ZY5DZ4ZZZYKX4MSRJSTN3Z6ACJ.json","graph_json":"https://pith.science/api/pith-number/ZY5DZ4ZZZYKX4MSRJSTN3Z6ACJ/graph.json","events_json":"https://pith.science/api/pith-number/ZY5DZ4ZZZYKX4MSRJSTN3Z6ACJ/events.json","paper":"https://pith.science/paper/ZY5DZ4ZZ"},"agent_actions":{"view_html":"https://pith.science/pith/ZY5DZ4ZZZYKX4MSRJSTN3Z6ACJ","download_json":"https://pith.science/pith/ZY5DZ4ZZZYKX4MSRJSTN3Z6ACJ.json","view_paper":"https://pith.science/paper/ZY5DZ4ZZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1201.3729&json=true","fetch_graph":"https://pith.science/api/pith-number/ZY5DZ4ZZZYKX4MSRJSTN3Z6ACJ/graph.json","fetch_events":"https://pith.science/api/pith-number/ZY5DZ4ZZZYKX4MSRJSTN3Z6ACJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZY5DZ4ZZZYKX4MSRJSTN3Z6ACJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZY5DZ4ZZZYKX4MSRJSTN3Z6ACJ/action/storage_attestation","attest_author":"https://pith.science/pith/ZY5DZ4ZZZYKX4MSRJSTN3Z6ACJ/action/author_attestation","sign_citation":"https://pith.science/pith/ZY5DZ4ZZZYKX4MSRJSTN3Z6ACJ/action/citation_signature","submit_replication":"https://pith.science/pith/ZY5DZ4ZZZYKX4MSRJSTN3Z6ACJ/action/replication_record"}},"created_at":"2026-05-18T04:03:08.725587+00:00","updated_at":"2026-05-18T04:03:08.725587+00:00"}