{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:IYBUZDWINN3DCHQRGVZTVEDT3A","short_pith_number":"pith:IYBUZDWI","schema_version":"1.0","canonical_sha256":"46034c8ec86b76311e1135733a9073d81011566be0a61ffb858cb47548570ba7","source":{"kind":"arxiv","id":"2412.11697","version":2},"attestation_state":"computed","paper":{"title":"On the second coefficient in the semi-classical expansion of Toeplitz Operators","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.CV","authors_text":"Chin-Chia Chang, Chin-Yu Hsiao, Hendrik Herrmann","submitted_at":"2024-12-16T12:20:32Z","abstract_excerpt":"Let $X$ be a compact strictly pseudoconvex embeddable CR manifold and let $A$ be the Toeplitz operator on $X$ associated with a Reeb vector field $\\mathcal{T}\\in\\mathscr{C}^\\infty(X,TX)$. Consider the operator $\\chi_k(A)$ defined by functional calculus of $A$, where $\\chi$ is a smooth function with compact support in the positive real line and $\\chi_k(\\lambda):=\\chi(k^{-1}\\lambda)$. It was established recently that $\\chi_k(A)(x,y)$ admits a full asymptotic expansion in $k$. The second coefficient of the expansion plays an important role in the further study of CR geometry. In this work, we cal"},"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":"2412.11697","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CV","submitted_at":"2024-12-16T12:20:32Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"b576ab1aa5304791e1238d7a68f0e47af87faa1667df375ad097beaa6dfde387","abstract_canon_sha256":"d4291bb9f9185c6fff4e50cf0ff8235dcded26a5e262d2fc2d0836a13c6e5088"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:45:30.319985Z","signature_b64":"mbkW3BPhZg3R15ui6b/3SCakjwv50Un/yk3W/gEkziq+Q1yQbYHlji3uGd8NHVhzkEwz9BF2Ipd0gS2vgVqSBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"46034c8ec86b76311e1135733a9073d81011566be0a61ffb858cb47548570ba7","last_reissued_at":"2026-07-05T11:45:30.319478Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:45:30.319478Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the second coefficient in the semi-classical expansion of Toeplitz Operators","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.CV","authors_text":"Chin-Chia Chang, Chin-Yu Hsiao, Hendrik Herrmann","submitted_at":"2024-12-16T12:20:32Z","abstract_excerpt":"Let $X$ be a compact strictly pseudoconvex embeddable CR manifold and let $A$ be the Toeplitz operator on $X$ associated with a Reeb vector field $\\mathcal{T}\\in\\mathscr{C}^\\infty(X,TX)$. Consider the operator $\\chi_k(A)$ defined by functional calculus of $A$, where $\\chi$ is a smooth function with compact support in the positive real line and $\\chi_k(\\lambda):=\\chi(k^{-1}\\lambda)$. It was established recently that $\\chi_k(A)(x,y)$ admits a full asymptotic expansion in $k$. The second coefficient of the expansion plays an important role in the further study of CR geometry. In this work, we cal"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.11697","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2412.11697/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2412.11697","created_at":"2026-07-05T11:45:30.319537+00:00"},{"alias_kind":"arxiv_version","alias_value":"2412.11697v2","created_at":"2026-07-05T11:45:30.319537+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.11697","created_at":"2026-07-05T11:45:30.319537+00:00"},{"alias_kind":"pith_short_12","alias_value":"IYBUZDWINN3D","created_at":"2026-07-05T11:45:30.319537+00:00"},{"alias_kind":"pith_short_16","alias_value":"IYBUZDWINN3DCHQR","created_at":"2026-07-05T11:45:30.319537+00:00"},{"alias_kind":"pith_short_8","alias_value":"IYBUZDWI","created_at":"2026-07-05T11:45:30.319537+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2505.13014","citing_title":"Spectral asymptotics of semi-classical Toeplitz operators on Levi non-degenerate CR manifolds","ref_index":13,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/IYBUZDWINN3DCHQRGVZTVEDT3A","json":"https://pith.science/pith/IYBUZDWINN3DCHQRGVZTVEDT3A.json","graph_json":"https://pith.science/api/pith-number/IYBUZDWINN3DCHQRGVZTVEDT3A/graph.json","events_json":"https://pith.science/api/pith-number/IYBUZDWINN3DCHQRGVZTVEDT3A/events.json","paper":"https://pith.science/paper/IYBUZDWI"},"agent_actions":{"view_html":"https://pith.science/pith/IYBUZDWINN3DCHQRGVZTVEDT3A","download_json":"https://pith.science/pith/IYBUZDWINN3DCHQRGVZTVEDT3A.json","view_paper":"https://pith.science/paper/IYBUZDWI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2412.11697&json=true","fetch_graph":"https://pith.science/api/pith-number/IYBUZDWINN3DCHQRGVZTVEDT3A/graph.json","fetch_events":"https://pith.science/api/pith-number/IYBUZDWINN3DCHQRGVZTVEDT3A/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IYBUZDWINN3DCHQRGVZTVEDT3A/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IYBUZDWINN3DCHQRGVZTVEDT3A/action/storage_attestation","attest_author":"https://pith.science/pith/IYBUZDWINN3DCHQRGVZTVEDT3A/action/author_attestation","sign_citation":"https://pith.science/pith/IYBUZDWINN3DCHQRGVZTVEDT3A/action/citation_signature","submit_replication":"https://pith.science/pith/IYBUZDWINN3DCHQRGVZTVEDT3A/action/replication_record"}},"created_at":"2026-07-05T11:45:30.319537+00:00","updated_at":"2026-07-05T11:45:30.319537+00:00"}