{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:CGRGN357YRH53IKBQAXDR35EBT","short_pith_number":"pith:CGRGN357","schema_version":"1.0","canonical_sha256":"11a266efbfc44fdda141802e38efa40cce25f3c8a74b530e7f84020b72fe16e4","source":{"kind":"arxiv","id":"1504.04256","version":1},"attestation_state":"computed","paper":{"title":"The Generalized Legendre transform and its applications to inverse spectral problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SG"],"primary_cat":"math.SP","authors_text":"Victor Guillemin, Zuoqin Wang","submitted_at":"2015-04-16T14:44:06Z","abstract_excerpt":"Let $M$ be a Riemannian manifold, $\\tau: G \\times M \\to M$ an isometric action on $M$ of an $n$-torus $G$ and $V: M \\to \\mathbb R$ a bounded $G$-invariant smooth function. By $G$-invariance the Schr\\\"odinger operator, $P=-\\hbar^2 \\Delta_M+V$, restricts to a self-adjoint operator on $L^2(M)_{\\alpha/\\hbar}$, $\\alpha$ being a weight of $G$ and $1/\\hbar$ a large positive integer. Let $[c_\\alpha, \\infty)$ be the asymptotic support of the spectrum of this operator. We will show that $c_\\alpha$ extend to a function, $W: \\mathfrak g^* \\to \\mathbb R$ and that, modulo assumptions on $\\tau$ and $V$ one c"},"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":"1504.04256","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2015-04-16T14:44:06Z","cross_cats_sorted":["math.SG"],"title_canon_sha256":"829a9a9100abe393baf03957ea68079524341352fda2061feb9d5cf38dd59c97","abstract_canon_sha256":"583cfb310bd07ed1e5fa4cfd575d19061ee8d9f97ca76cb11c792e2cff869d99"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:40.531095Z","signature_b64":"YiRSQqAn3MNiWLm/8WsfKsvxsCk4bWzUZtndZxQp5NWKyCOnn3GlsJUbAL6V3qwz/7vkYf7BS4ag9fFc0yQVBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"11a266efbfc44fdda141802e38efa40cce25f3c8a74b530e7f84020b72fe16e4","last_reissued_at":"2026-05-18T01:22:40.530406Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:40.530406Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Generalized Legendre transform and its applications to inverse spectral problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SG"],"primary_cat":"math.SP","authors_text":"Victor Guillemin, Zuoqin Wang","submitted_at":"2015-04-16T14:44:06Z","abstract_excerpt":"Let $M$ be a Riemannian manifold, $\\tau: G \\times M \\to M$ an isometric action on $M$ of an $n$-torus $G$ and $V: M \\to \\mathbb R$ a bounded $G$-invariant smooth function. By $G$-invariance the Schr\\\"odinger operator, $P=-\\hbar^2 \\Delta_M+V$, restricts to a self-adjoint operator on $L^2(M)_{\\alpha/\\hbar}$, $\\alpha$ being a weight of $G$ and $1/\\hbar$ a large positive integer. Let $[c_\\alpha, \\infty)$ be the asymptotic support of the spectrum of this operator. We will show that $c_\\alpha$ extend to a function, $W: \\mathfrak g^* \\to \\mathbb R$ and that, modulo assumptions on $\\tau$ and $V$ one c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.04256","kind":"arxiv","version":1},"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":"1504.04256","created_at":"2026-05-18T01:22:40.530509+00:00"},{"alias_kind":"arxiv_version","alias_value":"1504.04256v1","created_at":"2026-05-18T01:22:40.530509+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.04256","created_at":"2026-05-18T01:22:40.530509+00:00"},{"alias_kind":"pith_short_12","alias_value":"CGRGN357YRH5","created_at":"2026-05-18T12:29:14.074870+00:00"},{"alias_kind":"pith_short_16","alias_value":"CGRGN357YRH53IKB","created_at":"2026-05-18T12:29:14.074870+00:00"},{"alias_kind":"pith_short_8","alias_value":"CGRGN357","created_at":"2026-05-18T12:29:14.074870+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/CGRGN357YRH53IKBQAXDR35EBT","json":"https://pith.science/pith/CGRGN357YRH53IKBQAXDR35EBT.json","graph_json":"https://pith.science/api/pith-number/CGRGN357YRH53IKBQAXDR35EBT/graph.json","events_json":"https://pith.science/api/pith-number/CGRGN357YRH53IKBQAXDR35EBT/events.json","paper":"https://pith.science/paper/CGRGN357"},"agent_actions":{"view_html":"https://pith.science/pith/CGRGN357YRH53IKBQAXDR35EBT","download_json":"https://pith.science/pith/CGRGN357YRH53IKBQAXDR35EBT.json","view_paper":"https://pith.science/paper/CGRGN357","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1504.04256&json=true","fetch_graph":"https://pith.science/api/pith-number/CGRGN357YRH53IKBQAXDR35EBT/graph.json","fetch_events":"https://pith.science/api/pith-number/CGRGN357YRH53IKBQAXDR35EBT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CGRGN357YRH53IKBQAXDR35EBT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CGRGN357YRH53IKBQAXDR35EBT/action/storage_attestation","attest_author":"https://pith.science/pith/CGRGN357YRH53IKBQAXDR35EBT/action/author_attestation","sign_citation":"https://pith.science/pith/CGRGN357YRH53IKBQAXDR35EBT/action/citation_signature","submit_replication":"https://pith.science/pith/CGRGN357YRH53IKBQAXDR35EBT/action/replication_record"}},"created_at":"2026-05-18T01:22:40.530509+00:00","updated_at":"2026-05-18T01:22:40.530509+00:00"}