{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:CGRGN357YRH53IKBQAXDR35EBT","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"583cfb310bd07ed1e5fa4cfd575d19061ee8d9f97ca76cb11c792e2cff869d99","cross_cats_sorted":["math.SG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2015-04-16T14:44:06Z","title_canon_sha256":"829a9a9100abe393baf03957ea68079524341352fda2061feb9d5cf38dd59c97"},"schema_version":"1.0","source":{"id":"1504.04256","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.04256","created_at":"2026-05-18T01:22:40Z"},{"alias_kind":"arxiv_version","alias_value":"1504.04256v1","created_at":"2026-05-18T01:22:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.04256","created_at":"2026-05-18T01:22:40Z"},{"alias_kind":"pith_short_12","alias_value":"CGRGN357YRH5","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_16","alias_value":"CGRGN357YRH53IKB","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_8","alias_value":"CGRGN357","created_at":"2026-05-18T12:29:14Z"}],"graph_snapshots":[{"event_id":"sha256:7f4e65d098682c1855c298ebbe513ab02ccc96cbee55b0eb1ead697ff9849603","target":"graph","created_at":"2026-05-18T01:22:40Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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","authors_text":"Victor Guillemin, Zuoqin Wang","cross_cats":["math.SG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2015-04-16T14:44:06Z","title":"The Generalized Legendre transform and its applications to inverse spectral problems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.04256","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:ac90db8fbca463ada27fd148b431cf614d96a02f058d97228f78f87fd811a794","target":"record","created_at":"2026-05-18T01:22:40Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"583cfb310bd07ed1e5fa4cfd575d19061ee8d9f97ca76cb11c792e2cff869d99","cross_cats_sorted":["math.SG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2015-04-16T14:44:06Z","title_canon_sha256":"829a9a9100abe393baf03957ea68079524341352fda2061feb9d5cf38dd59c97"},"schema_version":"1.0","source":{"id":"1504.04256","kind":"arxiv","version":1}},"canonical_sha256":"11a266efbfc44fdda141802e38efa40cce25f3c8a74b530e7f84020b72fe16e4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"11a266efbfc44fdda141802e38efa40cce25f3c8a74b530e7f84020b72fe16e4","first_computed_at":"2026-05-18T01:22:40.530406Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:22:40.530406Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YiRSQqAn3MNiWLm/8WsfKsvxsCk4bWzUZtndZxQp5NWKyCOnn3GlsJUbAL6V3qwz/7vkYf7BS4ag9fFc0yQVBA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:22:40.531095Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.04256","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ac90db8fbca463ada27fd148b431cf614d96a02f058d97228f78f87fd811a794","sha256:7f4e65d098682c1855c298ebbe513ab02ccc96cbee55b0eb1ead697ff9849603"],"state_sha256":"4bbdeeb77536d1717c9251df6b95271e7025f5041d0bfa3661e66e98b8924761"}