{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2022:W64INKDN2JBQNMFXBWAJDIOGQD","short_pith_number":"pith:W64INKDN","schema_version":"1.0","canonical_sha256":"b7b886a86dd24306b0b70d8091a1c680d2b5291c35dd91e916a4cbb01f35eed0","source":{"kind":"arxiv","id":"2201.03140","version":3},"attestation_state":"computed","paper":{"title":"Propagation of singularities and Fredholm analysis for the time-dependent Schr\\\"odinger equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SP"],"primary_cat":"math.AP","authors_text":"Andrew Hassell, Jesse Gell-Redman, Sean Gomes","submitted_at":"2022-01-10T02:34:57Z","abstract_excerpt":"We study the time-dependent Schr\\\"odinger operator $P = D_t + \\Delta_g + V$ acting on functions defined on $\\mathbb{R}^{n+1}$, where, using coordinates $z \\in \\mathbb{R}^n$ and $t \\in \\mathbb{R}$, $D_t$ denotes $-i \\partial_t$, $\\Delta_g$ is the positive Laplacian with respect to a time dependent family of non-trapping metrics $g_{ij}(z, t) dz^i dz^j$ on $\\mathbb{R}^n$ which is equal to the Euclidean metric outside of a compact set in spacetime, and $V = V(z, t)$ is a potential function which is also compactly supported in spacetime. In this paper we introduce a new approach to studying $P$, b"},"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":"2201.03140","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2022-01-10T02:34:57Z","cross_cats_sorted":["math.SP"],"title_canon_sha256":"ab842ee4f74924a0de379ba83fe87df1f68e170b5c43853edb89c23e8f06f624","abstract_canon_sha256":"bff71278ae72171b61a05a13fb5e9a0827ca02c3dfe395cc93a399b4f66db8c9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:11:11.804821Z","signature_b64":"K0MZ0q3wyxEwNvsM0BMaEgAzEXucmCpbD2Ark0e62UjVKWTCuO5QtTu8KtUqh9BLyml7/h1HDRIuyqq7Eno3Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b7b886a86dd24306b0b70d8091a1c680d2b5291c35dd91e916a4cbb01f35eed0","last_reissued_at":"2026-07-05T07:11:11.804306Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:11:11.804306Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Propagation of singularities and Fredholm analysis for the time-dependent Schr\\\"odinger equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SP"],"primary_cat":"math.AP","authors_text":"Andrew Hassell, Jesse Gell-Redman, Sean Gomes","submitted_at":"2022-01-10T02:34:57Z","abstract_excerpt":"We study the time-dependent Schr\\\"odinger operator $P = D_t + \\Delta_g + V$ acting on functions defined on $\\mathbb{R}^{n+1}$, where, using coordinates $z \\in \\mathbb{R}^n$ and $t \\in \\mathbb{R}$, $D_t$ denotes $-i \\partial_t$, $\\Delta_g$ is the positive Laplacian with respect to a time dependent family of non-trapping metrics $g_{ij}(z, t) dz^i dz^j$ on $\\mathbb{R}^n$ which is equal to the Euclidean metric outside of a compact set in spacetime, and $V = V(z, t)$ is a potential function which is also compactly supported in spacetime. In this paper we introduce a new approach to studying $P$, b"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2201.03140","kind":"arxiv","version":3},"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/2201.03140/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":"2201.03140","created_at":"2026-07-05T07:11:11.804377+00:00"},{"alias_kind":"arxiv_version","alias_value":"2201.03140v3","created_at":"2026-07-05T07:11:11.804377+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2201.03140","created_at":"2026-07-05T07:11:11.804377+00:00"},{"alias_kind":"pith_short_12","alias_value":"W64INKDN2JBQ","created_at":"2026-07-05T07:11:11.804377+00:00"},{"alias_kind":"pith_short_16","alias_value":"W64INKDN2JBQNMFX","created_at":"2026-07-05T07:11:11.804377+00:00"},{"alias_kind":"pith_short_8","alias_value":"W64INKDN","created_at":"2026-07-05T07:11:11.804377+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.09280","citing_title":"The effect of geometric focusing on dispersive estimates for Schr\\\"odinger and wave equations","ref_index":9,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/W64INKDN2JBQNMFXBWAJDIOGQD","json":"https://pith.science/pith/W64INKDN2JBQNMFXBWAJDIOGQD.json","graph_json":"https://pith.science/api/pith-number/W64INKDN2JBQNMFXBWAJDIOGQD/graph.json","events_json":"https://pith.science/api/pith-number/W64INKDN2JBQNMFXBWAJDIOGQD/events.json","paper":"https://pith.science/paper/W64INKDN"},"agent_actions":{"view_html":"https://pith.science/pith/W64INKDN2JBQNMFXBWAJDIOGQD","download_json":"https://pith.science/pith/W64INKDN2JBQNMFXBWAJDIOGQD.json","view_paper":"https://pith.science/paper/W64INKDN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2201.03140&json=true","fetch_graph":"https://pith.science/api/pith-number/W64INKDN2JBQNMFXBWAJDIOGQD/graph.json","fetch_events":"https://pith.science/api/pith-number/W64INKDN2JBQNMFXBWAJDIOGQD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/W64INKDN2JBQNMFXBWAJDIOGQD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/W64INKDN2JBQNMFXBWAJDIOGQD/action/storage_attestation","attest_author":"https://pith.science/pith/W64INKDN2JBQNMFXBWAJDIOGQD/action/author_attestation","sign_citation":"https://pith.science/pith/W64INKDN2JBQNMFXBWAJDIOGQD/action/citation_signature","submit_replication":"https://pith.science/pith/W64INKDN2JBQNMFXBWAJDIOGQD/action/replication_record"}},"created_at":"2026-07-05T07:11:11.804377+00:00","updated_at":"2026-07-05T07:11:11.804377+00:00"}