{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:XA6YQ47B7EYUHW2WAYBJPZ4CS7","short_pith_number":"pith:XA6YQ47B","canonical_record":{"source":{"id":"1612.02937","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-12-09T08:21:35Z","cross_cats_sorted":["math.SP"],"title_canon_sha256":"b333054846cf204c7075e6d941c14ccbedb6ffc5a8f0e66d424391d9601204ae","abstract_canon_sha256":"d10f5027a1fdc2b362ec05a36d3c998b5aceaff0b9b8559b5ac09e54b0c7a72f"},"schema_version":"1.0"},"canonical_sha256":"b83d8873e1f93143db56060297e78297e2b512f6b7a1ee7b0c45e7fb43a3dae2","source":{"kind":"arxiv","id":"1612.02937","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.02937","created_at":"2026-05-18T00:55:27Z"},{"alias_kind":"arxiv_version","alias_value":"1612.02937v1","created_at":"2026-05-18T00:55:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.02937","created_at":"2026-05-18T00:55:27Z"},{"alias_kind":"pith_short_12","alias_value":"XA6YQ47B7EYU","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"XA6YQ47B7EYUHW2W","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"XA6YQ47B","created_at":"2026-05-18T12:30:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:XA6YQ47B7EYUHW2WAYBJPZ4CS7","target":"record","payload":{"canonical_record":{"source":{"id":"1612.02937","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-12-09T08:21:35Z","cross_cats_sorted":["math.SP"],"title_canon_sha256":"b333054846cf204c7075e6d941c14ccbedb6ffc5a8f0e66d424391d9601204ae","abstract_canon_sha256":"d10f5027a1fdc2b362ec05a36d3c998b5aceaff0b9b8559b5ac09e54b0c7a72f"},"schema_version":"1.0"},"canonical_sha256":"b83d8873e1f93143db56060297e78297e2b512f6b7a1ee7b0c45e7fb43a3dae2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:55:27.925897Z","signature_b64":"vq1XC4C7QrY2GYFjXH1hXvjLlD+jFdmLV+O7Wdyuoj1jasIH/xD8NHlGgruhHeZbj+m+VzT08CkRiheYZS62AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b83d8873e1f93143db56060297e78297e2b512f6b7a1ee7b0c45e7fb43a3dae2","last_reissued_at":"2026-05-18T00:55:27.925438Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:55:27.925438Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1612.02937","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:55:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7FM1dLzfeH57B7FiJ4Ac4fY6efhe9fJf1kAuyKe/nUvlIjATi5YACT15MScqqd8yN26qhivXmcxR8kCW0yETAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T18:48:00.955274Z"},"content_sha256":"cee3bda5a14bf97a3e5b5313b6df3e03df2ec7796e47bc89eb4fa1cbb0f0430c","schema_version":"1.0","event_id":"sha256:cee3bda5a14bf97a3e5b5313b6df3e03df2ec7796e47bc89eb4fa1cbb0f0430c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:XA6YQ47B7EYUHW2WAYBJPZ4CS7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Multidimensional Borg--Levinson theorems for unbounded potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SP"],"primary_cat":"math.AP","authors_text":"Valter Pohjola","submitted_at":"2016-12-09T08:21:35Z","abstract_excerpt":"We prove that the Dirichlet eigenvalues and Neumann boundary data of the corresponding eigenfunctions of the operator $-\\Delta + q$, determine the potential $q$, when $q \\in L^{n/2}(\\Omega,\\mathbb{R})$ and $n \\geq 3$. We also consider the case of incomplete spectral data, in the sense that the above spectral data is unknown for some finite number of eigenvalues. In this case we prove that the potential $q$ is uniquely determined for $q \\in L^p(\\Omega,\\mathbb{R})$ with $p=n/2$, for $n\\geq4$ and $p>n/2$, for $n=3$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.02937","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:55:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qSjZpkroy7ZI1QMuA1O0xX5CFrw8+0n9qG20T3iWFALymC7Wu8ZavRx1Id8SdEfeWUex1FxchyTRRZyRk1HoBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T18:48:00.955727Z"},"content_sha256":"6ca105d2cb68d37d9bfd573052f835b2339a5e2db5e9fb8087c6a351d283e642","schema_version":"1.0","event_id":"sha256:6ca105d2cb68d37d9bfd573052f835b2339a5e2db5e9fb8087c6a351d283e642"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XA6YQ47B7EYUHW2WAYBJPZ4CS7/bundle.json","state_url":"https://pith.science/pith/XA6YQ47B7EYUHW2WAYBJPZ4CS7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XA6YQ47B7EYUHW2WAYBJPZ4CS7/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-30T18:48:00Z","links":{"resolver":"https://pith.science/pith/XA6YQ47B7EYUHW2WAYBJPZ4CS7","bundle":"https://pith.science/pith/XA6YQ47B7EYUHW2WAYBJPZ4CS7/bundle.json","state":"https://pith.science/pith/XA6YQ47B7EYUHW2WAYBJPZ4CS7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XA6YQ47B7EYUHW2WAYBJPZ4CS7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:XA6YQ47B7EYUHW2WAYBJPZ4CS7","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":"d10f5027a1fdc2b362ec05a36d3c998b5aceaff0b9b8559b5ac09e54b0c7a72f","cross_cats_sorted":["math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-12-09T08:21:35Z","title_canon_sha256":"b333054846cf204c7075e6d941c14ccbedb6ffc5a8f0e66d424391d9601204ae"},"schema_version":"1.0","source":{"id":"1612.02937","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.02937","created_at":"2026-05-18T00:55:27Z"},{"alias_kind":"arxiv_version","alias_value":"1612.02937v1","created_at":"2026-05-18T00:55:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.02937","created_at":"2026-05-18T00:55:27Z"},{"alias_kind":"pith_short_12","alias_value":"XA6YQ47B7EYU","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"XA6YQ47B7EYUHW2W","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"XA6YQ47B","created_at":"2026-05-18T12:30:51Z"}],"graph_snapshots":[{"event_id":"sha256:6ca105d2cb68d37d9bfd573052f835b2339a5e2db5e9fb8087c6a351d283e642","target":"graph","created_at":"2026-05-18T00:55:27Z","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":"We prove that the Dirichlet eigenvalues and Neumann boundary data of the corresponding eigenfunctions of the operator $-\\Delta + q$, determine the potential $q$, when $q \\in L^{n/2}(\\Omega,\\mathbb{R})$ and $n \\geq 3$. We also consider the case of incomplete spectral data, in the sense that the above spectral data is unknown for some finite number of eigenvalues. In this case we prove that the potential $q$ is uniquely determined for $q \\in L^p(\\Omega,\\mathbb{R})$ with $p=n/2$, for $n\\geq4$ and $p>n/2$, for $n=3$.","authors_text":"Valter Pohjola","cross_cats":["math.SP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-12-09T08:21:35Z","title":"Multidimensional Borg--Levinson theorems for unbounded potentials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.02937","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:cee3bda5a14bf97a3e5b5313b6df3e03df2ec7796e47bc89eb4fa1cbb0f0430c","target":"record","created_at":"2026-05-18T00:55:27Z","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":"d10f5027a1fdc2b362ec05a36d3c998b5aceaff0b9b8559b5ac09e54b0c7a72f","cross_cats_sorted":["math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-12-09T08:21:35Z","title_canon_sha256":"b333054846cf204c7075e6d941c14ccbedb6ffc5a8f0e66d424391d9601204ae"},"schema_version":"1.0","source":{"id":"1612.02937","kind":"arxiv","version":1}},"canonical_sha256":"b83d8873e1f93143db56060297e78297e2b512f6b7a1ee7b0c45e7fb43a3dae2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b83d8873e1f93143db56060297e78297e2b512f6b7a1ee7b0c45e7fb43a3dae2","first_computed_at":"2026-05-18T00:55:27.925438Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:55:27.925438Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vq1XC4C7QrY2GYFjXH1hXvjLlD+jFdmLV+O7Wdyuoj1jasIH/xD8NHlGgruhHeZbj+m+VzT08CkRiheYZS62AQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:55:27.925897Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.02937","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cee3bda5a14bf97a3e5b5313b6df3e03df2ec7796e47bc89eb4fa1cbb0f0430c","sha256:6ca105d2cb68d37d9bfd573052f835b2339a5e2db5e9fb8087c6a351d283e642"],"state_sha256":"ca7850538f5beebbdd1f7e76ce61e3119d1e1d9b9dd4ca63ae2990f40ba5f70b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"svtTrttspb1nlzOMgXMtkntIKGVWGezRMTKiQ4UnX6Au+FVpW92Ax2TMyQTYi9cMf++JE4wF76/3yjUfa/xEAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T18:48:00.959038Z","bundle_sha256":"4b44d86113452ac2405883b3556778b7f1857db675835328444fdbfb8f56972f"}}