{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:UNM7L63R2CW5YNSG5BCZKAKIU7","short_pith_number":"pith:UNM7L63R","canonical_record":{"source":{"id":"1810.01024","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2018-10-02T00:53:32Z","cross_cats_sorted":["math.AP","physics.comp-ph"],"title_canon_sha256":"d4bc7ea4e609e3e5cb359520671dabd25bc753c3a3fdc1295de12298d823509b","abstract_canon_sha256":"caf7e0526c148ab9b0e34c550c5a144b1fbb9cd3a65264df493273730e1836e0"},"schema_version":"1.0"},"canonical_sha256":"a359f5fb71d0addc3646e845950148a7f60b37c5d098adb657f8e76423417142","source":{"kind":"arxiv","id":"1810.01024","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1810.01024","created_at":"2026-05-17T23:59:49Z"},{"alias_kind":"arxiv_version","alias_value":"1810.01024v2","created_at":"2026-05-17T23:59:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.01024","created_at":"2026-05-17T23:59:49Z"},{"alias_kind":"pith_short_12","alias_value":"UNM7L63R2CW5","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_16","alias_value":"UNM7L63R2CW5YNSG","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_8","alias_value":"UNM7L63R","created_at":"2026-05-18T12:32:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:UNM7L63R2CW5YNSG5BCZKAKIU7","target":"record","payload":{"canonical_record":{"source":{"id":"1810.01024","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2018-10-02T00:53:32Z","cross_cats_sorted":["math.AP","physics.comp-ph"],"title_canon_sha256":"d4bc7ea4e609e3e5cb359520671dabd25bc753c3a3fdc1295de12298d823509b","abstract_canon_sha256":"caf7e0526c148ab9b0e34c550c5a144b1fbb9cd3a65264df493273730e1836e0"},"schema_version":"1.0"},"canonical_sha256":"a359f5fb71d0addc3646e845950148a7f60b37c5d098adb657f8e76423417142","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:59:49.405681Z","signature_b64":"zzFpSL320XwqtcNWY5Gq0suxPr4gKifCj5wZA35WvCiES37xdZFxWZLvn7C1RM+z8/Ao5S1RMQp9/tKf/fMlAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a359f5fb71d0addc3646e845950148a7f60b37c5d098adb657f8e76423417142","last_reissued_at":"2026-05-17T23:59:49.405174Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:59:49.405174Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1810.01024","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:59:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rKKkziNvHtb5mqC8HRaVgKowC5IdhdWguTBDISZ6/qv8mzk00kBdG6bBDsWQVj+K1aEwJeCSQxL5pHAxMiqMCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T06:35:30.330806Z"},"content_sha256":"a722d646b85a59441f83c08e1c7cf144781f425af29793adf8e567e1dc032cfe","schema_version":"1.0","event_id":"sha256:a722d646b85a59441f83c08e1c7cf144781f425af29793adf8e567e1dc032cfe"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:UNM7L63R2CW5YNSG5BCZKAKIU7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Pointwise Products of Elliptic Eigenfunctions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","physics.comp-ph"],"primary_cat":"math.SP","authors_text":"Jianfeng Lu, Stefan Steinerberger","submitted_at":"2018-10-02T00:53:32Z","abstract_excerpt":"We consider eigenfunctions of Schr\\\"odinger operators on a $d-$dimensional bounded domain $\\Omega$ (or a $d-$dimensional compact manifold $\\Omega$) with Dirichlet conditions. These operators give rise to a sequence of eigenfunctions $(\\phi_n)_{n \\in \\mathbb{N}}$. We study the subspace of all pointwise products $$ A_n = \\mbox{span} \\left\\{ \\phi_i(x) \\phi_j(x): 1 \\leq i,j \\leq n\\right\\} \\subseteq L^2(\\Omega).$$ Clearly, that vector space has dimension $\\mbox{dim}(A_n) = n(n+1)/2$. We prove that products $\\phi_i \\phi_j$ of eigenfunctions are simple in a certain sense: for any $\\varepsilon > 0$, t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.01024","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":""},"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":"2026-05-17T23:59:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oMpDfBkYWM7vzyp3d5LJk4v+TFxX66FAihewaTEc1idUP5BDkGabwezMj5Vb1e3ftIMmxuYa9D7w4V2q01ZPAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T06:35:30.331495Z"},"content_sha256":"ea4ce745af2b5b14b7aab10fbbe9204ea82102688753bfb20a92c155a72a92e4","schema_version":"1.0","event_id":"sha256:ea4ce745af2b5b14b7aab10fbbe9204ea82102688753bfb20a92c155a72a92e4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UNM7L63R2CW5YNSG5BCZKAKIU7/bundle.json","state_url":"https://pith.science/pith/UNM7L63R2CW5YNSG5BCZKAKIU7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UNM7L63R2CW5YNSG5BCZKAKIU7/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":"2026-05-26T06:35:30Z","links":{"resolver":"https://pith.science/pith/UNM7L63R2CW5YNSG5BCZKAKIU7","bundle":"https://pith.science/pith/UNM7L63R2CW5YNSG5BCZKAKIU7/bundle.json","state":"https://pith.science/pith/UNM7L63R2CW5YNSG5BCZKAKIU7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UNM7L63R2CW5YNSG5BCZKAKIU7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:UNM7L63R2CW5YNSG5BCZKAKIU7","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":"caf7e0526c148ab9b0e34c550c5a144b1fbb9cd3a65264df493273730e1836e0","cross_cats_sorted":["math.AP","physics.comp-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2018-10-02T00:53:32Z","title_canon_sha256":"d4bc7ea4e609e3e5cb359520671dabd25bc753c3a3fdc1295de12298d823509b"},"schema_version":"1.0","source":{"id":"1810.01024","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1810.01024","created_at":"2026-05-17T23:59:49Z"},{"alias_kind":"arxiv_version","alias_value":"1810.01024v2","created_at":"2026-05-17T23:59:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.01024","created_at":"2026-05-17T23:59:49Z"},{"alias_kind":"pith_short_12","alias_value":"UNM7L63R2CW5","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_16","alias_value":"UNM7L63R2CW5YNSG","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_8","alias_value":"UNM7L63R","created_at":"2026-05-18T12:32:56Z"}],"graph_snapshots":[{"event_id":"sha256:ea4ce745af2b5b14b7aab10fbbe9204ea82102688753bfb20a92c155a72a92e4","target":"graph","created_at":"2026-05-17T23:59:49Z","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 consider eigenfunctions of Schr\\\"odinger operators on a $d-$dimensional bounded domain $\\Omega$ (or a $d-$dimensional compact manifold $\\Omega$) with Dirichlet conditions. These operators give rise to a sequence of eigenfunctions $(\\phi_n)_{n \\in \\mathbb{N}}$. We study the subspace of all pointwise products $$ A_n = \\mbox{span} \\left\\{ \\phi_i(x) \\phi_j(x): 1 \\leq i,j \\leq n\\right\\} \\subseteq L^2(\\Omega).$$ Clearly, that vector space has dimension $\\mbox{dim}(A_n) = n(n+1)/2$. We prove that products $\\phi_i \\phi_j$ of eigenfunctions are simple in a certain sense: for any $\\varepsilon > 0$, t","authors_text":"Jianfeng Lu, Stefan Steinerberger","cross_cats":["math.AP","physics.comp-ph"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2018-10-02T00:53:32Z","title":"On Pointwise Products of Elliptic Eigenfunctions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.01024","kind":"arxiv","version":2},"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:a722d646b85a59441f83c08e1c7cf144781f425af29793adf8e567e1dc032cfe","target":"record","created_at":"2026-05-17T23:59:49Z","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":"caf7e0526c148ab9b0e34c550c5a144b1fbb9cd3a65264df493273730e1836e0","cross_cats_sorted":["math.AP","physics.comp-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2018-10-02T00:53:32Z","title_canon_sha256":"d4bc7ea4e609e3e5cb359520671dabd25bc753c3a3fdc1295de12298d823509b"},"schema_version":"1.0","source":{"id":"1810.01024","kind":"arxiv","version":2}},"canonical_sha256":"a359f5fb71d0addc3646e845950148a7f60b37c5d098adb657f8e76423417142","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a359f5fb71d0addc3646e845950148a7f60b37c5d098adb657f8e76423417142","first_computed_at":"2026-05-17T23:59:49.405174Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:59:49.405174Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zzFpSL320XwqtcNWY5Gq0suxPr4gKifCj5wZA35WvCiES37xdZFxWZLvn7C1RM+z8/Ao5S1RMQp9/tKf/fMlAQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:59:49.405681Z","signed_message":"canonical_sha256_bytes"},"source_id":"1810.01024","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a722d646b85a59441f83c08e1c7cf144781f425af29793adf8e567e1dc032cfe","sha256:ea4ce745af2b5b14b7aab10fbbe9204ea82102688753bfb20a92c155a72a92e4"],"state_sha256":"1ace1bd9fbe389dadc415a63320a357fed771b1f03814f4d4b94f25d3230446c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fXBlD5wTHSdIK3WiNfpP+ZFUZCf3IuV/Pij6N79DTWHlkG5wRz322kS+Dp8lv5JH+tRbySnvsA/FaiLak77QDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T06:35:30.335186Z","bundle_sha256":"0ac9105a3121cf33d0709c336a92b07c956696939c10deedf7bfb8441f7ffa1c"}}