{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:LY7I7EVYLDJYYCFVG7WLPNAX7F","short_pith_number":"pith:LY7I7EVY","canonical_record":{"source":{"id":"2409.10387","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2024-09-16T15:21:55Z","cross_cats_sorted":["math-ph","math.CV","math.MP"],"title_canon_sha256":"31f9be8af292a4071e6693bdcde8f885836373578b02dc47b41977ce4e187e1a","abstract_canon_sha256":"57177d420bc838d3d7b44bc778c63325da0436dca712c8c064034cbc073a5d34"},"schema_version":"1.0"},"canonical_sha256":"5e3e8f92b858d38c08b537ecb7b417f95c2ae4497f9914ebb7b45f004682ee82","source":{"kind":"arxiv","id":"2409.10387","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2409.10387","created_at":"2026-07-05T09:07:39Z"},{"alias_kind":"arxiv_version","alias_value":"2409.10387v1","created_at":"2026-07-05T09:07:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2409.10387","created_at":"2026-07-05T09:07:39Z"},{"alias_kind":"pith_short_12","alias_value":"LY7I7EVYLDJY","created_at":"2026-07-05T09:07:39Z"},{"alias_kind":"pith_short_16","alias_value":"LY7I7EVYLDJYYCFV","created_at":"2026-07-05T09:07:39Z"},{"alias_kind":"pith_short_8","alias_value":"LY7I7EVY","created_at":"2026-07-05T09:07:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:LY7I7EVYLDJYYCFVG7WLPNAX7F","target":"record","payload":{"canonical_record":{"source":{"id":"2409.10387","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2024-09-16T15:21:55Z","cross_cats_sorted":["math-ph","math.CV","math.MP"],"title_canon_sha256":"31f9be8af292a4071e6693bdcde8f885836373578b02dc47b41977ce4e187e1a","abstract_canon_sha256":"57177d420bc838d3d7b44bc778c63325da0436dca712c8c064034cbc073a5d34"},"schema_version":"1.0"},"canonical_sha256":"5e3e8f92b858d38c08b537ecb7b417f95c2ae4497f9914ebb7b45f004682ee82","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:07:39.381832Z","signature_b64":"fgYeD5eSl7DSSiZI5Oq+Cd79WOumz96kMrAdm1BbEJYFOoaYBAn2sT67ptB7k/kVByGBps3012kLYOSM77/9AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5e3e8f92b858d38c08b537ecb7b417f95c2ae4497f9914ebb7b45f004682ee82","last_reissued_at":"2026-07-05T09:07:39.381358Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:07:39.381358Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2409.10387","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":"2026-07-05T09:07:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"W881LAMsN6SQsI8wvdMGchU/LarAFtSEMFrjdLIRm5v5dUOOZIl/o8oZd/bK5X0Avyz/p5/V5A5ZlZj/EhDpCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-10T20:18:26.536891Z"},"content_sha256":"9414df717de63fd9d6e364fb594b0537f841729b86c1ab938f41b310eaf68b98","schema_version":"1.0","event_id":"sha256:9414df717de63fd9d6e364fb594b0537f841729b86c1ab938f41b310eaf68b98"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:LY7I7EVYLDJYYCFVG7WLPNAX7F","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sharp decay rate for eigenfunctions of perturbed periodic Schr\\\"odinger operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.CV","math.MP"],"primary_cat":"math.SP","authors_text":"John N. Treuer, Rodrigo Matos, Wencai Liu","submitted_at":"2024-09-16T15:21:55Z","abstract_excerpt":"This paper investigates uniqueness results for perturbed periodic Schr\\\"odinger operators on $\\mathbb{Z}^d$. Specifically, we consider operators of the form $H = -\\Delta + V + v$, where $\\Delta$ is the discrete Laplacian, $V: \\mathbb{Z}^d \\rightarrow \\mathbb{R}$ is a periodic potential, and $v: \\mathbb{Z}^d \\rightarrow \\mathbb{C}$ represents a decaying impurity. We establish quantitative conditions under which the equation $-\\Delta u + V u + v u = \\lambda u$, for $\\lambda \\in \\mathbb{C}$, admits only the trivial solution $u \\equiv 0$. Key applications include the absence of embedded eigenvalue"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.10387","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2409.10387/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T09:07:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kVf3E8RKLey8ueQN52r04vCKro1WNAnsBAyRU3XMKLoyhF9k4kUZ5pQuO58qyEJR3fYhOt2pfnstJbpUlJw5DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-10T20:18:26.537258Z"},"content_sha256":"3f22b7c71caff16a841da0e1eea9b4c2e5921f1f9aaa685e7235baf4fffccbe3","schema_version":"1.0","event_id":"sha256:3f22b7c71caff16a841da0e1eea9b4c2e5921f1f9aaa685e7235baf4fffccbe3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LY7I7EVYLDJYYCFVG7WLPNAX7F/bundle.json","state_url":"https://pith.science/pith/LY7I7EVYLDJYYCFVG7WLPNAX7F/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LY7I7EVYLDJYYCFVG7WLPNAX7F/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":"2026-07-10T20:18:26Z","links":{"resolver":"https://pith.science/pith/LY7I7EVYLDJYYCFVG7WLPNAX7F","bundle":"https://pith.science/pith/LY7I7EVYLDJYYCFVG7WLPNAX7F/bundle.json","state":"https://pith.science/pith/LY7I7EVYLDJYYCFVG7WLPNAX7F/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LY7I7EVYLDJYYCFVG7WLPNAX7F/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:LY7I7EVYLDJYYCFVG7WLPNAX7F","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":"57177d420bc838d3d7b44bc778c63325da0436dca712c8c064034cbc073a5d34","cross_cats_sorted":["math-ph","math.CV","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2024-09-16T15:21:55Z","title_canon_sha256":"31f9be8af292a4071e6693bdcde8f885836373578b02dc47b41977ce4e187e1a"},"schema_version":"1.0","source":{"id":"2409.10387","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2409.10387","created_at":"2026-07-05T09:07:39Z"},{"alias_kind":"arxiv_version","alias_value":"2409.10387v1","created_at":"2026-07-05T09:07:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2409.10387","created_at":"2026-07-05T09:07:39Z"},{"alias_kind":"pith_short_12","alias_value":"LY7I7EVYLDJY","created_at":"2026-07-05T09:07:39Z"},{"alias_kind":"pith_short_16","alias_value":"LY7I7EVYLDJYYCFV","created_at":"2026-07-05T09:07:39Z"},{"alias_kind":"pith_short_8","alias_value":"LY7I7EVY","created_at":"2026-07-05T09:07:39Z"}],"graph_snapshots":[{"event_id":"sha256:3f22b7c71caff16a841da0e1eea9b4c2e5921f1f9aaa685e7235baf4fffccbe3","target":"graph","created_at":"2026-07-05T09:07:39Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2409.10387/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This paper investigates uniqueness results for perturbed periodic Schr\\\"odinger operators on $\\mathbb{Z}^d$. Specifically, we consider operators of the form $H = -\\Delta + V + v$, where $\\Delta$ is the discrete Laplacian, $V: \\mathbb{Z}^d \\rightarrow \\mathbb{R}$ is a periodic potential, and $v: \\mathbb{Z}^d \\rightarrow \\mathbb{C}$ represents a decaying impurity. We establish quantitative conditions under which the equation $-\\Delta u + V u + v u = \\lambda u$, for $\\lambda \\in \\mathbb{C}$, admits only the trivial solution $u \\equiv 0$. Key applications include the absence of embedded eigenvalue","authors_text":"John N. Treuer, Rodrigo Matos, Wencai Liu","cross_cats":["math-ph","math.CV","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2024-09-16T15:21:55Z","title":"Sharp decay rate for eigenfunctions of perturbed periodic Schr\\\"odinger operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.10387","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":"sha256:9414df717de63fd9d6e364fb594b0537f841729b86c1ab938f41b310eaf68b98","target":"record","created_at":"2026-07-05T09:07:39Z","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":"57177d420bc838d3d7b44bc778c63325da0436dca712c8c064034cbc073a5d34","cross_cats_sorted":["math-ph","math.CV","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2024-09-16T15:21:55Z","title_canon_sha256":"31f9be8af292a4071e6693bdcde8f885836373578b02dc47b41977ce4e187e1a"},"schema_version":"1.0","source":{"id":"2409.10387","kind":"arxiv","version":1}},"canonical_sha256":"5e3e8f92b858d38c08b537ecb7b417f95c2ae4497f9914ebb7b45f004682ee82","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5e3e8f92b858d38c08b537ecb7b417f95c2ae4497f9914ebb7b45f004682ee82","first_computed_at":"2026-07-05T09:07:39.381358Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:07:39.381358Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fgYeD5eSl7DSSiZI5Oq+Cd79WOumz96kMrAdm1BbEJYFOoaYBAn2sT67ptB7k/kVByGBps3012kLYOSM77/9AA==","signature_status":"signed_v1","signed_at":"2026-07-05T09:07:39.381832Z","signed_message":"canonical_sha256_bytes"},"source_id":"2409.10387","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9414df717de63fd9d6e364fb594b0537f841729b86c1ab938f41b310eaf68b98","sha256:3f22b7c71caff16a841da0e1eea9b4c2e5921f1f9aaa685e7235baf4fffccbe3"],"state_sha256":"5ea1366e6763ca93461a2ebbe14f08cb186aa8cc622f1ccfbfdb884f19f009a6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cGSEeMMTrCLr2IQUWAkjAtfvdLelqB0/z4Cl91LGGb+hxt/lWFguSGUmSaUZj1iBOCjPUUL/C0wI+f2hbP1iBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-10T20:18:26.539773Z","bundle_sha256":"23b186a1a34c142209c54360cc87ad19cc0343b997c38cd1e3030fe3db91e503"}}