{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2004:W2U7R4VL74MLA5MSVPUG7WC5RB","short_pith_number":"pith:W2U7R4VL","canonical_record":{"source":{"id":"math-ph/0409074","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math-ph","submitted_at":"2004-09-28T21:46:46Z","cross_cats_sorted":["math.MP","math.SP"],"title_canon_sha256":"5f5888cf2a61e82bc514cd98ca0db5a1c684169201ad06a4fb5044e856af4fbe","abstract_canon_sha256":"040adf36d39a07fdbb45e49219a5a5ec22c5dd3b4854523dec1c5df18bfc9cbd"},"schema_version":"1.0"},"canonical_sha256":"b6a9f8f2abff18b07592abe86fd85d88564575eb6b0fccc8a6c5ff3d68dbcb63","source":{"kind":"arxiv","id":"math-ph/0409074","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math-ph/0409074","created_at":"2026-05-18T02:30:23Z"},{"alias_kind":"arxiv_version","alias_value":"math-ph/0409074v2","created_at":"2026-05-18T02:30:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math-ph/0409074","created_at":"2026-05-18T02:30:23Z"},{"alias_kind":"pith_short_12","alias_value":"W2U7R4VL74ML","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_16","alias_value":"W2U7R4VL74MLA5MS","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_8","alias_value":"W2U7R4VL","created_at":"2026-05-18T12:25:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2004:W2U7R4VL74MLA5MSVPUG7WC5RB","target":"record","payload":{"canonical_record":{"source":{"id":"math-ph/0409074","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math-ph","submitted_at":"2004-09-28T21:46:46Z","cross_cats_sorted":["math.MP","math.SP"],"title_canon_sha256":"5f5888cf2a61e82bc514cd98ca0db5a1c684169201ad06a4fb5044e856af4fbe","abstract_canon_sha256":"040adf36d39a07fdbb45e49219a5a5ec22c5dd3b4854523dec1c5df18bfc9cbd"},"schema_version":"1.0"},"canonical_sha256":"b6a9f8f2abff18b07592abe86fd85d88564575eb6b0fccc8a6c5ff3d68dbcb63","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:30:23.186660Z","signature_b64":"7x4VmbpthqiSw9pez8LtksMd5k+3Z7A9Z9m563zoE1qI/zmkyhB80TbdAd0kYvUQSm4twrDrEXa8o8OsxCE/CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b6a9f8f2abff18b07592abe86fd85d88564575eb6b0fccc8a6c5ff3d68dbcb63","last_reissued_at":"2026-05-18T02:30:23.186238Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:30:23.186238Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math-ph/0409074","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":"2026-05-18T02:30:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wPmxu+uhMe9OFFOwivYpK+fkzgr2UAC0hLGuo7rtvIoC+oJO+iUqtuyuFeyAOU0Ckf2g4I7T3v+NOvXQ7Mh8Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T03:05:59.535528Z"},"content_sha256":"cdcd5d924bb9c351fa66fa22779e4d7284820d7ea63022c141ec5ddbeb8ccdaa","schema_version":"1.0","event_id":"sha256:cdcd5d924bb9c351fa66fa22779e4d7284820d7ea63022c141ec5ddbeb8ccdaa"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2004:W2U7R4VL74MLA5MSVPUG7WC5RB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Schroedinger Operators With Few Bound States","license":"","headline":"","cross_cats":["math.MP","math.SP"],"primary_cat":"math-ph","authors_text":"Barry Simon (Caltech), David Damanik (Caltech), Rowan Killip (UCLA)","submitted_at":"2004-09-28T21:46:46Z","abstract_excerpt":"We show that whole-line Schr\\\"odinger operators with finitely many bound states have no embedded singular spectrum. In contradistinction, we show that embedded singular spectrum is possible even when the bound states approach the essential spectrum exponentially fast.\n  We also prove the following result for one- and two-dimensional Schr\\\"odinger operators, $H$, with bounded positive ground states: Given a potential $V$, if both $H\\pm V$ are bounded from below by the ground-state energy of $H$, then $V\\equiv 0$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math-ph/0409074","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":"2026-05-18T02:30:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"N5n+Dlq3HDx09h+eFLSVS6w/e9BwLVju0bzNAIJwUdyGlVdbPiSzVnezAYKwD10HFgojQl7OwvNqa0e9HkgPDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T03:05:59.536152Z"},"content_sha256":"c9b527b59b51452281ac095c0160dd2e67a8f240494a12280263e70c571939b0","schema_version":"1.0","event_id":"sha256:c9b527b59b51452281ac095c0160dd2e67a8f240494a12280263e70c571939b0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/W2U7R4VL74MLA5MSVPUG7WC5RB/bundle.json","state_url":"https://pith.science/pith/W2U7R4VL74MLA5MSVPUG7WC5RB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/W2U7R4VL74MLA5MSVPUG7WC5RB/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":"2026-06-04T03:05:59Z","links":{"resolver":"https://pith.science/pith/W2U7R4VL74MLA5MSVPUG7WC5RB","bundle":"https://pith.science/pith/W2U7R4VL74MLA5MSVPUG7WC5RB/bundle.json","state":"https://pith.science/pith/W2U7R4VL74MLA5MSVPUG7WC5RB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/W2U7R4VL74MLA5MSVPUG7WC5RB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2004:W2U7R4VL74MLA5MSVPUG7WC5RB","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":"040adf36d39a07fdbb45e49219a5a5ec22c5dd3b4854523dec1c5df18bfc9cbd","cross_cats_sorted":["math.MP","math.SP"],"license":"","primary_cat":"math-ph","submitted_at":"2004-09-28T21:46:46Z","title_canon_sha256":"5f5888cf2a61e82bc514cd98ca0db5a1c684169201ad06a4fb5044e856af4fbe"},"schema_version":"1.0","source":{"id":"math-ph/0409074","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math-ph/0409074","created_at":"2026-05-18T02:30:23Z"},{"alias_kind":"arxiv_version","alias_value":"math-ph/0409074v2","created_at":"2026-05-18T02:30:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math-ph/0409074","created_at":"2026-05-18T02:30:23Z"},{"alias_kind":"pith_short_12","alias_value":"W2U7R4VL74ML","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_16","alias_value":"W2U7R4VL74MLA5MS","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_8","alias_value":"W2U7R4VL","created_at":"2026-05-18T12:25:52Z"}],"graph_snapshots":[{"event_id":"sha256:c9b527b59b51452281ac095c0160dd2e67a8f240494a12280263e70c571939b0","target":"graph","created_at":"2026-05-18T02:30:23Z","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 show that whole-line Schr\\\"odinger operators with finitely many bound states have no embedded singular spectrum. In contradistinction, we show that embedded singular spectrum is possible even when the bound states approach the essential spectrum exponentially fast.\n  We also prove the following result for one- and two-dimensional Schr\\\"odinger operators, $H$, with bounded positive ground states: Given a potential $V$, if both $H\\pm V$ are bounded from below by the ground-state energy of $H$, then $V\\equiv 0$.","authors_text":"Barry Simon (Caltech), David Damanik (Caltech), Rowan Killip (UCLA)","cross_cats":["math.MP","math.SP"],"headline":"","license":"","primary_cat":"math-ph","submitted_at":"2004-09-28T21:46:46Z","title":"Schroedinger Operators With Few Bound States"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math-ph/0409074","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:cdcd5d924bb9c351fa66fa22779e4d7284820d7ea63022c141ec5ddbeb8ccdaa","target":"record","created_at":"2026-05-18T02:30:23Z","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":"040adf36d39a07fdbb45e49219a5a5ec22c5dd3b4854523dec1c5df18bfc9cbd","cross_cats_sorted":["math.MP","math.SP"],"license":"","primary_cat":"math-ph","submitted_at":"2004-09-28T21:46:46Z","title_canon_sha256":"5f5888cf2a61e82bc514cd98ca0db5a1c684169201ad06a4fb5044e856af4fbe"},"schema_version":"1.0","source":{"id":"math-ph/0409074","kind":"arxiv","version":2}},"canonical_sha256":"b6a9f8f2abff18b07592abe86fd85d88564575eb6b0fccc8a6c5ff3d68dbcb63","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b6a9f8f2abff18b07592abe86fd85d88564575eb6b0fccc8a6c5ff3d68dbcb63","first_computed_at":"2026-05-18T02:30:23.186238Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:30:23.186238Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7x4VmbpthqiSw9pez8LtksMd5k+3Z7A9Z9m563zoE1qI/zmkyhB80TbdAd0kYvUQSm4twrDrEXa8o8OsxCE/CA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:30:23.186660Z","signed_message":"canonical_sha256_bytes"},"source_id":"math-ph/0409074","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cdcd5d924bb9c351fa66fa22779e4d7284820d7ea63022c141ec5ddbeb8ccdaa","sha256:c9b527b59b51452281ac095c0160dd2e67a8f240494a12280263e70c571939b0"],"state_sha256":"b3de7ec001abca646bafe37d206c9cbf3ffde8e826f2c23b3e6d8f4b9a037c07"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"S0XfeJSaP4nClvXH3IMQZNUkqmYliS04TFquM3w02rf8sd2Wty1gqwi4kb32OlUj0IRH6XHt78FOXHpF8rmhDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T03:05:59.538622Z","bundle_sha256":"85428e2a80fb55213220030a8aa682f98533676c3eefb8b74f665e47cb6d765a"}}