{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:UXYZU3B2TLZWHQR3IXP36J3VXB","short_pith_number":"pith:UXYZU3B2","canonical_record":{"source":{"id":"1510.08746","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"quant-ph","submitted_at":"2015-10-29T15:46:13Z","cross_cats_sorted":["cond-mat.other","math-ph","math.MP"],"title_canon_sha256":"42167eb79ba74890be78758d84b32a30bd818ca60a6584485c995d868951eece","abstract_canon_sha256":"ee1c081f1008e6b01536e4b8d19edb790ca978efb146f49aca13525219dd9375"},"schema_version":"1.0"},"canonical_sha256":"a5f19a6c3a9af363c23b45dfbf2775b84501abd4e981e239e5bff6cc33ae9567","source":{"kind":"arxiv","id":"1510.08746","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.08746","created_at":"2026-05-18T00:39:13Z"},{"alias_kind":"arxiv_version","alias_value":"1510.08746v2","created_at":"2026-05-18T00:39:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.08746","created_at":"2026-05-18T00:39:13Z"},{"alias_kind":"pith_short_12","alias_value":"UXYZU3B2TLZW","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"UXYZU3B2TLZWHQR3","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"UXYZU3B2","created_at":"2026-05-18T12:29:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:UXYZU3B2TLZWHQR3IXP36J3VXB","target":"record","payload":{"canonical_record":{"source":{"id":"1510.08746","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"quant-ph","submitted_at":"2015-10-29T15:46:13Z","cross_cats_sorted":["cond-mat.other","math-ph","math.MP"],"title_canon_sha256":"42167eb79ba74890be78758d84b32a30bd818ca60a6584485c995d868951eece","abstract_canon_sha256":"ee1c081f1008e6b01536e4b8d19edb790ca978efb146f49aca13525219dd9375"},"schema_version":"1.0"},"canonical_sha256":"a5f19a6c3a9af363c23b45dfbf2775b84501abd4e981e239e5bff6cc33ae9567","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:39:13.802837Z","signature_b64":"GIchKPqAj/QVw5doeVtR8whBjNC5y/Rk48YsjeiUdXKTu+uJzPXhZOMcT0oDO2NwMEFJ3Rf8Wgb466rskjhdCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a5f19a6c3a9af363c23b45dfbf2775b84501abd4e981e239e5bff6cc33ae9567","last_reissued_at":"2026-05-18T00:39:13.802224Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:39:13.802224Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1510.08746","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":"2026-05-18T00:39:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"u3L9RcytrFMdnm4ke1ndAiGwD4c5Hj8XFPaUdXYNUHSUlZeWBzEZoK4X2D70uDlAgFhDBGzXbx02t1IYUqQSAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T14:14:40.974066Z"},"content_sha256":"51eef85dce99833f1e91b5e3fcecefb76782754937ec922f08a0d8aae566df73","schema_version":"1.0","event_id":"sha256:51eef85dce99833f1e91b5e3fcecefb76782754937ec922f08a0d8aae566df73"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:UXYZU3B2TLZWHQR3IXP36J3VXB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Self-similar analogues of Stark ladders: a path to fractal potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.other","math-ph","math.MP"],"primary_cat":"quant-ph","authors_text":"E. Sadurn\\'i, S. Castillo","submitted_at":"2015-10-29T15:46:13Z","abstract_excerpt":"We treat the eigenvalue problem posed by self-similar potentials, i.e. homogeneous functions under a particular affine transformation, by means of symmetry techniques. We find that the eigenfunctions of such problems are localized, even when the potential does not rise to infinity in every direction. It is shown that the logarithm of the energy displays levels contained in families that are analogous to Wannier-Stark ladders. The position of each ladder is proved to be determined by the specific details of the potential and not by its transformation properties. This is done by direct computati"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.08746","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":"2026-05-18T00:39:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"k0XxPBk5AW9Ggpyf73M+PDMe43gsdtad5QaFwyG1k31PxtCXjpv2C7gLaPL/qydgwTA1exsj9rl/REBUFVskDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T14:14:40.974423Z"},"content_sha256":"16a336a1fde464553b0413367123f5e0154e262e7f99f21ea5bd368bdfb080d7","schema_version":"1.0","event_id":"sha256:16a336a1fde464553b0413367123f5e0154e262e7f99f21ea5bd368bdfb080d7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UXYZU3B2TLZWHQR3IXP36J3VXB/bundle.json","state_url":"https://pith.science/pith/UXYZU3B2TLZWHQR3IXP36J3VXB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UXYZU3B2TLZWHQR3IXP36J3VXB/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":"2026-06-09T14:14:40Z","links":{"resolver":"https://pith.science/pith/UXYZU3B2TLZWHQR3IXP36J3VXB","bundle":"https://pith.science/pith/UXYZU3B2TLZWHQR3IXP36J3VXB/bundle.json","state":"https://pith.science/pith/UXYZU3B2TLZWHQR3IXP36J3VXB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UXYZU3B2TLZWHQR3IXP36J3VXB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:UXYZU3B2TLZWHQR3IXP36J3VXB","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":"ee1c081f1008e6b01536e4b8d19edb790ca978efb146f49aca13525219dd9375","cross_cats_sorted":["cond-mat.other","math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"quant-ph","submitted_at":"2015-10-29T15:46:13Z","title_canon_sha256":"42167eb79ba74890be78758d84b32a30bd818ca60a6584485c995d868951eece"},"schema_version":"1.0","source":{"id":"1510.08746","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.08746","created_at":"2026-05-18T00:39:13Z"},{"alias_kind":"arxiv_version","alias_value":"1510.08746v2","created_at":"2026-05-18T00:39:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.08746","created_at":"2026-05-18T00:39:13Z"},{"alias_kind":"pith_short_12","alias_value":"UXYZU3B2TLZW","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"UXYZU3B2TLZWHQR3","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"UXYZU3B2","created_at":"2026-05-18T12:29:44Z"}],"graph_snapshots":[{"event_id":"sha256:16a336a1fde464553b0413367123f5e0154e262e7f99f21ea5bd368bdfb080d7","target":"graph","created_at":"2026-05-18T00:39:13Z","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 treat the eigenvalue problem posed by self-similar potentials, i.e. homogeneous functions under a particular affine transformation, by means of symmetry techniques. We find that the eigenfunctions of such problems are localized, even when the potential does not rise to infinity in every direction. It is shown that the logarithm of the energy displays levels contained in families that are analogous to Wannier-Stark ladders. The position of each ladder is proved to be determined by the specific details of the potential and not by its transformation properties. This is done by direct computati","authors_text":"E. Sadurn\\'i, S. Castillo","cross_cats":["cond-mat.other","math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"quant-ph","submitted_at":"2015-10-29T15:46:13Z","title":"Self-similar analogues of Stark ladders: a path to fractal potentials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.08746","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":"sha256:51eef85dce99833f1e91b5e3fcecefb76782754937ec922f08a0d8aae566df73","target":"record","created_at":"2026-05-18T00:39:13Z","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":"ee1c081f1008e6b01536e4b8d19edb790ca978efb146f49aca13525219dd9375","cross_cats_sorted":["cond-mat.other","math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"quant-ph","submitted_at":"2015-10-29T15:46:13Z","title_canon_sha256":"42167eb79ba74890be78758d84b32a30bd818ca60a6584485c995d868951eece"},"schema_version":"1.0","source":{"id":"1510.08746","kind":"arxiv","version":2}},"canonical_sha256":"a5f19a6c3a9af363c23b45dfbf2775b84501abd4e981e239e5bff6cc33ae9567","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a5f19a6c3a9af363c23b45dfbf2775b84501abd4e981e239e5bff6cc33ae9567","first_computed_at":"2026-05-18T00:39:13.802224Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:39:13.802224Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GIchKPqAj/QVw5doeVtR8whBjNC5y/Rk48YsjeiUdXKTu+uJzPXhZOMcT0oDO2NwMEFJ3Rf8Wgb466rskjhdCg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:39:13.802837Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.08746","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:51eef85dce99833f1e91b5e3fcecefb76782754937ec922f08a0d8aae566df73","sha256:16a336a1fde464553b0413367123f5e0154e262e7f99f21ea5bd368bdfb080d7"],"state_sha256":"91fd0659e6094829abadba9650cba52a2d41a6541c7e4978c953392c4098710a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"W6esvh4RaC6rxZXaK2/NNveLS+51siUwcm176befAmzuLCWAUVV0jh/ucelZbR2ZErX8quA0GqoYkmD9JGazBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T14:14:40.976378Z","bundle_sha256":"66c7043be92837da51a04dd888ab397b8ca991124eddbc197195ba4d74030a27"}}