{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:VAMUXSJ5ECMTXPGAGAKA2SWO34","short_pith_number":"pith:VAMUXSJ5","schema_version":"1.0","canonical_sha256":"a8194bc93d20993bbcc030140d4acedf146cdc14d114c09472187c95733f35e8","source":{"kind":"arxiv","id":"1507.08909","version":4},"attestation_state":"computed","paper":{"title":"Ballistic Motion in One-Dimensional Quasi-Periodic Discrete Schr\\\"odinger Equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MP","math.SP"],"primary_cat":"math-ph","authors_text":"Zhiyan Zhao","submitted_at":"2015-07-31T15:09:38Z","abstract_excerpt":"For the solution $q(t)=(q_n(t))_{n\\in\\mathbb Z}$ to one-dimensional discrete Schr\\\"odinger equation $${\\rm i}\\dot{q}_n=-(q_{n+1}+q_{n-1})+ V(\\theta+n\\omega) q_n, \\quad n\\in\\mathbb Z,$$ with $\\omega\\in\\mathbb R^d$ Diophantine, and $V$ a small real-analytic function on $\\mathbb T^d$, we consider the growth rate of the diffusion norm $\\|q(t)\\|_{D}:=\\left(\\sum_{n}n^2|q_n(t)|^2\\right)^{\\frac12}$ for any non-zero $q(0)$ with $\\|q(0)\\|_{D}<\\infty$. We prove that $\\|q(t)\\|_{D}$ grows {\\it linearly} with the time $t$ for any $\\theta\\in\\mathbb T^d$ if $V$ is sufficiently small."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1507.08909","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2015-07-31T15:09:38Z","cross_cats_sorted":["math.AP","math.MP","math.SP"],"title_canon_sha256":"76768cde0e51370dc3521db1ccb2d9c8dd36e16bf5d712edf05154a83e645dc7","abstract_canon_sha256":"87cc7b34a6d0219ceedd71abd09f138cdfc70e8cb77b98e7a41dbf9acc28e6f4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:18:57.307795Z","signature_b64":"Wo+IbslXRkgcV4/MxPuP8RfVUNkjwNo3R2wC0lGBzeSV6c7Sk/9cKIfUzfMMKWUYxXuEQKf6+YWGCCcuNJY3BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a8194bc93d20993bbcc030140d4acedf146cdc14d114c09472187c95733f35e8","last_reissued_at":"2026-05-18T01:18:57.307290Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:18:57.307290Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ballistic Motion in One-Dimensional Quasi-Periodic Discrete Schr\\\"odinger Equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MP","math.SP"],"primary_cat":"math-ph","authors_text":"Zhiyan Zhao","submitted_at":"2015-07-31T15:09:38Z","abstract_excerpt":"For the solution $q(t)=(q_n(t))_{n\\in\\mathbb Z}$ to one-dimensional discrete Schr\\\"odinger equation $${\\rm i}\\dot{q}_n=-(q_{n+1}+q_{n-1})+ V(\\theta+n\\omega) q_n, \\quad n\\in\\mathbb Z,$$ with $\\omega\\in\\mathbb R^d$ Diophantine, and $V$ a small real-analytic function on $\\mathbb T^d$, we consider the growth rate of the diffusion norm $\\|q(t)\\|_{D}:=\\left(\\sum_{n}n^2|q_n(t)|^2\\right)^{\\frac12}$ for any non-zero $q(0)$ with $\\|q(0)\\|_{D}<\\infty$. We prove that $\\|q(t)\\|_{D}$ grows {\\it linearly} with the time $t$ for any $\\theta\\in\\mathbb T^d$ if $V$ is sufficiently small."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.08909","kind":"arxiv","version":4},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1507.08909","created_at":"2026-05-18T01:18:57.307380+00:00"},{"alias_kind":"arxiv_version","alias_value":"1507.08909v4","created_at":"2026-05-18T01:18:57.307380+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.08909","created_at":"2026-05-18T01:18:57.307380+00:00"},{"alias_kind":"pith_short_12","alias_value":"VAMUXSJ5ECMT","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_16","alias_value":"VAMUXSJ5ECMTXPGA","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_8","alias_value":"VAMUXSJ5","created_at":"2026-05-18T12:29:44.643036+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/VAMUXSJ5ECMTXPGAGAKA2SWO34","json":"https://pith.science/pith/VAMUXSJ5ECMTXPGAGAKA2SWO34.json","graph_json":"https://pith.science/api/pith-number/VAMUXSJ5ECMTXPGAGAKA2SWO34/graph.json","events_json":"https://pith.science/api/pith-number/VAMUXSJ5ECMTXPGAGAKA2SWO34/events.json","paper":"https://pith.science/paper/VAMUXSJ5"},"agent_actions":{"view_html":"https://pith.science/pith/VAMUXSJ5ECMTXPGAGAKA2SWO34","download_json":"https://pith.science/pith/VAMUXSJ5ECMTXPGAGAKA2SWO34.json","view_paper":"https://pith.science/paper/VAMUXSJ5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1507.08909&json=true","fetch_graph":"https://pith.science/api/pith-number/VAMUXSJ5ECMTXPGAGAKA2SWO34/graph.json","fetch_events":"https://pith.science/api/pith-number/VAMUXSJ5ECMTXPGAGAKA2SWO34/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VAMUXSJ5ECMTXPGAGAKA2SWO34/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VAMUXSJ5ECMTXPGAGAKA2SWO34/action/storage_attestation","attest_author":"https://pith.science/pith/VAMUXSJ5ECMTXPGAGAKA2SWO34/action/author_attestation","sign_citation":"https://pith.science/pith/VAMUXSJ5ECMTXPGAGAKA2SWO34/action/citation_signature","submit_replication":"https://pith.science/pith/VAMUXSJ5ECMTXPGAGAKA2SWO34/action/replication_record"}},"created_at":"2026-05-18T01:18:57.307380+00:00","updated_at":"2026-05-18T01:18:57.307380+00:00"}