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For the unitary evolution $e^{-i tH}$, we prove that it exhibits ballistic transport in the sense that, for any $r > 0$, the weighted $\\ell^2-$norm $$\\|e^{-i tH}u\\|_r:=\\left(\\sum_{n\\in\\mathbb{Z}^d} (1+|n|^2)^{r} |(e^{-i tH}u)_n|^2\\right)^\\frac12 $$ grows at rate $\\simeq t"},"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":true},"canonical_record":{"source":{"id":"2507.04988","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2025-07-07T13:28:20Z","cross_cats_sorted":["math.AP","math.MP","math.SP"],"title_canon_sha256":"167b2048aee550311e0a9f4cf525a798a04b3972aaa4c87de5bb5d8328d668d0","abstract_canon_sha256":"122af008fb4e36255161340ec8f1bbf701db29bfc56631d152d302ba74be2303"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-04T01:09:37.801763Z","signature_b64":"g8WS7c0VGXOqArfNzS7iLX0setJzRz0HeREYxJ0oaqno76vytaAQn5ZDDOqxDs6LIDahiJzJiUHXjhM9LqxeBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aebf7619ab27fc7d0cd05b813a4aec740ae9bc60114683e9cd23d3b103b96e88","last_reissued_at":"2026-06-04T01:09:37.801176Z","signature_status":"signed_v1","first_computed_at":"2026-06-04T01:09:37.801176Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ballistic Transport for Discrete Multi-Dimensional Schr\\\"odinger Operators With Decaying Potential","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Discrete Schrödinger operators with potentials decaying faster than 1/|n| have purely absolutely continuous spectrum and support ballistic transport.","cross_cats":["math.AP","math.MP","math.SP"],"primary_cat":"math-ph","authors_text":"David Damanik (Rice University), Zhiyan Zhao (Universit\\'e C\\^ote d'Azur)","submitted_at":"2025-07-07T13:28:20Z","abstract_excerpt":"We consider the discrete Schr\\\"odinger operator $H = -\\Delta + V$ on $\\ell^2(\\mathbb{Z}^d)$ with a decaying potential, in arbitrary lattice dimension $d\\in\\mathbb{N}^*$, where $\\Delta$ is the standard discrete Laplacian and $V_n = o(|n|^{-1})$ as $|n| \\to \\infty$. %We prove the absence of singular continuous spectrum for $H$. For the unitary evolution $e^{-i tH}$, we prove that it exhibits ballistic transport in the sense that, for any $r > 0$, the weighted $\\ell^2-$norm $$\\|e^{-i tH}u\\|_r:=\\left(\\sum_{n\\in\\mathbb{Z}^d} (1+|n|^2)^{r} |(e^{-i tH}u)_n|^2\\right)^\\frac12 $$ grows at rate $\\simeq t"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove the absence of singular continuous spectrum for H. 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For the unitary evolution e^{-itH}, we prove that it exhibits ballistic transport in the sense that, for any r > 0, the weighted ℓ²-norm ||e^{-itH}u||_r grows at rate ≃ t^r as t→∞, provided that the initial state u is in the absolutely continuous subspace and satisfies ||u||_r < ∞.","one_line_summary":"Discrete Schrödinger operators on Z^d with V_n = o(|n|^{-1}) have purely absolutely continuous spectrum and exhibit ballistic transport where weighted position moments grow as t^r for AC initial states.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The potential satisfies V_n = o(|n|^{-1}) as |n| → ∞; this decay is invoked to apply compactness arguments and localized spectral projections that extend the free Laplacian result to the perturbed operator.","pith_extraction_headline":"Discrete Schrödinger operators with potentials decaying faster than 1/|n| have purely absolutely continuous spectrum and support ballistic transport."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.04988/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":2,"snapshot_sha256":"de65522ed34ee19f0100bfe8407ddf056156a1310e1cc00bce5cd70a6c8382cb"},"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":"2507.04988","created_at":"2026-06-04T01:09:37.801255+00:00"},{"alias_kind":"arxiv_version","alias_value":"2507.04988v6","created_at":"2026-06-04T01:09:37.801255+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.04988","created_at":"2026-06-04T01:09:37.801255+00:00"},{"alias_kind":"pith_short_12","alias_value":"V27XMGNLE76H","created_at":"2026-06-04T01:09:37.801255+00:00"},{"alias_kind":"pith_short_16","alias_value":"V27XMGNLE76H2DGQ","created_at":"2026-06-04T01:09:37.801255+00:00"},{"alias_kind":"pith_short_8","alias_value":"V27XMGNL","created_at":"2026-06-04T01:09:37.801255+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":2,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/V27XMGNLE76H2DGQLOATUSXMOQ","json":"https://pith.science/pith/V27XMGNLE76H2DGQLOATUSXMOQ.json","graph_json":"https://pith.science/api/pith-number/V27XMGNLE76H2DGQLOATUSXMOQ/graph.json","events_json":"https://pith.science/api/pith-number/V27XMGNLE76H2DGQLOATUSXMOQ/events.json","paper":"https://pith.science/paper/V27XMGNL"},"agent_actions":{"view_html":"https://pith.science/pith/V27XMGNLE76H2DGQLOATUSXMOQ","download_json":"https://pith.science/pith/V27XMGNLE76H2DGQLOATUSXMOQ.json","view_paper":"https://pith.science/paper/V27XMGNL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2507.04988&json=true","fetch_graph":"https://pith.science/api/pith-number/V27XMGNLE76H2DGQLOATUSXMOQ/graph.json","fetch_events":"https://pith.science/api/pith-number/V27XMGNLE76H2DGQLOATUSXMOQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/V27XMGNLE76H2DGQLOATUSXMOQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/V27XMGNLE76H2DGQLOATUSXMOQ/action/storage_attestation","attest_author":"https://pith.science/pith/V27XMGNLE76H2DGQLOATUSXMOQ/action/author_attestation","sign_citation":"https://pith.science/pith/V27XMGNLE76H2DGQLOATUSXMOQ/action/citation_signature","submit_replication":"https://pith.science/pith/V27XMGNLE76H2DGQLOATUSXMOQ/action/replication_record"}},"created_at":"2026-06-04T01:09:37.801255+00:00","updated_at":"2026-06-04T01:09:37.801255+00:00"}