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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 < ∞."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"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."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"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."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Discrete Schrödinger operators with potentials decaying faster than 1/|n| have purely absolutely continuous spectrum and support ballistic transport."}],"snapshot_sha256":"10873d22e660b7d764370eeecc8fb239897e39ba5377bf4cb7374abffbceb32e"},"formal_canon":{"evidence_count":2,"snapshot_sha256":"de65522ed34ee19f0100bfe8407ddf056156a1310e1cc00bce5cd70a6c8382cb"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2507.04988/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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","authors_text":"David Damanik (Rice University), Zhiyan Zhao (Universit\\'e C\\^ote d'Azur)","cross_cats":["math.AP","math.MP","math.SP"],"headline":"Discrete Schrödinger operators with potentials decaying faster than 1/|n| have purely absolutely continuous spectrum and support ballistic transport.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2025-07-07T13:28:20Z","title":"Ballistic Transport for Discrete Multi-Dimensional Schr\\\"odinger Operators With Decaying Potential"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.04988","kind":"arxiv","version":6},"verdict":{"created_at":"2026-05-19T06:22:42.199699Z","id":"b6f75443-9902-4f11-8dab-88f6b1592c5b","model_set":{"reader":"grok-4.3"},"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","pith_extraction_headline":"Discrete Schrödinger operators with potentials decaying faster than 1/|n| have purely absolutely continuous spectrum and support ballistic transport.","strongest_claim":"We prove the absence of singular continuous spectrum for H. 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