{"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. 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 < ∞.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Discrete Schrödinger operators with potentials decaying faster than 1/|n| have purely absolutely continuous spectrum and support ballistic transport.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"10873d22e660b7d764370eeecc8fb239897e39ba5377bf4cb7374abffbceb32e"},"source":{"id":"2507.04988","kind":"arxiv","version":6},"verdict":{"id":"b6f75443-9902-4f11-8dab-88f6b1592c5b","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T06:22:42.199699Z","strongest_claim":"We prove the absence of singular continuous spectrum for H. 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"}