{"paper":{"title":"Decay estimates for discrete bi-Laplace operators with potentials on the lattice $\\mathbb{Z}$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Sisi Huang, Xiaohua Yao","submitted_at":"2025-06-29T06:57:03Z","abstract_excerpt":"It is known that the discrete Laplace operator $\\Delta$ on the lattice $\\mathbb{Z}$ satisfies the following sharp time decay estimate: $$\\big\\|e^{it\\Delta}\\big\\|_{\\ell^1\\rightarrow\\ell^{\\infty}}\\lesssim|t|^{-\\frac{1}{3}},\\quad t\\neq0,$$ which is slower than the usual $ O(|t|^{-\\frac{1}{2}})$ decay in the continuous case on $\\mathbb{R}$. However, this paper shows that the discrete bi-Laplacian $\\Delta^2$ on $\\mathbb{Z}$ actually exhibits the same sharp decay estimate $|t|^{-\\frac{1}{4}}$ as its continuous counterpart.\n  In view of the free decay estimate, we further investigate the discrete bi-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.23119","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.23119/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":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}