{"paper":{"title":"On the lowest possible dimension of supports of solutions to the discrete Schrodinger equation","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Stanislav Krymskii","submitted_at":"2024-01-05T13:28:27Z","abstract_excerpt":"In this article we study the possible size of support of solutions to the discrete stationary Schrodinger equation $\\Delta u(x)+V(x)u(x)=0$ in $\\mathbb{Z}^d$. We show that for any nonzero solution to any discrete stationary Schrodinger equation the dimension of the support is at least $\\log_2(d)-7.$\n  In the related setting of $\\mathbb{Z}_2$-valued harmonic functions in $\\mathbb{Z}^d$ one can improve the estimate on support's dimension to $\\log_2(d).$\n  However, we also provide an example where a $\\mathbb{Z}_2$-valued harmonic function in $\\mathbb{Z}^d$ has a fractal-like support with dimensio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2401.02800","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/2401.02800/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"}