{"paper":{"title":"Discrete Schr\\\"odinger operators with random alloy-type potential","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP","math.SP"],"primary_cat":"math-ph","authors_text":"Alexander Elgart, Helge Kr\\\"uger, Ivan Veseli\\'c, Martin Tautenhahn","submitted_at":"2011-07-14T12:26:15Z","abstract_excerpt":"We review recent results on localization for discrete alloy-type models based on the multiscale analysis and the fractional moment method, respectively. The discrete alloy-type model is a family of Schr\\\"odinger operators $H_\\omega = - \\Delta + V_\\omega$ on $\\ell^2 (\\ZZ^d)$ where $\\Delta$ is the discrete Laplacian and $V_\\omega$ the multiplication by the function $V_\\omega (x) = \\sum_{k \\in \\ZZ^d} \\omega_k u(x-k)$. Here $\\omega_k$, $k \\in \\ZZ^d$, are i.i.d. random variables and $u \\in \\ell^1 (\\ZZ^d ; \\RR)$ is a so-called single-site potential. Since $u$ may change sign, certain properties of $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.2800","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":""},"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"}