{"paper":{"title":"Ground state energy of trimmed discrete Schr\\\"odinger operators and localization for trimmed Anderson models","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Abel Klein, Alexander Elgart","submitted_at":"2013-01-22T18:38:44Z","abstract_excerpt":"We consider discrete Schr\\\"odinger operators of the form $H=-\\Delta +V$ on $\\ell^2(\\Z^d)$, where $\\Delta$ is the discrete Laplacian and $V$ is a bounded potential. Given $\\Gamma \\subset \\Z^d$, the $\\Gamma$-trimming of $H$ is the restriction of $H$ to $\\ell^2(\\Z^d\\setminus\\Gamma)$, denoted by $H_\\Gamma$. We investigate the dependence of the ground state energy $E_\\Gamma(H)=\\inf \\sigma (H_\\Gamma)$ on $\\Gamma$. We show that for relatively dense proper subsets $\\Gamma$ of $\\Z^d$ we always have $E_\\Gamma(H)>E_\\emptyset(H)$. We use this lifting of the ground state energy to establish Wegner estimate"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.5268","kind":"arxiv","version":2},"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"}