{"paper":{"title":"Lifschitz tail for alloy-type models driven by the fractional Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.FA","math.MP","math.SP"],"primary_cat":"math.PR","authors_text":"Kamil Kaleta, Katarzyna Pietruska-Pa{\\l}uba","submitted_at":"2019-06-08T08:46:07Z","abstract_excerpt":"We establish precise asymptotics near zero of the integrated density of states for the random Schr\\\"{o}dinger operators $(-\\Delta)^{\\alpha/2} + V^{\\omega}$ in $L^2(\\mathbb R^d)$ for the full range of $\\alpha\\in(0,2]$ and a fairly large class of random nonnegative alloy-type potentials $V^{\\omega}$. The IDS exhibits the Lifschitz tail singularity. We prove the existence of the limit $$\\lim_{s\\to 0} s^{d/\\alpha}\\ln\\ell([0,s]) = -C \\left(\\lambda_d^{(\\alpha)}\\right)^{d/\\alpha},$$ with $C \\in (0,\\infty]$. The constant $C$ is is finite if and only if the common distribution of the lattice random var"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.03419","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"}