{"paper":{"title":"Persistence of Anderson localization in Schr\\\"odinger operators with decaying random potentials","license":"","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Abel Klein, Alexander Figotin, Fran\\c{c}ois Germinet, Peter M\\\"uller","submitted_at":"2006-04-09T19:40:32Z","abstract_excerpt":"We show persistence of both Anderson and dynamical localization in Schr\\\"odinger operators with non-positive (attractive) random decaying potential. We consider an Anderson-type Schr\\\"odinger operator with a non-positive ergodic random potential, and multiply the random potential by a decaying envelope function. If the envelope function decays slower than $|x|^{-2}$ at infinity, we prove that the operator has infinitely many eigenvalues below zero. For envelopes decaying as $|x|^{-\\alpha}$ at infinity, we determine the number of bound states below a given energy $E<0$, asymptotically as $\\alph"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math-ph/0604020","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"}