{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"0b9e227bad8b610dae71640d5c452a815dd555bedefc237e61e6ac5f11557df9","created":"2026-10-03T07:18:05Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["308175e9b01389474edb7a6a8c0580517ee278a7ce5dca35c2a56608c5e2ca41","5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"d0878df7d4d69acca0722376533b666dfca65941b7a69f4372fa40f5f59a18d3","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.anderson-delocalization.ac-spectrum-3d","kind":"well-posed","literature_status":"open","n":"1","parents":[{"note":"","parent_revision":"308175e9b01389474edb7a6a8c0580517ee278a7ce5dca35c2a56608c5e2ca41","relation":"special_case"}],"plain":"Prove that at least some electron states in a weakly disordered 3D lattice spread over the whole sample. This is the half of the mobility-edge problem that has no proof at all.","posed_since":"1958","precise":"For $H = \\Delta + \\lambda V$ on $\\ell^2(Z^3)$, with $\\Delta$ the nearest-neighbour hopping operator (spectrum $[-6, 6]$ at $\\lambda = 0$), i.i.d. bounded $V$ and $\\lambda$ small, prove that the almost-sure spectrum has a nonzero absolutely continuous component in some energy interval inside $(-6, 6)$, or at least that eigenfunctions in that interval are not exponentially localized. Answer: a proof.","problem_ref":null,"references":"","settled_by":"A proof of absolutely continuous spectrum, or of delocalized eigenfunctions in finite volume uniformly in the volume, for $d = 3$.","status_note":"Absolutely continuous spectrum at weak disorder is proven on tree graphs (Klein, 1998, and later work of Aizenman and Warzel), not on $Z^d$.","title":"Absolutely continuous spectrum of the 3D Anderson model at weak disorder","topic_ref":"401a78c75cbf7b8acc9217ce7c5ab1fcd800a8c8cb9b4f7b06d2e6e76f0bedcb"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"a055707375e6357e41e3b8c7e1407b9f497c085d5aff58464ad396159aa48cac","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"d2aca96288344a2d45fcf191f273799c7d2eb7ab7223962bb4a879063e7d50cf","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"tKfCDJycpf3qYIbma4VsBYUWrWSKXpK_mHyr2I-v46rO5EfzeD14_u-2U4nwRdNAG9uLYVfRrLyXXU_vhIr2CA"},"schema":"pubphys.envelope/1"},"record_hash":"a055707375e6357e41e3b8c7e1407b9f497c085d5aff58464ad396159aa48cac","leaf_index":1594}