{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"98b1cb1351fce192edb6001d2e6dd31b37b8cff48d9b0396db168ebd044b0538","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":["ai:gpt-5.5"],"kind":"seed"},"parents":["b8e12ec2ea1d269e7bb90c049f9bd056eb8851930b854ad7ee865b44ecea5f1d","ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b"],"salt":"f81310855cabc26fdf40b07272e276175639c6305be2847984897eeb044b3d27","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":["ai:gpt-5.5"],"external_id":"mathph.anderson-delocalization.quantum-diffusion","kind":"well-posed","literature_status":"open","n":"2","parents":[],"plain":"An electron in a weakly disordered 3D crystal should wander like a random walker, with its mean squared distance growing linearly in time forever. This is proven only up to a long but finite time.","posed_since":"","precise":"For $H = \\Delta + \\lambda V$ on $Z^3$, $\\Delta$ the nearest-neighbour hopping operator, show that $E \\sum_{x} \\mid x \\mid^{2} \\mid \\langle x \\mid \\mathrm{exp}(-itH) \\mid 0\\rangle \\mid^{2} \\sim D(\\lambda) t$ as $t \\to \\infty$ with a diffusion constant $D(\\lambda) > 0$ for small $\\lambda$ (or the analogous statement for an energy-localized initial state). Erdos, Salmhofer and Yau proved diffusion up to times $t \\sim \\lambda^{-2-\\kappa}$ for some $\\kappa > 0$. Answer: a proof for $t \\to \\infty$ at fixed small $\\lambda$.","problem_ref":"b8e12ec2ea1d269e7bb90c049f9bd056eb8851930b854ad7ee865b44ecea5f1d","references":"","settled_by":"A proof of diffusive spreading uniformly in time at fixed small $\\lambda$.","status_note":"Diffusion is controlled only up to times $\\lambda^{-2-\\kappa}$ (Erdos, Salmhofer and Yau, around 2007-2008).","title":"Quantum diffusion for all times in the weakly disordered 3D lattice","topic_ref":null},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"c6ccdbe0482f9498848fe4b51fa69af971da6a01fcb21e9bab0c8db6eef4cb54","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"da4c896f832ff7e898a396b101091afd30c0840fd9e6c529a030243e4a9d2bc2","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"eC2Va8JQaYRFOrYkzZ8GtAk_WOm9dpWu95W2S0dCNyiwi4XW_FE1ROdIRmHF0PvhuNh8tUr2a-_Z_gl7x0xzAw"},"schema":"pubphys.envelope/1"},"record_hash":"c6ccdbe0482f9498848fe4b51fa69af971da6a01fcb21e9bab0c8db6eef4cb54","leaf_index":3400}