{"paper":{"title":"The weak coupling limit for the random Schr\\\"odinger equation: The average wave function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MP"],"primary_cat":"math-ph","authors_text":"Lenya Ryzhik, Thomas Chen, Tomasz Komorowski","submitted_at":"2016-09-09T00:14:00Z","abstract_excerpt":"We consider the Schr\\\"odinger equation with a time-independent weakly random potential of a strength $\\epsilon\\ll 1$, with Gaussian statistics. We prove that when the initial condition varies on a scale much larger than the correlation length of the potential, the compensated wave function converges to a deterministic limit on the time scale $t\\sim\\epsilon^{-2}$. This is shown under the sharp assumption that the correlation function $R(x)$ of the random potential decays slower than $1/|x|^2$, which ensures that the effective potential is finite. When $R(x)$ decays slower than $1/|x|^2$ we esta"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.02623","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"}