{"paper":{"title":"Long time behavior of small solutions of NLS with non-generic potentials in one dimension","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Neba Polneau","submitted_at":"2026-07-01T22:16:46Z","abstract_excerpt":"We consider the one-dimensional cubic nonlinear Schr\\\"odinger equation with a non-generic real-valued external potential $V$. We prove almost global-in-time quantitative bounds for small solutions. More precisely, small initial data of size $\\varepsilon$ in a weighted Sobolev space give rise to solutions with the sharp decay rate $t^{-1/2}$ in $L^{\\infty}_x$ up to time $\\exp(\\frac{1}{c\\varepsilon^{2}})$. The main novelty of our result is that no additional symmetry assumption is imposed on $V$.\n  First, we use a modification of the standard distorted Fourier transform basis to resolve the poss"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01509","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.01509/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}