{"paper":{"title":"Dispersive estimates for four dimensional Schr\\\"{o}dinger and wave equations with obstructions at zero energy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"M. Burak Erdogan, Michael Goldberg, William R. Green","submitted_at":"2013-10-23T17:30:21Z","abstract_excerpt":"We investigate $L^1(\\mathbb R^4)\\to L^\\infty(\\mathbb R^4)$ dispersive estimates for the Schr\\\"odinger operator $H=-\\Delta+V$ when there are obstructions, a resonance or an eigenvalue, at zero energy. In particular, we show that if there is a resonance or an eigenvalue at zero energy then there is a time dependent, finite rank operator $F_t$ satisfying $\\|F_t\\|_{L^1\\to L^\\infty} \\lesssim 1/\\log t$ for $t>2$ such that $$\\|e^{itH}P_{ac}-F_t\\|_{L^1\\to L^\\infty} \\lesssim t^{-1},\\,\\,\\,\\,\\,\\text{for} t>2.$$ We also show that the operator $F_t=0$ if there is an eigenvalue but no resonance at zero ener"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.6302","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"}