{"paper":{"title":"On the $L^p$ boundedness of wave operators for four-dimensional Schr\\\"odinger Operators with a threshold eigenvalue","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Michael Goldberg, William R. Green","submitted_at":"2016-06-21T18:14:13Z","abstract_excerpt":"Let $H=-\\Delta+V$ be a Schr\\\"odinger operator on $L^2(\\mathbb R^4)$ with real-valued potential $V$, and let $H_0=-\\Delta$. If $V$ has sufficient pointwise decay, the wave operators $W_{\\pm}=s-\\lim_{t\\to \\pm\\infty} e^{itH}e^{-itH_0}$ are known to be bounded on $L^p(\\mathbb R^4)$ for all $1\\leq p\\leq \\infty$ if zero is not an eigenvalue or resonance, and on $\\frac43<p<4$ if zero is an eigenvalue but not a resonance. We show that in the latter case, the wave operators are also bounded on $L^p(\\mathbb R^4)$ for $1\\leq p\\leq \\frac43$ by direct examination of the integral kernel of the leading terms"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.06691","kind":"arxiv","version":2},"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"}