{"paper":{"title":"On the $L^p$ boundedness of wave operators for two-dimensional Schr\\\"odinger operators with threshold obstructions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Burak Erdogan, Michael Goldberg, William R. Green","submitted_at":"2017-06-05T20:32:29Z","abstract_excerpt":"Let $H=-\\Delta+V$ be a Schr\\\"odinger operator on $L^2(\\mathbb R^2)$ 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 $L^p(\\mathbb R^2)$ for all $1< p< \\infty$ if zero is not an eigenvalue or resonance. We show that if there is an s-wave resonance or an eigenvalue only at zero, then the wave operators are bounded on $L^p(\\mathbb R^2)$ for $1 < p<\\infty$. This result stands in contrast to results in higher dimensions, where the presence of zero energy obs"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.01530","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"}