{"paper":{"title":"Imbedded Singular Continuous Spectrum for Schr\\\"odinger Operators","license":"","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.SP","authors_text":"A. Kiselev","submitted_at":"2001-11-19T03:02:38Z","abstract_excerpt":"We construct examples of potentials $V(x)$ satisfying $|V(x)| \\leq \\frac{h(x)}{1+x},$ where the function $h(x)$ is growing arbitrarily slowly, such that the corresponding Schr\\\"odinger operator has imbedded singular continuous spectrum. This solves one of the fifteen \"twenty-first century\" problems for Schr\\\"odinger operators posed by Barry Simon. The construction also provides the first example of a Schr\\\"odinger operator for which M\\\"oller wave operators exist but are not asymptotically complete due to the presence of singular continuous spectrum."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0111200","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/math/0111200/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"}