{"paper":{"title":"Necessary and Sufficient Conditions in the Spectral Theory of Jacobi Matrices and Schr\\\"odinger Operators","license":"","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.SP","authors_text":"Barry Simon, David Damanik, Rowan Killip","submitted_at":"2003-09-12T06:11:37Z","abstract_excerpt":"We announce three results in the theory of Jacobi matrices and Schr\\\"odinger operators. First, we give necessary and sufficient conditions for a measure to be the spectral measure of a Schr\\\"odinger operator $-\\f{d^2}{dx^2} +V(x)$ on $L^2 (0,\\infty)$ with $V\\in L^2 (0,\\infty)$ and $u(0)=0$ boundary condition. Second, we give necessary and sufficient conditions on the Jacobi parameters for the associated orthogonal polynomials to have Szeg\\H{o} asymptotics. Finally, we provide necessary and sufficient conditions on a measure to be the spectral measure of a Jacobi matrix with exponential decay a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0309206","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"}