{"paper":{"title":"On a necessary aspect for the Riesz basis property for indefinite Sturm-Liouville problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.SP","authors_text":"Aleksey Kostenko","submitted_at":"2012-02-11T14:02:06Z","abstract_excerpt":"In 1996, H. Volkmer observed that the inequality \\[(\\int_{-1}^1\\frac{1}{|r|}|f'|dx)^2 \\le K^2 \\int_{-1}^1|f|^2dx\\int_{-1}^1\\Big|\\Big(\\frac{1}{r}f'\\Big)'\\Big|^2dx \\] is satisfied with some positive constant $K>0$ for a certain class of functions $f$ on $[-1,1]$ if the eigenfunctions of the problem \\[ -y\"=\\lambda\\, r(x)y,\\quad y(-1)=y(1)=0 \\] form a Riesz basis of the Hilbert space $L^2_{|r|}(-1,1)$. Here the weight $r\\in L^1(-1,1)$ is assumed to satisfy $xr(x)>0$ a.e. on $[-1,1]$.\n  We present two criteria in terms of Weyl-Titchmarsh $m$-functions for the Volkmer inequality to be valid. Using t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.2444","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"}