{"paper":{"title":"Divergence of spectral decompositions of Hill operators with two exponential term potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.CA","math.FA","math.MP"],"primary_cat":"math.SP","authors_text":"Boris Mityagin, Plamen Djakov","submitted_at":"2012-10-15T06:06:50Z","abstract_excerpt":"We consider the Hill operator $$ Ly = - y^{\\prime \\prime} + v(x)y, \\quad 0 \\leq x \\leq \\pi, $$ subject to periodic or antiperiodic boundary conditions ($bc$) with potentials of the form $$ v(x) = a e^{-2irx} + b e^{2isx}, \\quad a, b \\neq 0, r,s \\in \\mathbb{N}, r\\neq s. $$\n  It is shown that the system of root functions does not contain a basis in $L^2 ([0,\\pi], \\mathbb{C})$ if $bc$ are periodic or if $bc$ are antiperiodic and $r, s$ are odd or $r=1$ and $s \\geq 3. $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.3907","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"}