{"paper":{"title":"Homogenized spectral problems for exactly solvable operators: asymptotics of polynomial eigenfunctions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV","math.SP"],"primary_cat":"math.CA","authors_text":"Boris Shapiro, Julius Borcea, Rikard B{\\o}gvad","submitted_at":"2007-05-19T15:45:55Z","abstract_excerpt":"Consider a homogenized spectral pencil of exactly solvable linear differential operators $T_{\\la}=\\sum_{i=0}^k Q_{i}(z)\\la^{k-i}\\frac {d^i}{dz^i}$, where each $Q_{i}(z)$ is a polynomial of degree at most $i$ and $\\la$ is the spectral parameter. We show that under mild nondegeneracy assumptions for all sufficiently large positive integers $n$ there exist exactly $k$ distinct values $\\la_{n,j}$, $1\\le j\\le k$, of the spectral parameter $\\la$ such that the operator $T_{\\la}$ has a polynomial eigenfunction $p_{n,j}(z)$ of degree $n$. These eigenfunctions split into $k$ different families according"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0705.2822","kind":"arxiv","version":3},"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"}