{"paper":{"title":"On Pointwise Products of Elliptic Eigenfunctions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","physics.comp-ph"],"primary_cat":"math.SP","authors_text":"Jianfeng Lu, Stefan Steinerberger","submitted_at":"2018-10-02T00:53:32Z","abstract_excerpt":"We consider eigenfunctions of Schr\\\"odinger operators on a $d-$dimensional bounded domain $\\Omega$ (or a $d-$dimensional compact manifold $\\Omega$) with Dirichlet conditions. These operators give rise to a sequence of eigenfunctions $(\\phi_n)_{n \\in \\mathbb{N}}$. We study the subspace of all pointwise products $$ A_n = \\mbox{span} \\left\\{ \\phi_i(x) \\phi_j(x): 1 \\leq i,j \\leq n\\right\\} \\subseteq L^2(\\Omega).$$ Clearly, that vector space has dimension $\\mbox{dim}(A_n) = n(n+1)/2$. We prove that products $\\phi_i \\phi_j$ of eigenfunctions are simple in a certain sense: for any $\\varepsilon > 0$, t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.01024","kind":"arxiv","version":2},"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"}