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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1810.01024","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2018-10-02T00:53:32Z","cross_cats_sorted":["math.AP","physics.comp-ph"],"title_canon_sha256":"d4bc7ea4e609e3e5cb359520671dabd25bc753c3a3fdc1295de12298d823509b","abstract_canon_sha256":"caf7e0526c148ab9b0e34c550c5a144b1fbb9cd3a65264df493273730e1836e0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:59:49.405681Z","signature_b64":"zzFpSL320XwqtcNWY5Gq0suxPr4gKifCj5wZA35WvCiES37xdZFxWZLvn7C1RM+z8/Ao5S1RMQp9/tKf/fMlAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a359f5fb71d0addc3646e845950148a7f60b37c5d098adb657f8e76423417142","last_reissued_at":"2026-05-17T23:59:49.405174Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:59:49.405174Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\mbox{dim}(A_n) = n(n+1)/2$. 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