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Thus, the heat expansion is finite if and only if the potential u(x) is a rational solution of the KdV hierarchy decaying at infinity studied in [1,2]. Equivalently, one can characterize the corresponding operators L as the rank one bispectral family in [8]."},"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":"math-ph/0504046","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math-ph","submitted_at":"2005-04-14T21:02:51Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"067df00ed57ec7dea4816eaafb4a7f6d73e4509d916596e53c1dd8bc8c322a36","abstract_canon_sha256":"4ded2ef0d8a7a21ae77f36f121fd7d5241f6d401fd2a6231433923ecbb5045fd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:57:13.205007Z","signature_b64":"ygOit0fmxBghsy92pkisOoMsBKWf3xNnAx/HHF2IUj05dL8fdu5i5yqb7Vn/ch5DNcjupB4tK8RhKVxPdLn/AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"40f4acf6a36bbda9dc52b025df270220d65a6ddfd64f05386dad777986613352","last_reissued_at":"2026-05-18T03:57:13.204374Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:57:13.204374Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Finite heat kernel expansions on the real line","license":"","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Plamen Iliev","submitted_at":"2005-04-14T21:02:51Z","abstract_excerpt":"Let L=d^2/dx^2+u(x) be the one-dimensional Schrodinger operator and H(x,y,t) be the corresponding heat kernel. We prove that the nth Hadamard's coefficient H_n(x,y) is equal to 0 if and only if there exists a differential operator M of order 2n-1 such that L^{2n-1}=M^2. Thus, the heat expansion is finite if and only if the potential u(x) is a rational solution of the KdV hierarchy decaying at infinity studied in [1,2]. 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