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Zeta-invariants are defined by $Z_m(a)={\\rm Tr}[(a\\Lambda)^{2m}-(aD)^{2m}]$ for every smooth function $a$. In the case of a positive $a$, zeta-invariants are determined by the Steklov spectrum. We obtain some estimate from below for $Z_m(a)$ in th"},"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":"1611.05919","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2016-11-04T16:19:38Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"25302b744ecfb6f5528ff9707482f66f6d66a12c04becf2d5e6d45d20f67f6c3","abstract_canon_sha256":"9eba3dcc853bfbbee84709061460c3f9f92121a69e1e475baf5aa66e23c0a798"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:57:43.459752Z","signature_b64":"9nXTlgSw7qVzIL4ZP0HTuxKh/9KMS73KLqdhAZsWr0Lwz0ucLkRPng6BLp8yOlasBigQyWzz+pPB8jlTU4HtDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2f7927bf673e875bdf71b1c8e872d0c1dca8802b0ce604c1c11068971dda95ac","last_reissued_at":"2026-05-18T00:57:43.459346Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:57:43.459346Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Steklov zeta-invariants and a compactness theorem for isospectral families of planar domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.SP","authors_text":"Alexandre Jollivet, Vladimir Sharafutdinov","submitted_at":"2016-11-04T16:19:38Z","abstract_excerpt":"The inverse problem of recovering a smooth simply connected multisheet planar domain from its Steklov spectrum is equivalent to the problem of determination, up to a gauge transform, of a smooth positive function $a$ on the unit circle from the spectrum of the operator $a\\Lambda$, where $\\Lambda$ is the Dirichlet-to-Neumann operator of the unit disk. Zeta-invariants are defined by $Z_m(a)={\\rm Tr}[(a\\Lambda)^{2m}-(aD)^{2m}]$ for every smooth function $a$. In the case of a positive $a$, zeta-invariants are determined by the Steklov spectrum. 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