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This extends previous results obtained by the authors for the fractional Laplace operator (psi(x) = x^{alpha/2}) and for the Klein-Gordon square root operator (psi(x) = (1+x^2)^{1/2} - 1). The formula for the eigenvalues in (-a,a) is of the form lambda_n = psi(mu_n^2) + O(1/n), where mu_n is the solution of mu_n = (n pi)/(2 a) - the"},"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":"1507.05483","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2015-07-20T13:13:47Z","cross_cats_sorted":["math.AP","math.PR"],"title_canon_sha256":"a556e0396724c2454f601df9706ff278ccf62687185cb110bd09b9f75a83ce20","abstract_canon_sha256":"0b42cb5d2154c602142646c0800ecebf0686f7c13a6310c689281678e3ac8fed"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:50:52.465827Z","signature_b64":"/CV+M8NzCc7NggwScyzLlxfcZ2rPZc+S8z3OI27mNiL7QOscUPNB7V83h7USJKNUnZ6djMEr4gB31Gc+lWy/Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d5c78b146effb03a31f008fbd807cd79723883280af5d2d368bdc3fc03bb9a16","last_reissued_at":"2026-05-18T00:50:52.465353Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:50:52.465353Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotic estimate of eigenvalues of pseudo-differential operators in an interval","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.PR"],"primary_cat":"math.SP","authors_text":"Jacek Ma{\\l}ecki, Kamil Kaleta, Mateusz Kwa\\'snicki","submitted_at":"2015-07-20T13:13:47Z","abstract_excerpt":"We prove a two-term Weyl-type asymptotic law, with error term O(1/n), for the eigenvalues of the operator psi(-Delta) in an interval, with zero exterior condition, for complete Bernstein functions psi such that x psi'(x) converges to infinity as x goes to infinity. 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