{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:XTTEOUBFO6Y3UBNHQNVFGUEF5M","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"bb0ff504378f5d03aa9401105e58b2aa5028dab89c5972e3ff58ebda44e0d2b1","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2009-06-17T13:11:51Z","title_canon_sha256":"1f13b7b30ec72a3fd0d06c8fe32be61524e8b1d03a207f71e588e1eb0ac35f41"},"schema_version":"1.0","source":{"id":"0906.3113","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0906.3113","created_at":"2026-05-18T04:41:16Z"},{"alias_kind":"arxiv_version","alias_value":"0906.3113v2","created_at":"2026-05-18T04:41:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0906.3113","created_at":"2026-05-18T04:41:16Z"},{"alias_kind":"pith_short_12","alias_value":"XTTEOUBFO6Y3","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_16","alias_value":"XTTEOUBFO6Y3UBNH","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_8","alias_value":"XTTEOUBF","created_at":"2026-05-18T12:26:02Z"}],"graph_snapshots":[{"event_id":"sha256:7a168f72dcdca8e5a3d17dcd10d864d11dae5f390f28866647815f3ed4e6af74","target":"graph","created_at":"2026-05-18T04:41:16Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We study the spectral properties of the transition semigroup of the killed one-dimensional Cauchy process on the half-line (0,infty) and the interval (-1,1). This process is related to the square root of one-dimensional Laplacian A = -sqrt(-d^2/dx^2) with a Dirichlet exterior condition (on a complement of a domain), and to a mixed Steklov problem in the half-plane. For the half-line, an explicit formula for generalized eigenfunctions psi_lambda of A is derived, and then used to construct spectral representation of A. Explicit formulas for the transition density of the killed Cauchy process in ","authors_text":"Andrzej Stos, Jacek Ma{\\l}ecki, Mateusz Kwa\\'snicki, Tadeusz Kulczycki","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2009-06-17T13:11:51Z","title":"Spectral properties of the Cauchy process"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0906.3113","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:87561b9e90c9f6c7b61bcdd66bed6de350441d89b90871fcea07e9c38384dc8b","target":"record","created_at":"2026-05-18T04:41:16Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"bb0ff504378f5d03aa9401105e58b2aa5028dab89c5972e3ff58ebda44e0d2b1","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2009-06-17T13:11:51Z","title_canon_sha256":"1f13b7b30ec72a3fd0d06c8fe32be61524e8b1d03a207f71e588e1eb0ac35f41"},"schema_version":"1.0","source":{"id":"0906.3113","kind":"arxiv","version":2}},"canonical_sha256":"bce647502577b1ba05a7836a535085eb087530ce5c337a31c526e1d17a394bc2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bce647502577b1ba05a7836a535085eb087530ce5c337a31c526e1d17a394bc2","first_computed_at":"2026-05-18T04:41:16.412396Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:41:16.412396Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7fCceqtpD83b0PSxGYCZT0foOWMjf/aUizjfSrcb24IADmI9uWk4H5nFEk/arXKPkYC8fUyk4jwmpGh4zrDOCA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:41:16.412969Z","signed_message":"canonical_sha256_bytes"},"source_id":"0906.3113","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:87561b9e90c9f6c7b61bcdd66bed6de350441d89b90871fcea07e9c38384dc8b","sha256:7a168f72dcdca8e5a3d17dcd10d864d11dae5f390f28866647815f3ed4e6af74"],"state_sha256":"fcbd07491a2fb9df1df3cc9c68b9610c781941f96af68a6e21f075b0dcedb1bf"}