{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:X6UXDAL77SMYYZSJSKTT2M2LUG","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":"319f9dac361710b29bc7ab202b6bf4372843b32dbc34293c81f4f351e78ef173","cross_cats_sorted":["math.MP","quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2015-06-04T21:05:14Z","title_canon_sha256":"32bfe2acf65cf2c4dfb80bf8f6fdf46765683580abd3e24b92031a24c9147cc7"},"schema_version":"1.0","source":{"id":"1506.01728","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.01728","created_at":"2026-05-18T01:11:55Z"},{"alias_kind":"arxiv_version","alias_value":"1506.01728v2","created_at":"2026-05-18T01:11:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.01728","created_at":"2026-05-18T01:11:55Z"},{"alias_kind":"pith_short_12","alias_value":"X6UXDAL77SMY","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"X6UXDAL77SMYYZSJ","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"X6UXDAL7","created_at":"2026-05-18T12:29:47Z"}],"graph_snapshots":[{"event_id":"sha256:4b2174003d02283873fc25338cd019415ffc5a6d178a5658ca74ea419dc6fd1b","target":"graph","created_at":"2026-05-18T01:11:55Z","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 establish sharp (or `refined') comparison theorems for the Klein--Gordon equation. We show that the condition $V_a\\le V_b$, which leads to $E_a\\le E_b$, can be replaced by the weaker assumption $U_a\\le U_b$ which still implies the spectral ordering $E_a\\le E_b$. In the simplest case, for $d=1$, $U_i(x)=\\int_0^x V_i(t)dt$, $i=a$ or $b$, and for $d>1$, $U_i(r)=\\int_0^r V_i(t) t^{d-1}dt$, $i=a$ or $b$. We also consider sharp comparison theorems in the presence of a scalar potential $S$ (a `variable mass') in addition to the vector term $V$ (the time component of a $4$-vector). The theorems are","authors_text":"Petr Zorin, Richard L. Hall","cross_cats":["math.MP","quant-ph"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2015-06-04T21:05:14Z","title":"Sharp comparison theorems for the Klein--Gordon equation in $d$ dimensions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.01728","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":"sha256:2ca1be739cee11a5ebb6feb895dda773a1891e94f4d60bdd2f4550dc4df9d03e","target":"record","created_at":"2026-05-18T01:11:55Z","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":"319f9dac361710b29bc7ab202b6bf4372843b32dbc34293c81f4f351e78ef173","cross_cats_sorted":["math.MP","quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2015-06-04T21:05:14Z","title_canon_sha256":"32bfe2acf65cf2c4dfb80bf8f6fdf46765683580abd3e24b92031a24c9147cc7"},"schema_version":"1.0","source":{"id":"1506.01728","kind":"arxiv","version":2}},"canonical_sha256":"bfa971817ffc998c664992a73d334ba1ae461d4b03cc8438cc0789c8d47e040a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bfa971817ffc998c664992a73d334ba1ae461d4b03cc8438cc0789c8d47e040a","first_computed_at":"2026-05-18T01:11:55.146653Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:11:55.146653Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EjK5eTq9zDaZeBzfTnsX2tJxQ2popd5KMDmL9jgzNxokqRPBTLg3t8xqYFymv+y9KabHajGc7wVcfhgrQ7heAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:11:55.147076Z","signed_message":"canonical_sha256_bytes"},"source_id":"1506.01728","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2ca1be739cee11a5ebb6feb895dda773a1891e94f4d60bdd2f4550dc4df9d03e","sha256:4b2174003d02283873fc25338cd019415ffc5a6d178a5658ca74ea419dc6fd1b"],"state_sha256":"fe0558320e89ae116adcfecabb73aad44d293073a0ca501a0bcb49b1e729147b"}