{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:5POTVAEUWI5VUKVJTN5GVYGN7J","short_pith_number":"pith:5POTVAEU","schema_version":"1.0","canonical_sha256":"ebdd3a8094b23b5a2aa99b7a6ae0cdfa5e99a77a83a5789af6ebb81628687fff","source":{"kind":"arxiv","id":"2606.12455","version":1},"attestation_state":"computed","paper":{"title":"King Function for Shifted Gaussian: Laguerre Structure, Spectral Theory and Density","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.MP","math.SP","physics.plasm-ph"],"primary_cat":"math-ph","authors_text":"Yanpeng Wang, Zhe Gao","submitted_at":"2026-06-05T18:00:09Z","abstract_excerpt":"We study King function arising as radial kernels in the laboratory-frame spherical harmonic expansion of shifted Gaussian distributions. We first clarify their relation with the co-moving Laguerre hierarchy by means of a King--Laguerre expansion. We then derive the King differential equation and show that the associated self-adjoint operator in a Gaussian-weighted Hilbert space is unitarily equivalent to the free radial Schr\\\"odinger operator on the half-line. This yields the spectral representation and generalized eigenfunction. Finally, we prove that real-parameter King function, lies in 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":"2606.12455","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math-ph","submitted_at":"2026-06-05T18:00:09Z","cross_cats_sorted":["math.MP","math.SP","physics.plasm-ph"],"title_canon_sha256":"7f170fe4958b1ff8f3aa3ba23581cbf5fe21d6ee31f5af5029cfbddde243d46e","abstract_canon_sha256":"76c24ad4da796c06be35f6f4345cec017ffe0e5f7265577127a360b821fff075"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-12T00:07:51.169245Z","signature_b64":"rzq70Pj2RjJg9JkvNAq3/UuzHKQERUimGEPcgStSKMFpsD8nbAgzpa5YHzwgFK1e3TOirNw+tCfqeENGHLbnBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ebdd3a8094b23b5a2aa99b7a6ae0cdfa5e99a77a83a5789af6ebb81628687fff","last_reissued_at":"2026-06-12T00:07:51.168822Z","signature_status":"signed_v1","first_computed_at":"2026-06-12T00:07:51.168822Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"King Function for Shifted Gaussian: Laguerre Structure, Spectral Theory and Density","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.MP","math.SP","physics.plasm-ph"],"primary_cat":"math-ph","authors_text":"Yanpeng Wang, Zhe Gao","submitted_at":"2026-06-05T18:00:09Z","abstract_excerpt":"We study King function arising as radial kernels in the laboratory-frame spherical harmonic expansion of shifted Gaussian distributions. We first clarify their relation with the co-moving Laguerre hierarchy by means of a King--Laguerre expansion. We then derive the King differential equation and show that the associated self-adjoint operator in a Gaussian-weighted Hilbert space is unitarily equivalent to the free radial Schr\\\"odinger operator on the half-line. This yields the spectral representation and generalized eigenfunction. Finally, we prove that real-parameter King function, lies in the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.12455","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.12455/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2606.12455","created_at":"2026-06-12T00:07:51.168889+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.12455v1","created_at":"2026-06-12T00:07:51.168889+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.12455","created_at":"2026-06-12T00:07:51.168889+00:00"},{"alias_kind":"pith_short_12","alias_value":"5POTVAEUWI5V","created_at":"2026-06-12T00:07:51.168889+00:00"},{"alias_kind":"pith_short_16","alias_value":"5POTVAEUWI5VUKVJ","created_at":"2026-06-12T00:07:51.168889+00:00"},{"alias_kind":"pith_short_8","alias_value":"5POTVAEU","created_at":"2026-06-12T00:07:51.168889+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5POTVAEUWI5VUKVJTN5GVYGN7J","json":"https://pith.science/pith/5POTVAEUWI5VUKVJTN5GVYGN7J.json","graph_json":"https://pith.science/api/pith-number/5POTVAEUWI5VUKVJTN5GVYGN7J/graph.json","events_json":"https://pith.science/api/pith-number/5POTVAEUWI5VUKVJTN5GVYGN7J/events.json","paper":"https://pith.science/paper/5POTVAEU"},"agent_actions":{"view_html":"https://pith.science/pith/5POTVAEUWI5VUKVJTN5GVYGN7J","download_json":"https://pith.science/pith/5POTVAEUWI5VUKVJTN5GVYGN7J.json","view_paper":"https://pith.science/paper/5POTVAEU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.12455&json=true","fetch_graph":"https://pith.science/api/pith-number/5POTVAEUWI5VUKVJTN5GVYGN7J/graph.json","fetch_events":"https://pith.science/api/pith-number/5POTVAEUWI5VUKVJTN5GVYGN7J/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5POTVAEUWI5VUKVJTN5GVYGN7J/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5POTVAEUWI5VUKVJTN5GVYGN7J/action/storage_attestation","attest_author":"https://pith.science/pith/5POTVAEUWI5VUKVJTN5GVYGN7J/action/author_attestation","sign_citation":"https://pith.science/pith/5POTVAEUWI5VUKVJTN5GVYGN7J/action/citation_signature","submit_replication":"https://pith.science/pith/5POTVAEUWI5VUKVJTN5GVYGN7J/action/replication_record"}},"created_at":"2026-06-12T00:07:51.168889+00:00","updated_at":"2026-06-12T00:07:51.168889+00:00"}