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The operator is defined pathwise through the quasi-derivative realization of Sturm--Liouville operators with distributional potentials. Let $\\lambda_n$ be the Dirichlet eigenvalues, $\\lambda_n^+=\\max\\{\\lambda_n,0\\}$, and $k_n=\\sqrt{\\lambda_n^+}$. For every finite $p$, we prove the high-energy expansion \\[\n  k_n=\\frac{n\\pi}{L}\n  +\\frac{\\rho}{n\\pi}\\int_0^L\n  \\sin^2\\left(\\frac{n\\pi s}{L}\\righ"},"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":"2606.22426","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.SP","submitted_at":"2026-06-21T10:24:54Z","cross_cats_sorted":[],"title_canon_sha256":"5c5acfe65264e8268b8290eba2c9a6a9d3338de7630db3966158172d161557f3","abstract_canon_sha256":"a3e45ef7aa604c68bf804108c46aa64b239843e278513be1082f07dc404fef1d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-23T02:13:38.133535Z","signature_b64":"44ycVHOxGUIASoWpLq7rx6T3dCEAm/au/xs9yL0bXwjyX3P3E6wrYjcmeNZVcwLolqs/z6EiBKtq+tu5A5N6Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0c18987d892c6bfb786a138973f28ebe1bfe8319abfc4a11385ae59643667bd9","last_reissued_at":"2026-06-23T02:13:38.133117Z","signature_status":"signed_v1","first_computed_at":"2026-06-23T02:13:38.133117Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"High-energy asymptotics for finite-interval Schr\\\"odinger operators with Gaussian white-noise potential","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Wenwen Jian, Xiaoping Yuan, Yingdu Dong","submitted_at":"2026-06-21T10:24:54Z","abstract_excerpt":"We study the one-dimensional Schr\\\"odinger operator on a fixed interval with Gaussian white-noise potential, \\[\n  H_\\omega=-\\frac{\\dd^2}{\\dd x^2}+\\rho\\dot B_x(\\omega), \\] under Dirichlet boundary conditions. The operator is defined pathwise through the quasi-derivative realization of Sturm--Liouville operators with distributional potentials. Let $\\lambda_n$ be the Dirichlet eigenvalues, $\\lambda_n^+=\\max\\{\\lambda_n,0\\}$, and $k_n=\\sqrt{\\lambda_n^+}$. For every finite $p$, we prove the high-energy expansion \\[\n  k_n=\\frac{n\\pi}{L}\n  +\\frac{\\rho}{n\\pi}\\int_0^L\n  \\sin^2\\left(\\frac{n\\pi s}{L}\\righ"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.22426","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/2606.22426/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":"2606.22426","created_at":"2026-06-23T02:13:38.133177+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.22426v1","created_at":"2026-06-23T02:13:38.133177+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.22426","created_at":"2026-06-23T02:13:38.133177+00:00"},{"alias_kind":"pith_short_12","alias_value":"BQMJQ7MJFRV7","created_at":"2026-06-23T02:13:38.133177+00:00"},{"alias_kind":"pith_short_16","alias_value":"BQMJQ7MJFRV7W6DK","created_at":"2026-06-23T02:13:38.133177+00:00"},{"alias_kind":"pith_short_8","alias_value":"BQMJQ7MJ","created_at":"2026-06-23T02:13:38.133177+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/BQMJQ7MJFRV7W6DKCOEXH4UOXY","json":"https://pith.science/pith/BQMJQ7MJFRV7W6DKCOEXH4UOXY.json","graph_json":"https://pith.science/api/pith-number/BQMJQ7MJFRV7W6DKCOEXH4UOXY/graph.json","events_json":"https://pith.science/api/pith-number/BQMJQ7MJFRV7W6DKCOEXH4UOXY/events.json","paper":"https://pith.science/paper/BQMJQ7MJ"},"agent_actions":{"view_html":"https://pith.science/pith/BQMJQ7MJFRV7W6DKCOEXH4UOXY","download_json":"https://pith.science/pith/BQMJQ7MJFRV7W6DKCOEXH4UOXY.json","view_paper":"https://pith.science/paper/BQMJQ7MJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.22426&json=true","fetch_graph":"https://pith.science/api/pith-number/BQMJQ7MJFRV7W6DKCOEXH4UOXY/graph.json","fetch_events":"https://pith.science/api/pith-number/BQMJQ7MJFRV7W6DKCOEXH4UOXY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BQMJQ7MJFRV7W6DKCOEXH4UOXY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BQMJQ7MJFRV7W6DKCOEXH4UOXY/action/storage_attestation","attest_author":"https://pith.science/pith/BQMJQ7MJFRV7W6DKCOEXH4UOXY/action/author_attestation","sign_citation":"https://pith.science/pith/BQMJQ7MJFRV7W6DKCOEXH4UOXY/action/citation_signature","submit_replication":"https://pith.science/pith/BQMJQ7MJFRV7W6DKCOEXH4UOXY/action/replication_record"}},"created_at":"2026-06-23T02:13:38.133177+00:00","updated_at":"2026-06-23T02:13:38.133177+00:00"}