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Kac, who found a formula for \\[\\lim_{n\\rightarrow\\infty} \\frac{\\det(T_n(f))}{G(f)^n} \\] in terms of the values of $f$, where $G(f)$ is a constant. We extend this result in two ways: First, we consider shifting the index: Let \\[T_n(f;\\varepsilon) = -[\\delta_{j,j+1}+\\delta_{j+1,j}] + \\m"},"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":"1609.04125","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2016-09-14T03:47:35Z","cross_cats_sorted":[],"title_canon_sha256":"ea4461eecc257f8ba15543fb79422a0b4559dffddac11bbea36684a7793e2e5f","abstract_canon_sha256":"921aef6967bf6648a02246a8917deb772b0aef1c1fb6d71666c8455e82254fa3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:32.355494Z","signature_b64":"2aAc2mSqCqCIGpALaUqxfL5G0qwYlIW2eXrgee74UtgcRFS7Ccvho+vP2gImtSeaN5QPRQpz/F6/lK+2v42+CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0c59328c8ab1be9f38c6735dd0953d7cce5c903625d42d0cb97500e172a9447c","last_reissued_at":"2026-05-18T01:04:32.354702Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:32.354702Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotics of determinants of discrete Schr\\\"odinger operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Alain Bourget, Tyler McMillen","submitted_at":"2016-09-14T03:47:35Z","abstract_excerpt":"We consider the asymptotics of the determinants of large discrete Schr\\\"odinger operators, i.e. \"discrete Laplacian $+$ diagonal\": \\[T_n(f) = -[\\delta_{j,j+1}+\\delta_{j+1,j}] + \\mbox{diag}\\left(f\\left(\\frac{1}{n}\\right), f\\left(\\frac{2}{n}\\right),\\dots, f\\left(\\frac{n}{n}\\right)\\right) \\] We extend a result of M. Kac, who found a formula for \\[\\lim_{n\\rightarrow\\infty} \\frac{\\det(T_n(f))}{G(f)^n} \\] in terms of the values of $f$, where $G(f)$ is a constant. We extend this result in two ways: First, we consider shifting the index: Let \\[T_n(f;\\varepsilon) = -[\\delta_{j,j+1}+\\delta_{j+1,j}] + \\m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.04125","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"1609.04125","created_at":"2026-05-18T01:04:32.354834+00:00"},{"alias_kind":"arxiv_version","alias_value":"1609.04125v2","created_at":"2026-05-18T01:04:32.354834+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.04125","created_at":"2026-05-18T01:04:32.354834+00:00"},{"alias_kind":"pith_short_12","alias_value":"BRMTFDEKWG7J","created_at":"2026-05-18T12:30:07.202191+00:00"},{"alias_kind":"pith_short_16","alias_value":"BRMTFDEKWG7J6OGG","created_at":"2026-05-18T12:30:07.202191+00:00"},{"alias_kind":"pith_short_8","alias_value":"BRMTFDEK","created_at":"2026-05-18T12:30:07.202191+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/BRMTFDEKWG7J6OGGONO5BFJ5PT","json":"https://pith.science/pith/BRMTFDEKWG7J6OGGONO5BFJ5PT.json","graph_json":"https://pith.science/api/pith-number/BRMTFDEKWG7J6OGGONO5BFJ5PT/graph.json","events_json":"https://pith.science/api/pith-number/BRMTFDEKWG7J6OGGONO5BFJ5PT/events.json","paper":"https://pith.science/paper/BRMTFDEK"},"agent_actions":{"view_html":"https://pith.science/pith/BRMTFDEKWG7J6OGGONO5BFJ5PT","download_json":"https://pith.science/pith/BRMTFDEKWG7J6OGGONO5BFJ5PT.json","view_paper":"https://pith.science/paper/BRMTFDEK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1609.04125&json=true","fetch_graph":"https://pith.science/api/pith-number/BRMTFDEKWG7J6OGGONO5BFJ5PT/graph.json","fetch_events":"https://pith.science/api/pith-number/BRMTFDEKWG7J6OGGONO5BFJ5PT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BRMTFDEKWG7J6OGGONO5BFJ5PT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BRMTFDEKWG7J6OGGONO5BFJ5PT/action/storage_attestation","attest_author":"https://pith.science/pith/BRMTFDEKWG7J6OGGONO5BFJ5PT/action/author_attestation","sign_citation":"https://pith.science/pith/BRMTFDEKWG7J6OGGONO5BFJ5PT/action/citation_signature","submit_replication":"https://pith.science/pith/BRMTFDEKWG7J6OGGONO5BFJ5PT/action/replication_record"}},"created_at":"2026-05-18T01:04:32.354834+00:00","updated_at":"2026-05-18T01:04:32.354834+00:00"}