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We consider operators of the form $H=\\Delta+V$, $V$ multiplication by a positive sequence $\\{V(n), n \\in \\ZZ^d\\}$ with $V(n) \\rightarrow \\infty, |n| \\rightarrow \\infty $\non $\\ell^2(\\ZZ^d)$ and $\\pi_{\\lambda}$ the orthogonal projection of $\\ell^2(\\mathbb{Z}^d)$ on to the space of eigenfunctions of $H$ with eigenvalues $\\leq \\lambda$. We take $B$ to be a pseudo difference operator of order zero with symbol $b(x,n), (x,n) \\in \\TT^d\\times \\ZZ^d$ and show that for nice functions $f$ $$ \\lim_{\\lambda \\rightarrow \\infty} Tr(f("},"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":"1102.4131","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-02-21T04:53:01Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"da4ae97070f9ff170d0ada5a9ef13ba0448ecdfd432de9e44a9538e12f86181f","abstract_canon_sha256":"a965931326fcecf616822c5dc09c93845c113c6ef87007eb09bf0a33fd45224a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:51:12.953568Z","signature_b64":"gHWQmuJj/WJa5QAJT25/NeXA8NLPAuGhox5auKNr+RBWR7JdoTjbTrfan1pPYuyW/UmvKCP8+AOpNb/gg0MtDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"39a14733a4d9f1e2a19fda56afae282a3bde567b2b23ad35954de94b58196d60","last_reissued_at":"2026-05-18T03:51:12.952789Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:51:12.952789Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Szego limit theorem on the lattice","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Jitendriya Swain, M. Krishna","submitted_at":"2011-02-21T04:53:01Z","abstract_excerpt":"In this paper, we prove a Szeg\\\"{o} type limit theorem on $\\ell^2(\\ZZ^d)$. We consider operators of the form $H=\\Delta+V$, $V$ multiplication by a positive sequence $\\{V(n), n \\in \\ZZ^d\\}$ with $V(n) \\rightarrow \\infty, |n| \\rightarrow \\infty $\non $\\ell^2(\\ZZ^d)$ and $\\pi_{\\lambda}$ the orthogonal projection of $\\ell^2(\\mathbb{Z}^d)$ on to the space of eigenfunctions of $H$ with eigenvalues $\\leq \\lambda$. We take $B$ to be a pseudo difference operator of order zero with symbol $b(x,n), (x,n) \\in \\TT^d\\times \\ZZ^d$ and show that for nice functions $f$ $$ \\lim_{\\lambda \\rightarrow \\infty} Tr(f("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.4131","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":"1102.4131","created_at":"2026-05-18T03:51:12.952905+00:00"},{"alias_kind":"arxiv_version","alias_value":"1102.4131v2","created_at":"2026-05-18T03:51:12.952905+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.4131","created_at":"2026-05-18T03:51:12.952905+00:00"},{"alias_kind":"pith_short_12","alias_value":"HGQUOM5E3HY6","created_at":"2026-05-18T12:26:30.835961+00:00"},{"alias_kind":"pith_short_16","alias_value":"HGQUOM5E3HY6FIM7","created_at":"2026-05-18T12:26:30.835961+00:00"},{"alias_kind":"pith_short_8","alias_value":"HGQUOM5E","created_at":"2026-05-18T12:26:30.835961+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/HGQUOM5E3HY6FIM73JLK7LRIFI","json":"https://pith.science/pith/HGQUOM5E3HY6FIM73JLK7LRIFI.json","graph_json":"https://pith.science/api/pith-number/HGQUOM5E3HY6FIM73JLK7LRIFI/graph.json","events_json":"https://pith.science/api/pith-number/HGQUOM5E3HY6FIM73JLK7LRIFI/events.json","paper":"https://pith.science/paper/HGQUOM5E"},"agent_actions":{"view_html":"https://pith.science/pith/HGQUOM5E3HY6FIM73JLK7LRIFI","download_json":"https://pith.science/pith/HGQUOM5E3HY6FIM73JLK7LRIFI.json","view_paper":"https://pith.science/paper/HGQUOM5E","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1102.4131&json=true","fetch_graph":"https://pith.science/api/pith-number/HGQUOM5E3HY6FIM73JLK7LRIFI/graph.json","fetch_events":"https://pith.science/api/pith-number/HGQUOM5E3HY6FIM73JLK7LRIFI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HGQUOM5E3HY6FIM73JLK7LRIFI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HGQUOM5E3HY6FIM73JLK7LRIFI/action/storage_attestation","attest_author":"https://pith.science/pith/HGQUOM5E3HY6FIM73JLK7LRIFI/action/author_attestation","sign_citation":"https://pith.science/pith/HGQUOM5E3HY6FIM73JLK7LRIFI/action/citation_signature","submit_replication":"https://pith.science/pith/HGQUOM5E3HY6FIM73JLK7LRIFI/action/replication_record"}},"created_at":"2026-05-18T03:51:12.952905+00:00","updated_at":"2026-05-18T03:51:12.952905+00:00"}