{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:W5JRGRYQGXS2KKYENATBDQYHCL","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":"6490e3d6a9cea99108f9c34115640ecd5adb046f09dd358bd756656c9534cee2","cross_cats_sorted":["math.MP","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2017-06-20T10:30:50Z","title_canon_sha256":"b31ff998cb6eb31b3bde77402cf46187a845ade55b21986440a1a52ce1c9293c"},"schema_version":"1.0","source":{"id":"1706.06357","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1706.06357","created_at":"2026-05-18T00:42:03Z"},{"alias_kind":"arxiv_version","alias_value":"1706.06357v1","created_at":"2026-05-18T00:42:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.06357","created_at":"2026-05-18T00:42:03Z"},{"alias_kind":"pith_short_12","alias_value":"W5JRGRYQGXS2","created_at":"2026-05-18T12:31:53Z"},{"alias_kind":"pith_short_16","alias_value":"W5JRGRYQGXS2KKYE","created_at":"2026-05-18T12:31:53Z"},{"alias_kind":"pith_short_8","alias_value":"W5JRGRYQ","created_at":"2026-05-18T12:31:53Z"}],"graph_snapshots":[{"event_id":"sha256:3d2bd6d104f6196087e328f91bec680a82adbe55d5c1f19bd16626676dac469b","target":"graph","created_at":"2026-05-18T00:42:03Z","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":"For a general class of difference operators $H_\\varepsilon = T_\\varepsilon + V_\\varepsilon$ on $\\ell^2(\\varepsilon\\mathbb{Z}^d)$, where $V_\\varepsilon$ is a multi-well potential and $\\varepsilon$ is a small parameter, we analyze the asymptotic behavior as $\\varepsilon\\to 0$ of the (low-lying) eigenvalues and eigenfunctions. We show that the first $n$ eigenvalues of $H_\\varepsilon$ converge to the first $n$ eigenvalues of the direct sum of harmonic oscillators on $\\mathbb{R}^d$ located at the several wells. Our proof is microlocal.","authors_text":"Elke Rosenberger, Markus Klein","cross_cats":["math.MP","math.SP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2017-06-20T10:30:50Z","title":"Harmonic Approximation of Difference Operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.06357","kind":"arxiv","version":1},"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":"sha256:2d8b6ac3b1e745254156ba280c45bbaacd47061a1c9b01af4397bc4965feaebf","target":"record","created_at":"2026-05-18T00:42:03Z","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":"6490e3d6a9cea99108f9c34115640ecd5adb046f09dd358bd756656c9534cee2","cross_cats_sorted":["math.MP","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2017-06-20T10:30:50Z","title_canon_sha256":"b31ff998cb6eb31b3bde77402cf46187a845ade55b21986440a1a52ce1c9293c"},"schema_version":"1.0","source":{"id":"1706.06357","kind":"arxiv","version":1}},"canonical_sha256":"b75313471035e5a52b04682611c30712d3accce30e5816fc6ad880c8eb9e6c98","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b75313471035e5a52b04682611c30712d3accce30e5816fc6ad880c8eb9e6c98","first_computed_at":"2026-05-18T00:42:03.456796Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:42:03.456796Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7NDvDAucuhb2Y7UcmcZ9iXYrS9NHepO++1edc3yeBDE7+4sRdXrNhT62shJaTXHV6aT+f9NTZ/98V7QimSaBBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:42:03.457366Z","signed_message":"canonical_sha256_bytes"},"source_id":"1706.06357","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2d8b6ac3b1e745254156ba280c45bbaacd47061a1c9b01af4397bc4965feaebf","sha256:3d2bd6d104f6196087e328f91bec680a82adbe55d5c1f19bd16626676dac469b"],"state_sha256":"7431567a37f504f99b2c7ba6fd776057121627e10a13b252f39fcdaa831ceec3"}