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This operator has a discrete spectrum: eventually the eigenvalues are simple and $\\lambda_n = (2n + 1) + s^2 (\\kappa(n) / n) + \\rho(n)$, where $ \\kappa(n) = \\frac{1}{2\\pi} [(-1)^{n + 1} \\sin ( 2 b \\sqrt{2n} ) - \\frac{1}{2} \\sin ( 4 b \\sqrt{2n} ) ]$ and $ |\\rho(n) | \\leq C (\\log n) / (n^{3/2})$ If $s = i \\gamma$, $\\gamma$ real, the number $T(\\gamma)$ of non-real eigenvalues is finite, and $T(\\gamma) \\leq [ C (1 + | \\gamma |"},"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":"1407.4153","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2014-07-15T21:39:59Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"9bee63bfe7cde62530fe4393d10d8f4c7c3fa3f284715e99f8c790d83a3e6c37","abstract_canon_sha256":"dd137af925450d1e5b929386e120fc31bbd8cc58f0c215efd73bb953ff9bd1c1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:42:36.424042Z","signature_b64":"Enb2QAC3atbWOL17bMg+JnKGSLRVESXmBepV+tRh/zrrdx5DmMgOIKZZR0OLNtDPUfn5vrtuAmR4CbVu/Fn6Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"436b9c62c01e6eadd28493f1654db38d3645e57d6815c88e4cfd79eea1498927","last_reissued_at":"2026-05-18T01:42:36.423470Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:42:36.423470Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The spectrum of a Harmonic Oscillator Operator Perturbed by Point Interactions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.SP","authors_text":"Boris Mityagin","submitted_at":"2014-07-15T21:39:59Z","abstract_excerpt":"We consider the operator $ L = - (d/dx)^2 + x^2 y + w(x) y , y \\in L^2(\\mathbb{R}) $, where $ w(x) = s [ \\delta(x - b) - \\delta(x + b)], b \\neq 0,$ real, $s \\in \\mathbb{C}$. This operator has a discrete spectrum: eventually the eigenvalues are simple and $\\lambda_n = (2n + 1) + s^2 (\\kappa(n) / n) + \\rho(n)$, where $ \\kappa(n) = \\frac{1}{2\\pi} [(-1)^{n + 1} \\sin ( 2 b \\sqrt{2n} ) - \\frac{1}{2} \\sin ( 4 b \\sqrt{2n} ) ]$ and $ |\\rho(n) | \\leq C (\\log n) / (n^{3/2})$ If $s = i \\gamma$, $\\gamma$ real, the number $T(\\gamma)$ of non-real eigenvalues is finite, and $T(\\gamma) \\leq [ C (1 + | \\gamma |"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.4153","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":""},"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":"1407.4153","created_at":"2026-05-18T01:42:36.423548+00:00"},{"alias_kind":"arxiv_version","alias_value":"1407.4153v1","created_at":"2026-05-18T01:42:36.423548+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.4153","created_at":"2026-05-18T01:42:36.423548+00:00"},{"alias_kind":"pith_short_12","alias_value":"INVZYYWADZXK","created_at":"2026-05-18T12:28:33.132498+00:00"},{"alias_kind":"pith_short_16","alias_value":"INVZYYWADZXK3UUE","created_at":"2026-05-18T12:28:33.132498+00:00"},{"alias_kind":"pith_short_8","alias_value":"INVZYYWA","created_at":"2026-05-18T12:28:33.132498+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/INVZYYWADZXK3UUESPYWKTNTRU","json":"https://pith.science/pith/INVZYYWADZXK3UUESPYWKTNTRU.json","graph_json":"https://pith.science/api/pith-number/INVZYYWADZXK3UUESPYWKTNTRU/graph.json","events_json":"https://pith.science/api/pith-number/INVZYYWADZXK3UUESPYWKTNTRU/events.json","paper":"https://pith.science/paper/INVZYYWA"},"agent_actions":{"view_html":"https://pith.science/pith/INVZYYWADZXK3UUESPYWKTNTRU","download_json":"https://pith.science/pith/INVZYYWADZXK3UUESPYWKTNTRU.json","view_paper":"https://pith.science/paper/INVZYYWA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1407.4153&json=true","fetch_graph":"https://pith.science/api/pith-number/INVZYYWADZXK3UUESPYWKTNTRU/graph.json","fetch_events":"https://pith.science/api/pith-number/INVZYYWADZXK3UUESPYWKTNTRU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/INVZYYWADZXK3UUESPYWKTNTRU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/INVZYYWADZXK3UUESPYWKTNTRU/action/storage_attestation","attest_author":"https://pith.science/pith/INVZYYWADZXK3UUESPYWKTNTRU/action/author_attestation","sign_citation":"https://pith.science/pith/INVZYYWADZXK3UUESPYWKTNTRU/action/citation_signature","submit_replication":"https://pith.science/pith/INVZYYWADZXK3UUESPYWKTNTRU/action/replication_record"}},"created_at":"2026-05-18T01:42:36.423548+00:00","updated_at":"2026-05-18T01:42:36.423548+00:00"}