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We consider $\\mathcal{U}^\\varepsilon$ a tubular neighborhood of the surface $\\Sigma$ with a thickness parameter $\\varepsilon>0$, and we define the perturbed Dirac operator $\\mathfrak{D}^{\\varepsilon}_{M}=D_m +M\\beta \\mathbb{1}_{\\mathcal{U}^{\\varepsilon}},$ with $D_m$ the free Dirac operator, $M>0$, and $\\mathbb{1}_{\\mathcal{U }^{\\varepsilon}}$ the characteristic function of $\\mathcal{U}^{\\varepsilon}$. Then, in the norm resolvent sense, the Dirac operator $\\mathfrak{D}^{\\varepsilon}_M$ co"},"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":"2404.07784","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2024-04-11T14:25:00Z","cross_cats_sorted":[],"title_canon_sha256":"d6f0b42ed19594b53d129c499c961016dff2e2288818dbf86163c41d77a4d71f","abstract_canon_sha256":"f6344071da0072df53be6654b7b1139c94085e090ee05cab434442efcee63fc6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:06:57.676016Z","signature_b64":"7p7obRtbYia2vRkbrPCmkvyP6tAVShea1c4Ie1S6/rMszsxFsDkf5Gd2u7LXus6dp4topr/Tasy4AdgG6R5TBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1b9a2be1bccc5592ae042c8043bd9bf3c202380b4fbdd1f7428f22dcd917ec51","last_reissued_at":"2026-07-05T08:06:57.675515Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:06:57.675515Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the approximation of the Dirac operator coupled with confining Lorentz scalar $\\delta$-shell interactions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Mahdi Zreik","submitted_at":"2024-04-11T14:25:00Z","abstract_excerpt":"Let $\\Omega_+\\subset\\mathbb{R}^{3}$ be a fixed bounded domain with boundary $\\Sigma = \\partial\\Omega_{+}$. We consider $\\mathcal{U}^\\varepsilon$ a tubular neighborhood of the surface $\\Sigma$ with a thickness parameter $\\varepsilon>0$, and we define the perturbed Dirac operator $\\mathfrak{D}^{\\varepsilon}_{M}=D_m +M\\beta \\mathbb{1}_{\\mathcal{U}^{\\varepsilon}},$ with $D_m$ the free Dirac operator, $M>0$, and $\\mathbb{1}_{\\mathcal{U }^{\\varepsilon}}$ the characteristic function of $\\mathcal{U}^{\\varepsilon}$. Then, in the norm resolvent sense, the Dirac operator $\\mathfrak{D}^{\\varepsilon}_M$ co"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.07784","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/2404.07784/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":"2404.07784","created_at":"2026-07-05T08:06:57.675571+00:00"},{"alias_kind":"arxiv_version","alias_value":"2404.07784v1","created_at":"2026-07-05T08:06:57.675571+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2404.07784","created_at":"2026-07-05T08:06:57.675571+00:00"},{"alias_kind":"pith_short_12","alias_value":"DONCXYN4ZRKZ","created_at":"2026-07-05T08:06:57.675571+00:00"},{"alias_kind":"pith_short_16","alias_value":"DONCXYN4ZRKZFLQE","created_at":"2026-07-05T08:06:57.675571+00:00"},{"alias_kind":"pith_short_8","alias_value":"DONCXYN4","created_at":"2026-07-05T08:06:57.675571+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2505.22191","citing_title":"Approximation of Dirac operators with confining electrostatic and Lorentz scalar $\\delta$-shell potentials","ref_index":25,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/DONCXYN4ZRKZFLQEFSAEHPM36P","json":"https://pith.science/pith/DONCXYN4ZRKZFLQEFSAEHPM36P.json","graph_json":"https://pith.science/api/pith-number/DONCXYN4ZRKZFLQEFSAEHPM36P/graph.json","events_json":"https://pith.science/api/pith-number/DONCXYN4ZRKZFLQEFSAEHPM36P/events.json","paper":"https://pith.science/paper/DONCXYN4"},"agent_actions":{"view_html":"https://pith.science/pith/DONCXYN4ZRKZFLQEFSAEHPM36P","download_json":"https://pith.science/pith/DONCXYN4ZRKZFLQEFSAEHPM36P.json","view_paper":"https://pith.science/paper/DONCXYN4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2404.07784&json=true","fetch_graph":"https://pith.science/api/pith-number/DONCXYN4ZRKZFLQEFSAEHPM36P/graph.json","fetch_events":"https://pith.science/api/pith-number/DONCXYN4ZRKZFLQEFSAEHPM36P/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DONCXYN4ZRKZFLQEFSAEHPM36P/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DONCXYN4ZRKZFLQEFSAEHPM36P/action/storage_attestation","attest_author":"https://pith.science/pith/DONCXYN4ZRKZFLQEFSAEHPM36P/action/author_attestation","sign_citation":"https://pith.science/pith/DONCXYN4ZRKZFLQEFSAEHPM36P/action/citation_signature","submit_replication":"https://pith.science/pith/DONCXYN4ZRKZFLQEFSAEHPM36P/action/replication_record"}},"created_at":"2026-07-05T08:06:57.675571+00:00","updated_at":"2026-07-05T08:06:57.675571+00:00"}