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For $\\alpha>0$, let $H^\\Omega_\\alpha$ denote the Laplacian in $\\Omega$, $u\\mapsto -\\Delta u$, with the Robin boundary conditions $\\partial u/\\partial\\nu =\\alpha u$, where $\\nu$ is the exterior unit normal at the boundary of $\\Omega$. We show that, for any fixed $m\\in\\mathbb{N}$, the $m$th eigenvalue $E^\\Omega_m(\\alpha)$ of $H^\\Omega_\\alpha$ behaves as \\[ E^\\Omega_m(\\alpha)=-\\alpha^2+\\mu^D_m +\\mathcal{O}\\Big(\\dfrac{1}{\\sqrt\\alpha}\\Big) \\quad {as $\\alpha$ tends to $+\\infty$}, \\] where"},"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":"1411.1956","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2014-11-07T15:49:05Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"7f5796171b6529a095e14d01f91bc42f1f7ba2c4e546141ea186a4f4fb0f9352","abstract_canon_sha256":"bf502a448e69d1feb6a0bb9382edeca9d6beca030d22878a74a1bb4a97bf69e4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:26:46.814138Z","signature_b64":"8kt7YdwCI2H9H3hq4yTEzjj2d4w/Etk99N2onE0bh9vWJ5bql/lm/685+Y5bnA570CP2GO1/GaXDL6bpmthnAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"017c7b8c111c4fd3043441cf6467a8f62831d558d9e87ca5aabe019c29944340","last_reissued_at":"2026-05-18T02:26:46.813768Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:26:46.813768Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Robin eigenvalues of the Laplacian in the exterior of a convex polygon","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.SP","authors_text":"Konstantin Pankrashkin","submitted_at":"2014-11-07T15:49:05Z","abstract_excerpt":"Let $\\Omega\\subset \\mathbb{R}^2$ be the exterior of a convex polygon whose side lengths are $\\ell_1,...,\\ell_M$. For $\\alpha>0$, let $H^\\Omega_\\alpha$ denote the Laplacian in $\\Omega$, $u\\mapsto -\\Delta u$, with the Robin boundary conditions $\\partial u/\\partial\\nu =\\alpha u$, where $\\nu$ is the exterior unit normal at the boundary of $\\Omega$. We show that, for any fixed $m\\in\\mathbb{N}$, the $m$th eigenvalue $E^\\Omega_m(\\alpha)$ of $H^\\Omega_\\alpha$ behaves as \\[ E^\\Omega_m(\\alpha)=-\\alpha^2+\\mu^D_m +\\mathcal{O}\\Big(\\dfrac{1}{\\sqrt\\alpha}\\Big) \\quad {as $\\alpha$ tends to $+\\infty$}, \\] where"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.1956","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":"1411.1956","created_at":"2026-05-18T02:26:46.813826+00:00"},{"alias_kind":"arxiv_version","alias_value":"1411.1956v2","created_at":"2026-05-18T02:26:46.813826+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.1956","created_at":"2026-05-18T02:26:46.813826+00:00"},{"alias_kind":"pith_short_12","alias_value":"AF6HXDARDRH5","created_at":"2026-05-18T12:28:19.803747+00:00"},{"alias_kind":"pith_short_16","alias_value":"AF6HXDARDRH5GBBU","created_at":"2026-05-18T12:28:19.803747+00:00"},{"alias_kind":"pith_short_8","alias_value":"AF6HXDAR","created_at":"2026-05-18T12:28:19.803747+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/AF6HXDARDRH5GBBUIHHWIZ5I6Y","json":"https://pith.science/pith/AF6HXDARDRH5GBBUIHHWIZ5I6Y.json","graph_json":"https://pith.science/api/pith-number/AF6HXDARDRH5GBBUIHHWIZ5I6Y/graph.json","events_json":"https://pith.science/api/pith-number/AF6HXDARDRH5GBBUIHHWIZ5I6Y/events.json","paper":"https://pith.science/paper/AF6HXDAR"},"agent_actions":{"view_html":"https://pith.science/pith/AF6HXDARDRH5GBBUIHHWIZ5I6Y","download_json":"https://pith.science/pith/AF6HXDARDRH5GBBUIHHWIZ5I6Y.json","view_paper":"https://pith.science/paper/AF6HXDAR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1411.1956&json=true","fetch_graph":"https://pith.science/api/pith-number/AF6HXDARDRH5GBBUIHHWIZ5I6Y/graph.json","fetch_events":"https://pith.science/api/pith-number/AF6HXDARDRH5GBBUIHHWIZ5I6Y/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AF6HXDARDRH5GBBUIHHWIZ5I6Y/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AF6HXDARDRH5GBBUIHHWIZ5I6Y/action/storage_attestation","attest_author":"https://pith.science/pith/AF6HXDARDRH5GBBUIHHWIZ5I6Y/action/author_attestation","sign_citation":"https://pith.science/pith/AF6HXDARDRH5GBBUIHHWIZ5I6Y/action/citation_signature","submit_replication":"https://pith.science/pith/AF6HXDARDRH5GBBUIHHWIZ5I6Y/action/replication_record"}},"created_at":"2026-05-18T02:26:46.813826+00:00","updated_at":"2026-05-18T02:26:46.813826+00:00"}