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We obtain a leading asymptotic for the spectral counting function for $\\lambda^{-1}R(\\lambda)$ in an interval $[a_1, a_2)$ as $\\lambda \\to \\infty$, under the assumption that the measure of periodic billiards on $T^*M$ is zero. The asymptotic takes the form \\begin{equation*} N(\\lambda; a_1,a_2) = \\bigl(\\kappa(a_2)-\\kappa(a_1)\\bigr)\\mathsf{vol}'(\\partial M) \\lambda^{d-1}+o(\\lambda^{d-1}), \\end{equation*} where $\\kappa(a)$ is given explicitly by \\begin{e"},"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":"1505.04894","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2015-05-19T07:22:29Z","cross_cats_sorted":[],"title_canon_sha256":"d8dbedd8adfaf73ed7200a588d975281c7066a2bb9b830d00ce14dffcfa83906","abstract_canon_sha256":"7bb3e80e0d487c1140d38793c4163e8b89dcd6095b267450d5f8dd7fcef0c1dd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:41:45.939832Z","signature_b64":"v+qMoAVvP5oJBf2sPRMAi3TehSFaziOxJZH1XQUuVI+irxFrp/atkfueKe4h33X9CacehUrblOpw20drNPkJDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a8743fd829461c51b74333781e31b10423eeb77d92c0fa7eb48a043fca224c28","last_reissued_at":"2026-05-18T01:41:45.939329Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:41:45.939329Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Spectral asymptotics for the semiclassical Dirichlet to Neumann operator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Andrew Hassell, Victor Ivrii","submitted_at":"2015-05-19T07:22:29Z","abstract_excerpt":"Let $M$ be a compact Riemannian manifold with smooth boundary, and let $R(\\lambda)$ be the Dirichlet-to-Neumann operator at frequency $\\lambda$. We obtain a leading asymptotic for the spectral counting function for $\\lambda^{-1}R(\\lambda)$ in an interval $[a_1, a_2)$ as $\\lambda \\to \\infty$, under the assumption that the measure of periodic billiards on $T^*M$ is zero. The asymptotic takes the form \\begin{equation*} N(\\lambda; a_1,a_2) = \\bigl(\\kappa(a_2)-\\kappa(a_1)\\bigr)\\mathsf{vol}'(\\partial M) \\lambda^{d-1}+o(\\lambda^{d-1}), \\end{equation*} where $\\kappa(a)$ is given explicitly by \\begin{e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.04894","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":"1505.04894","created_at":"2026-05-18T01:41:45.939403+00:00"},{"alias_kind":"arxiv_version","alias_value":"1505.04894v2","created_at":"2026-05-18T01:41:45.939403+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.04894","created_at":"2026-05-18T01:41:45.939403+00:00"},{"alias_kind":"pith_short_12","alias_value":"VB2D7WBJIYOF","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_16","alias_value":"VB2D7WBJIYOFDN2D","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_8","alias_value":"VB2D7WBJ","created_at":"2026-05-18T12:29:44.643036+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/VB2D7WBJIYOFDN2DGN4B4MNRAQ","json":"https://pith.science/pith/VB2D7WBJIYOFDN2DGN4B4MNRAQ.json","graph_json":"https://pith.science/api/pith-number/VB2D7WBJIYOFDN2DGN4B4MNRAQ/graph.json","events_json":"https://pith.science/api/pith-number/VB2D7WBJIYOFDN2DGN4B4MNRAQ/events.json","paper":"https://pith.science/paper/VB2D7WBJ"},"agent_actions":{"view_html":"https://pith.science/pith/VB2D7WBJIYOFDN2DGN4B4MNRAQ","download_json":"https://pith.science/pith/VB2D7WBJIYOFDN2DGN4B4MNRAQ.json","view_paper":"https://pith.science/paper/VB2D7WBJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1505.04894&json=true","fetch_graph":"https://pith.science/api/pith-number/VB2D7WBJIYOFDN2DGN4B4MNRAQ/graph.json","fetch_events":"https://pith.science/api/pith-number/VB2D7WBJIYOFDN2DGN4B4MNRAQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VB2D7WBJIYOFDN2DGN4B4MNRAQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VB2D7WBJIYOFDN2DGN4B4MNRAQ/action/storage_attestation","attest_author":"https://pith.science/pith/VB2D7WBJIYOFDN2DGN4B4MNRAQ/action/author_attestation","sign_citation":"https://pith.science/pith/VB2D7WBJIYOFDN2DGN4B4MNRAQ/action/citation_signature","submit_replication":"https://pith.science/pith/VB2D7WBJIYOFDN2DGN4B4MNRAQ/action/replication_record"}},"created_at":"2026-05-18T01:41:45.939403+00:00","updated_at":"2026-05-18T01:41:45.939403+00:00"}