{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:57XQJHPLBZECQITIOEU3DJGEOB","short_pith_number":"pith:57XQJHPL","schema_version":"1.0","canonical_sha256":"efef049deb0e482822687129b1a4c4705aa452ac37c2466d27d6b7ab2e98fa5a","source":{"kind":"arxiv","id":"2511.12561","version":2},"attestation_state":"computed","paper":{"title":"On Rellich-type asymptotics for eigenfunctions on rank one symmetric spaces of noncompact type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Eigenfunctions of the Laplace-Beltrami operator on exterior domains in rank-one symmetric spaces satisfy sharp quantitative L^p growth estimates in geodesic annuli.","cross_cats":["math.CA"],"primary_cat":"math.AP","authors_text":"Pritam Ganguly","submitted_at":"2025-11-16T11:31:16Z","abstract_excerpt":"We study eigenfunctions of the Laplace--Beltrami operator \\(\\Delta_X\\) in exterior domains \\(\\Omega\\) of rank-one Riemannian symmetric spaces of noncompact type \\(X\\), a class that includes all hyperbolic spaces. Extending the classical \\(L^2\\) Rellich theorem for the Euclidean Laplacian, we analyze the asymptotic behaviour and \\(L^p\\)-integrability of solutions to the Helmholtz equation\n  \\[\n  \\Delta_X f + (\\lambda^2 + \\rho^2) f = 0 \\quad \\text{in } \\Omega,\n  \\]\n  where \\(\\lambda \\in \\mathbb{C}\\setminus i\\mathbb{Z}\\) and \\(\\rho\\) denotes the half-sum of positive roots.\n  We establish sharp Re"},"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":true,"formal_links_present":true},"canonical_record":{"source":{"id":"2511.12561","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2025-11-16T11:31:16Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"502e9db583977886fbfb93691c87090aa88e23736ae8a58a96e6c1f78705a41a","abstract_canon_sha256":"a1b9b2c570c4d3a90460ad04d0d589a402fa70fb2087dfd39bb480192e65616f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:09:33.224100Z","signature_b64":"NDW+txjLSuz4J4o++b0YvUHciuQ/uhe8EJoKapFR9IBoClDDWB4ps1Hjy+L58b4ZyD55CsfSCcXMUQGhHyajBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"efef049deb0e482822687129b1a4c4705aa452ac37c2466d27d6b7ab2e98fa5a","last_reissued_at":"2026-05-18T03:09:33.223630Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:09:33.223630Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Rellich-type asymptotics for eigenfunctions on rank one symmetric spaces of noncompact type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Eigenfunctions of the Laplace-Beltrami operator on exterior domains in rank-one symmetric spaces satisfy sharp quantitative L^p growth estimates in geodesic annuli.","cross_cats":["math.CA"],"primary_cat":"math.AP","authors_text":"Pritam Ganguly","submitted_at":"2025-11-16T11:31:16Z","abstract_excerpt":"We study eigenfunctions of the Laplace--Beltrami operator \\(\\Delta_X\\) in exterior domains \\(\\Omega\\) of rank-one Riemannian symmetric spaces of noncompact type \\(X\\), a class that includes all hyperbolic spaces. Extending the classical \\(L^2\\) Rellich theorem for the Euclidean Laplacian, we analyze the asymptotic behaviour and \\(L^p\\)-integrability of solutions to the Helmholtz equation\n  \\[\n  \\Delta_X f + (\\lambda^2 + \\rho^2) f = 0 \\quad \\text{in } \\Omega,\n  \\]\n  where \\(\\lambda \\in \\mathbb{C}\\setminus i\\mathbb{Z}\\) and \\(\\rho\\) denotes the half-sum of positive roots.\n  We establish sharp Re"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We establish sharp Rellich-type quantitative L^p-growth estimates in geodesic annuli, which yield the nonexistence of nontrivial L^p(Ω)-solutions in the optimal range 1 ≤ p ≤ 2 for spectral parameters satisfying |Im(λ)| ≤ (2/p - 1)ρ.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The domain Ω is an exterior domain in a rank-one Riemannian symmetric space X of noncompact type, and the spectral parameter λ satisfies |Im(λ)| ≤ (2/p - 1)ρ with λ not in iℤ; the analysis relies on the standard structure of the Laplace-Beltrami operator and the half-sum of positive roots ρ.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Sharp quantitative L^p growth estimates are established for Helmholtz eigenfunctions on rank-one symmetric spaces, yielding nonexistence of nontrivial L^p solutions for |Im(λ)| ≤ (2/p - 1)ρ and refined uniqueness theorems.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Eigenfunctions of the Laplace-Beltrami operator on exterior domains in rank-one symmetric spaces satisfy sharp quantitative L^p growth estimates in geodesic annuli.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"dc0138c80c5588645caf88eb8a0e5308aa4aecff73dc0d74f0cafb5b909e51e8"},"source":{"id":"2511.12561","kind":"arxiv","version":2},"verdict":{"id":"c1f0ccf1-6be8-4798-a2f2-8b20bf51ed8a","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-17T22:20:31.404172Z","strongest_claim":"We establish sharp Rellich-type quantitative L^p-growth estimates in geodesic annuli, which yield the nonexistence of nontrivial L^p(Ω)-solutions in the optimal range 1 ≤ p ≤ 2 for spectral parameters satisfying |Im(λ)| ≤ (2/p - 1)ρ.","one_line_summary":"Sharp quantitative L^p growth estimates are established for Helmholtz eigenfunctions on rank-one symmetric spaces, yielding nonexistence of nontrivial L^p solutions for |Im(λ)| ≤ (2/p - 1)ρ and refined uniqueness theorems.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The domain Ω is an exterior domain in a rank-one Riemannian symmetric space X of noncompact type, and the spectral parameter λ satisfies |Im(λ)| ≤ (2/p - 1)ρ with λ not in iℤ; the analysis relies on the standard structure of the Laplace-Beltrami operator and the half-sum of positive roots ρ.","pith_extraction_headline":"Eigenfunctions of the Laplace-Beltrami operator on exterior domains in rank-one symmetric spaces satisfy sharp quantitative L^p growth estimates in geodesic annuli."},"references":{"count":39,"sample":[{"doi":"","year":1970,"title":"Agmon, Lower bounds for solutions of Schr¨ odinger equations,J","work_id":"f852ffd5-a1e4-4425-ba84-699a10851628","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2023,"title":"W. Ballman, M. Mukherjee, P. Polymerakis, On the spectrum of certain Hadamard manifoldsSIGMA Symmetry Integrability Geom. Methods Appl.19 (2023), Paper No. 050, 19 pp","work_id":"033b36a9-e85f-44d2-ab4e-57b6d2cbee8f","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"A. Banerjee, N. Garofalo, An observation on eigenfunctions of the Laplacian,La Matematica3 (2024), no. 4, 1451–1455","work_id":"d459b0c3-7ed8-4bdb-8ed6-c09e2f653056","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"A. Banerjee, N. Garofalo, A Rellich type estimate for a subelliptic Helmholtz equation with mixed homogeneities, arXiv:2311.11559,J. d’Analyse Mathematique, to appear","work_id":"7b75d287-b5c3-4c12-a5cf-23259068cc38","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"A. Banerjee, N. 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