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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","authors_text":"Pritam Ganguly","cross_cats":["math.CA"],"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.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2025-11-16T11:31:16Z","title":"On Rellich-type asymptotics for eigenfunctions on rank one symmetric spaces of noncompact type"},"references":{"count":39,"internal_anchors":0,"resolved_work":39,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"Agmon, Lower bounds for solutions of Schr¨ odinger equations,J","work_id":"f852ffd5-a1e4-4425-ba84-699a10851628","year":1970},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"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","year":2023},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"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","year":2024},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"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","year":null},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"A. Banerjee, N. Garofalo, Absence ofL p spectrum for asymptotically flat diffusions in region with cavities, arXiv:2507.10728 (2025)","work_id":"8e991fe5-0048-40af-b0a8-475f6cb50f20","year":2025}],"snapshot_sha256":"cd707b7f8cc52a8ebb623176c4419bfb6f8b2a3d11c3d7718b015fdf715d8f90"},"source":{"id":"2511.12561","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-17T22:20:31.404172Z","id":"c1f0ccf1-6be8-4798-a2f2-8b20bf51ed8a","model_set":{"reader":"grok-4.3"},"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","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.","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.","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."}},"verdict_id":"c1f0ccf1-6be8-4798-a2f2-8b20bf51ed8a"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:5276a519fa6c2ed5c8116d0276d171f3793c578dd2c3b33604341a1ad3f083e8","target":"record","created_at":"2026-05-18T03:09:33Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"a1b9b2c570c4d3a90460ad04d0d589a402fa70fb2087dfd39bb480192e65616f","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2025-11-16T11:31:16Z","title_canon_sha256":"502e9db583977886fbfb93691c87090aa88e23736ae8a58a96e6c1f78705a41a"},"schema_version":"1.0","source":{"id":"2511.12561","kind":"arxiv","version":2}},"canonical_sha256":"efef049deb0e482822687129b1a4c4705aa452ac37c2466d27d6b7ab2e98fa5a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"efef049deb0e482822687129b1a4c4705aa452ac37c2466d27d6b7ab2e98fa5a","first_computed_at":"2026-05-18T03:09:33.223630Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:09:33.223630Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NDW+txjLSuz4J4o++b0YvUHciuQ/uhe8EJoKapFR9IBoClDDWB4ps1Hjy+L58b4ZyD55CsfSCcXMUQGhHyajBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:09:33.224100Z","signed_message":"canonical_sha256_bytes"},"source_id":"2511.12561","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5276a519fa6c2ed5c8116d0276d171f3793c578dd2c3b33604341a1ad3f083e8","sha256:3a6a3b565d5e6ff0347a6fba6c2a93aa4a3381ab1880d74f9bd486f2c7bd080c"],"state_sha256":"d3d73b52bc376d048f28ee9758d05be198465235b61e2dd59e264675c67378e1"}