{"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. Garofalo, Absence ofL p spectrum for asymptotically flat diffusions in region with cavities, arXiv:2507.10728 (2025)","work_id":"8e991fe5-0048-40af-b0a8-475f6cb50f20","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":39,"snapshot_sha256":"cd707b7f8cc52a8ebb623176c4419bfb6f8b2a3d11c3d7718b015fdf715d8f90","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"10959194a8f202916498ea4478a2adbf2c1618c20d8455a12ed029385c51425c"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}