Pith Number

pith:57XQJHPL

pith:2025:57XQJHPLBZECQITIOEU3DJGEOB

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On Rellich-type asymptotics for eigenfunctions on rank one symmetric spaces of noncompact type

Pritam Ganguly

Eigenfunctions of the Laplace-Beltrami operator on exterior domains in rank-one symmetric spaces satisfy sharp quantitative L^p growth estimates in geodesic annuli.

arxiv:2511.12561 v2 · 2025-11-16 · math.AP · math.CA

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Claims

C1strongest 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.

C2weakest 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.

C3one 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.

References

39 extracted · 39 resolved · 0 Pith anchors

[1] Agmon, Lower bounds for solutions of Schr¨ odinger equations,J 1970

[2] 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 2023

[3] A. Banerjee, N. Garofalo, An observation on eigenfunctions of the Laplacian,La Matematica3 (2024), no. 4, 1451–1455 2024

[4] 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

[5] A. Banerjee, N. Garofalo, Absence ofL p spectrum for asymptotically flat diffusions in region with cavities, arXiv:2507.10728 (2025) 2025

Formal links

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First computed	2026-05-18T03:09:33.223630Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

efef049deb0e482822687129b1a4c4705aa452ac37c2466d27d6b7ab2e98fa5a

Aliases

arxiv: 2511.12561 · arxiv_version: 2511.12561v2 · doi: 10.48550/arxiv.2511.12561 · pith_short_12: 57XQJHPLBZEC · pith_short_16: 57XQJHPLBZECQITI · pith_short_8: 57XQJHPL

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/57XQJHPLBZECQITIOEU3DJGEOB \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: efef049deb0e482822687129b1a4c4705aa452ac37c2466d27d6b7ab2e98fa5a

Canonical record JSON

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      "math.CA"
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    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.AP",
    "submitted_at": "2025-11-16T11:31:16Z",
    "title_canon_sha256": "502e9db583977886fbfb93691c87090aa88e23736ae8a58a96e6c1f78705a41a"
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