Two Growth Filtrations for Drift Laplacians: Compatibility, Rigidity, and an Inverse Hermite/Laguerre Theorem
Abstract
Let (Mn,g) be a complete, connected, noncompact Riemannian manifold, and let f≥0 be a proper C2 weight. We assume polynomial bounds on f, |∇ f|, and Δf (Assumptions H1-H3), with growth exponent α>0. We study the drift Laplacian Lf=Δ−⟨∇ f,∇·⟩ on L2(M,e−fdV). We assume that its eigenfunctions have finite polynomial growth orders γk, measured on the level sets of f1/α. We also assume that γk→∞ in spectral order (Assumption H4). We compare this growth filtration with the filtration by eigenvalue λk. For α>1, we study the relation λkasympγk(2α−2)/α, called compatibility. An explicit rotationally symmetric example shows that this relation can fail for the full spectrum, even when α=2. We prove discreteness and weighted Agmon estimates for α>1. A moment estimate bounds the concentration radius in terms of the growth order and a finite-scale prefactor. A lower localization condition gives one direction of the spectral comparison. For exact warped products with a one-dimensional base, we impose the radial growth condition and a regularity assumption on the radial drift. These give α=2 and λkasympγk in the radial sector. For general (M,g), we assume a transitive isometric symmetry of the level sets instead of a warped-product structure. The invariant-sector growth condition and radial-drift regularity then give α=2 and λkasympγkasymp k. Finally, exact polynomiality of the invariant eigenfunctions implies α=2 without the drift regularity assumptions. The reduced equation is then Hermite or generalized Laguerre after normalization. The arguments use neither curvature bounds nor soliton equations.
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BibTeX
@misc{matinpour2026growth,
title = {{Two Growth Filtrations for Drift Laplacians: Compatibility, Rigidity, and an Inverse Hermite/Laguerre Theorem}},
author = {Elham Matinpour},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25289},
eprint = {2609.25289},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25289},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Two Growth Filtrations for Drift Laplacians: Compatibility, Rigidity, and an Inverse Hermite/Laguerre Theorem
AU - Elham Matinpour
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.25289
UR - https://arxiv.org/abs/2609.25289
AB - Let (M^n,g) be a complete, connected, noncompact Riemannian manifold, and let f≥0 be a proper C^2 weight. We assume polynomial bounds on f, |∇ f|, and Δf (Assumptions H1-H3), with growth exponent α>0. We study the drift Laplacian L_f=Δ−⟨∇ f,∇·⟩ on L^2(M,e^−fdV). We assume that its eigenfunctions have finite polynomial growth orders γ_k, measured on the level sets of f^1/α. We also assume that γ_k→∞ in spectral order (Assumption H4). We compare this growth filtration with the filtration by eigenvalue λ_k. For α>1, we study the relation λ_kasympγ_k^(2α−2)/α, called compatibility. An explicit rotationally symmetric example shows that this relation can fail for the full spectrum, even when α=2. We prove discreteness and weighted Agmon estimates for α>1. A moment estimate bounds the concentration radius in terms of the growth order and a finite-scale prefactor. A lower localization condition gives one direction of the spectral comparison. For exact warped products with a one-dimensional base, we impose the radial growth condition and a regularity assumption on the radial drift. These give α=2 and λ_kasympγ_k in the radial sector. For general (M,g), we assume a transitive isometric symmetry of the level sets instead of a warped-product structure. The invariant-sector growth condition and radial-drift regularity then give α=2 and λ_kasympγ_kasymp k. Finally, exact polynomiality of the invariant eigenfunctions implies α=2 without the drift regularity assumptions. The reduced equation is then Hermite or generalized Laguerre after normalization. The arguments use neither curvature bounds nor soliton equations.
ER -