Two Growth Filtrations for Drift Laplacians: Compatibility, Rigidity, and an Inverse Hermite/Laguerre Theorem
Let $(M^n,g)$ be a complete, connected, noncompact Riemannian manifold, and let $f\ge0$ be a proper $C^2$ weight. We assume polynomial bounds on $f$, $|\nabla f|$, and $Δf$ (Assumptions H1-H3), with growth exponent $α>0$. We study the drift Laplacian $L_f=Δ-\langle\nabla f,\nabla\cdot\rangle$ on $L^2(M,e^{-f}dV)$. 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 $γ_k\to\infty$ in spectral order (Assumption H4). We compare this growth filtration with the filtration by eigenvalue $λ_k$. For $α>1$, we study the relation $λ_k\asympγ_k^{(2α-2)/α}$, 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 $λ_k\asympγ_k$ 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 $λ_k\asympγ_k\asymp k$. 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.