发表机构
Auburn University（奥本大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究完备非紧黎曼流形上漂移拉普拉斯算子的特征函数增长阶与特征值过滤的相容性，在对称性条件下证明刚性结果并导出逆Hermite/Laguerre定理。
AI 中文摘要
设$(M^n,g)$为完备、连通、非紧的黎曼流形，$f\ge0$为恰当的$C^2$权函数。我们假设$f$、$|\nabla f|$和$\Delta f$满足多项式界（假设H1-H3），增长指数$\alpha>0$。我们研究$L^2(M,e^{-f}dV)$上的漂移拉普拉斯算子$L_f=\Delta-\langle\nabla f,\nabla\cdot\rangle$。我们假设其特征函数在$f^{1/\alpha}$的水平集上具有有限多项式增长阶$\gamma_k$，并假设$\gamma_k$按谱序趋于无穷（假设H4）。我们将此增长过滤与按特征值$\lambda_k$的过滤进行比较。对于$\alpha>1$，我们研究关系$\lambda_k\asymp\gamma_k^{(2\alpha-2)/\alpha}$，称为相容性。一个显式的旋转对称例子表明，即使当$\alpha=2$时，该关系对全谱也可能失效。我们证明了$\alpha>1$时的离散性和加权Agmon估计。一个矩估计以增长阶和有限尺度前因子为界约束了集中半径。一个下局部化条件给出了谱比较的一个方向。对于具有一维基底的精确扭曲积，我们施加径向增长条件和径向漂移的正则性假设。这些条件在径向扇区给出$\alpha=2$和$\lambda_k\asymp\gamma_k$。对于一般$(M,g)$，我们假设水平集具有传递等距对称性，而非扭曲积结构。不变扇区增长条件和径向漂移正则性随后给出$\alpha=2$和$\lambda_k\asymp\gamma_k\asymp k$。最后，不变特征函数的精确多项式性蕴含$\alpha=2$，无需漂移正则性假设。归一化后，约化方程即为Hermite或广义Laguerre方程。论证既不使用曲率界，也不使用孤子方程。
英文摘要
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.
