Asymptotic behaviour of Dirichlet eigenvalues for homogeneous H\"{o}rmander operators and algebraic geometry approach

TL;DR

This paper analyzes the asymptotic behavior of Dirichlet eigenvalues for homogeneous Hörmander operators using algebraic geometry, heat kernel estimates, and convex geometry, revealing explicit growth rates as eigenvalue index increases.

Contribution

It introduces a novel approach combining algebraic geometry and refined analysis to determine the asymptotic eigenvalue distribution for homogeneous Hörmander operators.

Findings

Eigenvalues grow like k^{2/Q_0} (ln k)^{-2d_0/Q_0} as k→∞

Established explicit asymptotic formulas for Dirichlet eigenvalues

Provided optimal bounds for the index Q_0 based on homogeneous dimension

Abstract

We study the Dirichlet eigenvalue problem of homogeneous H\"{o}rmander operators $\triangle_{X}=\sum_{j=1}^{m}X_{j}^{2}$ on a bounded open domain containing the origin, where $X_1, X_2, \ldots, X_m$ are linearly independent smooth vector fields in $\mathbb{R}^n$ satisfying H\"{o}rmander's condition and a suitable homogeneity property with respect to a family of non-isotropic dilations. Suppose that $\Omega$ is an open bounded domain in $\mathbb{R}^n$ containing the origin. We use the Dirichlet form to study heat semigroups and subelliptic heat kernels. Then, by utilizing subelliptic heat kernel estimates, the resolution of singularities in algebraic geometry, and employing some refined analysis involving convex geometry, we establish the explicit asymptotic behavior $\lambda_k \approx k^{\frac{2}{Q_0}}(\ln k)^{-\frac{2d_0}{Q_0}}$ as $k \to +\infty$, where $\lambda_k$ denotes the $k$-th Dirichlet eigenvalue of $\triangle_{X}$ on $\Omega$, $Q_0$ is a positive rational number, and $d_0$ is a non-negative integer. Furthermore, we provide optimal bounds of index $Q_0$, which depend on the homogeneous dimension associated with the vector fields $X_1, X_2, \ldots, X_m$.