Asymptotic behaviour of Dirichlet eigenvalues for homogeneous H\"{o}rmander operators and algebraic geometry approach
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 a smooth open bounded domain in $\mathbb{R}^n$ containing the origin. Combining the subelliptic heat kernel estimates, the resolution of singularities in algebraic geometry and some refined analysis involving convex geometry, we establish the explicit asymptotic behaviour $\lambda_k\approx k^{\frac{2}{Q_{0}}}(\ln k)^{-\frac{2d_{0}}{Q_{0}}}$ 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 also give the optimal bounds of index $Q_{0}$, which depends on the homogeneous dimension associated with vector fields $X_{1}, X_{2},\ldots, X_{m}$.