MiS Preprint Repository

Let $X=(X_1, X_2, \cdots, X_m)$ be a system of real smooth vector fields defined in an open domain $\tilde{\Omega}\subset \mathbb{R}^n$, $\Omega \subset\subset\tilde{\Omega}$ be a bounded open subset in $\mathbb{R}^n$ with smooth boundary $\partial\Omega$, $\triangle_{X}=\sum_{j=1}^{m}X_j^2$. In this paper, if $\lambda_j$ is the $j^{th}$ Dirichlet eigenvalue for the degenerate elliptic operator $-\triangle_{X}$ (or the degenerate Schrödinger operator $-\triangle_{X}+V$) on $\Omega$, we deduce respectively that the lower estimates for the sums $\sum_{j=1}^{k}\lambda_j$ in both cases for the operator $-\triangle_{X}$ to be finitely degenerate (i.e. the Hörmander condition is satisfied) or infinitely degenerate (i.e. the Hörmander condition is not satisfied).