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 $\tilde{\mathrm{\Omega }}\subset {\mathbb{R}}^n$, $\mathrm{\Omega }\subset \subset \tilde{\mathrm{\Omega }}$ be a bounded open subset in ${\mathbb{R}}^n$ with smooth boundary $\partial \mathrm{\Omega }$, ${\mathrm{△}}_X=\sum _{j=1}^m{X}_j^2$. In this paper, if ${\lambda }_j$ is the $j^{th}$ Dirichlet eigenvalue for the degenerate elliptic operator $-{\mathrm{△}}_X$ (or the degenerate Schrödinger operator $-{\mathrm{△}}_X+V$) on $\mathrm{\Omega }$, we deduce respectively that the lower estimates for the sums $\sum _{j=1}^k{\lambda }_j$ in both cases for the operator $-{\mathrm{△}}_X$ to be finitely degenerate (i.e. the Hörmander condition is satisfied) or infinitely degenerate (i.e. the Hörmander condition is not satisfied).