Spectral Asymptotics for Operators of Hormander Type

An asymptotic equality of the form $\operatorname{Tr}_{L^2} e^{-t(L+V)}=Ct^{-\alpha}+o(t^{-\alpha})$ as $t\rightarrow 0$ is given for the trace of the heat semigroup generated by operators on compact manifolds of the form $L+V=-\sum_{i=1}^{m}X_i^2 +\sum_{i,j=1}^mc_{ij}[X_i,X_j]+\sum_{i=1}^m \gamma_iX_i+V$ for smooth real potentials $(V)$ which satisfy H\"{o}rmander's bracket-generating condition. In the self-adjoint case, a Weyl law is proved for the spectra of such operators. Analogous results are proved for the Dirichlet boundary value problem.