Spectral Asymptotics for Operators of Hormander Type

Abstract

An asymptotic equality of the form ${\mathrm{Tr}}_{L^2}{e}^{-t(L+V)}=C{t}^{-\alpha }+o(t^{-\alpha })$ as $t\to 0$ is given for the trace of the heat semigroup generated by operators on compact manifolds of the form $L+V=-{\sum }_{i=1}^m{X}_i^2+{\sum }_{i,j=1}^m{c}_{ij}[X_i,X_j]+{\sum }_{i=1}^m{\gamma }_i{X}_i+V$ for smooth real potentials $(V)$ which satisfy Hörmander'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.

arXiv comment16 pages, submitted