The classical Weyl law says that if $\Delta$ is the Laplace operator on functions on a compact Riemannian manifold $(M^n,g)$, $n>2$, then its $k$th eigenvalue satisfies the asymptotic formula $$\lambda_k\sim C k^{2/n}, k\to \infty, $$ where $C$ is a constant which can be written explicitly (see e.g. p. 3 here http://www2.unine.ch/repository/default/content/sites/math/files/shared/documents/articles/2010/Paper_Dido_Conference.pdf).

**My question is what is known about the growth of eigenvalues of more general elliptic (pseudo) differential operators on compact manifolds acting on functions or, more generally, sections of vector bundles (e.g. on differential forms).**