# Estimates of eigenvalues of elliptic operators on compact manifolds

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).

• For self-adjoint and elliptic operator acting on functions, try to look at section 15 of Shubin's book "Pseudo differential Operators and Spectral Theory " – shu Mar 12 '15 at 10:17
• For second order, self adjoint elliptic operator acting on sections of vector bundles, try to look at Cor 2.43 of Berline-Getzler-Vergne's book "Heat Kernels and Dirac Operators". – shu Mar 12 '15 at 10:22
• The passage from scalar to vector bundle Laplacians is not difficult. The pseudodifferentia case has a few peculiarities. Hormander's results are the sharpest. They are discussed in Theorems 2.1, 2.3 of paper arxiv.org/pdf/1406.0934v1.pdf (They are proved in the appendix of the paper.) The references in the paper are also useful. – Liviu Nicolaescu Mar 12 '15 at 11:20