I am looking at page 32 (beginning of Chapter 5) [here](https://web.math.princeton.edu/~chang/zur.pdf). We are given a formally self-adjoint, metrically defined differential operator $A$ on $(M^n,g)$ of order $2l$ with positive definite leading symbol, such that $A\_{\tilde{g}}=c^{-2l}A\_g$ whenever $\bar{g}=c^2g$ is a change of metric for some $c>0$ constant. We are told that the asymptotic behaviour of eigenvalues of $A$ satisfies Weyl's formula
\begin{equation}\tag{1}
\lambda\_j\sim C(g,A)\,j^\frac{2l}{n}~~~\mathrm{as}~j\rightarrow\infty,
\end{equation}
but I am struggling to find a reference for this result. It is also stated [here](http://www.ams.org/journals/proc/1991-113-03/S0002-9939-1991-1050018-8/S0002-9939-1991-1050018-8.pdf) without proof, but even in the suggested references I still can't find anything. 


Lots of the literature (for instance [this](https://projecteuclid.org/download/pdf_1/euclid.dmj/1077314178) paper of Gilkey) also only states (1) for (higher order) Laplace-type operators, i.e. operators for which the leading symbol is (a power of) the metric. Am I correct in saying the conditions above are more general? Thanks in advance.


EDIT: by the conditions above I mean in particular the scaling property $A\_{\tilde{g}}=c^{-2l}A\_g$. Does this imply, or is it implied by, $A$ having its leading symbol being a power of the metric? I guess this is a second question; answers to either would be welcomed.



Reference for Weyl's law for higher order operators on closed Riemannian manifolds
One possible reference is Seeley's paper on [Complex powers of Elliptic Operators](http://inspirehep.net/record/51406?ln=en), where Seeley did it for the Laplacian (page 6). But the discussion carries over to all elliptic $\Psi DO$s without much difficulty. For a "modern" expository article, see [this](http://www.uni-math.gwdg.de/iwitt/SpecGeo2014/ht2.pdf). Seeley's argument would imply a term of type $j^{2l/n}$, just as the article claimed. A standard textbook is [Shubin's book](http://www.springer.com/gp/book/9783540411956), where the result is proved in page 130. 


