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Let $D$ be a first-order self-adjoint elliptic operator on a closed Riemannian manifold $M$. Then $D$ has discrete spectrum in $\mathbb{R}$, and there is an orthonormal basis for $L^2(M)$ consisting of eigenfunctions of $D$.

Question 1: Do the eigenvalues $\lambda_i$ of $D$ satisfy an asymptotic estimate of the form $$|\lambda_i|\sim i^\alpha$$ for some $\alpha>0$?

Now suppose $M$ is a compact manifold with boundary, and consider the Dirichlet eigenfunctions and eigenvalues of $D$, i.e. smooth $u$ and $\lambda\in\mathbb{R}$ such that $$Du=\lambda u,$$ $$u|_{\partial M}=0.$$

Question 2: Do we still have a decomposition of $L^2(M)$ into eigenspaces of $D$? Do the Dirichlet eigenvalues $\lambda_i$ of $D$ satisfy an asymptotic estimate of the form $$|\lambda_i|\sim i^\alpha$$ for some $\alpha>0$?

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    $\begingroup$ I'm not sure whether I understand the second question correctly. Dirichlet boundary conditions on all of $\partial M$ for a first-order differential operator result in an overdetermined system (Consider for instance $i$ times the first derivative on $[0,1]$. The only function $u$ that satisfies Dirichlet boundary conditions both in $0$ and $1$ as well as the eigenvalue equation $iu' = \lambda u$ for any $\lambda$ is $u=0$.) $\endgroup$
    – Jochen Glueck
    Commented Aug 11, 2021 at 13:57
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    $\begingroup$ (By the way, I think this "overdetermined" property also implies that the resolvent set of the operator is empty, so it cannot be self-adjoint.) $\endgroup$
    – Jochen Glueck
    Commented Aug 11, 2021 at 14:00
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    $\begingroup$ This result is called Weyl's law when $D$ is the Laplace-Beltrami operator, and it takes the form $\lambda_i ~ \frac{(2\pi)^2}{(\omega_d vol(M))^{2/d}} i^{\frac{2}{d}}$. It applies in the boundaryless, Neumann and Dirichlet cases. I'm sure someone has generalised it to first-order self-adjoint elliptic operators (probably with a different constant), possibly under addition conditions. $\endgroup$
    – user7868
    Commented Aug 12, 2021 at 0:54
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    $\begingroup$ The answer to question 1 is "yes" when $D$ is a Dirac operator (see e.g. the book by Berline-Getzler-Vergne). For question 2, one can take suitable global boundary conditions instead (like Atiyah-Patodi-Singer or MIT-bag conditions). I am not so sure about more general selfadjoint elliptic operators. $\endgroup$
    – Sebastian Goette
    Commented Aug 14, 2021 at 20:37

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