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Let $M$ be a compact Riemannian manifold. Let $E$ be a vector bundle over $M$ equipped with a Hermitian (or Euclidean) metric on its fibers. Let $D$ be a linear elliptic differential operator acting on smooth sections of $E$. Assume that $D$ is symmetric, i.e. $$\int_M(\phi, D\psi)dvol=\int_M (D\phi,\psi)dvol.$$

Is it true that there exists an orthonormal eigen basis in the space of $L^2$-sections of $E$? If yes, is it true that each eigenvalue has finite multiplicity, and eigenvalues have no accumulation point on the real line?

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    $\begingroup$ I don't have a canonical reference, but by my recollection the underlying logic is that $D$ has a compact resolvent and the properties that you want follow from the spectral properties of (normal/self-adjoint) compact operators. The compactness follows from the representation of the resolvent as an integral kernel, possibly checking that it is Hilbert-Schmidt. $\endgroup$
    – Igor Khavkine
    Commented Mar 16, 2023 at 10:39

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As Igor pointed out, the argument is that the resolvent is compact. If your manifold is compact, then Sobolev space embeddings are compact (you can show this for instance by using pseudodifferential calculus) and hence the resolvent considered as an operator $L^2 \to L^2$ is compact. The spectral theorem for compact operators applied to the resolvent gives you what you want.

One good reference is the book by Shubin ('Pseudodifferential operators and spectral theory' or something like that).

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