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?**