Higher order Weyl coefficients for the operator curl

We establish refined spectral asymptotics for the operator curl acting on a connected oriented closed Riemannian 3-manifold. We treat positive and negative eigenvalues separately and obtain explicit formulae for the first six global Weyl coefficients. With local Weyl coefficients we compute the first four coefficients as well as the sixth one, and we determine the fifth up to a universal constant. As a consequence, we prove that the eta function of curl (both in its local and global versions) is holomorphic in the complex half-plane $\mathrm{Re}s>-2$. Finally, under appropriate assumptions on the geodesic flow, we improve Bär’s asymptotic formulae for the positive and negative counting functions, refining the remainder to $o({\lambda }^2)$.