Weyl's law for singular Riemannian manifolds


In this paper, we study the asymptotic growth of the eigenvalues of the Laplace-Beltrami operator on singular Riemannian manifolds, where all geometrical invariants appearing in classical spectral asymptotics are unbounded, and the total volume can be infinite. Under suitable assumptions on the curvature blow-up, we show how the singularity influences the Weyl’s asymptotics and the localization of the eigenfunctions for large frequencies.
As a consequence of our results, we identify a class of singular structures such that the corresponding Laplace-Beltrami operator has the following non-classical Weyl’s law: $$N(\lambda) \sim \frac{\omega_n}{(2\pi)^n} \lambda^{n/2} v(\lambda),$$ where $v$ is slowly varying at infinity in the sense of Karamata. Finally, for any non-decreasing slowly varying function $v$, we construct singular Riemannian structures admitting the above Weyl’s law.
A key tool in our arguments is a universal estimate for the remainder of the heat trace on Riemannian manifolds, which is of independent interest.