Weyl's law for singular Riemannian manifolds

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. Our main motivation comes from the construction of singular Riemannian metrics with prescribed non-classical Weyl's law. Namely, for any non-decreasing slowly varying function $\upsilon$ we construct a singular Riemannian structure whose spectrum is discrete and satisfies $$N(\lambda) \sim \frac{\omega_n}{(2\pi)^n} \lambda^{n/2} \upsilon(\lambda).$$ Examples of slowly varying functions are $\log\lambda$, its iterations $\log_k \lambda = \log_{k-1}\log\lambda$, any rational function with positive coefficients of $\log_k \lambda$, and functions with non-lo\-ga\-rithmic growth such as $\exp\left((\log \lambda)^{\alpha_1} \dots (\log_k \lambda)^{\alpha_k} \right)$ for $\alpha_i \in (0,1)$. A key tool in our arguments is a new quantitative estimate for the remainder of the heat trace and the Weyl's function on Riemannian manifolds, which is of independent interest.