The critical case for the concentration of eigenfunctions on singular Riemannian manifolds

We consider a compact Riemannian manifold with boundary with a certain class of critical singular Riemannian metrics that are singular at the boundary. The corresponding Laplace-Beltrami operator can be seen as a Grushin-type operator plus a potential. We show in the critical case that the average density of eigenfunctions for the Laplace-Beltrami operator with eigenvalues below $\lambda>0$ is distributed over all length scales between $\lambda^{-1/2}$ and $1$ near the boundary. We give a precise description of this distribution as $\lambda\to\infty$.