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                <title xml:lang="en">The critical case for the concentration of eigenfunctions on singular Riemannian manifolds</title>
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                <term xml:lang="en">Grushin</term>
                <term xml:lang="en">sub-Riemannian geometry</term>
                <term xml:lang="en">Weyl's law</term>
                <term xml:lang="en">singular Riemannian metric</term>
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              <p>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 $\lambda&amp;gt;0$ is distributed over all length scales between $\lambda^{-1/2}$ and $1$ near the boundary. We give a precise description of this distribution as $\lambda\to\infty$.</p>
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