Resistance-form discretization conjecture for eigenmaps on the real line

Let $g$ be a smooth density on $\mathbb{R}$ and let $\mathcal{L}_{(g)}$ be the Laplacian associated with the Dirichlet form

$$\mathcal{E}_g(f)=\int_{\mathbb{R}}(f'(x))^2g(x)\,dx.$$

Resistance-form discretization conjecture. Numerically accurate locally uniform approximations to the Laplacian eigenmap can be obtained by discretizing the left- and right-hand sides in the weak definition of the Dirichlet-form Laplacian using the theory of resistance forms. This proposes an alternative to direct random averaging based on the associated weak formulation; analogous half-line and interval versions require boundary conditions.