On the Dual Geometry of Laplacian Eigenfunctions

We discuss the geometry of Laplacian eigenfunctions $-\Delta \phi = \lambda \phi$ on compact manifolds $(M,g)$ and combinatorial graphs $G=(V,E)$. The 'dual' geometry of Laplacian eigenfunctions is well understood on $\mathbb{T}^d$ (identified with $\mathbb{Z}^d$) and $\mathbb{R}^n$ (which is self-dual). The dual geometry is of tremendous role in various fields of pure and applied mathematics. The purpose of our paper is to point out a notion of similarity between eigenfunctions that allows to reconstruct that geometry. Our measure of 'similarity' $\alpha(\phi_{\lambda}, \phi_{\mu})$ between eigenfunctions $\phi_{\lambda}$ and $\phi_{\mu}$ is given by a global average of local correlations $$\alpha(\phi_{\lambda}, \phi_{\mu})^2 = \| \phi_{\lambda} \phi_{\mu} \|_{L^2}^{-2}\int_{M}{ \left( \int_{M}{ p(t,x,y)( \phi_{\lambda}(y) - \phi_{\lambda}(x))( \phi_{\mu}(y) - \phi_{\mu}(x)) dy} \right)^2 dx},$$ where $p(t,x,y)$ is the classical heat kernel and $e^{-t \lambda} + e^{-t \mu} = 1$. This notion recovers all classical notions of duality but is equally applicable to other (rough) geometries and graphs; many numerical examples in different continuous and discrete settings illustrate the result.