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.