On the Dual Geometry of Laplacian Eigenfunctions

Abstract

We discuss the geometry of Laplacian eigenfunctions $-\hat{I}”\ddot{I}•=\hat{I}»\ddot{I}•$ 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' $\hat{I}\pm (\ddot{I}{•}_{\hat{I}}»,\ddot{I}{•}_{\hat{I}}¼)$ between eigenfunctions $\ddot{I}{•}_{\hat{I}}»$ and $\ddot{I}{•}_{\hat{I}}¼$ is given by a global average of local correlations $$\hat{I}\pm (\ddot{I}{•}_{\hat{I}}»,\ddot{I}{•}_{\hat{I}}¼{)}^2=\Vert \ddot{I}{•}_{\hat{I}}»\ddot{I}{•}_{\hat{I}}¼{\Vert }_{L^2}^{-2}{\int }_M{\left({\int }_{M}p(t,x,y)(\ddot{I}{•}_{\hat{I}}»(y)-\ddot{I}{•}_{\hat{I}}»(x))(\ddot{I}{•}_{\hat{I}}¼(y)-\ddot{I}{•}_{\hat{I}}¼(x))dy\right)}^{2}dx,$$ where $p(t,x,y)$ is the classical heat kernel and $e^{-t\hat{I}»}+e^{-t\hat{I}¼}=1$. 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.