The product of two high-frequency Graph Laplacian eigenfunctions is smooth

Abstract

In the continuous setting, we expect the product of two oscillating functions to oscillate even more (generically). On a graph G=(V,E), there are only |V| eigenvectors of the Laplacian L=D−A, so one oscillates ‘the most’. The purpose of this short note is to point out an interesting phenomenon: if ϕ1,ϕ2 are delocalized eigenvectors of L corresponding to large eigenvalues, then their (pointwise) product ϕ1⋅ϕ2 is smooth (in the sense of small Dirichlet energy): highly oscillatory functions have largely matching oscillation patterns.