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 $\phi_1, \phi_2$ are delocalized eigenvectors of $L$ corresponding to large eigenvalues, then their (pointwise) product $\phi_1 \cdot \phi_2$ is smooth (in the sense of small Dirichlet energy): highly oscillatory functions have largely matching oscillation patterns.