On the spectral resolution of products of Laplacian eigenfunctions

Abstract

We study products of eigenfunctions of the Laplacian $-\Delta {\varphi }_{\lambda }=\lambda {\varphi }_{\lambda }$ on compact manifolds. If ${\varphi }_{\mu },{\varphi }_{\lambda }$ are two eigenfunctions and $\mu \le \lambda$, then one would perhaps expect their product ${\varphi }_{\mu }{\varphi }_{\lambda }$ to be mostly a linear combination of eigenfunctions with eigenvalue close to $\lambda$. This can faily quite dramatically: on ${\mathbb{T}}^2$, we see that

has half of its $L^2-$mass at eigenvalue 1. Conversely, the product

lives at eigenvalue

and the heuristic is valid. We show that the main reason is that in the first example 'the waves point in the same direction': if the heuristic fails and multiplication carries $L^2-$mass to lower frequencies, then ${\varphi }_{\mu }$ and ${\varphi }_{\lambda }$ are strongly correlated at scale $\sim {\lambda }^{-1/2}$ (the shorter wavelength)

where $p(t,x,y)$ is the classical heat kernel and $t\sim {\lambda }^{-1}$. This turns out to be a fairly fundamental principle and is even valid for the Hadamard product of eigenvectors of a Graph Laplacian.