On the spectral resolution of products of Laplacian eigenfunctions

Abstract

We study products of eigenfunctions of the Laplacian $-\Delta \phi\_{\lambda} = \lambda \phi\_{\lambda}$ on compact manifolds. If $\phi\_{\mu}, \phi\_{\lambda}$ are two eigenfunctions and $\mu \leq \lambda$, then one would perhaps expect their product $\phi\_{\mu}\phi\_{\lambda}$ 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 $$ 2\sin{(n x)} \sin{((n+1) x)} = \cos{(x)} - \cos{( (2n+1) x)} $$ has half of its $L^2-$mass at eigenvalue 1. Conversely, the product $$ \sin{(n x)} \sin{(m y)} $$ lives at eigenvalue $$ \max{\left{m^2,n^2\right}} \leq m^2 + n^2 \leq 2\max{\left{m^2,n^2\right}} $$ 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 $\phi\_{\mu}$ and $\phi\_{\lambda}$ are strongly correlated at scale $\sim \lambda^{-1/2}$ (the shorter wavelength) $$ \left| \int\_{M}{ p(t,x,y)( \phi\_{\lambda}(y) - \phi\_{\lambda}(x))( \phi\_{\mu}(y) - \phi\_{\mu}(x)) dy} \right|{L^2\_x} \gtrsim | \phi{\mu}\phi\_{\lambda}|\_{L^2}, $$ 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.