On the Spectral Resolution of Products of Laplacian Eigenfunctions

Abstract

We study products of eigenfunctions of the Laplacian $-Δϕ_λ = λϕ_λ$ on compact manifolds. If $ϕ_μ, ϕ_λ$ are two eigenfunctions and $μ\leq λ$, then one would perhaps expect their product $ϕ_μϕ_λ$ to be mostly a linear combination of eigenfunctions with eigenvalue close to $λ$. 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 $$ \sin{(n x)} \sin{(m y)} \qquad \mbox{lives at eigenvalue} \quad \max{\left\{m^2,n^2\right\}} \leq m^2 + n^2 \leq 2\max{\left\{m^2,n^2\right\}}$$ 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 $ϕ_μ$ and $ϕ_λ$ are strongly correlated at scale $ \sim λ^{-1/2}$ (the shorter wavelength) $$ \left\| \int_{M}{ p(t,x,y)( ϕ_λ(y) - ϕ_λ(x))( ϕ_μ(y) - ϕ_μ(x)) dy} \right\|_{L^2_x} \gtrsim \| ϕ_μϕ_λ\|_{L^2},$$ where $p(t,x,y)$ is the classical heat kernel and $t \sim λ^{-1}$. This turns out to be a fairly fundamental principle and is even valid for the Hadamard product of eigenvectors of a Graph Laplacian.