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 $$2\sin (nx)\sin ((n+1)x)=\cos (x)-\cos ((2n+1)x)$$ 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 ${\varphi }_{\mu }$ and ${\varphi }_{\lambda }$ are strongly correlated at scale $\sim {\lambda }^{-1/2}$ (the shorter wavelength) $${\Vert {\int }_{M}p(t,x,y)({\varphi }_{\lambda }(y)-{\varphi }_{\lambda }(x))({\varphi }_{\mu }(y)-{\varphi }_{\mu }(x))dy\Vert }_{L_x^2}\gtrsim \Vert {\varphi }_{\mu }{\varphi }_{\lambda }{\Vert }_{L^2},$$ 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.