On the Spectral Resolution of Products of Laplacian Eigenfunctions

We study products of eigenfunctions of the Laplacian $-\Delta \phi_{\lambda} = \lambda \phi_{\lambda}$oncompactmanifolds\mathrm{.}If$\phi_{\mu}, \phi_{\lambda}$aretwoeigenfunctionsand$\mu \leq \lambda$, 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 $$\sin (nx)\sin (my)livesateigenvalue\max \{m^2,n^2\}\le m^2+n^2\le 2\max \{m^2,n^2\}$$ 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 $\sim {\lambda }^{-1\mathrm{/}2}$ (the shorter wavelength) $${\parallel {\int }_{M}p(t,x,y)({\varphi }_{\lambda }(y)-{\varphi }_{\lambda }(x))({\varphi }_{\mu }(y)-{\varphi }_{\mu }(x))dy\parallel }_{L_x^2}\gtrsim \mathrm{\parallel }{\varphi }_{\mu }{\varphi }_{\lambda }{\mathrm{\parallel }}_{L^2},$$ 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.