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.