On the Spectral Resolution of Products of Laplacian Eigenfunctions

TL;DR

This paper investigates how products of Laplacian eigenfunctions on compact manifolds distribute across spectral frequencies, revealing that their spectral resolution depends on the correlation of the eigenfunctions at small scales.

Contribution

It introduces a fundamental principle linking eigenfunction correlation at small scales to the spectral resolution of their products, applicable to both manifolds and graph Laplacians.

Findings

Eigenfunction products can concentrate at unexpected frequencies.

Correlation at the wavelength scale influences spectral distribution.

The principle applies to both continuous manifolds and discrete graphs.

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 $$ \sin{(n x)} \sin{(m y)} \qquad \mbox{lives at eigenvalue} \quad \max{\left\{m^2,n^2\right\}} \leq m^2 + n^2 \leq 2\max{\left\{m^2,n^2\right\}}$$ 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 $\phi_{\mu}$ 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.