On the spectral resolution of products of Laplacian eigenfunctions

  • Stefan Steinerberger

    Yale University, New Haven, USA
On the spectral resolution of products of Laplacian eigenfunctions cover

A subscription is required to access this article.

Abstract

We study products of eigenfunctions of the Laplacian on compact manifolds. If are two eigenfunctions and , then one would perhaps expect their product to be mostly a linear combination of eigenfunctions with eigenvalue close to . This can faily quite dramatically: on , we see that

has half of its mass at eigenvalue 1. Conversely, the product

lives at eigenvalue

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 mass to lower frequencies, then and are strongly correlated at scale (the shorter wavelength)

where is the classical heat kernel and . This turns out to be a fairly fundamental principle and is even valid for the Hadamard product of eigenvectors of a Graph Laplacian.

Cite this article

Stefan Steinerberger, On the spectral resolution of products of Laplacian eigenfunctions. J. Spectr. Theory 9 (2019), no. 4, pp. 1367–1384

DOI 10.4171/JST/279