Polygons and multi-product of eigenfunctions

TL;DR

This paper investigates the inner products of eigenfunctions on compact Riemannian manifolds, revealing that their main concentration relates to configurations of polygons with side lengths determined by eigenvalues, after averaging.

Contribution

It introduces a novel geometric interpretation of eigenfunction inner products through polygon configurations, advancing understanding of eigenfunction interactions on manifolds.

Findings

Main concentration linked to polygon configurations with side lengths from eigenvalues

Rapid decay of mass in non-polygonal frequency regimes

Provides geometric criteria for eigenfunction product behavior

Abstract

Let $M$ be a compact Riemannian manifold without boundary, with $L^2$-normalized Laplace-Beltrami eigenfunctions $\{e_j\}_j$, which satisfy $\Delta_g e_j = -\lambda_j^2 e_j$. We study the following inner product of eigenfunctions \[ \langle e_{i_1} e_{i_2} \ldots e_{i_k}, e_{i_{k+1}} \rangle = \int e_{i_1} e_{i_2}\ldots e_{i_k} \overline{e_{i_{k+1}}} \, dV. \] We show that, after a mild averaging in the frequency variables, the main $\ell^2$-concentration of this inner product is determined by the measure of a set of configurations of $(k+1)$-gons whose side lengths are the frequencies $\lambda_{i_1}, \lambda_{i_2}, \dots, \lambda_{i_{k+1}}$. We prove that a rapidly vanishing proportion of this mass lies in the regime where $\lambda_{i_1}, \lambda_{i_2}, \dots, \lambda_{i_{k+1}}$ cannot occur as the side lengths of any $(k+1)$-gon.