Approximating Pointwise Products of Laplacian Eigenfunctions

TL;DR

This paper demonstrates that products of Laplacian eigenfunctions on bounded domains can be approximated by low-dimensional subspaces, enabling efficient computations in applications like electronic structure calculations.

Contribution

The authors prove that all pointwise products of Laplacian eigenfunctions can be approximated within any desired accuracy by low-dimensional spaces, with bounds on the dimension depending polynomially on the eigenfunction index.

Findings

Pointwise products of eigenfunctions are low-rank.

Approximation spaces have dimension roughly n^{1+δ} for accuracy ε.

Results apply to products of arbitrary length and in H^{-1} norm.

Abstract

We consider Laplacian eigenfunctions on a $d-$dimensional bounded domain $M$ (or a $d-$dimensional compact manifold $M$) with Dirichlet conditions. These operators give rise to a sequence of eigenfunctions $(e_\ell)_{\ell \in \mathbb{N}}$. We study the subspace of all pointwise products $$ A_n = \mbox{span} \left\{ e_i(x) e_j(x): 1 \leq i,j \leq n\right\} \subseteq L^2(M).$$ Clearly, that vector space has dimension $\mbox{dim}(A_n) = n(n+1)/2$. We prove that products $e_i e_j$ of eigenfunctions are simple in a certain sense: for any $\varepsilon > 0$, there exists a low-dimensional vector space $B_n$ that almost contains all products. More precisely, denoting the orthogonal projection $\Pi_{B_n}:L^2(M) \rightarrow B_n$, we have $$ \forall~1 \leq i,j \leq n~ \qquad \|e_ie_j - \Pi_{B_n}( e_i e_j) \|_{L^2} \leq \varepsilon$$ and the size of the space $\mbox{dim}(B_n)$ is relatively small: for every $\delta > 0$, $$ \mbox{dim}(B_n) \lesssim_{M,\delta} \varepsilon^{-\delta} n^{1+\delta}.$$ We obtain the same sort of bounds for products of arbitrary length, as well for approximation in $H^{-1}$ norm. Pointwise products of eigenfunctions are low-rank. This has implications, among other things, for the validity of fast algorithms in electronic structure computations.