Approximating Pointwise Products of Laplacian Eigenfunctions

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.