On Pointwise Products of Elliptic Eigenfunctions

We consider eigenfunctions of Schr"odinger operators on a $d-$dimensional bounded domain $\mathrm{\Omega }$ (or a $d-$dimensional compact manifold $\mathrm{\Omega }$) with Dirichlet conditions. These operators give rise to a sequence of eigenfunctions $({\varphi }_n{)}_{n\in \mathbb{N}}$. We study the subspace of all pointwise products $$A_n=span\{{\varphi }_i(x){\varphi }_j(x):1\le i,j\le n\}\subseteq L^2(\mathrm{\Omega })\mathrm{.}$$ Clearly, that vector space has dimension $dim(A_n)=n(n+1)\mathrm{/}2$. We prove that products ${\varphi }_i{\varphi }_j$ of eigenfunctions are simple in a certain sense: for any $\epsilon >0$, there exists a low-dimensional vector space $B_n$ that almost contains all products. More precisely, denoting the orthogonal projection $\Pi_{B_n}:L^2(\Omega) \rightarrow B_n$,wehave$$ \forall~1 \leq i,j \leq n~ \qquad |\phi_i\phi_j - \Pi_{B_n}( \phi_i \phi_j) |{L^2} \leq \varepsilonand the size of the space $dim(B_n)$ is relatively small \mbox{dim}(B_n) \lesssim \left( \frac{1}{\varepsilon} \max{1 \leq i \leq n} |\phi_i|_{L^{\infty}} \right)^d n.$$ In the generic delocalized setting, this bound grows linearly up to logarithmic factors: pointwise products of eigenfunctions are low-rank. This has implications, among other things, for the validity of fast algorithms in electronic structure computations.