On Pointwise Products of Elliptic Eigenfunctions

We consider eigenfunctions of Schr\"odinger operators on a $d-$dimensional bounded domain $\Omega$ (or a $d-$dimensional compact manifold $\Omega$) with Dirichlet conditions. These operators give rise to a sequence of eigenfunctions $(\phi_n)_{n \in \mathbb{N}}$. We study the subspace of all pointwise products $$ A_n = \mbox{span} \left\{ \phi_i(x) \phi_j(x): 1 \leq i,j \leq n\right\} \subseteq L^2(\Omega).$$ Clearly, that vector space has dimension $\mbox{dim}(A_n) = n(n+1)/2$. We prove that products $\phi_i \phi_j$ 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 $\Pi_{B_n}:L^2(\Omega) \rightarrow B_n$, we have $$ \forall~1 \leq i,j \leq n~ \qquad \|\phi_i\phi_j - \Pi_{B_n}( \phi_i \phi_j) \|_{L^2} \leq \varepsilon$$ and the size of the space $\mbox{dim}(B_n)$ 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.