On Pointwise Products of Elliptic Eigenfunctions

Abstract

We consider eigenfunctions of Schrödinger 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 $({\varphi }_n{)}_{n\in \mathbb{N}}$. We study the subspace of all pointwise products $$ A_n = \mbox{span} \left\{ ϕ_i(x) ϕ_j(x): 1 \leq i,j \leq n\right\} \subseteq L^2(Ω).$$ Clearly, that vector space has dimension $\mbox{dim}(A_n) = n(n+1)/2$. 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 ${\Pi }_{B_n}:L^2(\Omega )\to B_n$, we have $$\forall 1\le i,j\le n\Vert {\varphi }_i{\varphi }_j-{\Pi }_{B_n}({\varphi }_i{\varphi }_j){\Vert }_{L^2}\le \epsilon$$ and the size of the space $\mbox{dim}(B_n)$ is relatively small $$ \mbox{dim}(B_n) \lesssim \left( \frac{1}{\varepsilon} \max_{1 \leq i \leq n} \|ϕ_i\|_{L^{\infty}} \right)^d n.$$ 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.

The result is superseded by a more recent preprint joint with Christopher D. Sogge, arXiv:1811.10447