# Approximating pointwise products of Laplacian eigenfunctions

Journal Article (Journal Article)

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 ) . We study the subspace of all pointwise products A =span{e (x)e (x):1≤i,j≤n}⊆L (M). Clearly, that vector space has dimension dim(A )=n(n+1)/2. We prove that products e e of eigenfunctions are simple in a certain sense: for any ε>0, there exists a low-dimensional vector space B that almost contains all products. More precisely, denoting the orthogonal projection Π :L (M)→B , we have ∀1≤i,j≤n‖e e −Π (e e )‖ ≤ε and the size of the space dim(B ) is relatively small: for every δ>0, dim(B )≲ ε n . We obtain the same sort of bounds for products of arbitrary length, as well for approximation in H norm. Pointwise products of eigenfunctions are low-rank. This has implications, among other things, for the validity of fast algorithms in electronic structure computations. ℓ ℓ∈N n i j n i j n B n n i j B n i j L n n M,δ 2 2 2 −δ 1+δ −1

### Full Text

### Duke Authors

### Cited Authors

- Lu, J; Sogge, CD; Steinerberger, S

### Published Date

- November 1, 2019

### Published In

### Volume / Issue

- 277 / 9

### Start / End Page

- 3271 - 3282

### Electronic International Standard Serial Number (EISSN)

- 1096-0783

### International Standard Serial Number (ISSN)

- 0022-1236

### Digital Object Identifier (DOI)

- 10.1016/j.jfa.2019.05.025

### Citation Source

- Scopus