Approximating pointwise products of Laplacian eigenfunctions
Approximating pointwise products of Laplacian eigenfunctions
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DOI：
10.1016/j.jfa.2019.05.025
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发表时间：
2018-11
影响因子：
1.7
通讯作者：
Jianfeng Lu;C. Sogge;S. Steinerberger
中科院分区：
文献类型：
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作者：
Jianfeng Lu;C. Sogge;S. Steinerberger
考虑了具有Dirichlet条件的d维有界域M（或d维紧流形M）上的拉普拉斯特征函数。这些算子得到一个特征函数序列（e∈n） （e∈n）。我们研究了所有点积的子空间an = span {e i (x) e j (x): 1≤i， j≤n}的子空间l2 (M)。显然，这个向量空间的维数是dim (A n)= n (n+ 1)/2。证明了特征函数的积ei e j在一定意义上是简单的：对于任意ε> 0，存在一个几乎包含所有积的低维向量空间B n。更精确地说，表示正交投影Π B n: l2 (M)→B n，我们有∀1≤i， j≤n‖e i e j−Π B n （e i e j）‖l2≤ε，并且空间dim （B n）的大小相对较小：对于每一个δ> 0, dim (B n) > M, δ ε−δ n 1+ δ。对于任意长度的乘积，我们得到了同样的界，对于H−1范数的近似，我们也得到了同样的界。特征函数的点积是低秩的。除其他事项外，这对电子结构计算中快速算法的有效性具有影响。
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 ℓ) ℓ∈ N. We study the subspace of all pointwise products A n= span {e i (x) e j (x): 1≤ i, j≤ n}⊆ L 2 (M). Clearly, that vector space has dimension 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 ε> 0, there exists a low-dimensional vector space B n that almost contains all products. More precisely, denoting the orthogonal projection Π B n: L 2 (M)→ B n, we have∀ 1≤ i, j≤ n‖ e i e j− Π B n (e i e j)‖ L 2≤ ε and the size of the space dim (B n) is relatively small: for every δ> 0, dim (B n)≲ M, δ ε− δ n 1+ δ. 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.