On the expansion of solutions of Laplace-like equations into traces of separable higher dimensional functions

Abstract

This paper deals with the equation $-\Delta u+\mu u=f$ on high-dimensional spaces ${\mathbb{R}}^m$ where $\mu$ is a positive constant. If the right-hand side $f$ is a rapidly converging series of separable functions, the solution $u$ can be represented in the same way. These constructions are based on approximations of the function $1/r$ by sums of exponential functions. The aim of this paper is to prove results of similar kind for more general right-hand sides $f(x)=F(Tx)$ that are composed of a separable function on a space of a dimension $n$ greater than $m$ and a linear mapping given by a matrix $T$ of full rank. These results are based on the observation that in the high-dimensional case, for $\omega$ in most of the ${\mathbb{R}}^n$, the euclidian norm of the vector $T^t\omega$ in the lower dimensional space ${\mathbb{R}}^m$ behaves like the euclidian norm of $\omega$.

Version 4 coincides almost completely with the published version of the paper. The last Version 5 is slightly updated and contains some additional remarks on the asymptotic behavior for large dimensions m and ratios n/m

References in corpus (1)

• On the approximation of electronic wavefunctions by anisotropic Gauss and Gauss-Hermite functions

Cited by in corpus (2)

• A measure concentration effect for matrices of high, higher, and even higher dimension
• An iterative method for the solution of Laplace-like equations in high and very high space dimensions