The Efficient Computation of Certain Determinants Arising in the Treatment of Schrödinger's Equations

Abstract

The solution of Schrödinger's equation leads to a high number N of independent variables. Furthermore, the restriction to (anti)symmetric functions implies some complications. We propose a sparse-grid approximation which leads to a set of non-orthogonal basis. Due to the antisymmetry, scalar products are expressed by sums of N×N-determinants. Because of the sparsity of the sparse-grid approximation, these determinants can be reduced from N×N to a much smaller size K×K. The sums over all permutations reduce to the quantities det _K (α₁,…,α _K ):=∑≤i ₁,i ₂,…,i _K ≤Ndet(a _iα ,i _β ^(αβ))_{α,β=1,…, K} to be determined, where a _i , _j ^(αβ) are certain one-dimensional scalar products involving (sparse-grid) basis functions ϕα_β. We propose a method to evaluate this expression such that the asymptotics of the computational cost with respect to N is O(N ³) for fixed K, while the storage requirements increase only with the factor N ². Furthermore, we describe a parallel version (N processors) with full speed up.