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 detK(α1,…,αK):=∑≤i1,i2,…,iK≤Ndet(aiα,iβ(αβ))α,β=1,…,K to be determined, where ai,j(αβ) are certain one-dimensional scalar products involving (sparse-grid) basis functions ϕαβ. We propose a method to evaluate this expression such that the asymptotics of the computational cost with respect to N is O(N3) for fixed K, while the storage requirements increase only with the factor N2. Furthermore, we describe a parallel version (N processors) with full speed up.