An error analysis of the multi-configuration time-dependent Hartree method of quantum dynamics

¹ Dipartimento di Matematica ed Informatica, Università degli studi di Salerno, Via ponte don Melillo, 84084 Fisciano (SA), Italy. dajconte@unisa.it
² Universität Tübingen, Mathematisches Institut, Auf der Morgenstelle 10, 72076 Tübingen, Germany. lubich@na.uni-tuebingen.de

Received: 27 January 2009
Revised: 1 October 2009

Abstract

This paper gives an error analysis of the multi-configuration time-dependent Hartree (MCTDH) method for the approximation of multi-particle time-dependent Schrödinger equations. The MCTDH method approximates the multivariate wave function by a linear combination of products of univariate functions and replaces the high-dimensional linear Schrödinger equation by a coupled system of ordinary differential equations and low-dimensional nonlinear partial differential equations. The main result of this paper yields an L² error bound of the MCTDH approximation in terms of a best-approximation error bound in a stronger norm and of lower bounds of singular values of matrix unfoldings of the coefficient tensor. This result permits us to establish convergence of the MCTDH method to the exact wave function under appropriate conditions on the approximability of the wave function, and it points to reasons for possible failure in other cases.