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

Published online by Cambridge University Press:  23 February 2010

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

This paper gives an error analysis of themulti-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 L 2 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.

Keywords

MCTDH methodmulti-dimensional quantum dynamicslow-rank approximation

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 4 , July 2010, pp. 759 - 780
Copyright
© EDP Sciences, SMAI, 2010

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