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

Dajana Conte; Christian Lubich

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 44, Issue: 4, page 759-780
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Conte, Dajana, and Lubich, Christian. "An error analysis of the multi-configuration time-dependent Hartree method of quantum dynamics." ESAIM: Mathematical Modelling and Numerical Analysis 44.4 (2010): 759-780. <http://eudml.org/doc/250714>.

@article{Conte2010,
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. },
author = {Conte, Dajana, Lubich, Christian},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {MCTDH method; multi-dimensional quantum dynamics; low-rank approximation; low-rank approximation},
language = {eng},
month = {6},
number = {4},
pages = {759-780},
publisher = {EDP Sciences},
title = {An error analysis of the multi-configuration time-dependent Hartree method of quantum dynamics},
url = {http://eudml.org/doc/250714},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Conte, Dajana
AU - Lubich, Christian
TI - An error analysis of the multi-configuration time-dependent Hartree method of quantum dynamics
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/6//
PB - EDP Sciences
VL - 44
IS - 4
SP - 759
EP - 780
AB - 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.
LA - eng
KW - MCTDH method; multi-dimensional quantum dynamics; low-rank approximation; low-rank approximation
UR - http://eudml.org/doc/250714
ER -

References

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