# Regularity of the multi-configuration time-dependent Hartree approximation in quantum molecular dynamics

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

- Volume: 41, Issue: 2, page 315-331
- ISSN: 0764-583X

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topKoch, Othmar, and Lubich, Christian. "Regularity of the multi-configuration time-dependent Hartree approximation in quantum molecular dynamics." ESAIM: Mathematical Modelling and Numerical Analysis 41.2 (2007): 315-331. <http://eudml.org/doc/250059>.

@article{Koch2007,

abstract = {
We discuss the multi-configuration
time-dependent Hartree (MCTDH) method for the approximation
of the time-dependent Schrödinger equation in quantum molecular dynamics.
This method approximates the high-dimensional nuclear
wave function by a linear combination of products of functions depending
only on a single degree of freedom. The
equations of motion, obtained via the Dirac-Frenkel
time-dependent variational principle,
consist of a coupled system of low-dimensional
nonlinear partial differential equations and ordinary differential equations.
We show that, with a smooth and bounded potential, the MCTDH equations
are well-posed and retain high-order Sobolev regularity globally in time, that is,
as long as the density matrices appearing in the method formulation remain invertible.
In particular, the solutions are regular enough to ensure
local quasi-optimality of the approximation and to admit an efficient numerical treatment.
},

author = {Koch, Othmar, Lubich, Christian},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {MCTDH method; wavepacket propagation; nonlinear evolution equation; well-posedness; regularity.},

language = {eng},

month = {6},

number = {2},

pages = {315-331},

publisher = {EDP Sciences},

title = {Regularity of the multi-configuration time-dependent Hartree approximation in quantum molecular dynamics},

url = {http://eudml.org/doc/250059},

volume = {41},

year = {2007},

}

TY - JOUR

AU - Koch, Othmar

AU - Lubich, Christian

TI - Regularity of the multi-configuration time-dependent Hartree approximation in quantum molecular dynamics

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2007/6//

PB - EDP Sciences

VL - 41

IS - 2

SP - 315

EP - 331

AB -
We discuss the multi-configuration
time-dependent Hartree (MCTDH) method for the approximation
of the time-dependent Schrödinger equation in quantum molecular dynamics.
This method approximates the high-dimensional nuclear
wave function by a linear combination of products of functions depending
only on a single degree of freedom. The
equations of motion, obtained via the Dirac-Frenkel
time-dependent variational principle,
consist of a coupled system of low-dimensional
nonlinear partial differential equations and ordinary differential equations.
We show that, with a smooth and bounded potential, the MCTDH equations
are well-posed and retain high-order Sobolev regularity globally in time, that is,
as long as the density matrices appearing in the method formulation remain invertible.
In particular, the solutions are regular enough to ensure
local quasi-optimality of the approximation and to admit an efficient numerical treatment.

LA - eng

KW - MCTDH method; wavepacket propagation; nonlinear evolution equation; well-posedness; regularity.

UR - http://eudml.org/doc/250059

ER -

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## Citations in EuDML Documents

top- Othmar Koch, Christof Neuhauser, Mechthild Thalhammer, Error analysis of high-order splitting methods for nonlinear evolutionary Schrödinger equations and application to the MCTDHF equations in electron dynamics
- Dajana Conte, Christian Lubich, An error analysis of the multi-configuration time-dependent Hartree method of quantum dynamics

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