Numerical integrators for quantum dynamics close to the adiabatic limit
type: Item
instance of: scholarly article
title: Numerical integrators for quantum dynamics close to the adiabatic limit
zbMATH Open document ID: 1029.65069
author: Tobias Jahnke
author: Christian Lubich
published in: Numerische Mathematik
publication date: 2003-08-19T00:00:00Z
review text: The authors consider the numerical integration of singularly perturbed Schrödinger equations \(i\dot\psi={1\over\varepsilon}H\psi\) where \(H\) is a time-dependent real symmetric matrix. Their goal is to develop time-reversible methods allowing for larger step sizes (in particular with \(h>\varepsilon\)) than traditional approaches. These methods are based on a reformulation of the equation: after a diagonalisation of \(H\) the authors introduce a new dependent variable \(\eta\) which represents essentially the coefficient vector of \(\psi\) with respect to an eigenbasis. This function is smoother than the original \(\psi\) and satisfies a uniform estimate on bounded time intervals. In high-dimensional problems further model reductions are possible by considering only the eigenspaces of some lower eigenvalues. The proposed methods are similar to midpoint schemes but with an additional averaging factor. They differ in some intermediate approximation steps which are performed in a more and more complicated manner in order to obtain higher order global error estimates (which are derived rigorously in the adiabatic case). All methods require only one evaluation and diagonalisation of \(H\) per time step. They are tested and compared with other approaches at a simple model problem. Finally, the authors discuss the problem of non-adiabatic behaviour near avoided crossings of energy levels and present an adaptive strategy for the step size reduction in such situations.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 81Q05
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 65L70
zbMATH DE Number: 1967702
zbMATH Keywords: Schrödinger equation
zbMATH Keywords: time-reversible integrator
zbMATH Keywords: adiabatic limit
zbMATH Keywords: quantum dynamics
zbMATH Keywords: singular perturbation
zbMATH Keywords: numerical examples
zbMATH Keywords: midpoint schemes
zbMATH Keywords: error estimates
reviewed by: Werner M. Seiler
MaRDI profile type: Publication
full work available at URL: s00211-002-0421-1
OpenAlex ID: W2005812552
DOI: 10.1007/S00211-002-0421-1
description: scientific article; zbMATH DE number 1967702