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Title: Functional differentiability in time-dependent quantum mechanics

Abstract

In this work, we investigate the functional differentiability of the time-dependent many-body wave function and of derived quantities with respect to time-dependent potentials. For properly chosen Banach spaces of potentials and wave functions, Fréchet differentiability is proven. From this follows an estimate for the difference of two solutions to the time-dependent Schrödinger equation that evolve under the influence of different potentials. Such results can be applied directly to the one-particle density and to bounded operators, and present a rigorous formulation of non-equilibrium linear-response theory where the usual Lehmann representation of the linear-response kernel is not valid. Further, the Fréchet differentiability of the wave function provides a new route towards proving basic properties of time-dependent density-functional theory.

Authors:
;  [1]
  1. Institut für Theoretische Physik, Universität Innsbruck, 6020 Innsbruck (Austria)
Publication Date:
OSTI Identifier:
22415556
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Chemical Physics; Journal Volume: 142; Journal Issue: 12; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BANACH SPACE; DENSITY FUNCTIONAL METHOD; EQUILIBRIUM; KERNELS; MANY-BODY PROBLEM; MATHEMATICAL SOLUTIONS; PARTICLES; POTENTIALS; QUANTUM MECHANICS; SCHROEDINGER EQUATION; TIME DEPENDENCE; WAVE FUNCTIONS

Citation Formats

Penz, Markus, E-mail: markus.penz@uibk.ac.at, and Ruggenthaler, Michael, E-mail: michael.ruggenthaler@uibk.ac.at. Functional differentiability in time-dependent quantum mechanics. United States: N. p., 2015. Web. doi:10.1063/1.4916390.
Penz, Markus, E-mail: markus.penz@uibk.ac.at, & Ruggenthaler, Michael, E-mail: michael.ruggenthaler@uibk.ac.at. Functional differentiability in time-dependent quantum mechanics. United States. doi:10.1063/1.4916390.
Penz, Markus, E-mail: markus.penz@uibk.ac.at, and Ruggenthaler, Michael, E-mail: michael.ruggenthaler@uibk.ac.at. Sat . "Functional differentiability in time-dependent quantum mechanics". United States. doi:10.1063/1.4916390.
@article{osti_22415556,
title = {Functional differentiability in time-dependent quantum mechanics},
author = {Penz, Markus, E-mail: markus.penz@uibk.ac.at and Ruggenthaler, Michael, E-mail: michael.ruggenthaler@uibk.ac.at},
abstractNote = {In this work, we investigate the functional differentiability of the time-dependent many-body wave function and of derived quantities with respect to time-dependent potentials. For properly chosen Banach spaces of potentials and wave functions, Fréchet differentiability is proven. From this follows an estimate for the difference of two solutions to the time-dependent Schrödinger equation that evolve under the influence of different potentials. Such results can be applied directly to the one-particle density and to bounded operators, and present a rigorous formulation of non-equilibrium linear-response theory where the usual Lehmann representation of the linear-response kernel is not valid. Further, the Fréchet differentiability of the wave function provides a new route towards proving basic properties of time-dependent density-functional theory.},
doi = {10.1063/1.4916390},
journal = {Journal of Chemical Physics},
number = 12,
volume = 142,
place = {United States},
year = {Sat Mar 28 00:00:00 EDT 2015},
month = {Sat Mar 28 00:00:00 EDT 2015}
}