# 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:

- 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}

}