Functional differentiability in time-dependent quantum mechanics

TL;DR

This paper establishes the mathematical foundation for the differentiability of solutions in time-dependent quantum mechanics, enabling rigorous analysis of response functions and density-functional theory.

Contribution

It proves Fréchet differentiability of the wave function and related quantities with respect to potentials, advancing the mathematical rigor in time-dependent quantum theory.

Findings

Proves Fréchet differentiability of wave functions in suitable Banach spaces.

Provides estimates for differences between solutions under different potentials.

Lays groundwork for rigorous non-equilibrium linear-response theory and density-functional theory.

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 Fr\'echet differentiability is proven. From this follows an estimate for the difference of two solutions to the time-dependent Schr\"odinger 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 Fr\'echet differentiability of the wave function provides a new route towards proving basic properties of time-dependent density-functional theory.