Functional differentiability in time-dependent quantum mechanics

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.