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Functional differentiability in time-dependent quantum mechanics.

Markus Penz1, Michael Ruggenthaler1

  • 1Institut für Theoretische Physik, Universität Innsbruck, 6020 Innsbruck, Austria.

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Summary
This summary is machine-generated.

This study proves the functional differentiability of quantum wave functions concerning time-dependent potentials. This finding rigorously formulates non-equilibrium linear-response theory and aids time-dependent density-functional theory.

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Area of Science:

  • Quantum Mechanics
  • Theoretical Chemistry
  • Mathematical Physics

Background:

  • Investigating the behavior of quantum systems under time-varying external influences is crucial.
  • Understanding the mathematical properties of wave functions and derived quantities is fundamental.
  • Existing theories for non-equilibrium systems have limitations, particularly concerning linear-response kernels.

Purpose of the Study:

  • To establish the functional differentiability of the time-dependent many-body wave function and related quantities.
  • To rigorously formulate non-equilibrium linear-response theory.
  • To provide a foundation for proving properties within time-dependent density-functional theory.

Main Methods:

  • Utilizing Banach spaces for potentials and wave functions.
  • Proving Fréchet differentiability for the wave function and derived quantities.
  • Developing estimates for solutions to the time-dependent Schrödinger equation under varying potentials.

Main Results:

  • Fréchet differentiability of the time-dependent many-body wave function with respect to time-dependent potentials is proven.
  • An estimate for the difference between solutions evolving under different potentials is derived.
  • The results offer a rigorous formulation of non-equilibrium linear-response theory, applicable to one-particle densities and bounded operators.

Conclusions:

  • The proven differentiability provides a rigorous mathematical framework for non-equilibrium quantum dynamics.
  • This work offers a new pathway for advancing time-dependent density-functional theory.
  • The findings are broadly applicable to quantum systems subjected to time-varying external fields.