On the mapping of time-dependent densities onto potentials in quantum mechanics

Roi Baer*

*Corresponding author for this work

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40 Scopus citations

Abstract

The mapping of time-dependent densities on potentials in systems of identical quantum mechanical particles is examined. This mapping is of significance ever since Runge and Gross [Phys. Rev. Lett. 52, 997 (1984)] established its uniqueness, forming the theoretical basis for time-dependent density functional theory. Beyond uniqueness there are two important issues: existence, often called v-representability, and stability. We show that v-representability for localized densities in turn-on situations is not assured and we give a simple example of nonexistence. As for stability, we discuss an inversion procedure and by computing its Lyapunov exponents we demonstrate that the mapping is unstable with respect to fluctuations in the initial state. We argue that such instabilities will plague any inversion procedure.

Original languageAmerican English
Article number044103
JournalJournal of Chemical Physics
Volume128
Issue number4
DOIs
StatePublished - 2008

Bibliographical note

Funding Information:
We thank Dr. Yair Kurzweil for important contributions to this work. We also thank one of the (anonymous) referees for pointing out some issues concerning relations between stability and v-representability in DFT. This work was funded by a grant from the Israel Science Foundation.

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