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Long-Time Behaviour of Time-Dependent Density Functional Theory

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Abstract

The density functional theory (DFT) is a remarkably successful theory of electronic structure of matter. At the foundation of this theory lies the Kohn–Sham (KS) equation. In this paper, we describe the long-time behaviour of the time-dependent KS equation. Assuming weak self-interactions, we prove global existence and scattering in (almost) the full “short-range” regime. This is achieved with new and simple techniques, naturally compatible with the structure of the DFT and involving commutator vector fields and non-abelian versions of Sobolev–Klainerman-type spaces and inequalities.

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Notes

  1. The kth Gâteaux derivative could be defined by induction either as

    $$\begin{aligned} d^k g_1 (\rho ) \xi _1 \xi _2 \dots \xi _k:= d(d^{k-1} g_1 (\rho ) \xi _1 \xi _2 \dots \xi _{k-1})\xi _k, \end{aligned}$$

    or

    $$\begin{aligned} d^k g_1 (\rho ) \xi _1 \xi _2 \dots \xi _k:= \prod _1^k{\partial }_{s_j}|_{s_i=0\forall i}g(\rho +\sum _1^k s_j \xi _j).\end{aligned}$$

  2. There is no restriction on q if \(d=1,2\).

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Acknowledgements

The authors are grateful to Ilias Chenn, Stéphane Nonnenmacher, Benjamin Schlein and Heinz Siedentop for useful discussions. We also thank the anonymous referee for helpful comments. The work on this paper was supported in part by NSERC Grant No. NA7901 (IMS), by a start-up grant from the University of Toronto and NSERC Grant No. 06487 (FP).

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  1. Department of Mathematics, University of Toronto, 40 St. George street, Toronto, ON, M5S 2E4, Canada

    Fabio Pusateri & Israel Michael Sigal

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  1. Fabio Pusateri
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Correspondence to Fabio Pusateri.

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Pusateri, F., Sigal, I.M. Long-Time Behaviour of Time-Dependent Density Functional Theory. Arch Rational Mech Anal 241, 447–473 (2021). https://doi.org/10.1007/s00205-021-01656-1

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