The Kohn-Sham density functional theory [7, 15, 16, 18, 19] provides the
most popular class of approximate models for the quantum many-body problem in
material science. Yet except for some existence results for the Kohn-Sham equation
[2, 17, 21], we know very little about the mathematical nature of the problem, such
as the uniqueness of solutions; compactness of the various operators associated with
the Kohn-Sham equation; whether these operators behave more like differential
operators of some order, or like integral operators; how the physical nature of the
underlying material, for example, whether it is a metal or an insulator, is reflected at
the mathematical level. From a practical viewpoint, one most interesting question is
how to connect the Kohn-Sham density functional theory, which describes materials
at the level of their electronic structure, to the more conventional macroscopic
continuum models, such as the models of nonlinear elasticity and the Landau-
Lifshitz theory of magnetic materials.
It is not a surprise that we know so little about the mathematical properties of
the Kohn-Sham equation. After all, this is a rather unconventional set of nonlocal
and nonlinear system for a large collection of coupled one-particle wave functions,
the number of wave functions is equal to the number of electrons in the mate-
rial. Consequently, in the continuum limit, the number of unknown wave functions
goes to infinity. In addition, the Fermionic nature of the electrons poses further
mathematical challenges.
The present paper is the third of a series of papers that are devoted to the
study of the electronic structure of smoothly deformed crystals or crystals in an
external field, by analyzing various quantum mechanics models at different levels
of complexity, including the Kohn-Sham density functional theory, Thomas-Fermi
type of models and tight-binding models. Our overall objective is to establish the
microscopic foundation of the continuum theories of solids, such as the nonlinear
elasticity theory and the theory of magnetic materials, in terms of quantum me-
chanics and to examine the boundary where the continuum theories break down.
In this regard, our objective is a lot like the one in [11, 12], except that [11] and
[12] considered only classical models of atoms in a crystal, and in this series we
consider quantum mechanics models and focus on the behavior of the electrons and
spins. In particular, in analogy with the stability condition for phonons and elas-
tic waves established in [11, 12], we will establish stability conditions for charge
density waves and spin waves.
In [10], we considered a non-interacting quantum mechanical model for multi-
electrons and we examined the structure of the subspace spanned by the wave func-
tions. In particular, we extended the construction of Wannier functions for perfect
crystals to smoothly deformed crystals. We also established for this construction
an extension of the classical Cauchy-Born rule which historically was developed in
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