On stability of solid state in the
Schrödinger–Poisson–Newton model

A. Komech

Faculty of Mathematics of Vienna University
and Institute for Information Transmission Problems RAS

e-mail<EMAIL_ADDRESS>

E. Kopylova

Faculty of Mathematics of Vienna University
and Institute for Information Transmission Problems RAS

e-mail<EMAIL_ADDRESS>
Resume

We survey our recent results on stability of 3D crystals in the
Schrödinger–Poisson–Newton model. We establish orbital stability for the
ground state in the case of finite periodic crystal and linear stability in
the case of infinite crystals under novel Jellium and Wiener conditions on the
charge density of ions. The corresponding examples are given.

For the finite crystals, the electron field is described a) by one-particle
Schrödinger equation, or b) by $N$-particle Schrödinger equation in the space
of antisymmetric wave functions respecting the Pauli exclusion principle. The
proofs rely on positivity of the Hessian of Hamiltonian functional in the
directions orthogonal to the manifold of ground states, and the Pauli
exclusion principle plays the key role in the proof of this positivity.

The problem of spatial periodicity of the ground states is reduced to a
generalisation of Newton-Girard formulas, and examples of non-periodic ground
states are constructed.

In the case of infinite crystals the proofs rely on our novel spectral theory
of Hamiltonian operators, which is a special version of the
Gohberg–Krein–Langer theory of selfadjoint operators in the Hilbert spaces
with indefinite metric. We establish the existence of the ground states and
the dispersive decay for the linearised dynamics.

Key words and phrases: crystal; ion; lattice; Schrödinger–Poisson equations;
Newton equations; ground state; linearization; linear stability; orbital
stability; dispersion relation; dispersive decay; analytic set; Hamiltonian
equation; Hamiltonian structure; Hamiltonian operator; spectral resolution;
selfadjoint operator; indefinite metric; spectral resolution; energy
conservation; charge conservation; $U(1)$-invariance; Hessian; Fourier
transform; Bloch transform; limiting absorption principle; discrete spectrum;
singular spectrum.

AMS subject classification: 35L10, 34L25, 47A40, 81U05

###### Contents

  1. 0.1 Preface
  2. I Orbital stability of finite crystals
    1. 1 One-particle Schrödinger theory
      1. 1.1 Introduction
      2. 1.2 Reduction to the integral equation
      3. 1.3 Global dynamics
      4. 1.4 Conservation laws
      5. 1.5 Jellium ground states
        1. 1.5.1 Description of all jellium ground states
        2. 1.5.2 Jellium and Wiener conditions. Examples
        3. 1.5.3 Non-periodic ground states
        4. 1.5.4 On the problem of periodicity
      6. 1.6 The orbital stability of the ground state
        1. 1.6.1 The Taylor expansion of the Hamiltonian functional
        2. 1.6.2 The null space of the energy Hessian
        3. 1.6.3 The positivity of the Hessian
        4. 1.6.4 The orbital stability
    2. 2 N-particle Schrödinger theory
      1. 2.1 Introduction
      2. 2.2 Function spaces and integral equation
      3. 2.3 Main assumptions
      4. 2.4 Global dynamics
      5. 2.5 Conservation laws
      6. 2.6 Jellium fermionic ground states
        1. 2.6.1 Uniform electronic charge density
        2. 2.6.2 Description of all ground states
      7. 2.7 The orbital stability of the ground state
        1. 2.7.1 The Taylor expansion of energy functional
        2. 2.7.2 The null space of Hessian
        3. 2.7.3 The positivity of Hessian
        4. 2.7.4 The orbital stability
  3. II Linear stability of infinite crystals
    1. 3 On existence of ground states
      1. 3.1 Introduction
      2. 3.2 3D lattice
        1. 3.2.1 Energy per cell
        2. 3.2.2 Compactness of minimizing sequence
        3. 3.2.3 Variation of the energy
        4. 3.2.4 The variational identity
        5. 3.2.5 The Schrödinger equation
        6. 3.2.6 Smoothness of ground state
      3. 3.3 2D lattice
        1. 3.3.1 The energy per cell
        2. 3.3.2 Compactness of minimizing sequence
        3. 3.3.3 The Poisson equation
        4. 3.3.4 Variation of the energy
        5. 3.3.5 The variational identity
        6. 3.3.6 The Schrödinger equation
        7. 3.3.7 Smoothness of ground state
      4. 3.4 1D lattice
        1. 3.4.1 The existence of ground state
        2. 3.4.2 The bound for the potential
    2. 4 Linear Stability
      1. 4.1 Introduction
      2. 4.2 Linearized dynamics
      3. 4.3 The Hamiltonian structure and the domain
      4. 4.4 Generator in the Fourier–Bloch transform
        1. 4.4.1 The discrete Fourier transform
        2. 4.4.2 Generator in the discrete Fourier transform
        3. 4.4.3 Generator in the Bloch transform
      5. 4.5 The positivity of energy
      6. 4.6 Weak solutions and linear stability
        1. 4.6.1 Weak solutions
        2. 4.6.2 Reduction to mild solution
        3. 4.6.3 Reduction to selfadjoint generator
        4. 4.6.4 Linear stability in the energy space
    3. 5 Dispersive decay
      1. 5.1 Introduction
      2. 5.2 Dispersion relations
      3. 5.3 Dispersion decay
        1. 5.3.1 Discrete spectral component
        2. 5.3.2 Continuous spectral component
      4. 5.4 Spectral properties of the selfadjoint generator

####  0.1 Preface

The first mathematical results on the stability of matter were obtained by
Dyson and Lenard in [22, 23] where the energy bound from below was
established. The thermodynamic limit for the Coulomb systems was studied first
by Lebowitz and Lieb [48, 49], see the survey and further development in [58].
These results were extended by Catto, Le Bris, Lions and others to Thomas-
Fermi and Hartree-Fock models [15, 16, 17]. Further results in this direction
were established by Cancés, Lahbabi, Lewin, Sabin, Stoltz, and others [10, 13,
47, 51, 52]. All these results concern either the convergence of the ground
state of finite particle systems in the thermodynamic limit or the existence
of the ground state for infinite particle systems.

In the Hartree-Fock model, the crystal ground state was constructed for the
first time by Catto, Le Bris, and Lions [16, 17]. For the Thomas-Fermi model,
see [15].

In [13], Cancés and Stoltz have established the well-posedness for the
dynamics of local perturbations of the crystal ground state in the random
phase approximation for the reduced Hartree-Fock equations with the Coulomb
pairwise interaction potential $w(x-y)=1/|x-y|$. The space-periodic nuclear
potential in the equation [13, (3)] does not depend on time that corresponds
to the fixed nuclei positions. The nonlinear Hartree-Fock dynamics for
crystals with the Coulomb potential and without the random phase approximation
was not studied previously, see the discussion in [47] and in the
introductions of [10, 13].

In [10], E. Cancès, S. Lahbabi, and M. Lewin have considered the random
reduced HF model of crystal when the ions charge density and the electron
density matrix are random processes, and the action of the lattice
translations on the probability space is ergodic. The authors obtain suitable
generalizations of the Hoffmann-Ostenhof and Lieb-Thirring inequalities for
ergodic density matrices, and construct a random potential which is a solution
to the Poisson equation with the corresponding stationary stochastic charge
density. The main result is the coincidence of this model with the
thermodynamic limit in the case of the short range Yukawa interaction.

In [51], Lewin and Sabin have established the well-posedness for the reduced
von Neumann equation, describing the Fermi gas, with density matrices of
infinite trace and pair-wise interaction potentials $w\in L^{1}({\mathbb
R}^{3})$. Moreover, the authors prove the asymptotic stability of translation-
invariant stationary states for 2D Fermi gas [52].

Traditional one-electron Bethe–Bloch–Sommerfeld mathematical model of crystals
reduces to the linear Schrödinger equation with a space-periodic static
potential, which corresponds to the standing ions. The corresponding spectral
theory is well developed, see [69] and the references therein. The scattering
theory for short-range and long-range perturbations of such ‘periodic
operators’ was constructed in [26, 27].

In [3], Anikin, Dobrokhotov and Katsnelson studied the semiclassical
asymptotic approximation of the spectrum of the two-dimensional Schrödinger
operator with a potential periodic in x and increasing at infinity in y. The
authors showed that the lower part of the spectrum has a band structure (where
bands can overlap) and calculate their widths and dispersion relations between
energy and quasimomenta. The key role in the obtained asymptotic approximation
is played by librations, i.e., unstable periodic trajectories of the
Hamiltonian system with an inverted potential. An effective numerical
algorithm for computing the widths of bands was presented. An applications to
quantum dimers was discussed.

However, the dynamical stability of crystals with moving ions was never
considered previously. This stability is necessary for a rigorous analysis of
fundamental quantum phenomena in the solid state physics: heat conductivity,
electric conductivity, thermoelectronic emission, photoelectric effect,
Compton effect, etc., see [8]. Here we survey our recent results on stability
of 3D crystals in the Schrödinger–Poisson–Newton model.

In the present book, we establish orbital stability for the ground state in
the case of finite periodic crystal and linear stability for infinite crystals
under novel Jellium and Wiener conditions on the charge density of ions. The
corresponding examples are given.

We establish orbital stability for the ground state in the case of finite
periodic crystal and linear stability in the case of infinite crystals under
novel Jellium and Wiener conditions on the charge density of ions. The
corresponding examples are given.

For the finite crystals, the electron field is described a) by one-particle
Schrödinger equation, or b) by $N$-particle Schrödinger equation in the space
of antisymmetric wave functions respecting the Pauli exclusion principle. The
proofs rely on positivity of the Hessian of Hamiltonian functional in the
directions orthogonal to the manifold of ground states, and the Pauli
exclusion principle plays the key role in the proof of this positivity (see
Remark 2.7.2).

The problem of spatial periodicity of the ground states is reduced to a
generalisation of Newton-Girard formulas, and examples of non-periodic ground
states are constructed (see Sections 1.5.3 and 1.5.4).

In the case of infinite crystals the proofs rely on our novel spectral theory
of Hamiltonian operators [36, 37], which is a special version of the
Gohberg–Krein theory of selfadjoint operators in the Hilbert spaces with
indefinite metric [30, 45, 46]. We establish the existence of the ground
states and the well-posedness and dispersive decay for the linearised
dynamics.
Acknowledgments. The authors are grateful to Herbert Spohn for helpful
discussions and remarks.

The authors are indebted to the Faculty of Mathematics of Vienna University,
and the Institute for the Information Transmission Problems of the Russian
Academy of Sciences for providing congenial facilities for the work.

The work was supported in part by the Department of Mechanics and Mathematics
of Moscow State University, and by the Austrian Science Fund (FWF) (project
nos. P28152 and P27492).

Alexander Komech and Elena Kopylova

Vienna, 15.01.2021

##  Part I Orbital stability of finite crystals

In the first part of present book, we consider finite crystals under periodic
boundary conditions with one ion per cell of a lattice.

The electron cloud is described by one-particle or $N$-particle Schrödinger
equations while the ions are described as classical charged particles moving
in an electrostatic potential created by the charge densities of electrons and
ions.

We construct the global dynamics and prove the conservation of energy and
charge. Our main result is the orbital stability of every ground state with
periodic arrangement of ions under novel ‘Jellium’ and ‘Wiener’ conditions on
the ion charge density.

The presentation mainly relies on our papers [41, 42] with suitable
extensions.

###  Chapter 1 One-particle Schrödinger theory

Abstract

In this chapter we describe the electron cloud in the crystal by one-particle
Schrödinger equation and moving ions. The ions are described as classical
particles that corresponds to the Born-Oppenheimer approximation. The ions
interact with the electron cloud via the scalar potential, which is a solution
to the corresponding Poisson equation.

We construct global dynamics, prove the conservation of energy and charge, and
give the description of all ground states. Our main result is the orbital
stability of every ground state with periodic arrangement of ions under novel
‘Jellium’ and ‘Wiener’ conditions on the ion charge density (1.1.11) and
(1.1.13).

Moreover, we construct examples of non-periodic ground states. In these
examples the Wiener condition fails. This suggests that the periodicity should
hold under the Wiener condition, but this is still an open problem.

The model with the one-particle Schrödinger equation does not respect the
Pauli exclusion principle for electrons. In the next chapter we extend the
orbital stability to the model with $N$-particle Schrödinger equation
respecting the Pauli exclusion principle.

####  1.1 Introduction

We consider crystals which occupy the finite torus ${\mathbb T}:={\mathbb
R}^{3}/N{\mathbb Z}^{3}$ and have one ion per cell of the cubic lattice
${\Gamma}:={\mathbb Z}^{3}/N{\mathbb Z}^{3}$, where $N\in{\mathbb N}$. The
cubic lattice is chosen for the simplicity of notations. We denote by
$\sigma(x)$ the charge density of one ion,

$\sigma\in C^{2}({{\mathbb T}}),\qquad\int_{{{\mathbb T}}}\sigma(x)dx=eZ>0,$
(1.1.1)

where $e>0$ is the elementary charge. Let $\psi(x,t)$ be the wave function of
the electron field, $q(n,t)$ denotes the ion displacement from the reference
position $n\in\Gamma$, and $\phi(x)$ be the electrostatic potential generated
by the ions and electrons. We assume $\hbar=c={\rm m}=1$, where $c$ is the
speed of light and ${\rm m}$ is the electron mass. Then the considered coupled
equations read

$\displaystyle i\partial_{t}\psi(x,t)\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!-\frac{1}{2}\Delta\psi(x,t)-e\phi(x,t)\psi(x,t),\qquad
x\in{{\mathbb T}},$ (1.1.2) $\displaystyle-\Delta\phi(x,t)\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\rho(x,t):=\sum_{n\in{\Gamma}}\sigma(x-n-q(n,t))-e|\psi(x,t)|^{2},\qquad
x\in{{\mathbb T}},$ (1.1.3) $\displaystyle M\ddot{q}(n,t)\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!-(\nabla\phi(x,t),\sigma(x-n-q(n,t))),\qquad
n\in{\Gamma}.$ (1.1.4)

Here the brackets $(\cdot,\cdot)$ stand for the scalar product on the real
Hilbert space $L^{2}({{\mathbb T}})$ and for its different extensions, and
$M>0$ is the mass of one ion. All derivatives here and below are understood in
the sense of distributions. Similar finite periodic approximations of crystals
are treated in all textbooks on quantum theory of solid state [9, 34, 75].
However, the stability of ground states in this model was never discussed.

Obviously,

$\displaystyle\int_{{\mathbb T}}\rho(x,t)dx=0$ (1.1.5)

by the Poisson equation (1.1.3). Hence, the potential $\phi(x,t)$ can be
eliminated from the system (1.1.2)–(1.1.4) using the operator
$G:=(-\Delta)^{-1}$, see (1.2.1) for a more precise definition. Substituting
$\phi(\cdot,t)=G\rho(\cdot,t)$ into equations (1.1.2) and (1.1.4), we can
write the system as

$\dot{X}(t)=F(X(t)),\qquad t\in{\mathbb R},$ (1.1.6)

where $X(t)=(\psi(\cdot,t),q(\cdot,t),p(\cdot,t))$ with
$p(\cdot,t):=\dot{q}(\cdot,t)$. The system (1.1.2)–(1.1.4) is equivalent, up
to a gauge transform (see the next section), to equation (2.1.9) with the
normalization

$\|\psi(\cdot,t)\|_{L^{2}({{\mathbb T}})}^{2}=ZN^{3},\qquad t\in{\mathbb R},$
(1.1.7)

which follows from (1.1.5). If the integral (1.1.1) vanishes, we have $Z=0$
and $\psi(x,t)\equiv 0$.

We will identify the complex functions $\psi(x)$ with two real functions
$\psi_{1}(x):={\rm Re{\hskip 1.42262pt}}\psi(x)$ and $\psi_{2}(x):={\rm
Im{\hskip 1.42262pt}}\psi(x)$. Now equation (1.1.6) is equivalent to the
Hamiltonian system

$\partial_{t}\psi_{1}(x,t)=\frac{1}{2}\partial_{\psi_{2}(x)}E,~{}~{}\partial_{t}\psi_{2}(x,t)=-\frac{1}{2}\partial_{\psi_{1}(x)}E,~{}~{}\partial_{t}q(n,t)=\partial_{p(n)}E,~{}~{}\partial_{t}p(n,t)=-\partial_{q(n)}E$
(1.1.8)

together with the normalisation condition (1.1.7). Here the Hamiltonian
functional (energy) reads

$E(\psi,q,p)=\frac{1}{2}\int_{{{\mathbb
T}}}|\nabla\psi(x)|^{2}dx+\frac{1}{2}(\rho,G\rho)+\sum_{n\in{\Gamma}}\frac{p^{2}(n)}{2M},$
(1.1.9)

where $q:=(q(n):~{}~{}n\in{\Gamma})\in[{{\mathbb T}}]^{\overline{N}}$,
$p:=(p(n):~{}~{}n\in{\Gamma})\in{\mathbb R}^{3\overline{N}}$ with
$\overline{N}:=N^{3}$, and

$\rho(x):=\sum_{n\in{\Gamma}}\sigma(x-n-q(n))-e|\psi(x)|^{2},\qquad
x\in{{\mathbb T}}.$ (1.1.10)

We prove the global well-posedness of the dynamics and the energy and charge
conservations (1.2.13). Our main goal is the stability of ground states, i.e.,
solutions to (1.1.2)–(1.1.4) with minimal (zero) energy (1.1.9). We will
consider only ${\Gamma}$-periodic ground states. Nonperiodic ground states
exist for some degenerate densities $\sigma$, see Remark 1.1.3 ii) and Section
1.5.3.

We will see that ${\Gamma}$-periodic ground states can be stable depending on
the choice of the ion density $\sigma$. We study special densities $\sigma$
satisfying some conditions below. Namely, we will assume the following
condition on the ion charge density,

$\mbox{\bf The Jellium
Condition:}~{}~{}~{}~{}~{}\hat{\sigma}(\xi):=\int_{{\mathbb T}}e^{i\xi
x}\sigma(x)dx=0,\quad\xi\in{\Gamma^{*}_{1}}\setminus
0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(1.1.11)

where ${\Gamma^{*}_{1}}:=2\pi{\mathbb Z}^{3}$. This condition immediately
implies that the periodized ion charge density is a positive constant
everywhere on the torus:

$\rho^{i}(x):=\sum_{n\in{\Gamma}}\sigma(x-n)\equiv eZ,\qquad x\in{{\mathbb
T}}.$ (1.1.12)

The simplest example of such a $\sigma$ is a constant over the unit cell of a
given lattice, which is what physicists usually call Jellium [29]. We give
further examples in Section 1.5.2. Here we study this model in the rigorous
context of the Schrödinger-Poisson equations.

Furthermore, we will assume a spectral property of the Wiener type

$\mbox{\bf The Wiener Condition:}~{}~{}~{}\Sigma(\theta):=\sum_{m\in{\mathbb
Z}^{3}}\Big{[}\frac{\xi\otimes\xi}{|\xi|^{2}}|\hat{\sigma}(\xi)|^{2}\Big{]}_{\xi=\theta+2\pi
m}>0,\ \quad\theta\in\Pi^{*}_{N}\setminus{\Gamma^{*}_{1}},$ (1.1.13)

where the Brillouin zone $\Pi^{*}_{N}$ is defined by

$\Pi^{*}_{N}:=\\{\xi=(\xi^{1},\xi^{2},\xi^{3})\in{\Gamma^{*}_{N}}:0\leq\xi^{j}\leq
2\pi,~{}~{}j=1,2,3\\},\quad{\Gamma^{*}_{N}}:=\frac{2\pi}{N}{\mathbb Z}^{3}.$
(1.1.14)

This condition is an analog of the Fermi Golden Rule for crystals. It is
independent of (1.1.11). We have introduced conditions of type (1.1.11) and
(1.1.13) for the first time in [40] in the framework of infinite crystals.

######  Remark 1.1.1.

i) The series (1.1.13) converges for
$\theta\in{\Gamma^{*}_{N}}\setminus{\Gamma^{*}_{1}}$ by the Parseval identity
since $\sigma\in L^{2}({\mathbb T})$ by (1.1.1).
ii) The matrix $\Sigma(\theta)$ is $\Gamma^{*}_{1}$-periodic outside
$\Gamma^{*}_{1}$. Thus, (1.1.13) means that $\Sigma(\theta)$ is a positive
matrix for $\theta\in\Pi^{*}_{N}\setminus 0$, where $\Pi^{*}_{N}$ is the
‘discrete Brillouin zone’ $\Gamma_{N}^{*}/\Gamma^{*}_{1}$.

The series (1.1.13) is a nonnegative matrix. Hence, the Wiener condition holds
‘generically’.

######  Example 1.1.2.

(1.1.13) holds if

$\hat{\sigma}(\xi)\neq
0,\qquad\xi\in{\Gamma^{*}_{N}}\setminus{\Gamma^{*}_{1}},$ (1.1.15)

i.e., (1.1.11) are the only zeros of $\hat{\sigma}(\xi)$.

However, (1.1.13) does not hold for the simplest Jellium model, when $\sigma$
is constant on the unit cell, see (1.5.5) and (1.5.6).

The energy (1.1.9) is nonnegative, and its minimum is zero. We show in Lemma
1.5.1 that under Jellium condition (1.1.11) all ${\Gamma}$-periodic ground
states are zero energy stationary solutions of the form

$S_{\alpha,r}=(\psi^{\alpha},\overline{r},0),\qquad\alpha\in[0,2\pi],\quad
r\in{{\mathbb T}},$ (1.1.16)

where $\psi^{\alpha}(x)\equiv e^{i\alpha}\sqrt{Z}$ and
$\overline{r}\in[{{\mathbb T}}]^{\overline{N}}$ is defined by

$\overline{r}(n)=r,\qquad n\in{\Gamma}.$ (1.1.17)

The corresponding electronic charge density reads

$\rho^{e}(x):=-e|\psi^{\alpha}(x)|^{2}\equiv-eZ,\qquad x\in{{\mathbb T}}.$
(1.1.18)

Hence, the corresponding total charge density (1.1.10) identically vanishes by
(1.1.12). Let us emphasize that both ionic and electronic charge densities are
uniform for the ground state under the Jellium condition.

Our main result (Theorem 1.6.7) is the stability of the real 4-dimensional
‘solitary manifold’

${\cal S}=\\{S_{\alpha,r}:~{}\alpha\in[0,2\pi],~{}r\in{{\mathbb T}}\\}.$
(1.1.19)

The stability means that any solution
$X(t)=(\psi(\cdot,t),q(\cdot,t),p(\cdot,t))$ to (1.1.6) with initial data,
lying in the vicinity of the manifold ${\cal S}$, is close to it uniformly in
time. This is the ‘orbital stability’ in the sense of [31], since the manifold
${\cal S}=S^{1}\times{{\mathbb T}}\times\\{0\\}$ is the orbit of the symmetry
group $U(1)\times{{\mathbb T}}$. Obviously,

$E(S)=0,\qquad S\in{\cal S}.$ (1.1.20)

Let us comment on our approach. We prove the local well-posedness for the
system (1.1.8) by the contraction mapping principle. The global well-posedness
for the equation (1.1.6) and the charge and energy conservation follow by the
Galerkin approximations and the uniqueness of solutions. We need the charge
conservation to return back from (1.1.8) to the system (1.1.2)–(1.1.4).

The orbital stability of the solitary manifold ${\cal S}$ is deduced from the
lower energy estimate

$E(X)\geq\nu\,d^{2}(X,{\cal S})\qquad{\rm if}\qquad d(X,{\cal
S})\leq\delta,\quad X\in{\cal M},$ (1.1.21)

where ${\cal M}$ is the manifold defined by the normalization (1.1.7) (see
Definition 1.6.4); $\nu,\delta>0$ and ‘$d$’ is the distance in the ‘energy
norm’. This estimate obviously implies the stability of the solitary manifold
${\cal S}$. We deduce (1.1.21) from the positivity of the Hessian
$E^{\prime\prime}(S)$ for $S\in{\cal S}$ in the orthogonal directions to
${\cal S}$ on the manifold ${\cal M}$. The Jellium and Wiener conditions are
sufficient for this positivity. We expect that these conditions are also
necessary; however, this is still an open problem. Anyway, the positivity can
break down when these conditions fail. We have shown this in [40, Lemma 10.1]
in the context of infinite crystals, however the proof extends directly to the
finite crystals.

######  Remarks 1.1.3.

i) In the case of infinite crystal, corresponding to $N=\infty$, the orbital
stability seems impossible. Namely, for $N=\infty$ the estimates (1.3.6),
(1.6.37) and (1.6.40) break down, as well as the estimate of type (1.1.21)
which is due to the discrete spectrum of the energy Hessian
$E^{\prime\prime}(S)$ on the compact torus.
ii) We show that the identity of type (1.1.12) holds for a wide set of
arrangements of ions which are not $\Gamma_{1}$-periodic, if $\sigma$ satisfy
additional spectral conditions. The corresponding examples are given, but in
all our examples the Wiener condition breaks down. We suppose that the Wiener
condition provides the periodicity (1.1.17), however this is a challenging
open problem, see Section 1.5.4. We prove the orbital stability only for
$\Gamma_{1}$-periodic ground states.

This chapter is organized as follows. In Section 2 we eliminate the potential
and reduce the dynamics to the integral equation. In Sections 3 we prove the
well-posedness. In Section 4 we prove the stability of the solitary manifold
${\cal S}$ establishing the lower estimate for the energy. In Appendices we
prove the conservation of the energy and charge, describe all ground states
and give some examples.

####  1.2 Reduction to the integral equation

The operator $G:=(-\Delta)^{-1}$ is well defined in the Fourier series:

$\rho(x)=\sum_{\xi\in{\Gamma^{*}_{N}}}\hat{\rho}(\xi)e^{i\xi x},\qquad
G\rho:=\sum_{\xi\in{\Gamma^{*}_{N}}\setminus
0}\frac{\hat{\rho}(\xi)}{\xi^{2}}e^{i\xi x}.$ (1.2.1)

The Poisson equation (1.1.3) implies that
$\hat{\rho}(0,t)=\displaystyle\int\rho(x,t)\,dx=0$, which is equivalent to
(1.1.7). Hence, $\phi(\cdot,t)=G\rho(\cdot,t)$ up to an additive constant
$C(t)$ which can be compensated by a gauge transform

$\psi(x,t)\mapsto\psi(x,t)\exp(-ie\displaystyle\int_{0}^{t}C(s)ds).$ (1.2.2)

The system (1.1.8) can be written as

$\dot{X}(t)=JE^{\prime}(X(t)),\qquad
X(t):=(\psi_{1}(t),\psi_{2}(t),q(t),p(t)),$ (1.2.3)

where

$J=\left(\begin{array}[]{rrrr}0&1/2&0&0\\\ -1/2&0&0&0\\\ 0&0&0&1\\\
0&0&-1&0\end{array}\right).$ (1.2.4)

For $\psi,\varphi\in L^{2}({\mathbb T})$ denote

$(\psi,\varphi):=\int_{{\mathbb
T}}\psi(\overline{x})\cdot\varphi(\overline{x})d\overline{x},$ (1.2.5)

where $\cdot$ is the inner product of the corresponding vectors in ${\mathbb
R}^{2}$. In particular,

$(1,i)=0.$ (1.2.6)

######  Definition 1.2.1.

i) Denote the real Hilbert spaces

${\cal X}:=L^{2}({{\mathbb T}})\oplus{\mathbb R}^{3\overline{N}}\oplus{\mathbb
R}^{3\overline{N}},\qquad{\cal W}:=H^{1}({{\mathbb T}})\oplus{\mathbb
R}^{3\overline{N}}\oplus{\mathbb R}^{3\overline{N}}.$ (1.2.7)

ii) ${\cal V}:=H^{1}({{\mathbb T}})\times[{{\mathbb
T}}]^{\overline{N}}\times{\mathbb R}^{3\overline{N}}$ is the Hilbert manifold
endowed with the metric

$d_{{\cal V}}(X,X^{\prime}):=\|\psi-\psi^{\prime}\|_{H^{1}({{\mathbb
T}})}+|q-q^{\prime}|+|p-p^{\prime}|,\qquad X=(\psi,q,p),\quad
X^{\prime}=(\psi^{\prime},q^{\prime},p^{\prime})$ (1.2.8)

and with the ‘quasinorm’

$|X|_{{\cal V}}:=\|\psi\|_{H^{1}({{\mathbb T}})}+|p|,\qquad X=(\psi,q,p).$
(1.2.9)

The linear space ${\cal W}$ is isomorphic to the tangent space to the Hilbert
manifold ${\cal V}$ at each point $X\in{\cal V}$. Denote by the brackets
$\langle\cdot,\cdot\rangle$ the scalar product in ${\cal X}$:

$\langle
Y,Y^{\prime}\rangle:=(\varphi,\varphi^{\prime})+\varkappa\varkappa^{\prime}+\pi\pi^{\prime},\qquad
Y=(\varphi,\varkappa,\pi),\quad
Y^{\prime}=(\varphi^{\prime},\varkappa^{\prime},\pi^{\prime}).$ (1.2.10)

The total charge of electrons is defined (up to a factor) by

$Q(X):=\int|\psi(x)|^{2}dx,\qquad X=(\psi,q,p)\in{\cal X}.$ (1.2.11)

Obviously,

$|X|_{\cal V}^{2}\leq C[E(X)+Q(X)],\qquad X\in{\cal V},$ (1.2.12)

The system (1.2.3) is a nonlinear finite-dimensional perturbation of the free
Schrödinger equation. We will prove that a solution $X\in C({\mathbb R},{\cal
V})$ exists and is unique for any initial state $X(0)\in{\cal V}$, and the
energy and the electronic charge are conserved,

$E(X(t))=E(X(0)),\quad Q(X(t))=Q(X(0)),\qquad t\in{\mathbb R}.$ (1.2.13)

The energy (1.1.9) and the charge are well defined and continuous on ${\cal
V}$ in the metric $d_{\cal V}$ by the estimate (1.3.14) below. The charge
conservation holds by the Noether theory [4, 31, 35] due to the
$U(1)$-invariance of the Hamiltonian functional:

$E(e^{i\alpha}\psi,q,p)=E(\psi,q,p),\qquad(\psi,q,p)\in{\cal
V},\quad\alpha\in{\mathbb R}.$ (1.2.14)

We rewrite the system (1.2.3) in the integral form

$\left\\{\begin{array}[]{lll}\psi(t)&=&e^{-iH_{0}t}\psi(0)+ie\displaystyle\int_{0}^{t}e^{-iH_{0}(t-s)}[\phi(s)\psi(s)]ds,\\\
q(n,t)&=&q(n,0)+\frac{1}{M}\displaystyle\int_{0}^{t}p(n,s)ds\mod N{\mathbb
Z}^{3},\\\
p(n,t)&=&p(n,0)-\displaystyle\int_{0}^{t}(\nabla\phi(s),\sigma(\cdot-
n-q(n,s)))ds,\end{array}\right|$ (1.2.15)

where $H_{0}:=-\frac{1}{2}\Delta$ and $\phi(s):=G\rho(s)$. In the vector form
(1.2.15) reads

$X(t)=e^{-At}X(0)+\int_{0}^{t}e^{-A(t-s)}N(X(s))ds,\qquad
A=\left(\begin{array}[]{ccc}iH_{0}&0&0\\\ 0&0&0\\\ 0&0&0\end{array}\right),$
(1.2.16)

where

$N(X)=(ie\phi\psi~{},p,~{}f),\qquad f(n):=-(\nabla\phi,\sigma(\cdot-
n-q(n))),\qquad\phi:=G\rho,$ (1.2.17)

and $\rho$, $G\rho$ are defined by (1.1.10) and (1.2.1) respectively.

####  1.3 Global dynamics

In this section we prove the well-posedness of the dynamics.

######  Theorem 1.3.1.

(Global well-posedness). Let (1.1.1) hold and $X(0)\in{\cal V}$. Then
i) Equation (1.2.3) admits a unique solution $X\in C({\mathbb R},{\cal V})$,
and the maps $U(t):X(0)\mapsto X(t)$ are continuous in ${\cal V}$ for
$t\in{\mathbb R}$.
ii) The conservation laws (1.2.13) hold.
iii) $X$ is the solution to (1.1.2)–(1.1.4) if

$Q(X(0))=Z\overline{N}^{3}.$ (1.3.1)

First, let us prove the local well-posedness.

######  Proposition 1.3.2.

(Local well-posedness). Let (1.1.1) hold and $|X(0)|_{\cal V}\leq R$. Then
there exists $\tau=\tau(R)>0$ such that equation (1.2.3) has a unique solution
$X\in C([-\tau,\tau],{\cal V})$, and the maps $U(t):X(0)\mapsto X(t)$ are
continuous in ${\cal V}$ for $t\in[-\tau,\tau]$.

In the next two lemmas we prove the boundedness and the local Lipschitz
continuity of the nonlinearity $N:{\cal V}\to{\cal W}$. With this proviso
Proposition 1.3.2 follows from the integral form (1.2.16) of the equation
(1.2.3) by the contraction mapping principle, since $e^{-At}$ is an isometry
of ${\cal W}$.

######  Lemma 1.3.3.

For any $R>0$ and $|X|_{\cal V}\leq R$

$\|N(X)\|_{\cal W}\leq C(R)$ (1.3.2)

###### Proof.

First, we estimate the potential $\phi(\cdot,t):=G\rho(\cdot,t)$. Namely, we
split $\rho(x,t)$ as

$\rho(x,t)=\rho^{i}(x,t)+\rho^{e}(x,t),$ (1.3.3)

where $\rho^{i}$ and $\rho^{e}$ are the ion and electron charge densities
respectively,

$\rho^{i}(x,t)=\sum_{n\in{\Gamma}}\sigma(x-n-q(n,t)),\qquad\rho^{e}(x,t)=-e|\psi(x,t)|^{2}.$

Applying the Cauchy-Schwarz inequality to the second formula (1.2.1), we
obtain that

$\|\phi\|_{C({{\mathbb T}})}\leq
C\|\hat{\rho}\|_{L^{2}({\Gamma^{*}_{N}})}=C\|\rho\|_{L^{2}({{\mathbb T}})}\leq
C(\|\rho^{i}\|_{L^{2}({{\mathbb T}})}+e\|\psi\|_{L^{4}({{\mathbb
T}})}^{2})\leq C_{1}(1+\|\psi\|_{H^{1}({{\mathbb T}})}^{2})$ (1.3.4)

since $H^{1}({{\mathbb T}})\subset L^{6}({{\mathbb T}})$ by the Sobolev
embedding theorem. On the other hand, the Hölder inequality implies that

$\|\nabla\rho^{e}\|_{L^{3/2}({{\mathbb T}})}\leq
e\|\nabla|\psi|^{2}\|_{L^{3/2}({{\mathbb T}})}\leq
C_{1}\|\psi\|_{L^{6}({{\mathbb T}})}\|\nabla\psi\|_{L^{2}({{\mathbb T}})}\leq
C_{2}\|\psi\|_{H^{1}({{\mathbb T}})}^{2}.$ (1.3.5)

Therefore, we get by the Hausdorff-Young and the Hölder inequalities [32]

$\displaystyle\|\nabla\phi\|_{L^{3}({{\mathbb T}})}\\!\\!$ $\displaystyle\leq$
$\displaystyle\\!\\!C\|\widehat{\nabla\phi}\|_{L^{3/2}({\Gamma^{*}_{N}})}\leq
C_{1}\|\xi\hat{\rho}\|_{L^{3}({\Gamma^{*}_{N}})}\Big{[}\sum_{\xi\in{\Gamma^{*}_{N}}\setminus
0}|\xi|^{-6}\Big{]}^{1/3}\leq C_{2}\|\nabla\rho\|_{L^{3/2}({{\mathbb T}})}$
$\displaystyle\leq$
$\displaystyle\\!\\!C_{2}(\|\nabla\rho^{i}\|_{L^{3/2}({{\mathbb
T}})}+\|\nabla\rho^{e}\|_{L^{3/2}({{\mathbb T}})})\leq
C_{3}(1+\|\psi\|_{H^{1}({{\mathbb T}})}^{2}).$ (1.3.6)

Now (1.3.4) and (1.3.6) imply by the Hölder inequality

$\displaystyle\|\psi\phi\|_{L^{2}({{\mathbb T}})}$ $\displaystyle\leq$
$\displaystyle\|\phi\|_{C({{\mathbb T}})}\cdot\|\psi\|_{L^{2}({{\mathbb
T}})}\leq C(1+\|\psi\|_{H^{1}({{\mathbb T}})}^{3})$
$\displaystyle\|\nabla\psi\phi\|_{L^{2}({{\mathbb T}})}$ $\displaystyle\leq$
$\displaystyle\|\phi\|_{C({{\mathbb T}})}\|\nabla\psi\|_{L^{2}({{\mathbb
T}})}\leq C(1+\|\psi\|_{H^{1}({{\mathbb T}})}^{3})$
$\displaystyle\|\psi\nabla\phi\|_{L^{2}({{\mathbb T}})}$ $\displaystyle\leq$
$\displaystyle C\|\psi\|_{L^{6}({{\mathbb
T}})}\cdot\|\nabla\phi\|_{L^{3}({{\mathbb T}})}\leq
C_{1}(1+\|\psi\|_{H^{1}({{\mathbb T}})}^{3}).$ (1.3.7)

Hence,

$\|\phi\psi\|_{H^{1}({{\mathbb T}})}\leq C(1+\|\psi\|_{H^{1}({{\mathbb
T}})}^{3}).$ (1.3.8)

Finally, (1.3.6) and (1.1.1) imply that

$|f(n)|\leq\|\phi\|_{C({{\mathbb T}})}\|\nabla\sigma\|_{L^{1}({{\mathbb
T}})}\leq C(1+\|\psi\|_{H^{1}({{\mathbb T}})}^{2}),\qquad n\in{\Gamma}.$
(1.3.9)

At last, (1.3.2) holds by (1.3.8) and (1.3.9). ∎

It remains to prove that the nonlinearity is locally Lipschitz.

######  Lemma 1.3.4.

For any $R>0$ and $X_{1},X_{2}\in{\cal V}$

$\|N(X_{1})-N(X_{2})\|_{\cal W}\leq C^{\prime}(R)d_{\cal
V}(X_{1},X_{2})\qquad{\rm if}\quad|X_{1}|_{\cal V},|X_{2}|_{\cal V}\leq R.$
(1.3.10)

###### Proof.

Writing $X_{k}=(\psi_{k},q_{k},p_{k})$ and $\phi_{k}=G\rho_{k}$, we obtain
that

$\|\phi_{1}\psi_{1}-\phi_{2}\psi_{2}\|_{H^{1}({{\mathbb
T}})}\leq\|(\phi_{1}-\phi_{2})\psi_{1}\|_{H^{1}({{\mathbb
T}})}+\|\phi_{2}(\psi_{1}-\psi_{2})\|_{H^{1}({{\mathbb T}})}.$ (1.3.11)

Similarly to (1.3.4) - (1.3.6) we obtain

$\displaystyle\|\phi_{2}(\psi_{1}-\psi_{2})\|_{H^{1}({{\mathbb T}})}$
$\displaystyle\leq$ $\displaystyle\|\phi_{2}\|_{C({{\mathbb
T}})}\|\psi_{1}-\psi_{2}\|_{H^{1}({{\mathbb
T}})}+\|\nabla\phi_{2}\|_{L^{3}({{\mathbb
T}})}\|\psi_{1}-\psi_{2}\|_{L^{6}({{\mathbb T}})}$ (1.3.12)
$\displaystyle\leq$ $\displaystyle
C(1+R^{2})\|\psi_{1}-\psi_{2}\|_{H^{1}({{\mathbb T}})}\leq C(R)d_{\cal
V}(X_{1},X_{2})$

Further, similarly to (1.3.5),

$\|\nabla(\rho^{e}_{1}-\rho^{e}_{2})\|_{L^{3/2}({{\mathbb T}})}\leq
C\|\psi_{1}-\psi_{2}\|_{H^{1}({{\mathbb T}})}[\|\psi_{1}\|_{H^{1}({{\mathbb
T}})}+\|\psi_{2}\|_{H^{1}({{\mathbb T}})}].$ (1.3.13)

Moreover, $|\sigma(x)-\sigma(x-a)|\leq C|a|$, where $|a|:=\min_{r\in a}|r|$
for $a\in{\mathbb T}$. Hence, similarly to (1.3.6),

$\displaystyle\|(\phi_{1}-\phi_{2})\psi_{1}\|_{H^{1}({{\mathbb
T}})}\leq\|\phi_{1}-\phi_{2}\|_{C({{\mathbb T}})}\|\psi_{1}\|_{H^{1}({{\mathbb
T}})}+\|\nabla(\phi_{1}-\phi_{2})\|_{L^{3}({{\mathbb
T}})}\|\psi_{1}\|_{L^{6}({{\mathbb T}})}$ $\displaystyle\leq
CR\Big{[}\|\rho_{1}^{i}-\rho_{2}^{i}\|_{L^{2}({{\mathbb
T}})}+\|\rho_{1}^{e}-\rho_{2}^{e}\|_{L^{2}({{\mathbb
T}})}+\|\nabla(\rho_{1}^{i}-\rho_{2}^{i})\|_{L^{3/2}({{\mathbb
T}})}+\|\nabla(\rho_{1}^{e}-\rho_{2}^{e})\|_{L^{3/2}({{\mathbb T}})}\Big{]}$
$\displaystyle\leq
C_{1}R(|q_{1}-q_{2}|+R\|\psi_{1}-\psi_{2}\|_{H^{1}({{\mathbb T}})})\leq
C(R)d_{\cal V}(X_{1},X_{2}).$

Now (1.3.11) and (1.3.12) give

$\|\phi_{1}\psi_{1}-\phi_{2}\psi_{2}\|_{H^{1}({{\mathbb T}})}\leq C(R)d_{\cal
V}(X_{1},X_{2}).$ (1.3.14)

Similarly,

$\displaystyle|(\nabla\phi_{1},\sigma(\cdot-n-
q_{1}(n)))-(\nabla\phi_{2},\sigma(\cdot-n-q_{2}(n)))|$
$\displaystyle\leq|(\nabla\phi_{1}-\nabla\phi_{2},\sigma(\cdot-n-
q_{1}(n)))|+|(\nabla\phi_{2},\sigma(\cdot-n-q_{1}(n))-\sigma(\cdot-n-
q_{2}(n)))|$ $\displaystyle\leq C(\|\phi_{1}-\phi_{2}\|_{C({{\mathbb
T}})}+\|\phi_{2}\|_{C({{\mathbb T}})}|q_{1}-q_{2}|\leq C(R)d_{\cal
V}(X_{1},X_{2}).$ (1.3.15)

This estimate and (1.3.14) imply (1.3.10). ∎

Now Proposition 1.3.2 follows from Lemmas 1.3.3 and 1.3.4.
Proof of Theorem 1.3.1. The local solution $X\in C([-\tau,\tau],{\cal V})$ of
(1.2.3) exists and is unique by Proposition 1.3.2. On the other hand, the
conservation laws (1.2.13) (proved in Proposition 1.4.2 iii)) together with
(1.2.12) imply a priori bound

$|X(t)|_{\cal V}^{2}\leq C[E(X(0))+Q(X(0))],\qquad t\in[-\tau,\tau].$ (1.3.16)

Hence, the local solution admits an extension to the global one $X\in
C({\mathbb R},{\cal V})$. Further, the normalisation (1.3.1) implies that
$Q(X(t))=Z\overline{N}^{3}$ for all $t\in{\mathbb R}$ by the charge
conservation (1.2.13). Hence, (1.2.3) implies (1.1.2)–(1.1.4). $\Box$

####  1.4 Conservation laws

We deduce the conservation laws (1.2.13) by the Galerkin approximations [60].

######  Definition 1.4.1.

i)${\cal V}_{m}$ with $m\in{\mathbb N}$ denotes finite dimensional submanifold
of ${\cal V}$ formed by

$(\sum_{k\in{\Gamma^{*}_{N}}(m)}C_{k}e^{ikx},q,p),\qquad q\in{{\mathbb
T}}^{\overline{N}},\quad p\in{\mathbb R}^{3\overline{N}}.$ (1.4.1)

where ${\Gamma^{*}_{N}}(m):=\\{k\in{\Gamma^{*}_{N}}:k^{2}\leq m\\}$.
ii) ${\cal W}_{m}$ with $m\in{\mathbb N}$ denotes the finite dimensional
linear subspace of ${\cal W}$ spanned by

$(\sum_{k\in{\Gamma^{*}_{N}}(m)}C_{k}e^{ikx},\varkappa,v),\qquad\varkappa\in{\mathbb
R}^{3\overline{N}},\quad v\in{\mathbb R}^{3\overline{N}}.$ (1.4.2)

Obviously, ${\cal V}_{1}\subset{\cal V}_{2}\subset...$, the union
$\cup_{m}{\cal V}_{m}$ is dense in ${\cal V}$, and ${\cal W}_{m}$ are
invariant with respect to $A$ and $J$. Let us denote by $P_{m}$ the orthogonal
projector ${\cal X}\to{\cal W}_{m}$. This projector is also orthogonal in
${\cal W}$. Let us approximate the system (1.2.3) by the finite-dimensional
Hamiltonian systems on the manifold ${\cal V}_{m}$,

$\dot{X}_{m}(t)=JE_{m}^{\prime}(X_{m}(t)),\qquad t\in{\mathbb R},$ (1.4.3)

where $E_{m}:=E|_{{\cal V}_{m}}$ and
$X_{m}(t)=(\psi_{m}(t),q_{m}(t),p_{m}(t))\in C({\mathbb R},{\cal V}_{m})$. The
equation (1.4.3) can be also written as

$\langle\dot{X}_{m}(t),Y\rangle=-\langle E^{\prime}(X_{m}(t)),JY\rangle,\qquad
Y\in{\cal W}_{m}.$ (1.4.4)

This form of the equation (1.4.3) holds since $E_{m}:=E|_{{\cal V}_{m}}$ and
${\cal W}_{m}$ is invariant with respect to $J$. Equivalently,

$\dot{X}_{m}(t)=-A\,X_{m}(t)+P_{m}N(X_{m}(t)).$ (1.4.5)

The Hamiltonian form guarantees the energy and charge conservation (1.2.13):

$E(X_{m}(t))=E(X_{m}(0)),\quad Q(X_{m}(t))=Q(X_{m}(0)),\qquad t\in{\mathbb
R}.$ (1.4.6)

Indeed, the energy conservation holds by the Hamiltonian form (1.4.3), while
the charge conservation holds by the Noether theory [4, 31, 35] due to the
$U(1)$-invariance of $E_{m}$, see (1.2.14).

The equation (1.4.5) admits a unique local solution for every initial state
$X_{m}(0)\in{\cal V}_{m}$ since the right hand side is locally bounded and
Lipschitz continuous. The global solutions exist by (1.2.12) and the energy
and charge conservation (1.4.6).

Finally, we take any $X(0)\in{\cal V}$ and choose a sequence

$X_{m}(0)\to X(0),\qquad m\to\infty,$ (1.4.7)

where the convergence holds in the metric of ${\cal V}$. Therefore,

$E(X_{m}(0))\to E(X(0)),\qquad Q(X_{m}(0))\to Q(X(0)).$ (1.4.8)

Hence, (1.4.6) and (1.2.12) imply the basic uniform bound

$R:=\sup_{m\in{\mathbb N}}\,\,\sup_{t\in{\mathbb R}}|X_{m}(t)|_{\cal
V}<\infty.$ (1.4.9)

Therefore, (1.4.5) and Lemma 1.3.3 imply the second basic uniform bound

$\sup_{m\in{\mathbb N}}\,\sup_{t\in{\mathbb R}}\,\|\dot{X}_{m}(t)\|_{{\cal
W}^{-1}}<C(R),$ (1.4.10)

since the operator $A:{\cal W}\to{\cal W}^{-1}$ is bounded, and the projector
$P_{m}$ is also a bounded operator in ${\cal W}\subset{\cal W}^{-1}$. Hence,
the Galerkin approximations $X_{m}(t)$ are uniformly Lipschitz-continuous with
values in ${\cal V}^{-1}$:

$\sup_{m\in{\mathbb N}}\,d_{{\cal V}^{-1}}(X_{m}(t),X_{m}(s))\leq
C(R)|t-s|,\qquad s,t\in{\mathbb R}.$ (1.4.11)

Let us show that the uniform estimates (1.4.9) and (1.4.11) provide a
compactness of the Galerkin approximations and the conservation laws. Let us
recall that ${\cal X}:={\cal V}^{0}$ and ${\cal V}:={\cal V}^{1}$.

######  Proposition 1.4.2.

Let (1.1.1) hold and $X(0)\in{\cal V}$. Then
i) There exists a subsequence $m^{\prime}\to\infty$ such that

$X_{m^{\prime}}(t)\stackrel{{\scriptstyle{\cal
X}}}{{-\\!\\!\\!\\!\longrightarrow}}X(t),\qquad m^{\prime}\to\infty,\qquad
t\in{\mathbb R},$ (1.4.12)

where $X(\cdot)\in C({\mathbb R},{\cal X})$.
ii) Every limit function $X(\cdot)$ is a solution to (1.2.16), and
$X(\cdot)\in C({\mathbb R},{\cal V})$.
iii) The conservation laws (1.2.13) hold.

###### Proof.

i) The convergence (1.4.12) follows from (1.4.9) and (1.4.10) by the Dubinsky
‘theorem on three spaces’ [20] (Theorem 5.1 of [60]). Namely, the embedding
${\cal V}\subset{\cal X}$ is compact by the Sobolev theorem [1], and hence,
(1.4.12) holds by (1.4.9) for $t\in D$, where $D$ is a countable dense set.
Finally, let us use the interpolation inequality and (1.4.9), (1.4.11): for
any $\varepsilon>0$

$d_{\cal X}(X_{m}(t),X_{m}(s))\leq\varepsilon d_{\cal
V}(X_{m}(t),X_{m}(s))+C(\varepsilon)d_{{\cal V}^{-1}}(X_{m}(t),X_{m}(s))\leq
2\varepsilon R+C(\varepsilon,R)|t-s|.$ (1.4.13)

This inequality implies the equicontinuity of the Galerkin approximations with
the values in ${\cal X}$. Hence, convergence (1.4.12) holds for all
$t\in{\mathbb R}$ since it holds for the dense set of $t\in D$. The same
equicontinuity also implies the continuity of the limit function $X\in
C({\mathbb R},{\cal X})$.
ii) Integrating equation (1.4.5), we obtain

$\int_{0}^{t}\langle\dot{X}_{m}(t),Y\rangle\,ds=-\int_{0}^{t}\langle
X_{m}(s),AY)\,ds+\int_{0}^{t}\langle N(X_{m}(s)),Y\rangle\,ds,\qquad Y\in{\cal
W}_{m},$ (1.4.14)

Below we will write $m$ instead of $m^{\prime}$. To prove (1.2.16) it suffices
to check that in the limit $m\to\infty$, we get

$\int_{0}^{t}\langle\dot{X}(t),Y\rangle\,ds=-\int_{0}^{t}\langle
X(s),AY\rangle\,ds+\int_{0}^{t}\langle N(X(s)),Y\rangle\,ds,\qquad Y\in{\cal
W}_{n},\qquad n\in{\mathbb N}.$ (1.4.15)

The convergence of the left hand side and of the first term on the right hand
side of (1.4.14) follow from (1.4.12) and (1.4.7) since $AY\in{\cal W}_{m}$.

It remains to consider the last integral of (1.4.14). The integrand is
uniformly bounded by (1.4.9) and Lemma 1.3.3. Hence, it suffices to check the
pointwise convergence

$\langle N(X_{m}(t),Y\rangle-\\!\\!\\!-\\!\\!\\!\\!\to\langle
N(X(t),Y\rangle,\quad m\to\infty,\qquad Y\in{\cal W}_{n}$ (1.4.16)

for any $t\in{\mathbb R}$. Here
$N(X_{m}(t))=(ie\phi_{m}(t)\psi_{m}(t),p_{m}(t),f_{m}(t))$ according to the
notations (1.2.17), and $Y=(\varphi,\varkappa,v)\in{\cal W}_{n}$. Hence,
(1.4.16) reads

$ie(\phi_{m}(t)\psi_{m}(t),\varphi)+p_{m}(t)\varkappa+f_{m}(t)v\,\to\,ie(\phi(t)\psi(t),\varphi)+p(t)\varkappa+f(t)v,\quad
m\to\infty.$ (1.4.17)

The convergence of $p_{m}(s)\varkappa$ follows from (1.4.12) (with
$m^{\prime}=m$) . To prove the convergence of two remaining terms, we first
show that

$\phi_{m}(t):=G\rho_{m}\stackrel{{\scriptstyle C({{\mathbb
T}})}}{{-\\!\\!-\\!\\!\\!\\!\longrightarrow}}\phi(t):=G\rho,\quad m\to\infty.$
(1.4.18)

Indeed, (1.4.12) implies that

$\psi_{m}(t)\stackrel{{\scriptstyle L^{2}({{\mathbb
T}})}}{{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!\\!\longrightarrow}}\psi(t),\qquad
q_{m}(t)\to q(t),\quad m\to\infty.$ (1.4.19)

The sequence $\psi_{m}(t)$ is bounded in $H^{1}({{\mathbb T}})$ by (1.4.9).
Hence, $\psi(t)\in H^{1}({{\mathbb T}})$ and the sequence $\rho_{m}(t)$ is
bounded in the Sobolev space $W^{1,3/2}({{\mathbb T}})$ by (1.3.5). Therefore,
the sequence $\rho_{m}(t)$ is precompact in $L^{2}({{\mathbb T}})$ by the
Sobolev compactness theorem. Hence,

$\rho_{m}\stackrel{{\scriptstyle L^{2}({{\mathbb
T}})}}{{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!\\!\longrightarrow}}\rho,\quad
m\to\infty$ (1.4.20)

by (1.4.19). Therefore, (1.4.18) holds since the operator $G:L^{2}({{\mathbb
T}})\to C({{\mathbb T}})$ is continuous. From (1.4.18) and (1.4.19) it follows
that

$\phi_{m}(t)\psi_{m}(t)\stackrel{{\scriptstyle L^{2}({{\mathbb
T}})}}{{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!\\!\longrightarrow}}{\hskip
1.42262pt}\phi(t)\psi(t),\quad f_{m}(t)\to f(t),\quad m\to\infty,$ (1.4.21)

which proves (1.4.17). Now (1.4.15) is proved for $Y\in{\cal V}_{n}$ with any
$n\in{\mathbb N}$. Hence, $X(t)$ is a solution to (1.2.3). Finally,
$\|N(X(\cdot))\|_{\cal W}$ is a bounded function by (1.4.9) and Lemma 1.3.3.
Hence, (1.2.16) implies that $X(\cdot)\in C({\mathbb R},{\cal V})$.
iii) The conservation laws (1.4.6) and the convergences (1.4.7), (1.4.12)
imply that

$E(X(t))\leq E(X(0)),\quad Q(X(t))\leq Q(X(0)),\qquad t\in{\mathbb R}.$
(1.4.22)

The last inequality holds by the first convergence of (1.4.19). The first
inequality follows from the representation

$E(X_{m}(t))=\frac{1}{2}\|\nabla\psi_{m}(t)\|_{L^{2}({{\mathbb
T}})}^{2}+\frac{1}{2}\|\sqrt{G}\rho_{m}(t)\|_{L^{2}({{\mathbb
T}})}^{2}+\sum_{n\in\Gamma_{n}}\frac{p_{m}^{2}(n,t)}{2M}.$ (1.4.23)

Namely, the last two terms on the right hand side converge by (1.4.20) and
(1.4.12). Moreover, the first term is bounded by (1.4.9). Hence, the first
convergence of (1.4.19) implies the weak convergence

$\nabla\psi_{m}(t)\stackrel{{\scriptstyle L^{2}_{w}({{\mathbb
T}})}}{{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!\\!\longrightarrow}}\nabla\psi(t)$
(1.4.24)

by the Banach theorem. Now the first inequality of (1.4.22) follows by the
property of the weak convergence in the Hilbert space. Finally, the opposite
inequalities to (1.4.22) are also true by the uniqueness of solutions
$X(\cdot)\in C({\mathbb R},{\cal V})$, which is proved in Proposition 1.3.2. ∎

####  1.5 Jellium ground states

We describe all solutions to (1.1.2)–(1.1.4) with minimal energy (1.1.9), give
some examples of ion densities illustrating the Jellium and the Wiener
conditions, and show the existence of non-periodic ground states.

#####  1.5.1 Description of all jellium ground states

The following lemma gives the description of all ground states of the system
(1.1.2)–(1.1.4).

######  Lemma 1.5.1.

Let the Jellium condition (1.1.11) hold. Then all solutions to (1.1.2)–(1.1.4)
of minimal (zero) energy are $(e^{i\alpha}\sqrt{Z},q^{*},0)$ with
$\alpha\in[0,2\pi]$ and $q^{*}\in{\mathbb T}^{\overline{N}}$ satisfying the
identity

$\sum_{n\in{\Gamma}}\sigma(x-q^{*}(n))\equiv eZ,\qquad x\in{{\mathbb T}},$
(1.5.1)

###### Proof.

First, let us note that the ${\Gamma}$-periodic solutions (1.1.16) have the
zero energy, and the identity (1.5.1) holds for $q^{*}=\overline{r}$ by
(1.1.12).

Further, for any solution with zero energy (1.1.9) all summands on the right
hand side of (1.1.9) vanish. The first integral vanishes only for constant
functions. Hence, the normalization condition (1.1.7) gives

$\psi(x,t)\equiv\psi_{\alpha(t)}(x)\equiv
e^{i\alpha(t)}\sqrt{Z},\qquad\alpha(t)\in{\mathbb R}.$ (1.5.2)

Then

$\rho^{e}(x,t):=-e|\psi_{\alpha(t)}(x)|^{2}\equiv-eZ,\qquad x\in{{\mathbb
T}},\,\,\,\,t\in{\mathbb R},$ (1.5.3)

similarly to (1.1.18). Further, the second summand of (1.1.9) vanishes only
for $\rho(x,t)\equiv 0$. Hence, $\rho^{i}(x,t)\equiv eZ$ that is equivalent to
(1.5.1) with $q(n,t)$ instead of $q^{*}(n)$ by (1.5.3). However,
$M\partial_{t}q(n,t)=p(n,t)\equiv 0$ for the zero energy (1.1.9). Hence,

$q(n,t)\equiv q^{*}(n),\qquad t\in{\mathbb R},$ (1.5.4)

where $q^{*}$ satisfies (1.5.1). Moreover, the Poisson equation (1.1.3) with
$\rho(x,t)\equiv 0$ implies that $\phi(x,t)\equiv 0$ after a gauge
transformation (1.2.2). Hence, finally, substituting (1.5.2) into (1.1.2) with
$\phi(x,t)\equiv 0$, we obtain that $\alpha(t)\equiv{\rm const}$. ∎

This lemma implies that all ${\Gamma}$-periodic ground states are given by
(1.1.16).

#####  1.5.2 Jellium and Wiener conditions. Examples

The Wiener condition (1.1.13) for the ground states (1.1.16) holds under the
generic assumption (1.1.15). On the other hand, (1.1.13) does not hold for the
simplest Jellium model, when $\sigma(x)$ is the characteristic function

$\sigma(x)=\sigma_{1}(x):=\left\\{\begin{array}[]{ll}eZ,&x\in\Pi\\\
0,&x\in{{\mathbb T}}\setminus\Pi\end{array}\right|,$ (1.5.5)

where $\Pi:=[-1/2,1/2]^{3}$. Indeed, in this case the Fourier transform

$\hat{\sigma}_{1}(\xi)=eZ\hat{\chi}_{1}(\xi_{1})\hat{\chi}_{1}(\xi_{2})\hat{\chi}_{1}(\xi_{3});\qquad\hat{\chi}_{1}(s)=\frac{2\sin
s/2}{s},\quad s\in{\mathbb R}\setminus 0,$ (1.5.6)

where $\chi_{1}(s)$ is the characteristic function of the interval
$[-1/2,1/2]$. In this case we have for $\theta=(0,\theta_{2},\theta_{3})$,

$\Sigma(\theta)=\sum_{m\in{\mathbb
Z}^{3}:\,m_{1}=0}\Big{[}\frac{\xi\otimes\xi}{|\xi|^{2}}|\hat{\sigma}(\xi)|^{2}\Big{]}_{\xi=\theta+2\pi
m},$ (1.5.7)

which is a degenerate matrix since $\xi_{1}=0$ in each summand. Hence,
(1.1.13) fails. Similarly, the Wiener condition fails for
$\sigma_{k}(x)=eZ\chi_{k}(x_{1})\chi_{k}(x_{2})\chi_{k}(x_{3})$ where
$\chi_{k}=\chi_{1}*...*\chi_{1}$ ($k$ times) with $k=2,3,...$, since in this
case

$\hat{\sigma}_{k}(\xi)=eZ\hat{\chi}_{k}(\xi_{1})\hat{\chi}_{k}(\xi_{2})\hat{\chi}_{k}(\xi_{3});\qquad\hat{\chi}_{k}(s)=\Big{[}\frac{2\sin
s/2}{s}\Big{]}^{k},\quad s\in{\mathbb R}\setminus 0.$ (1.5.8)

#####  1.5.3 Non-periodic ground states

It is easy to construct ground states which are not $\Gamma_{1}$-periodic in
the case of characteristic function (1.5.5). Namely, the identity (1.5.1)
obviously holds for periodic arrangement of ions (1.1.17). Now let us modify
this periodic arrangement as follows:

$q^{*}(n)=(r_{1},r_{2},r_{3}+\tau(n_{1},n_{2})),\qquad n\in{\Gamma},$ (1.5.9)

where $\tau(n_{1},n_{2})$ is an arbitrary point of the circle ${\mathbb
R}/N{\mathbb Z}$. Now (1.5.1) obviously holds for any arrangement of ions
(1.5.9).

Next lemma gives a more general spectral assumptions on $\sigma$ which provide
ground states with non-periodic ion arrangements. For example, let us assume
that

$\sigma(\xi)=0,\quad\xi_{3}\in 2\pi{\mathbb Z}\setminus 0,\qquad{\rm
and}\qquad\sigma(\xi_{1},\xi_{2},0)=0,\quad(\xi_{1},\xi_{2})\in 2\pi{\mathbb
Z}^{2}\setminus 0.$ (1.5.10)

In particular, this holds for the densities (1.5.8) with all $k=1,...$

######  Lemma 1.5.2.

Let $\sigma$ satisfy the spectral condition (1.5.10). Then there exist ground
states which are not $\Gamma_{1}$-periodic.

###### Proof.

In the Fourier transform (1.5.1) reads

$\int_{{\mathbb T}}e^{i\xi
x}\sum_{n\in{\Gamma}}\sigma(x-n-q^{*}(n))dx=\hat{\sigma}(\xi)\sum_{n\in{\Gamma}}e^{i\xi(n+q^{*}(n))}=\left\\{\begin{array}[]{rl}eZ\overline{N},&\xi=0\\\
0,&\xi\in{\Gamma^{*}_{N}}\setminus 0.\end{array}\right.$ (1.5.11)

These identities hold for any density $\sigma$ satisfying (1.1.11) if

$\sum_{n\in{\Gamma}}e^{i\xi(n+q^{*}(n))}=0,\qquad\xi\in{\Gamma^{*}_{N}}\setminus{\Gamma^{*}_{1}}.$
(1.5.12)

In particular, $q^{*}=\overline{r}$ satisfies the system (1.5.12) since then

$\sum_{n\in{\Gamma}}e^{i\xi(n+q^{*}(n))}=\sum_{n\in{\Gamma}}e^{i\xi(n+r)}=e^{i\xi
r}\sum_{n\in{\Gamma}}e^{i\xi
n}=0,\qquad\xi\in{\Gamma^{*}_{N}}\setminus{\Gamma^{*}_{1}}.$ (1.5.13)

Indeed,

$\sum_{n\in{\Gamma}}e^{i\xi
n}=\sum_{n\in{\Gamma}}e^{i(\xi_{1}n_{1}+\xi_{2}n_{2}+\xi_{3}n_{3})}=\sum_{n_{1}=0}^{N-1}e^{i\xi_{1}n_{1}}\sum_{n_{2}=0}^{N-1}e^{i\xi_{2}n_{2}}\sum_{n_{3}=0}^{N-1}e^{i\xi_{3}n_{3}}=0$
(1.5.14)

since at least one $\xi_{k}\not\in 2\pi{\mathbb Z}$ for
$\xi\in{\Gamma^{*}_{N}}\setminus{\Gamma^{*}_{1}}$. Now we modify
$\overline{r}$ as follows:

$q^{*}(n):=(a_{1}(n_{1},n_{2})),a_{2}(n_{1},n_{2}),r_{3}),\qquad
n\in{\Gamma},$ (1.5.15)

where $a_{1}(n_{1},n_{2}))$ and $a_{2}(n_{1},n_{2})$ are arbitrary points of
the circle ${\mathbb R}/N{\mathbb Z}$. Obviously, $q^{*}(n)$ can be non-
periodic in $n_{1}$ and $n_{2}$.

On the other hand, (1.5.11) holds. Indeed, for $\xi_{3}\in 2\pi{\mathbb Z}$
this follows from (1.5.10), while for $\xi_{3}\not\in 2\pi{\mathbb Z}$ we have

$\sum_{n\in{\Gamma}}e^{i\xi(n+q^{*}(n))}=\sum_{n_{1},n_{2}=0}^{N-1}e^{i(\xi_{1}(n_{1}+a_{1}(n_{1},n_{2})+\xi_{2}(n_{2}+a_{2}(n_{1},n_{2})}\sum_{n_{3}=0}^{N-1}e^{i\xi_{3}n_{3}}=0.$
(1.5.16)

∎

#####  1.5.4 On the problem of periodicity

In the non-periodic examples above the Wiener condition (1.1.13) breaks down
for the case $\theta_{3}=0$ by (1.5.10). We suppose that the Wiener condition
provides only periodic ground states, however this is an open problem. For
densities $\sigma$ satisfying a more strong condition (1.1.15), the identity
(1.5.1) is equivalent to the system (1.5.12) by (1.5.11). The system (1.5.12)
can be written as an ‘algebraic system’

$\sum_{n\in{\Gamma}}w_{1}^{m_{1}}(n)w_{2}^{m_{2}}(n)w_{3}^{m_{3}}(n)=0,\qquad
m=(m_{1},m_{2},m_{3})\in{\mathbb Z}^{3}\setminus N{\mathbb Z}^{3}$ (1.5.17)

for

$w_{j}(n_{1},n_{2},n_{3}):=e^{i\displaystyle\frac{2\pi}{N}[n_{j}+q^{*}_{j}(n)]},\qquad
n\in{\Gamma},\quad j=1,2,3.$ (1.5.18)

The ${\Gamma}$-periodicity of $q^{*}$ is equivalent to the fact that only
solutions are

$w_{j}(n_{1},n_{2},n_{3})=C_{j}\lambda^{n_{j}},\qquad j=1,2,3,$ (1.5.19)

where $\lambda:=e^{i\displaystyle\frac{2\pi}{N}}$.

######  Remark 1.5.3.

For the corresponding 1D analog

$\sum_{n_{1}=0}^{N-1}w_{1}^{m_{1}}(n_{1})=0,\qquad m=1,...,N-1,$ (1.5.20)

the only solutions are $w_{1}(n_{1}):=C_{1}\lambda^{n_{1}}$ that follows
easily from the Newton-Girard formulas guessed by Girard in 1629 and
rediscovered (without a proof) by Newton in 1666. The formulas were proved by
Euler in 1747, see [24, 74].

####  1.6 The orbital stability of the ground state

In this section we expand the energy into the Taylor series and prove the
orbital stability checking the positivity of the energy Hessian.

#####  1.6.1 The Taylor expansion of the Hamiltonian functional

We will deduce the lower estimate (1.1.21) using the Taylor expansion of
$E(S+Y)$ for $S=S_{\alpha,r}:=(\psi^{\alpha},\overline{r},0)\in{\cal S}$ and
$Y=(\varphi,\varkappa,\pi)\in{\cal W}$:

$E(S+Y)=E(S)+\langle E^{\prime}(S),Y\rangle+\frac{1}{2}\langle
Y,E^{\prime\prime}(S)Y\rangle+R(S,Y)=\frac{1}{2}\langle
Y,E^{\prime\prime}(S)Y\rangle+R(S,Y)$ (1.6.1)

since $E(S)=0$ and $E^{\prime}(S)=0$. First, we expand the charge density
(1.1.10) corresponding to
$S+Y=(\psi^{\alpha}+\varphi,\overline{r}+\varkappa,\pi)$:

$\rho(x)=\rho^{(0)}(x)+\rho^{(1)}(x)+\rho^{(2)}(x),\qquad x\in{{\mathbb T}},$
(1.6.2)

where $\rho^{(0)}$ and $\rho^{(1)}$ are respectively the terms of zero and
first order in $Y$, while $\rho^{(2)}$ is the remainder. However,
$\rho^{(0)}(x)$ is the total charge density of the ground state which is
identically zero by (1.1.12) and (1.1.18):

$\rho^{(0)}(x)=\rho^{i}_{0}(x)-e|\psi^{\alpha}(x)|^{2}\equiv 0,\qquad
x\in{{\mathbb T}}.$ (1.6.3)

Thus, $\rho=\rho^{(1)}+\rho^{(2)}$. Expanding (1.1.10), we obtain

$\displaystyle\rho^{(1)}(x)\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\\!\\!\sigma^{(1)}(x)-2e\psi^{\alpha}(x)\cdot\varphi(x),\quad\sigma^{(1)}(x)=-\sum_{n\in{\Gamma}}\varkappa(n)\cdot\nabla\sigma(x-n-r),\vskip
3.0pt plus 1.0pt minus 1.0pt$ (1.6.4) $\displaystyle\rho^{(2)}(x)\\!\\!\\!\\!$
$\displaystyle\\!\\!=\\!\\!\\!\\!$
$\displaystyle\\!\\!\sigma^{(2)}(x)-e|\varphi(x)|^{2},~{}~{}\sigma^{(2)}(x)=\frac{1}{2}\sum_{n\in{\Gamma}}\int_{0}^{1}\\!\\!(1\\!-\\!s)[\varkappa(n)\cdot\nabla]^{2}\sigma(x-n-
r-s\varkappa(n))ds,$ (1.6.5)

where $\cdot$ means the inner product of real vectors, and in particular,
$\psi^{\alpha}(x)\cdot\varphi(x)$ is the inner product of the corresponding
vectors in ${\mathbb R}^{2}$. Substituting $\psi=\psi^{\alpha}+\varphi$ and
$\rho=\rho^{(1)}+\rho^{(2)}$ into (1.1.9), we obtain that the quadratic part
of (1.6.1) reads

$\frac{1}{2}\langle Y,E^{\prime\prime}(S)Y\rangle=\frac{1}{2}\int_{{{\mathbb
T}}}|\nabla\varphi(x)|^{2}dx+\frac{1}{2}(\rho^{(1)},G\rho^{(1)})+K(\pi),~{}~{}~{}~{}~{}~{}K(\pi):=\displaystyle\sum_{n}\frac{\pi^{2}(n)}{2M}~{}~{}$
(1.6.6)

and the remainder equals

$R(S,Y)=\frac{1}{2}(2\rho^{(1)}+\rho^{(2)},G\rho^{(2)}).$ (1.6.7)

#####  1.6.2 The null space of the energy Hessian

In this section we calculate the null space

${\cal K}(S):={\rm Ker{\hskip 1.42262pt}}\,E^{\prime\prime}(S)\Big{|}_{\cal
W},\qquad S\in{\cal S}$ (1.6.8)

under the Wiener condition.

######  Lemma 1.6.1.

Let the Jellium and the Wiener conditions (1.1.11), (1.1.13) hold and
$S\in{\cal S}$. Then

${\cal K}(S)=\\{(C,\overline{s},0):~{}~{}C\in{\mathbb C},~{}~{}s\in{\mathbb
R}^{3}\\},$ (1.6.9)

where $\overline{s}\in{\mathbb R}^{3\overline{N}}$ is defined similarly to
(1.1.17): $\overline{s}(n)\equiv s$.

###### Proof.

All summands of the energy (1.6.6) are nonnegative. Hence, this expression is
zero if and only if all the summands vanish:

$\varphi(x)\equiv
C,\quad(\rho^{(1)},G\rho^{(1)})=\|\sqrt{G}[\sigma^{(1)}-2e\psi^{\alpha}\cdot\varphi]\|_{L^{2}({{\mathbb
T}})}^{2}=0,\quad\pi=0.$ (1.6.10)

Note that $\sqrt{G}\psi^{\alpha}\cdot\varphi=\sqrt{G}\psi^{\alpha}\cdot C=0$
since the operator $G$ annihilates the constant functions by (1.2.1). Hence,
(1.6.10) implies that

$\sqrt{G}\sigma^{(1)}=0.$ (1.6.11)

On the other hand, (1.6.4) gives in the Fourier transform

$\hat{\sigma}^{(1)}(\xi)=\hat{\sigma}(\xi)\xi\cdot\sum_{n\in{\Gamma}}ie^{i\xi(n+r)}\varkappa(n)=i\hat{\sigma}(\xi)\xi\cdot
e^{i\xi r}\hat{\varkappa}(\xi),\qquad\xi\in{\Gamma^{*}_{N}},$ (1.6.12)

where $\hat{\varkappa}(\xi):=\sum_{n\in{\Gamma}}ie^{i\xi n}\varkappa(n)$ is a
$2\pi{\mathbb Z}^{3}$-periodic function on ${\Gamma^{*}_{N}}$. Hence,
definition (1.2.1) and the Jellium condition (1.1.11) imply that

$\displaystyle 0=\|\sqrt{G}\sigma^{(1)}\|_{L^{2}({{\mathbb T}})}^{2}$
$\displaystyle=$ $\displaystyle
N^{-3}\sum_{{\Gamma^{*}_{N}}\setminus{\Gamma^{*}_{1}}}|\hat{\sigma}(\xi)\frac{\xi\hat{\varkappa}(\xi)}{|\xi|}|^{2}$
(1.6.13) $\displaystyle=$ $\displaystyle
N^{-3}\sum_{\theta\in\Pi^{*}_{N}\setminus{\Gamma^{*}_{1}}}\langle\hat{\varkappa}(\theta),\sum_{m\in{\mathbb
Z}^{3}}\Big{[}\frac{\xi\otimes\xi}{|\xi|^{2}}|\hat{\sigma}(\xi)|^{2}\Big{]}_{\xi=\theta+2\pi
m}\hat{\varkappa}(\theta)\rangle$ $\displaystyle=$ $\displaystyle
N^{-3}\sum_{\theta\in\Pi^{*}_{N}\setminus{\Gamma^{*}_{1}}}\langle\hat{\varkappa}(\theta),\Sigma(\theta)\hat{\varkappa}(\theta)\rangle.$

As a result,

$\hat{\varkappa}(\theta)=0,\qquad\theta\in\Pi^{*}_{N}\setminus{\Gamma^{*}_{1}}$
(1.6.14)

by the Wiener condition (1.1.13). On the other hand,
$\hat{\varkappa}(0)\in{\mathbb R}^{3}$ remains arbitrary. Respectively,
$\varkappa=\overline{s}$ with an arbitrary $s\in{\mathbb R}^{3}$. ∎

######  Remark 1.6.2.

The key point of the proof is the explicit calculation (1.6.12) in the Fourier
transform. This calculation relies on the invariance of the Hessian
$E^{\prime\prime}(S)$ with respect to ${\Gamma}$-translations which is due to
the periodicity of the ions arrangement of the ground state.

######  Remark 1.6.3.

(Beyond the Wiener condition.) If the Wiener condition (1.1.13) fails, the
dimension of the space

$V:=\\{v\in{\mathbb
R}^{3\overline{N}}:~{}~{}v(n)=\sum_{\theta\in\Pi^{*}_{N}\setminus{\Gamma^{*}_{1}}}e^{-i\theta
n}\hat{v}(\theta),\qquad\hat{v}(\theta)\in{\mathbb
C}^{3},~{}~{}\Sigma(\theta)\hat{v}(\theta)\equiv\,0\\}$ (1.6.15)

is positive. The above calculations show that in this case

${\cal K}(S)=\\{(C,\overline{s}+v,0):~{}~{}C\in{\mathbb C},~{}~{}s\in{{\mathbb
T}},~{}~{}v\in V\\}.$ (1.6.16)

The subspace $V\subset{\mathbb R}^{3\overline{N}}$ is orthogonal to the $3D$
subspace $\\{\overline{s}:s\in{\mathbb R}^{3}\\}\subset{\mathbb
R}^{3\overline{N}}$ by the Parseval theorem. Hence, $\dim{\cal K}(S)=5+d$,
where $d:=\dim V>0$. Thus, $\dim{\cal K}(S)>5$. Under the Wiener condition
$V=0$, and (1.6.16) coincides with (1.6.9).

#####  1.6.3 The positivity of the Hessian

Denote by $N_{S}{\cal S}$ the normal subspace to ${\cal S}$ at a point $S$:

$N_{S}{\cal S}:=\\{Y\in{\cal W}:\langle Y,\tau\rangle=0,~{}~{}\tau\in
T_{S}{\cal S}\\},$ (1.6.17)

where $T_{S}{\cal S}$ is the tangent space to ${\cal S}$ at the point $S$ and
$\langle\cdot,\cdot\rangle$ stands for the scalar product (1.2.10).

######  Definition 1.6.4.

Denote by ${\cal M}$ the Hilbert manifold

${\cal M}:=\\{X\in{\cal V}:Q(X)=ZN^{3}\\}.$ (1.6.18)

Obviously, ${\cal S}\subset{\cal M}$, and a tangent space to ${\cal M}$ at a
point $S=(\psi^{\alpha},\overline{r},0)$ is given by

$T_{S}{\cal M}=\\{(\varphi,\varkappa,\pi)\in{\cal
W}:\varphi\bot\psi^{\alpha}\\},$ (1.6.19)

since $DQ(\psi^{\alpha},\overline{r},0)=(\psi^{\alpha},0,0)$.

######  Lemma 1.6.5.

Let the Jellium condition (1.1.11) hold and $S=S_{\alpha,r}\in{\cal S}$. Then
the Wiener condition (1.1.13) is necessary and sufficient for the positivity
of the Hessian $E^{\prime\prime}(S)$ in the orthogonal directions to ${\cal
S}$ on ${\cal M}$, i.e.,

$E^{\prime\prime}(S)\Big{|}_{N_{S}{\cal S}\cap T_{S}{\cal M}}>0.$ (1.6.20)

###### Proof.

i) Sufficiency. Differentiating
$S_{\alpha,r}=(e^{i\alpha}\psi_{0},\overline{r},0)\in{\cal S}$ in the
parameters $\alpha\in[0,2\pi]$ and $r\in{{\mathbb T}}$, we obtain

$T_{S}{\cal S}=\\{(iC\psi^{\alpha},\overline{s},0):~{}~{}C\in{\mathbb
R},~{}~{}s\in{\mathbb R}^{3}\\}.$ (1.6.21)

Hence, (1.6.9) implies that

$K(S):={\cal K}(S)\cap N_{S}{\cal
S}=\\{(C\psi^{\alpha},0,0):~{}~{}C\in{\mathbb R}\\}$ (1.6.22)

by (1.2.6) and (1.2.10). Therefore,

${\cal K}(S)\cap N_{S}{\cal S}\cap T_{S}{\cal M}=K(S)\cap T_{S}{\cal
M}=(0,0,0),$ (1.6.23)

since the vector $(\psi^{\alpha},0,0)$ is orthogonal to $T_{S}{\cal M}$ by
(1.6.19). Now (1.6.20) follows since $E^{\prime\prime}(S)\geq 0$ by (1.6.6).
ii) Necessity. If the Wiener condition (1.1.13) fails, the null space ${\cal
K}(S)$ is given by (1.6.16). Hence, (1.6.21) implies that now

$K(S)=\\{(C\psi^{\alpha},v,0):~{}~{}C\in{\mathbb R},~{}~{}v\in V\\}.$ (1.6.24)

However, $(\psi^{\alpha},\psi^{\alpha})>0$. Hence, (1.6.19) implies that
$(\psi^{\alpha},v,0)\not\in T_{S}{\cal M}$ and the intersection

$K(S)\cap T_{S}{\cal M}=\\{0,v,0):~{}~{}v\in V\\}$ (1.6.25)

is the nontrivial subspace of the dimension $d>0$. Thus, the Hessian
$E^{\prime\prime}(S)$ vanishes on this nontrivial subspace of $N_{S}{\cal
S}\cap T_{S}{\cal M}$. ∎

######  Remark 1.6.6.

The positivity of type (1.6.20) breaks down for the submanifold ${\cal
S}(r):=\\{S_{\alpha,r}:\alpha\in[0,2\pi]\\}$ with a fixed $r\in{{\mathbb T}}$
instead of the solitary manifold ${\cal S}$. Indeed, in this case the
corresponding tangent space is smaller,

$T_{S}{\cal S}(r)=\\{(iC\psi^{\alpha},0,0):~{}~{}C\in{\mathbb R}\\},$ (1.6.26)

and hence, the normal subspace $N_{S}{\cal S}(r)$ is larger, containing all
vectors $(0,\overline{s},0)$ generating the shifts of the torus. However, all
these vectors also belong to the null space (1.6.9) and to $T_{S}{\cal M}$.
Respectively, the null space of the Hessian $E^{\prime\prime}(S)$ in
$T_{S}{\cal M}\cap N_{S}{\cal S}(r)$ is 3-dimensional.

#####  1.6.4 The orbital stability

Here we prove our main result.

######  Theorem 1.6.7.

Let the conditions (1.1.11), (1.1.13) and (1.1.1) hold, and ${\cal S}$ is the
solitary manifold (1.1.19). Then for any $\varepsilon>0$ there exists
$\delta=\delta(\varepsilon)>0$ such that for $X(0)\in{\cal M}$ with $d_{\cal
V}(X(0),{\cal S})<\delta$ we have

$d_{\cal V}(X(t),{\cal S})<\varepsilon,\qquad t\in{\mathbb R}$ (1.6.27)

for the corresponding solution $X(t)\in C({\mathbb R},{\cal V})$ to
(1.1.2)–(1.1.4).

For the proof is suffices to check the lower energy estimate (1.1.21):

$E(X)\geq\nu\,d^{2}_{\cal V}(X,{\cal S})\quad{\rm if}\quad d_{\cal V}(X,{\cal
S})\leq\delta,\quad X\in{\cal M}$ (1.6.28)

with some $\nu,\delta>0$. This estimate implies Theorem 1.6.7, since the
energy is conserved along all trajectories. First, we prove similar lower
bound for the energy Hessian.

######  Lemma 1.6.8.

Let all conditions of Theorem 1.6.7 hold. Then for each $S\in{\cal S}$

$\langle Y,E^{\prime\prime}(S)Y\rangle>\nu\|Y\|_{\cal W}^{2},\qquad Y\in
N_{S}{\cal S}\cap T_{S}{\cal M},$ (1.6.29)

where $\nu>0$.

###### Proof.

It suffices to prove (1.6.29) for $S=(\psi_{0},0,0)$. Note that
$E^{\prime\prime}(S)$ is not complex linear due to the integral in (1.1.9).
Hence, we express the action of $E^{\prime\prime}(S)$ in $\varphi_{1}(x):={\rm
Re{\hskip 1.42262pt}}\varphi(x)$ and $\varphi_{1}(x):={\rm Im{\hskip
1.42262pt}}\varphi(x)$: formulas (1.6.4) and (1.6.6) imply that

$E^{\prime\prime}(S)Y=\left(\begin{array}[]{cccl}2H_{0}+4e^{2}\psi_{0}G\psi_{0}&0&2L&0\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ 0&2H_{0}&0&0\vskip 6.0pt plus 2.0pt minus
2.0pt\\\ 2L^{{\hskip 1.42262pt}*}&0&T&0\\\ 0&0&0&M^{-1}\\\
\end{array}\right)Y\qquad{\rm for}\quad
Y=\left(\begin{array}[]{c}\varphi_{1}\\\ \varphi_{2}\\\ \varkappa\\\
\pi\end{array}\right),$ (1.6.30)

where $H_{0}=-\frac{1}{2}\Delta$ , the operator $L$ corresponds to the matrix

$L(x,n):=e\psi_{0}(x)G\nabla\sigma(x-n):~{}~{}~{}~{}x\in{\mathbb
R}^{3},~{}n\in\Gamma,$ (1.6.31)

and $T$ corresponds to the real matrix with the entries

$T(n-n^{\prime}):=-\displaystyle\langle
G\nabla\otimes\nabla\sigma(x-n^{\prime}),\sigma(x-n)\rangle,\quad
n,n^{\prime}\in\Gamma.$ (1.6.32)

Thus, $E^{\prime\prime}(S)$ is the selfadjoint operator in ${\cal X}$ with the
discrete spectrum, and (1.6.20) implies that the minimal eigenvalue of
$E^{\prime\prime}(S)$ in the invariant space $N_{S}{\cal S}\cap T_{S}{\cal M}$
is positive. Therefore, (1.6.29) follows. ∎

The positivity (1.6.29) implies the lower energy estimate (1.6.28), since the
higher-order terms in (1.6.1) are negligible by the following lemma.

######  Lemma 1.6.9.

Let $\sigma(x)$ satisfy (1.1.1). Then the remainder (1.6.7) admits the
estimate

$|R(S,Y)|\leq C\|Y\|_{\cal W}^{3}\quad\,\,\,{\rm for}\quad\,\,\,\|Y\|_{\cal
W}\leq 1.$ (1.6.33)

###### Proof.

It suffices to prove the estimates

$\|\sqrt{G}\rho^{(1)}\|_{L^{2}({{\mathbb T}})}\leq C_{1}\|Y\|_{\cal
W},\quad\|\sqrt{G}\rho^{(2)}\|_{L^{2}({{\mathbb T}})}\leq C_{2}\|Y\|_{\cal
W}^{2}\quad\,\,\,{\rm for}\,\,\,\quad\|Y\|_{\cal W}\leq 1.$ (1.6.34)

Then (1.6.33) will follow from (1.6.7).
i) By (1.6.4) we have for $Y=(\varphi,\varkappa,\pi)$

$\sqrt{G}\rho^{(1)}=\sqrt{G}\sigma^{(1)}-2e\sqrt{G}\psi^{\alpha}(x)\cdot\varphi(x).$
(1.6.35)

The operator $\sqrt{G}$ is bounded in $L^{2}({\mathbb R}^{3})$ by the
definition (1.2.1). Hence,

$\|\sqrt{G}\sigma^{(1)}\|_{L^{2}({{\mathbb T}})}\leq C|\varkappa|$ (1.6.36)

by (1.6.4). Applying to the second term the Cauchy-Schwarz and Hausdorff-Young
inequalities, we obtain

$\|\sqrt{G}\psi^{\alpha}(x)\cdot\varphi\|_{L^{2}({{\mathbb T}})}\leq
C\Big{[}\sum_{\xi\in{\Gamma^{*}_{N}}}\frac{|\hat{\varphi}(\xi)|^{2}}{|\xi|^{2}}\Big{]}^{1/2}\leq
C\|\hat{\varphi}\|_{L^{4}({\Gamma^{*}_{N}})}\Big{[}\sum_{\xi\in{\Gamma^{*}_{N}}}|\xi|^{-4}\Big{]}^{1/2}\leq
C\|\varphi\|_{L^{4/3}({{\mathbb T}})}.$ (1.6.37)

Hence, the first inequality (1.6.34) is proved.
ii) Now we prove the second inequality (1.6.34). By (1.6.5) we have for
$Y=(\varphi,\varkappa,\pi)$

$\sqrt{G}\rho^{(2)}(x)=\sqrt{G}\sigma^{(2)}(x)-e\sqrt{G}|\varphi(x)|^{2}.$
(1.6.38)

Similarly to (1.6.36)

$\|\sqrt{G}\sigma^{(2)}\|_{L^{2}({{\mathbb T}})}\leq C|\varkappa|^{2}.$
(1.6.39)

Finally, denoting $\beta(x):=|\varphi(x)|^{2}$, we obtain similarly to
(1.6.37)

$\displaystyle\|\sqrt{G}|\varphi(x)|^{2}\|_{L^{2}({{\mathbb T}})}$
$\displaystyle\leq$ $\displaystyle
C\Big{[}\sum_{\xi\in{\Gamma^{*}_{N}}}\frac{|\hat{\beta}(\xi)|^{2}}{|\xi|^{2}}\Big{]}^{1/2}\leq
C\|\hat{\beta}\|_{L^{4}({\Gamma^{*}_{N}})}\Big{[}\sum_{\xi\in{\Gamma^{*}_{N}}}|\xi|^{-4}\Big{]}^{1/2}$
(1.6.40) $\displaystyle\leq$ $\displaystyle C_{1}\|\beta\|_{L^{4/3}({{\mathbb
T}})}=C_{1}\|\varphi\|_{L^{8/3}({{\mathbb T}})}^{2}\leq
C_{2}\|\varphi\|_{H^{1}({{\mathbb T}})}^{2}$

by the Sobolev embedding theorem [1]. Now the lemma is proved. ∎

###  Chapter 2 N-particle Schrödinger theory

Abstract

In this chapter, we consider the coupled Schrödinger–Poisson–Newton equations
for finite crystals with moving ions and antisymmetric $N$-particle wave
functions. This antisymmetry expresses the Pauli exclusion principle. The
dynamics of electron field is described by the many-particle Schrödinger
equation.

We construct global dynamics, prove the conservation of energy and charge, and
give the description of all ground states. Our main result is the orbital
stability of every ground state with periodic arrangement of ions under the
same ‘Jellium’ and Wiener-type conditions on the ion charge density (1.1.11),
(1.1.13). The Pauli exclusion principle plays the key role in the proof of
stability, see Remark 2.7.2.

####  2.1 Introduction

As in Chapter I, we consider crystals which occupy the finite torus ${\mathbb
T}:={\mathbb R}^{3}/N{\mathbb Z}^{3}$ and have one ion per cell of the cubic
lattice $\Gamma:={\mathbb Z}^{3}/N{\mathbb Z}^{3}$, where $N\in{\mathbb N}$
and $N>1$. We also assume the conditions (1.1.1). Now we denote

$\overline{\mathbb T}:={\mathbb
T}^{\overline{N}}:=\\{\overline{x}=(x_{1},...,x_{\overline{N}}):x_{j}\in{\mathbb
T},\quad j=1,...,\overline{N}\\},\qquad\overline{N}:=N^{3}.$ (2.1.1)

######  Definition 2.1.1.

${\cal F}$ is the ‘fermionic’ Hilbert space of complex antisymmetric functions
$\psi(x_{1},...,x_{\overline{N}})$ on $\overline{\mathbb T}$ with the norm

$\|\psi\|_{\cal F}^{2}:=\|\nabla^{\otimes}\psi\|_{L^{2}(\overline{\mathbb
T})}^{2}+\|\psi\|_{L^{2}(\overline{\mathbb T})}^{2},$ (2.1.2)

where $\nabla^{\otimes}$ denotes the gradient with respect to
$\overline{x}\in\overline{\mathbb T}$.

Let $\psi(\cdot,t)\in{\cal F}$ for $t\in{\mathbb R}$ be the antisymmetric wave
function of the fermionic electron field, $q(n,t)$ denotes the ion
displacement from the reference position $n\in\Gamma$, and $\phi(x,t)$ be the
electrostatic potential generated by the ions and electrons. We assume
$\hbar=c={\rm m}=1$, where $c$ is the speed of light and ${\rm m}$ is the
electron mass. Let us denote the ‘second quantized’ operators on ${\cal F}$,

$\Delta^{\otimes}:=\sum_{j=1}^{\overline{N}}\Delta_{x_{j}};\qquad\phi^{\otimes}(\overline{x},t):=\sum_{j=1}^{\overline{N}}\phi(x_{j},t).$
(2.1.3)

The coupled Schrödinger-Poisson-Newton equations read as follows

$\displaystyle i\dot{\psi}(\overline{x},t)\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!-\frac{1}{2}\Delta^{\otimes}\psi(\overline{x},t)-e\phi^{\otimes}(\overline{x},t)\psi(\overline{x},t),\qquad\overline{x}\in\overline{\mathbb
T},$ (2.1.4) $\displaystyle-\Delta\phi(x,t)\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\rho(x,t):=\sum_{n\in\Gamma}\sigma(x-n-q(n,t))+\rho^{e}(x,t),\qquad
x\in{\mathbb T},$ (2.1.5) $\displaystyle M\ddot{q}(n,t)\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!-({\bf\nabla}\phi(x,t),\sigma(x-n-q(n,t))),\qquad
n\in\Gamma.$ (2.1.6)

Here the brackets $(\cdot,\cdot)$ stand for the scalar product on the real
Hilbert space $L^{2}({\mathbb T})$ and for its different extensions, $M>0$ is
the mass of one ion, and the electronic charge density is defined by

$\rho^{e}(x,t):=-e\int_{\overline{\mathbb
T}}\sum_{j=1}^{\overline{N}}\delta(x-y_{j})|\psi(\overline{y},t)|^{2}\,d\overline{y},~{}~{}x\in{\mathbb
T}.$ (2.1.7)

Similar finite periodic approximations of crystals are treated in all
textbooks on quantum theory of solid state [9, 34, 75]. However, the stability
of ground states in this model was newer discussed.

The normalization (1.1.7) now reads,

$\|\psi(\cdot,t)\|_{L^{2}(\overline{\mathbb T})}^{2}=Z,\qquad t\in{\mathbb
R}.$ (2.1.8)

Hence, as in Chapter I, the potential $\phi(x,t)$ can be eliminated from the
system (2.1.4)–(2.1.6) using the operator $G:=(-\Delta)^{-1}$ defined in
(1.2.1). Substituting $\phi(\cdot,t)=G\rho(\cdot,t)$ into the remaining
equations (2.1.4) and (2.1.6) we can write these equations as

$\dot{X}(t)=F(X(t)),\qquad t\in{\mathbb R},$ (2.1.9)

where $X(t)=(\psi(\cdot,t),q(\cdot,t),p(\cdot,t))$ with
$p(\cdot,t):=\dot{q}(\cdot,t)$. We will identify complex functions
$\psi(\overline{x})$ with two real functions $\psi_{1}(\overline{x}):={\rm
Re{\hskip 1.42262pt}}\psi(\overline{x})$ and $\psi_{2}(\overline{x}):={\rm
Im{\hskip 1.42262pt}}\psi(\overline{x})$. With these notations, equation
(2.1.9) is equivalent to the Hamiltonian system

$\partial_{t}\psi_{1}(\overline{x},t)=\frac{1}{2}\partial_{\psi_{2}(\overline{x})}E,~{}~{}\partial_{t}\psi_{2}(\overline{x},t)=-\frac{1}{2}\partial_{\psi_{1}(\overline{x})}E,~{}~{}\partial_{t}q(n,t)=\partial_{p(n)}E,~{}~{}\partial_{t}p(n,t)=-\partial_{q(n)}E$
(2.1.10)

together with the normalisation condition (2.1.8). Now the Hamiltonian
functional (energy) reads

$E(\psi,q,p)=\frac{1}{2}\int_{\overline{\mathbb T}}|\nabla^{\otimes}{\hskip
1.42262pt}\psi(\overline{x})|^{2}\,d\overline{x}+\frac{1}{2}(\rho,G\rho)+\sum_{n\in\Gamma}\frac{p^{2}(n)}{2M}.$
(2.1.11)

As in (1.1.10), the total charge density $\rho(x)$ is the sum of the ion and
electronic charge densities,

$\rho(x):=\rho^{i}(x)+\rho^{e}(x),\quad\rho^{i}(x):=\sum\limits_{n\in\Gamma}\sigma(x-n-q(n)),\qquad
x\in{\mathbb T}.$ (2.1.12)

Similarly to notations of Chapter I, we denote the Hilbert manifolds

${\cal V}:={\cal F}\otimes\overline{\mathbb T}\otimes{\mathbb
R}^{3\overline{N}},\qquad{\cal M}:=\\{X\in{\cal V}:Q(X)=Z\\}.$ (2.1.13)

We prove the global well-posedness of the dynamics: for any $X(0)\in{\cal M}$
there exists a unique solution $X(t)\in C({\mathbb R},{\cal V})$ to (2.1.10),
and the energy and charge conservations hold:

$E(X(t))=E(X(0)),\quad Q(X(t))=Q(X(0)),\qquad t\in{\mathbb R}.$ (2.1.14)

Our main goal is the stability of ground states, i.e., solutions to (2.1.10)
with minimal energy (2.1.11) and with $\Gamma$-periodic arrangement of ions
under the same Jellium and the Wiener conditions (1.1.11) and (1.1.13) onto
the ion densities $\sigma(x)$. Nonperiodic arrangements exist for some
degenerate densities $\sigma$ as in Section 1.5.3.

The energy (2.1.11) is nonnegative, and its minimum is zero. We show that
under the Jellium condition all ground states with $\Gamma$-periodic
arrangement of ions have the form

$S(t):=(\psi_{0}e^{-i\omega_{0}t},\overline{r},0),~{}~{}~{}~{}~{}r\in{\mathbb
T}.$ (2.1.15)

Here

$\overline{r}\in\overline{\mathbb T}:\quad\overline{r}(n)=r,\,\,n\in\Gamma,$
(2.1.16)

while $\psi_{0}\in{\cal F}$ is a normalised eigenfunction

$-\frac{1}{2}\Delta^{\otimes}{\hskip
1.42262pt}\psi_{0}(\overline{x})=\omega_{0}\psi_{0}(\overline{x}),\qquad\overline{x}\in\overline{\mathbb
T};\qquad\|\psi_{0}\|_{L^{2}(\overline{\mathbb T})}=1,$ (2.1.17)

which corresponds to the minimal eigenvalue $\omega_{0}:=\min{\hskip
1.42262pt}{\hskip 1.42262pt}\mathop{\rm
Spec\,}\nolimits(-\frac{1}{2}\Delta^{\otimes}).$

We establish the stability of the real 4-dimensional ‘solitary manifold’

${\cal
S}=\\{S_{\alpha,r}=(\psi^{\alpha},\overline{r},0):~{}\psi^{\alpha}(\overline{x})\equiv
e^{i\alpha}\psi_{0}(\overline{x}),~{}~{}\alpha\in[0,2\pi];~{}~{}r\in{\mathbb
T}\\},$ (2.1.18)

where $\psi_{0}$ is a fixed eigenfunction, satisfying the additional
restriction (2.3.3). The normalization (2.1.8) and the identity (2.1.17) imply
that

$E(S)=\omega_{0}Z,\qquad S\in{\cal S}.$ (2.1.19)

The main result of this chapter is the following theorem.

######  Theorem 2.1.2.

Let the Jellium and Wiener conditions (1.1.11) and (1.1.13) hold as well as
(2.3.3). Then for any $\varepsilon>0$ there exists
$\delta=\delta(\varepsilon)>0$ such that for $X(0)\in{\cal M}$ with $d_{\cal
V}(X(0),{\cal S})<\delta$ we have

$d_{\cal V}(X(t),{\cal S})<\varepsilon,\qquad t\in{\mathbb R},$ (2.1.20)

where $X\in C({\mathbb R},{\cal V})$ is the corresponding solution to
(2.1.10).

In Chapter I, we have proved similar result for the system (1.1.2)–(1.1.4)
with the one-particle Schrödinger equation instead of (2.1.4). The general
plan of analysis in the present chapter is also similar. However, now all
estimates and the uniformity (1.1.18) require completely new arguments.

Let us comment on our approach. We prove the local well-posedness for the
system (2.1.10) by the contraction mapping principle. The global well-
posedness we deduce from the energy conservation which follows by the Galerkin
approximations.

The orbital stability of the solitary manifold ${\cal S}$ is deduced from the
lower energy estimate

$E(X)-\omega_{0}Z\geq\nu\,d^{2}(X,{\cal S})\qquad{\rm if}\qquad d(X,{\cal
S})\leq\delta,\qquad X\in{\cal M},$ (2.1.21)

where $\nu,\delta>0$ and ‘$d$’ is the distance in the ‘energy norm’. We deduce
this estimate from the positivity of the Hessian $E^{\prime\prime}(S)$ for
$S\in{\cal S}$ in the orthogonal directions to ${\cal S}$ on the manifold
${\cal M}$. We show that the Wiener condition (1.1.13) is necessary for this
positivity under the Jellium condition (1.1.11). We expect that this condition
is also necessary for the positivity of $E^{\prime\prime}(S)$; however, this
is still an open challenging problem.

This chapter is organized as follows. In Section 2 we introduce function
spaces. In Section 3 we collect all our assumptions. In Section 4 we describe
all fermionic jellium ground states and give basic examples. In Section 5 we
prove the stability of the solitary manifold ${\cal S}$ establishing the
positivity the energy Hessian. In Appendices we construct the global dynamics.

####  2.2 Function spaces and integral equation

Equation (2.1.9) with the normalization (2.1.8) is equivalent, up to a gauge
transform, to the system (2.1.4)–(2.1.6). The system (2.1.10) can be written
as

$\dot{X}(t)=JE^{\prime}(X(t)),$ (2.2.1)

where $J$ is the matrix (1.2.4). For $\psi,\varphi\in L^{2}(\overline{\mathbb
T})$ denote

$(\psi,\varphi):={\rm Re{\hskip 1.42262pt}}\int_{\overline{\mathbb
T}}\psi(\overline{x})\cdot\varphi(\overline{x})d\overline{x},$ (2.2.2)

where $\cdot$ is the inner product of the corresponding vectors in ${\mathbb
R}^{2}$.

######  Definition 2.2.1.

i) Denote the real Hilbert spaces

${\cal X}:=L^{2}(\overline{\mathbb T})\oplus{\mathbb
R}^{3\overline{N}}\oplus{\mathbb R}^{3\overline{N}},\qquad{\cal W}:={\cal
F}\oplus{\mathbb R}^{3\overline{N}}\oplus{\mathbb R}^{3\overline{N}}.$

ii) ${\cal V}:={\cal F}\otimes\overline{\mathbb T}\otimes{\mathbb
R}^{3\overline{N}}$ is the real Hilbert manifold endowed with the metric

$d_{{\cal V}}(X,X^{\prime}):=\|\psi-\psi^{\prime}\|_{{\cal
F}}+|q-q^{\prime}|+|p-p^{\prime}|,\qquad X=(\psi,q,p),\quad
X^{\prime}=(\psi^{\prime},q^{\prime},p^{\prime})$ (2.2.3)

and with the ‘quasinorm’

$|X|_{{\cal V}}:=\|\psi\|_{{\cal F}}+|p|,\qquad X=(\psi,q,p).$ (2.2.4)

The linear space ${\cal W}$ is isomorphic to the tangent space to the Hilbert
manifold ${\cal V}$ at each point $X\in{\cal V}$. As in Chapter I, we will use
notation (1.2.10) and the inequality (1.2.12). The system (2.2.1) in the
integral form reads similarly to (1.2.15),

$\left\\{\begin{array}[]{lll}\psi(t)&=&e^{-iH_{0}t}\psi(0)+ie\displaystyle\int_{0}^{t}e^{-iH_{0}(t-s)}[\phi^{\otimes}(s)\psi(s)]ds,\\\
\\\ q(n,t)&=&q(n,0)+\frac{1}{M}\displaystyle\int_{0}^{t}p(n,s)ds\mod N{\mathbb
Z}^{3},\\\ \\\
p(n,t)&=&p(n,0)-\displaystyle\int_{0}^{t}(\nabla\phi(s),\sigma(\cdot-
n-q(n,s)))ds,\end{array}\right|$ (2.2.5)

where $H_{0}=-\frac{1}{2}\Delta^{\otimes}$ and $\phi(s):=G\rho(s)$. In the
vector form (2.2.5) reads

$X(t)=e^{-tA}X(0)+\int_{0}^{t}e^{-(t-s)A}N(X(s))ds,\qquad
A=\left(\begin{array}[]{ccc}iH_{0}&0&0\\\ 0&0&0\\\ 0&0&0\end{array}\right),$
(2.2.6)

where

$N(X)=(ie\phi^{\otimes}{\hskip
1.42262pt}\psi~{},p,~{}f),\,\,\,f(n):=-(\nabla\phi,\sigma(\cdot-
n-q(n))),\,\,\,\phi:=G\rho,$ (2.2.7)

and $\rho$ is now defined by (2.1.12), (2.1.7).

####  2.3 Main assumptions

We assume the same Jellium and the Wiener conditions (1.1.11) and (1.1.13)
onto the ion densities $\sigma(x)$. However, now we need an additional
condition for the orbital stability of the ground state (2.1.15). Namely,
recall that the exterior product of functions $f_{j}\in L^{2}({\mathbb T})$ is
defined by

$[\Lambda_{j=1}^{\overline{N}}f_{j}](\overline{x}):=\frac{1}{\sqrt{\overline{N}!}}\sum_{\pi\in
S_{\overline{N}}}(-1)^{|\pi|}\prod\limits_{j=1}^{\overline{N}}f_{j}(x_{\pi(j)}),\qquad\overline{x}=(x_{1},...,x_{\overline{N}})\in\overline{\mathbb
T},$ (2.3.1)

where $S_{\overline{N}}$ is the symmetric group and $|\pi|$ denotes the sign
(or parity) of a transposition $\pi$. Every eigenfunction (2.1.17) admits an
expansion in the exterior products

$\psi_{0}(\overline{x})=\sum_{\overline{k}}C(\overline{k})\wedge_{j=1}^{\overline{N}}e^{ik_{j}x_{j}}.\qquad
k_{j}\in\Gamma^{*}_{N}:=\frac{2\pi}{N}{\mathbb Z}^{3}.$ (2.3.2)

Here $\overline{k}:=\\{k_{1},...,k_{\overline{N}}\\}$, where $k_{j}$ are
different for distinct $j$, and
$\frac{1}{2}\sum_{j=1}^{\overline{N}}k_{j}^{2}=\omega_{0}$. We will consider
the eigenfunctions (2.3.2) with the additional restriction

$\\#(\overline{k}\setminus\overline{k}^{\prime})\geq 2\quad{\rm
if}\quad\overline{k}\neq\overline{k}^{\prime}.$ (2.3.3)

This condition implies that the corresponding electronic charge density is
uniform (see Lemma 1.6.9),

$\rho^{e}(x)\equiv-eZ,\qquad x\in{\mathbb T}.$ (2.3.4)

This identity plays a crucial role in our approach. It implies that the
corresponding total charge density (2.1.12) identically vanishes by (1.1.12).
Let us emphasize that both ionic and electronic charge densities are uniform
for the ground state under the Jellium condition together with (2.3.3).

####  2.4 Global dynamics

Here we prove the global well-posedness of the system (2.2.1).

######  Theorem 2.4.1.

Let (1.1.1) hold and $X(0)\in{\cal M}$. Then
i) There exists a unique solution $X(t)\in C({\mathbb R},{\cal V})$ to
(2.2.1).
ii) The energy and charge conservations (2.1.14) hold.

First we construct the local solutions by contraction arguments. To construct
the global solutions we will prove energy conservation using the Galerkin
approximations.

######  Theorem 2.4.2.

(Local well-posedness). Let (1.1.1) hold and
$X(0)=(\psi_{0},q_{0},p_{0})\in{\cal V}$ with $|X(0)|_{\cal V}\leq R$. Then
there exists $\tau=\tau(R)>0$ such that equation (2.2.1) has a unique solution
$X\in C([-\tau,\tau],{\cal V})$, and the maps $U(t):X(0)\mapsto X(t)$ are
continuous in ${\cal V}$ for $t\in[-\tau,\tau]$.

In the next two propositions we prove the boundedness and the local Lipschitz
continuity of the nonlinearity $N:{\cal V}\to{\cal W}$ defined in (2.2.7).
With this proviso Theorem 1.3.2 follows from the integral form (2.2.6) of the
equation (2.2.1) by the contraction mapping principle, since $e^{-At}$ is an
isometry of ${\cal W}$. First, we prove the boundedness of $N$.

######  Proposition 2.4.3.

For any $R>0$ and $X=(\psi,q,p)\in{\cal V}$

$\|N(X)\|_{\cal W}\leq C(R)\qquad{\rm for}\quad|X|_{\cal V}\leq R.$ (2.4.1)

###### Proof.

We need appropriate bounds for the charge density $\rho$ and for the
corresponding potential $\phi$.

######  Lemma 2.4.4.

The charge density (2.1.12) admits the bounds

$\|\rho\|_{L^{3}({\mathbb T})}+\|\nabla\rho\|_{L^{3/2}({\mathbb T})}\leq
C(1+\|\psi\|_{\cal F}^{2}).$ (2.4.2)

###### Proof.

We have $\rho(x)=\rho^{i}(x)+\rho^{e}(x)$ by (2.1.12). The bound (2.4.2) for
$\rho^{i}$ holds by (1.1.1). It remains to prove the bound for $\rho^{e}$.
Definition (2.1.7) implies that

$\rho^{e}(x)=-e\sum_{j=1}^{\overline{N}}\int_{{\mathbb
T}^{\overline{N}-1}}|\psi(\overline{y})|^{2}\Big{|}_{y_{j}=x}\,dx_{1}...\widehat{dy_{j}}...dy_{\overline{N}},\quad
x\in{\mathbb T},$ (2.4.3)

where the hat means that this differential is omitted. Differentiating, we
obtain that

$\nabla\rho^{e}(x)=-e\sum_{j=1}^{\overline{N}}\int_{{\mathbb
T}^{\overline{N}-1}}\nabla_{y_{j}}|\psi(\overline{y})|^{2}\Big{|}_{y_{j}=x}\,dy_{1}...\widehat{dy_{j}}...dy_{\overline{N}},\quad
x\in{\mathbb T}.$ (2.4.4)

Applying the Cauchy–Schwarz inequality to (2.4.3), we get

$\displaystyle\|\rho^{e}(x)\|_{L^{3}({\mathbb T})}$ $\displaystyle\leq$
$\displaystyle C\sum_{j=1}^{\overline{N}}\int_{{\mathbb
T}^{\overline{N}-1}}[\int_{\mathbb
T}|\psi(\overline{y})|^{6}dy_{j}]^{1/3}\,dy_{1}...\widehat{dy_{j}}...dy_{\overline{N}}$
(2.4.5) $\displaystyle\leq$ $\displaystyle\int_{{\mathbb
T}^{\overline{N}-1}}[\int_{\mathbb
T}|\nabla_{y_{j}}\psi(\overline{y})|^{2}dy_{j}]\,dy_{1}...\widehat{dy_{j}}...dy_{\overline{N}}\leq
C\|\psi\|_{H^{1}(\overline{\mathbb T})}^{2}$

by the Sobolev embedding theorem [1, Theorem 5.4, Part I]. Similarly, (2.4.4)
implies that

$\displaystyle\|\nabla\rho^{e}(x)\|_{L^{3/2}({\mathbb T})}$
$\displaystyle\leq$ $\displaystyle C\sum_{j=1}^{\overline{N}}\int_{{\mathbb
T}^{\overline{N}-1}}[\int_{\mathbb
T}|\psi(\overline{y})\nabla_{y_{j}}\overline{\psi}(\overline{y})|^{3/2}dy_{j}]^{2/3}\,dy_{1}...\widehat{dy_{j}}...dy_{\overline{N}}$
(2.4.6) $\displaystyle\leq$ $\displaystyle\int_{{\mathbb
T}^{\overline{N}-1}}[\int_{\mathbb
T}|\psi(\overline{y})|^{6}dy_{j}]^{1/6}[\int_{\mathbb
T}|\nabla_{y_{j}}\psi(\overline{y})|^{2}dy_{j}]^{1/2}\,dy_{1}...\widehat{dy_{j}}...dy_{\overline{N}}$
$\displaystyle\leq$ $\displaystyle\int_{{\mathbb
T}^{\overline{N}-1}}[\int_{\mathbb
T}|\nabla_{y_{j}}\psi(\overline{y})|^{2}dy_{j}]\,dy_{1}...\widehat{dy_{j}}...dy_{\overline{N}}\leq
C\|\psi\|_{H^{1}(\overline{\mathbb T})}^{2}$

by the Hölder inequality and the Sobolev embedding theorem. ∎

######  Lemma 2.4.5.

The potential $\phi:=G\rho$ admits the bound

$\|\phi\|_{C({\mathbb T})}+\|\nabla\phi\|_{L^{3}({\mathbb T})}\leq
C(1+\|\psi\|_{\cal F}^{2}).$ (2.4.7)

###### Proof.

Applying the Hölder and Hausdorff–Young inequalities to (1.2.1), we obtain
that

$\|\phi\|_{C({\mathbb T})}\leq
C\|\frac{\tilde{\rho}(\xi)}{\xi^{2}}\|_{L^{1}(\Gamma^{*}_{N}\setminus 0)}\leq
C_{1}\|\xi\tilde{\rho}\|_{L^{3}(\Gamma^{*}_{N})}\Big{[}\sum_{\xi\in\Gamma^{*}_{N}\setminus
0}|\xi|^{-9/2}\Big{]}^{2/3}\leq C_{2}\|\nabla\rho\|_{L^{3/2}({\mathbb T})}.$
(2.4.8)

Similarly,

$\|\nabla\phi\|_{L^{3}({\mathbb T})}\leq
C\|\frac{\tilde{\rho}(\xi)}{|\xi|}\|_{L^{3/2}(\Gamma^{*}_{N}\setminus 0)}\leq
C_{1}\|\xi\tilde{\rho}\|_{L^{3}(\Gamma^{*}_{N})}\Big{[}\sum_{\xi\in\Gamma^{*}_{N}\setminus
0}|\xi|^{-6}\Big{]}^{1/3}\leq C_{2}\|\nabla\rho\|_{L^{3/2}({\mathbb T})}.$
(2.4.9)

Now the bound (2.4.7) follows from (2.4.2). ∎

Now we can prove the estimate (1.3.2). First, we will prove

$\|\phi^{\otimes}{\hskip 1.42262pt}\psi\|_{\cal F}\leq C(1+\|\psi\|_{\cal
F}^{3})$ (2.4.10)

in the notation (2.1.3). According to definition (2.1.2) it suffices to check
that

$\|\phi^{\otimes}{\hskip 1.42262pt}\psi\|_{L^{2}(\overline{\mathbb
T})}+\|\phi^{\otimes}\,\nabla^{\otimes}{\hskip
1.42262pt}\psi\|_{L^{2}(\overline{\mathbb
T})}+\|\psi\nabla^{\otimes}\,\phi^{\otimes}\|_{L^{2}(\overline{\mathbb
T})}\leq C(1+\|\psi\|_{\cal F}^{3}).$ (2.4.11)

The first two summands admit the needed estimate by (2.4.7). The third summand
requires some additional argument. Namely,

$\displaystyle\|\psi\nabla^{\otimes}\,\phi^{\otimes}{\hskip
1.42262pt}\|_{L^{2}(\overline{\mathbb T})}^{2}$ $\displaystyle=$
$\displaystyle\int_{\overline{\mathbb
T}}|\sum_{j=1}^{\overline{N}}\nabla\phi(x_{j})\psi(\overline{x})|^{2}\,d\overline{x}\leq
C\sum_{1}^{\overline{N}}\int_{\overline{\mathbb
T}}|\nabla\phi(x_{j})\psi(\overline{x})|^{2}\,d\overline{x}$ (2.4.12)
$\displaystyle=$ $\displaystyle C\sum_{j=1}^{\overline{N}}\int_{{\mathbb
T}^{\overline{N}-1}}\Big{[}\int_{{\mathbb
T}}|\nabla\phi(x_{j})\psi(\overline{x})|^{2}\,dx_{j}\Big{]}dx_{1}...\widehat{dx_{j}}...dx_{\overline{N}}.$

The inner integral is estimated as follows

$\int_{\mathbb
T}|\nabla\phi(x_{j})\psi(\overline{x})|^{2}\,dx_{j}\leq\|\nabla\phi\|_{L^{3}({\mathbb
T})}^{2}\Big{[}\int_{\mathbb
T}|\psi(\overline{x})|^{6}\,dx_{j}\Big{]}^{1/3}\leq
C\|\nabla\phi\|_{L^{3}({\mathbb T})}^{2}\int_{\mathbb
T}|\nabla_{x_{j}}\psi(\overline{x})|^{2}\,dx_{j}$ (2.4.13)

by the Hölder inequality and the Sobolev embedding theorem. Substituting this
estimate into (2.4.12), we obtain

$\|\psi\nabla^{\otimes}\,\phi^{\otimes}\|_{L^{2}(\overline{\mathbb T})}\leq
C\|\nabla\phi\|_{L^{3}({\mathbb T})}\|\psi\|_{\cal F}.$ (2.4.14)

This and (2.4.7) imply (2.4.11) for the third summand. Finally, using (2.2.7),
(2.4.7) and (1.1.1),

$|f(n)|\leq\|\phi\|_{C({\mathbb T})}\|\nabla\sigma\|_{L^{1}({\mathbb T})}\leq
C(1+\|\psi\|_{\cal F}^{2}),\qquad n\in\Gamma.$ (2.4.15)

Hence, (2.4.10) and (2.4.15) imply (2.4.1). Proposition 2.4.3 is proved. ∎

It remains to prove that the nonlinearity is locally Lipschitz.

######  Proposition 2.4.6.

For any $R>0$ and $X_{1},X_{2}\in{\cal V}$ with $|X_{1}|_{\cal
V},|X_{2}|_{\cal V}\leq R$

$\|N(X_{1})-N(X_{2})\|_{\cal W}\leq C(R)d_{\cal V}(X_{1},X_{2}).$ (2.4.16)

###### Proof.

Writing $X_{k}=(\psi_{k},q_{k},p_{k})$ and $\phi_{k}=G\rho_{k}$, we obtain
that

$\|\phi^{\otimes}_{1}{\hskip 1.42262pt}\psi_{1}-\phi^{\otimes}_{2}{\hskip
1.42262pt}\psi_{2}\|_{\cal
F}\leq\|(\phi^{\otimes}_{1}-\phi^{\otimes}_{2}){\hskip
1.42262pt}\psi_{1}\|_{\cal F}+\|\phi^{\otimes}_{2}{\hskip
1.42262pt}(\psi_{1}-\psi_{2})\|_{\cal F}.$ (2.4.17)

Using (2.4.14) and (2.4.7), we obtain

$\displaystyle\|\phi^{\otimes}_{2}{\hskip
1.42262pt}(\psi_{1}-\psi_{2})\|_{\cal F}$ $\displaystyle\leq$
$\displaystyle(\|\phi_{2}\|_{C({\mathbb
T})}+\|\nabla\phi_{2}\|_{L^{3}({\mathbb T})})\|\psi_{1}-\psi_{2}\|_{\cal F}$
(2.4.18) $\displaystyle\leq$ $\displaystyle
C(1+R^{2})\|\psi_{1}-\psi_{2}\|_{\cal F}\leq C(R)d_{\cal V}(X_{1},X_{2}).$

Further, (2.4.8) and (2.4.9) give that

$\|\phi_{1}-\phi_{2}\|_{C({\mathbb
T})}+\|\nabla\phi_{1}-\nabla\phi_{2}\|_{L^{3}({\mathbb
T})}\leq\|\nabla\rho_{1}-\nabla\rho_{2}\|_{L^{3/2}({\mathbb T})}$ (2.4.19)

However, $|\sigma(x)-\sigma(x-a)|\leq C|a|$, where $|a|:=\min_{r\in a}|r|$ for
$a\in{\mathbb T}$ (by definition, $a\subset{\mathbb R}^{3}$ is a class of
equivalence mod $N{\mathbb Z}^{3}$). Therefore, as in (2.4.2),

$\|\nabla(\rho_{1}-\rho_{2})\|_{L^{3/2}({\mathbb T})}\leq
CR(|q_{1}-q_{2}|+\|\psi_{1}-\psi_{2}\|_{\cal F}).$ (2.4.20)

Hence,

$\displaystyle\|(\phi^{\otimes}_{1}-\phi^{\otimes}_{2})\psi_{1}\|_{\cal F}$
$\displaystyle\leq$ $\displaystyle(\|\phi_{1}-\phi_{2}\|_{C({\mathbb
T})}+\|\nabla(\phi_{1}-\phi_{2})\|_{L^{3}({\mathbb T})})\|\psi_{1}\|_{\cal F}$
(2.4.21) $\displaystyle\leq$ $\displaystyle
CR\|\nabla(\rho_{1}-\rho_{2})\|_{L^{3/2}({\mathbb T})}\|\psi_{1}\|_{\cal
F}\leq C(R)d_{\cal V}(X_{1},X_{2}).$

Now (2.4.17)–(2.4.21) give

$\|\phi^{\otimes}_{1}{\hskip 1.42262pt}\psi_{1}-\phi^{\otimes}_{2}{\hskip
1.42262pt}\psi_{2}\|_{H^{1}({\mathbb T})}\leq C(R)d_{\cal V}(X_{1},X_{2}).$
(2.4.22)

Similarly, (2.4.19), (2.4.20) and (2.4.7) imply

$\displaystyle\|\langle\nabla\phi_{1},\sigma(\cdot-n-
q_{1}(n))\rangle-\langle\nabla\phi_{2},\sigma(\cdot-n-q_{2}(n))\rangle\|$
(2.4.23) $\displaystyle\leq$
$\displaystyle\|\langle\nabla(\phi_{1}-\phi_{2}),\sigma(\cdot-n-
q_{1}(n))\rangle\|+\|\langle\nabla\phi_{2},\sigma(\cdot-n-
q_{1}(n))-\sigma(\cdot-n-q_{2}(n))\rangle\|$ $\displaystyle\leq$
$\displaystyle C(\|\phi_{1}-\phi_{2}\|_{C({\mathbb
T})}+\|\phi_{2}\|_{C({\mathbb T})})|q_{1}-q_{2}|)\leq C(R)d_{\cal
V}(X_{1},X_{2}).$

This estimate together with (2.4.22) prove (2.4.16). ∎

Now Theorem 2.4.2 follows from Propositions 2.4.3 and 2.4.6.
Proof of Theorem 2.4.1. The local solution $X\in C([-\tau,\tau],{\cal V})$ to
(2.2.1) exists and is unique by Theorem 2.4.2. On the other hand, the
conservation laws (2.1.14) (proved in Proposition 2.5.2 iii) below) together
with (1.2.12) imply a priori bound

$|X(t)|_{\cal V}^{2}\leq C[E(X(0))+Q(X(0))],\qquad t\in[-\tau,\tau].$ (2.4.24)

Hence, the local solution admits an extension to the global solution $X\in
C({\mathbb R},{\cal V})$. $\Box$

######  Remark 2.4.7.

The condition $X(0)\in{\cal M}$ implies that $X(t)\in{\cal M}$ for all
$t\in{\mathbb R}$ by the charge conservation (2.1.14). Hence, (2.2.1) implies
(2.1.4)–(2.1.6) with the potential $\phi(\cdot,t)=G\rho(\cdot,t)$.

####  2.5 Conservation laws

We deduce the conservation laws (2.1.14) by the Galerkin approximations [60].

######  Definition 2.5.1.

i)${\cal V}_{m}$ with $m\in{\mathbb N}$ denotes the finite-dimensional
submanifold of the Hilbert manifold ${\cal V}$

${\cal
V}_{m}:=\\{(\sum_{\overline{k}}C(\overline{k})\Lambda_{j=1}^{\overline{N}}e^{ik_{j}x_{j}},q,p):k_{j}\in\Gamma^{*}_{N},~{}~{}C(\overline{k})\in{\mathbb
C},~{}~{}\sum\limits_{j=1}^{\overline{N}}k_{j}^{2}\leq
m,~{}q\in\overline{\mathbb T},~{}p\in{\mathbb R}^{3\overline{N}}\\},$ (2.5.1)

where $\overline{k}:=(k_{1},...,k_{\overline{N}})$.
ii) ${\cal W}_{m}$ with $m\in{\mathbb N}$ denotes the finite-dimensional
linear subspace of the Hilbert space ${\cal W}$

${\cal
W}_{m}:=\\{(\sum_{\overline{k}}C(\overline{k})\Lambda_{j=1}^{\overline{N}}e^{ik_{j}x_{j}},\varkappa,v):k_{j}\in\Gamma^{*}_{N},~{}~{}C(\overline{k})\in{\mathbb
C},~{}~{}\sum\limits_{j=1}^{\overline{N}}k_{j}^{2}\leq
m,~{}\varkappa\in{\mathbb R}^{3\overline{N}},~{}v\in{\mathbb
R}^{3\overline{N}}\\}.$ (2.5.2)

Obviously, ${\cal V}_{1}\subset{\cal V}_{2}\subset...$, the union
$\cup_{m}{\cal V}_{m}$ is dense in ${\cal V}$, and ${\cal W}_{m}$ are
invariant with respect to $H$ and $J$. Let us denote by $P_{m}$ the orthogonal
projector ${\cal X}\to{\cal W}_{m}$. This projector is also orthogonal in the
Hilbert space ${\cal W}$. Let us approximate the system (2.2.1) by finite-
dimensional Hamiltonian systems on the manifold ${\cal V}_{m}$,

$\dot{X}_{m}(t)=JE_{m}^{\prime}(X_{m}(t)),\qquad t\in{\mathbb R},$ (2.5.3)

where $E_{m}:=E|_{{\cal V}_{m}}$ and
$X_{m}(t)=(\psi_{m}(t),q_{m}(t),p_{m}(t))\in C({\mathbb R},{\cal V}_{m})$. The
equation (2.5.3) can be also written as

$\langle\dot{X}_{m}(t),Y\rangle=-\langle E^{\prime}(X_{m}(t)),JY\rangle,\qquad
Y\in{\cal W}_{m}.$ (2.5.4)

This form of the equation (2.5.3) holds since $E_{m}:=E|_{{\cal V}_{m}}$ and
${\cal W}_{m}$ is invariant with respect to $J$. Equivalently,

$\dot{X}_{m}(t)=HX_{m}(t)+P_{m}N(X_{m}(t)).$ (2.5.5)

The Hamiltonian form guarantees the energy and charge conservation (2.1.14):

$E(X_{m}(t))=E(X_{m}(0)),\quad Q(X_{m}(t))=Q(X_{m}(0)),\qquad t\in{\mathbb
R}.$ (2.5.6)

Indeed, the energy conservation holds by the Hamiltonian form (2.5.3), while
the charge conservation holds by the Noether theory [4, 31, 35] due to the
$U(1)$-invariance of $E_{m}$, see (1.2.14).

The equation (2.5.5) admits a unique local solution for every initial state
$X_{m}(0)\in{\cal V}_{m}$ since the right hand side is locally bounded and
Lipschitz continuous. The global solutions exist by (1.2.12) and the energy
and charge conservation (2.5.6).

Finally, we take any $X(0)\in{\cal V}$ and choose a sequence

$X_{m}(0)\to X(0),\qquad m\to\infty,$ (2.5.7)

where the convergence holds in the metric of ${\cal V}$. Therefore,

$E(X_{m}(0))\to E(X(0)),\qquad Q(X_{m}(0))\to Q(X(0)).$ (2.5.8)

Hence, (2.5.6) and (1.2.12) imply the basic uniform bound

$R:=\sup_{m\in{\mathbb N}}\,\,\sup_{t\in{\mathbb R}}|X_{m}(t)|_{\cal
V}<\infty.$ (2.5.9)

Therefore, (2.5.5) and Proposition 2.4.3 imply the second basic uniform bound

$\sup_{m\in{\mathbb N}}\,\sup_{t\in{\mathbb R}}\,\|\dot{X}_{m}(t)\|_{{\cal
W}^{-1}}<C(R),$ (2.5.10)

since the operator $H:{\cal W}\to{\cal W}^{-1}$ is bounded, and the projector
$P_{m}$ is also a bounded operator in ${\cal W}\subset{\cal W}^{-1}$. Hence,
the Galerkin approximations $X_{m}(t)$ are uniformly Lipschitz-continuous with
values in ${\cal V}^{-1}$:

$\sup_{m\in{\mathbb N}}\,d_{{\cal V}^{-1}}(X_{m}(t),X_{m}(s))\leq
C(R)|t-s|,\qquad s,t\in{\mathbb R}.$ (2.5.11)

Let us show that the uniform estimates (2.5.9) and (2.5.11) imply a
compactness of the Galerkin approximations and the conservation laws.

######  Proposition 2.5.2.

Let (1.1.1) hold and $X(0)\in{\cal V}$. Then
i) There exists a subsequence $m^{\prime}\to\infty$ such that

$X_{m^{\prime}}(t)\stackrel{{\scriptstyle{\cal
X}}}{{-\\!\\!\\!\\!\longrightarrow}}X(t),\qquad m^{\prime}\to\infty,\qquad
t\in{\mathbb R},$ (2.5.12)

where $X(\cdot)\in C({\mathbb R},{\cal X})$.
ii) Every limit function $X(\cdot)$ is a solution to (2.2.1), and $X(\cdot)\in
C({\mathbb R},{\cal V})$.
iii) The conservation laws (2.1.14) hold.

###### Proof.

i) The convergence (2.5.12) follows from (2.5.9) and (2.5.10) by the Dubinsky
‘theorem on three spaces’ [20] (Theorem 5.1 of [60]). Namely, the embedding
${\cal V}\subset{\cal X}$ is compact by the Sobolev theorem, and hence,
(2.5.12) holds by (2.5.9) for $t\in D$, where $D$ is a countable dense set.
Finally, let us use the interpolation inequality and (2.5.9), (2.5.11): for
any $\varepsilon>0$

$d_{\cal X}(X_{m}(t),X_{m}(s))\leq\varepsilon d_{\cal
V}(X_{m}(t),X_{m}(s))+C(\varepsilon)d_{{\cal V}^{-1}}(X_{m}(t),X_{m}(s))\leq
2\varepsilon R+C(\varepsilon,R)|t-s|.$ (2.5.13)

This inequality implies the equicontinuity of the Galerkin approximations with
values in ${\cal X}$. Hence, convergence (2.5.12) holds for all $t\in{\mathbb
R}$ since it holds for the dense set of $t\in D$. The same equicontinuity also
implies the continuity of the limit function $X\in C({\mathbb R},{\cal X})$.
ii) Integrating equation (2.5.5), we obtain

$\int_{0}^{t}\langle\dot{X}_{m}(t),Y\rangle\,ds=\int_{0}^{t}\langle
X_{m}(s),HY)\,ds+\int_{0}^{t}\langle N(X_{m}(s)),Y\rangle\,ds,\qquad Y\in{\cal
W}_{m},$ (2.5.14)

Below we will write $m$ instead of $m^{\prime}$. To prove (2.2.6) it suffices
to check that in the limit $m\to\infty$, we get

$\int_{0}^{t}\langle\dot{X}(t),Y\rangle\,ds=\int_{0}^{t}(X(s),HY)\,ds+\int_{0}^{t}\langle
N(X(s)),Y\rangle\,ds,\qquad Y\in{\cal W}_{n},\qquad n\in{\mathbb N}.$ (2.5.15)

The convergence of the left hand side and of the first term on the right hand
side of (2.5.14) follow from (2.5.12) and (2.5.7) since $HY\in{\cal W}_{m}$.

It remains to consider the last integral in (2.5.14). The integrand is
uniformly bounded by (2.5.9) and Proposition 2.4.3. Hence, it suffices to
check the pointwise convergence

$\langle N(X_{m}(s),Y\rangle-\\!\\!\\!-\\!\\!\\!\\!\to\langle
N(X(s),Y\rangle,\qquad Y\in{\cal W}_{n}$ (2.5.16)

for any $s\in{\mathbb R}$. Here
$N(X_{m}(s))=(ie\phi^{\otimes}_{m}(s)\psi_{m}(s),p_{m}(s),f_{m}(s))$ according
to the notation (2.2.7), and $Y=(\varphi,\varkappa,v)\in{\cal W}_{n}$. Hence,
(2.5.16) reads

$ie[\phi^{\otimes}_{m}(s)\psi_{m}(s),\varphi]+p_{m}(s)\varkappa+f_{m}(s)v\,\to\,ie(\phi^{\otimes}(s){\hskip
1.42262pt}\psi(s),\varphi)+p(s)\varkappa+f(s)v,$ (2.5.17)

where $[\cdot,\cdot]$ is the scalar product in $L^{2}(\overline{\mathbb T})$.
The convergence of $p_{m}(s)\varkappa$ follows from (2.5.12) (with
$m^{\prime}=m$) . To prove the convergence of the two remaining terms we first
show that

$\phi_{m}(s):=G\rho_{m}\stackrel{{\scriptstyle C({{\mathbb
T}})}}{{-\\!\\!-\\!\\!\\!\\!\longrightarrow}}\phi(s):=G\rho.$ (2.5.18)

Indeed, (2.5.12) implies that

$\psi_{m}(s)\stackrel{{\scriptstyle L^{2}(\overline{\mathbb
T})}}{{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!\\!\longrightarrow}}\psi(s),\qquad
q_{m}(s)\to q(s)$ (2.5.19)

Further, the sequence $\psi_{m}(s)$ is bounded in $H^{1}(\overline{\mathbb
T})$ by (2.5.9). Hence, the sequence $\rho_{m}(s)$ is bounded in the Sobolev
space $W^{1,3/2}({\mathbb T})$ by (2.4.2). Therefore, the sequence
$\rho_{m}(s)$ is precompact in $L^{2}({\mathbb T})$ by the Sobolev compactness
theorem. Hence,

$\rho_{m}\stackrel{{\scriptstyle L^{2}({{\mathbb
T}})}}{{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!\\!\longrightarrow}}\rho$ (2.5.20)

by (2.5.19). Therefore, (2.5.18) holds since the operator $G:L^{2}({\mathbb
T})\to C({\mathbb T})$ is continuous. Finally, (2.5.18) and (2.5.19) imply
that

$\phi^{\otimes}_{m}(s)\psi_{m}(s)\stackrel{{\scriptstyle
L^{2}(\overline{\mathbb
T})}}{{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!\\!\longrightarrow}}{\hskip
1.42262pt}\phi^{\otimes}(s)\psi(s),\qquad f_{m}(s)\to f(s),$ (2.5.21)

which proves (2.5.17). Now (2.5.15) is proved for $Y\in{\cal V}_{n}$ with any
$n\in{\mathbb N}$. Hence, $X(t)$ is a solution to (2.2.1). Finally, $N(X(t))$
is bounded in ${\cal W}$ by (2.5.9) and Proposition 2.4.3. Hence, (2.2.6)
implies that $X(\cdot)\in C({\mathbb R},{\cal V})$.
iii) The conservation laws (2.5.6) and the convergences (2.5.7), (2.5.12)
imply that

$E(X(t))\leq E(X(0)),\quad Q(X(t))\leq Q(X(0)),\qquad t\in{\mathbb R}.$
(2.5.22)

The last inequality holds by the first convergence of (2.5.19). The first
inequality follows from the representation

$E(X_{m}(t))=\frac{1}{2}\|\nabla\psi_{m}(t)\|_{L^{2}({{\mathbb
T}})}^{2}+\frac{1}{2}\|\sqrt{G}\rho_{m}(t)\|_{L^{2}({{\mathbb
T}})}^{2}+\sum_{n\in\Gamma_{n}}\frac{p_{m}^{2}(n,t)}{2M}.$ (2.5.23)

Namely, the last two terms on the right hand side converge by (2.5.20) and
(2.5.12). Moreover, the first term is bounded by (2.5.9). Hence, the first
convergence of (2.5.19) implies the weak convergence

$\nabla\psi_{m}(t)\stackrel{{\scriptstyle L^{2}_{w}({{\mathbb
T}})}}{{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!\\!\longrightarrow}}\nabla\psi(t)$
(2.5.24)

by the Banach theorem. Now the first inequality of (2.5.22) follows by the
property of the weak convergence in the Hilbert space. Finally, the opposite
inequalities to (2.5.22) are also true by the uniqueness of solutions
$X(\cdot)\in C({\mathbb R},{\cal V})$, which is proved in Proposition 2.4.2. ∎

####  2.6 Jellium fermionic ground states

In this section we check the key identity (2.3.4) and construct all solutions
to (2.2.1) with minimal energy (2.1.11). Furthermore we give examples
illustrating the Jellium and the Wiener conditions.

#####  2.6.1 Uniform electronic charge density

Let us establish the identity (2.3.4).

######  Lemma 2.6.1.

Let the condition (2.3.3) hold for an eigenfunction (2.3.2), and

$\int_{\overline{\mathbb T}}|\psi_{0}(\overline{x})|^{2}d\overline{x}=Z.$
(2.6.1)

Then the identity (2.3.4) holds.

###### Proof.

By the antisymmetry of $\psi_{0}(x_{1},...,x_{\overline{N}})$ it remains to
prove that

$\int_{\overline{\mathbb
T}}\delta(x-x_{1})|\psi_{0}(\overline{x})|^{2}d\overline{x}=Z/\overline{N},\qquad
x\in{\mathbb T}.$ (2.6.2)

Substitute here the expansion (2.3.2). The normalization condition (2.6.1)
gives

$\sum_{\overline{k}}|C(\overline{k})|^{2}\overline{N}^{\overline{N}}=Z.$
(2.6.3)

Further,

$\displaystyle\int_{\overline{\mathbb
T}}\delta(x-x_{1})|\psi_{0}(\overline{x})|^{2}d\overline{x}=\frac{1}{\overline{N}!}\sum_{\overline{k}}\Big{\\{}|C(\overline{k})|^{2}\int_{\overline{\mathbb
T}}\delta(x\\!-\\!x_{1})\Big{[}\sum_{\pi,\pi^{\prime}\in
S_{\overline{N}}}(-1)^{|\pi|+|\pi^{\prime}|}\prod\limits_{j=1}^{\overline{N}}e^{i[k_{\pi(j)}-k_{\pi^{\prime}(j)}]x_{j}}\Big{]}d\overline{x}\Big{\\}}$
(2.6.4) $\displaystyle+$ $\displaystyle\frac{1}{\overline{N}!}{\rm Re{\hskip
1.42262pt}}\sum_{\overline{k}\neq\overline{k}^{\prime}}\Big{\\{}C(\overline{k})\overline{C}(\overline{k}^{\prime})\Big{[}\sum_{\pi,\pi^{\prime}\in
S_{\overline{N}}}(-1)^{|\pi|+|\pi^{\prime}|}\int_{\overline{\mathbb
T}}\delta(x\\!-\\!x_{1})\prod\limits_{j=1}^{\overline{N}}e^{i[k_{\pi(j)}-k^{\prime}_{\pi^{\prime}(j)}]x_{j}}d\overline{x}\Big{]}\Big{\\}}.$

The integrals in the last line vanish since
$k_{\pi(j)}-k^{\prime}_{\pi^{\prime}(j)}\neq 0$ at least for one $j\neq 1$ by
(2.3.3). On the other hand, the integrals in the first line do not vanish only
in the case when $k_{\pi(j)}\equiv k_{\pi^{\prime}(j)}$ for $j\neq 1$, i.e.,
when $\pi=\pi^{\prime}$. Hence,

$\int_{\overline{\mathbb
T}}\delta(x-x_{1})|\psi_{0}(\overline{x})|^{2}d\overline{x}=\overline{N}^{\overline{N}-1}\sum_{\overline{k}}|C(\overline{k})|^{2}\int_{{\mathbb
T}}\delta(x\\!-\\!x_{1})dx_{1}=\overline{N}^{\overline{N}-1}\sum_{\overline{k}}|C(\overline{k})|^{2}=Z/\overline{N}$
(2.6.5)

by (2.6.3).

∎

######  Remark 2.6.2.

Similar calculations show that the uniformity (2.6.2) can break down for the
wave functions (2.3.2) if the condition (2.3.3) fails.

#####  2.6.2 Description of all ground states

The following lemma gives the description all ground states of the system
(2.2.1).

######  Lemma 2.6.3.

Let the Jellium condition (1.1.11) hold. Then all solutions to (2.1.4)–(2.1.6)
of minimal energy are
$(e^{i\alpha}\psi_{0}(\overline{x})e^{-i\omega_{0}t},q^{*},0)$ with
$\alpha\in[0,2\pi]$ and $q^{*}\in{\mathbb T}^{\overline{N}}$ satisfying the
identity

$\sum_{n\in{\Gamma}}\sigma(x-q^{*}(n))\equiv eZ,\qquad x\in{{\mathbb T}},$
(2.6.6)

###### Proof.

First, let us note that the ${\Gamma}$-periodic solutions (2.1.15) have the
minimal energy, and the identity (2.6.6) holds for $q^{*}=\overline{r}$ by
(1.1.12).

Further, a solution is of minimal energy (2.1.11) if the first summand on the
right hand side of (2.1.11) takes the minimal values $\omega_{0}$ while the
second and the third summands should vanish. Hence, the normalization
condition (2.1.8) gives

$\psi(\overline{x},t)\equiv
e^{i\alpha(t)}\psi_{0}(\overline{x}),\qquad\alpha(t)\in{\mathbb R},$ (2.6.7)

while

$\rho(x)\equiv 0,\qquad M\partial_{t}q(n,t)=p(n,t)\equiv 0.$ (2.6.8)

Therefore, (2.3.4) implies

$\rho^{i}(x,t)\equiv eZ,\qquad q(n,t)\equiv q^{*}(n),$ (2.6.9)

which is equivalent to (2.6.6).

Moreover, the Poisson equation (2.1.5) with $\rho(x,t)\equiv 0$ implies that
$\phi(x,t)\equiv 0$ after a gauge transformation (1.2.2). Hence, finally,
substituting (2.6.7) into (2.1.4) with $\phi(x,t)\equiv 0$, we obtain that
$-\dot{\alpha}(t)\equiv\omega_{0}$. ∎

This lemma implies that all ${\Gamma}$-periodic ground states are given by
(2.1.15). Recall that the examples of charge densities satisfying Jellium and
the Wiener conditions are given in Section 1.5.2.

####  2.7 The orbital stability of the ground state

In this section we expand the energy into the Taylor series and prove the
orbital stability checking the positivity of the energy Hessian.

#####  2.7.1 The Taylor expansion of energy functional

We will deduce the lower estimate (2.1.21) using the Taylor expansion of
$E(S+Y)$ at $S=S_{\alpha,r}=(\psi^{\alpha},\overline{r},0)\in{\cal S}$ and
$Y=(\varphi,\varkappa,\pi)\in{\cal W}$. Further calculations are very close to
the one of Section 1.6.1:

$E(S+Y)=E(S)+\langle E^{\prime}(S),Y\rangle+\frac{1}{2}\langle
Y,E^{\prime\prime}(S)Y\rangle+R(S,Y)=\omega_{0}Z+\frac{1}{2}\langle
Y,E^{\prime\prime}(S)Y\rangle+R(S,Y)$ (2.7.1)

since $E(S)=\omega_{0}Z$ by (2.1.19), and $E^{\prime}(S)=0$. Here
$E^{\prime}(S)$ and $E^{\prime\prime}(S)$ stand for the Gâteaux differentials.
Let us recall that $\psi^{\alpha}=e^{i\alpha}\psi_{0}(x)$ where $\psi_{0}(x)$
is given by (2.3.2), and the condition (2.3.3) holds.

First, we expand the charge density (2.1.12), (2.1.7) corresponding to
$S+Y=(\psi^{\alpha}+\varphi,\overline{r}+\varkappa,\pi)$:

$\rho(x)=\rho^{(0)}(x)+\rho^{(1)}(x)+\rho^{(2)}(x),\qquad x\in{\mathbb T},$
(2.7.2)

where $\rho^{(0)}$ and $\rho^{(1)}$ are respectively the terms of zero and
first order in $Y$, while $\rho^{(2)}$ is the remainder. However,
$\rho^{(0)}(x)$ is the total charge density of the ground state which is
identically zero by (1.1.12) and (2.3.4):

$\rho^{(0)}(x)=\rho^{i}_{0}(x)-e|\psi^{\alpha}(x)|^{2}\equiv 0,\qquad
x\in{{\mathbb T}}.$ (2.7.3)

Thus, $\rho=\rho^{(1)}+\rho^{(2)}$. Expanding (2.1.12) and (2.1.7), we obtain

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\rho^{(1)}(x)\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\\!\\!\sigma^{(1)}(x)-2e\,\Sigma(\psi^{\alpha})\varphi(x),\quad\sigma^{(1)}(x)=-\sum_{n\in\Gamma}\varkappa(n)\cdot\nabla\sigma(x-n-r),\quad$
(2.7.4)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\rho^{(2)}(x)\\!\\!\\!\\!$
$\displaystyle\\!\\!=\\!\\!\\!\\!$
$\displaystyle\\!\\!\sigma^{(2)}(x)\\!-\\!e\,\Sigma(\varphi)\varphi(x),~{}\sigma^{(2)}(x)\\!=\\!\frac{1}{2}\sum_{n\in\Gamma}\int_{0}^{1}\\!\\!(1\\!-\\!s)[\varkappa(n)\cdot\nabla]^{2}\sigma(x\\!-\\!n\\!-\\!r-\\!s\varkappa(n))ds,$
(2.7.5)

where $\Sigma(\psi)$ denotes the real-linear operator defined by

$\Sigma(\psi)\varphi(x):=\sum\limits_{j=1}^{\overline{N}}\int_{\overline{\mathbb
T}}\delta(x-y_{j})\psi(\overline{y})\cdot\varphi(\overline{y})\,d\overline{y},\qquad
x\in{\mathbb T},$ (2.7.6)

where $\cdot$ is the inner product of the corresponding vectors
$\psi(\overline{y})=(\psi_{1}(\overline{y}),\psi_{2}(\overline{y}))\in{\mathbb
R}^{2}$ and
$\varphi(\overline{y})=(\varphi_{1}(\overline{y}),\varphi_{2}(\overline{y}))\in~{}{\mathbb
R}^{2}$. The operator $\Sigma(\psi^{\alpha})$ is continuous
$L^{2}(\overline{\mathbb T})\to L^{2}({\mathbb T})$, which follows similarly
to (2.4.5). Obviously,

$\Sigma(\psi^{\alpha})\varphi(x)=\Sigma(\psi^{\alpha}_{1})\varphi_{1}(x)+\Sigma(\psi^{\alpha}_{2})\varphi_{2}(x).$
(2.7.7)

Note that $\Sigma(\varphi)\varphi\in L^{2}({\mathbb T})$ for $\varphi\in{\cal
F}$ by estimate (2.4.2).

Substituting $\psi=\psi^{\alpha}+\varphi$ and $\rho=\rho^{(1)}+\rho^{(2)}$
into (2.1.11), we obtain that the quadratic part of (2.7.1) reads

$\frac{1}{2}\langle
Y,E^{\prime\prime}(S)Y\rangle=\frac{1}{2}\int_{\overline{\mathbb
T}}|\nabla\varphi(\overline{x})|^{2}]d\overline{x}+\frac{1}{2}(\rho^{(1)},G\rho^{(1)})+K(\pi),~{}~{}~{}~{}~{}~{}K(\pi):=\displaystyle\sum_{n}\frac{\pi^{2}(n)}{2M}~{}~{}$
(2.7.8)

and the remainder equals

$R(S,Y)=\frac{1}{2}(2\rho^{(1)}+\rho^{(2)},G\rho^{(2)}).$ (2.7.9)

#####  2.7.2 The null space of Hessian

In this section we calculate the null space

${\cal K}(S):={\rm Ker{\hskip
1.42262pt}}\,\Big{[}E^{\prime\prime}(S)\Big{|}_{\cal W}\Big{]},\qquad
S\in{\cal S}.$ (2.7.10)

######  Lemma 2.7.1.

Let the Jellium condition (1.1.11) and the Wiener condition (1.1.13) hold, and
$S\in{\cal S}$. Then

${\cal K}(S)=\\{(0,\overline{s},0):~{}~{}s\in{\mathbb R}^{3}\\},$ (2.7.11)

where $\overline{s}\in{\mathbb R}^{3\overline{N}}$ is defined similarly to
(1.1.17): $\overline{s}(n)\equiv s$.

###### Proof.

All the summands of the energy (2.7.8) are nonnegative. Hence, this expression
is zero if and only if all the summands vanish: in the notation (2.7.4)

$\varphi(\overline{x})\equiv
C,\quad(\rho^{(1)},G\rho^{(1)})=\|\sqrt{G}[\sigma^{(1)}-2e\Sigma(\psi^{\alpha})\varphi]\|_{L^{2}({\mathbb
T})}^{2}=0,\quad\pi=0.$ (2.7.12)

Here $C=0$ by the antisymmetry of $\varphi$ since $N>1$. Therefore,
$\Sigma(\psi^{\alpha})\varphi=0$, and hence, (2.7.12) implies that

$\sqrt{G}\sigma^{(1)}=0.$ (2.7.13)

On the other hand, in the Fourier transform (2.7.4) reads

$\tilde{\sigma}^{(1)}(\xi)=\tilde{\sigma}(\xi)\xi\cdot\sum_{n\in\Gamma}ie^{i\xi[n+r]}\varkappa(n)=i\tilde{\sigma}(\xi)\xi\cdot
e^{i\xi r}\hat{\varkappa}(\xi),\qquad\xi\in\Gamma^{*}_{N},$ (2.7.14)

where $\hat{\varkappa}(\xi):=\sum_{n\in\Gamma}e^{i\xi n}\varkappa(n)$ is a
$2\pi{\mathbb Z}^{3}$-periodic function on $\Gamma^{*}_{N}$. Hence, Definition
(1.2.1) and the Jellium condition (1.1.11) imply that

$\displaystyle 0=\|\sqrt{G}\sigma^{(1)}\|_{L^{2}({\mathbb T})}^{2}$
$\displaystyle=$ $\displaystyle
N^{-3}\sum_{\Gamma^{*}_{N}\setminus\Gamma_{1}^{*}}|\tilde{\sigma}(\xi)\frac{\xi\hat{\varkappa}(\xi)}{|\xi|}|^{2}$
(2.7.15) $\displaystyle=$ $\displaystyle
N^{-3}\sum_{\theta\in\Pi^{*}\setminus\Gamma_{1}^{*}}\langle\hat{\varkappa}(\theta),\sum_{m\in{\mathbb
Z}^{3}}\Big{[}\frac{\xi\otimes\xi}{|\xi|^{2}}|\tilde{\sigma}(\xi)|^{2}\Big{]}_{\xi=\theta+2\pi
m}\hat{\varkappa}(\theta)\rangle$ $\displaystyle=$ $\displaystyle
N^{-3}\sum_{\theta\in\Pi^{*}\setminus\Gamma_{1}^{*}}\langle\hat{\varkappa}(\theta),\Sigma(\theta)\hat{\varkappa}(\theta)\rangle.$

As a result,

$\hat{\varkappa}(\theta)=0,\qquad\theta\in\Pi^{*}\setminus\Gamma_{1}^{*}$
(2.7.16)

by the Wiener condition (1.1.13). On the other hand,
$\hat{\varkappa}(0)\in{\mathbb R}^{3}$ remains arbitrary. Respectively,
$\varkappa=\overline{s}$ with an arbitrary $s\in{\mathbb R}^{3}$.∎

######  Remarks 2.7.2.

i) The annihilation of $\varphi(x)\equiv C$ in (2.7.12) is the main point
where we use the antisymmetry of wave functions (‘Pauli exclusion principle’).
ii) The key point of the proof is the explicit calculation (2.7.14) in the
Fourier transform. This calculation relies on the invariance of the Hessian
$E^{\prime\prime}(S)$ with respect to $\Gamma$-translations which is due to
the periodicity of the ions arrangement of the ground state.

######  Remark 2.7.3.

Beyond the Wiener condition If the Wiener condition (1.1.13) fails, the
dimension of the space

$V:=\\{v\in{\mathbb
R}^{3\overline{N}}:~{}~{}v(n)=\sum_{\theta\in\Pi^{*}\setminus\Gamma_{1}^{*}}e^{-i\theta
n}\hat{v}(\theta),\qquad\hat{v}(\theta)\in{\mathbb
C}^{3},~{}~{}\Sigma(\theta)\hat{v}(\theta)=0\\}$ (2.7.17)

is positive. The above calculations show that in this case

${\cal K}(S)=\\{(0,\overline{s}+v,0):~{}~{}s\in{\mathbb R}^{3},~{}~{}v\in
V\\}.$ (2.7.18)

The subspace $V\subset{\mathbb R}^{3\overline{N}}$ is orthogonal to the $3D$
subspace $\\{\overline{s}:s\in{\mathbb R}^{3}\\}\subset{\mathbb
R}^{3\overline{N}}$ by the Parseval theorem. Hence, $\dim{\cal K}(S)=3+d$,
where $d:=\dim V>0$. Thus, $\dim{\cal K}(S)>3$. Under the Wiener condition
$V=0$, and (2.7.18) coincides with (2.7.11).

#####  2.7.3 The positivity of Hessian

As in Section 1.6.3, denote by $N_{S}{\cal S}$ the normal subspace to ${\cal
S}$ at a point $S$:

$N_{S}{\cal S}:=\\{Y\in{\cal W}:\langle Y,\tau\rangle=0,~{}~{}\tau\in
T_{S}{\cal S}\\},$ (2.7.19)

where $T_{S}{\cal S}$ is the tangent space to ${\cal S}$ at the point $S$ and
$\langle\cdot,\cdot\rangle$ stands for the scalar product (1.2.10). Obviously,
${\cal S}\subset{\cal M}$, and the tangent space to ${\cal M}$ at a point
$S=(\psi^{\alpha},\overline{r},0)$ is given by

$T_{S}{\cal M}=\\{(\varphi,\varkappa,\pi)\in{\cal
W}:\varphi\bot\psi^{\alpha}\\},$ (2.7.20)

since $DQ(\psi^{\alpha},\overline{r},0)=2(\psi^{\alpha},0,0)$.

######  Lemma 2.7.4.

Let the Jellium condition (1.1.11) hold, and $S=S_{\alpha,r}\in{\cal S}$. Then
the Wiener condition (1.1.13) is necessary and sufficient for the positivity
of the Hessian $E^{\prime\prime}(S)$ in the orthogonal directions to ${\cal
S}$ on ${\cal M}$, i.e.,

$E^{\prime\prime}(S)\Big{|}_{N_{S}{\cal S}\cap T_{S}{\cal M}}>0.$ (2.7.21)

###### Proof.

i) Sufficiency. Differentiating
$S_{\alpha,r}=(e^{i\alpha}\psi_{0},\overline{r},0)\in{\cal S}$ in the
parameters $\alpha\in[0,2\pi]$ and $r\in{\mathbb T}$, we obtain

$T_{S}{\cal S}=\\{(iC\psi^{\alpha},\overline{s},0):~{}~{}C\in{\mathbb
R},~{}~{}s\in{\mathbb R}^{3}\\}.$ (2.7.22)

Hence, (2.7.11) implies that

${\cal K}(S)\cap N_{S}{\cal S}=(0,0,0).$ (2.7.23)

Now (2.7.21) follows since $E^{\prime\prime}(S)\geq 0$ by (2.7.8).
ii) Necessity. If the Wiener condition (1.1.13) fails, the space ${\cal
K}(SN)$ is given by (2.7.18), and hence, (2.7.22) implies that now

${\cal K}(S)\cap N_{S}{\cal S}=\\{0,v,0):~{}~{}C\in{\mathbb R},~{}~{}v\in
V\\}\subset T_{S}{\cal M}.$ (2.7.24)

Therefore, the Hessian $E^{\prime\prime}(S)$ vanishes on the nontrivial space
${\cal K}(S)\cap N_{S}{\cal S}\subset T_{S}{\cal M}$ of the dimension $d>0$.
Respectively, the positivity (2.7.21) breaks down. ∎

######  Remark 2.7.5.

The positivity of type (2.7.21) breaks down for the submanifold ${\cal
S}(r):=\\{S_{\alpha,r}:~{}\alpha\in[0,2\pi]\\}$ with a fixed $r\in{\mathbb T}$
instead of the solitary manifold ${\cal S}$. Indeed, then the corresponding
tangent space is smaller:

$T_{S}{\cal S}(r)=\\{(iC\psi^{\alpha},0,0):\,\,C\in{\mathbb R}\\}.$ (2.7.25)

Hence, the normal subspace $N_{S}{\cal S}(r)$ is larger, in particular
containing all the vectors $(0,\overline{s},0)$ generating the shifts of the
torus. However, all these vectors also belong to the null space (2.7.11) and
to $T_{S}{\cal M}$. Respectively, the null space of the Hessian
$E^{\prime\prime}(S)$ in $T_{S}{\cal M}\cap N_{S}{\cal S}(r)$ is at least
3-dimensional.

#####  2.7.4 The orbital stability

Here we prove Theorem 2.1.2 which is the main result of this chapter. For the
proof is suffices to check the lower energy estimate (2.1.21):

$E(X)-\omega_{0}Z\geq\nu\,d^{2}_{\cal V}(X,{\cal S})\quad{\rm if}\quad d_{\cal
V}(X,{\cal S})\leq\delta,\quad X\in{\cal M}$ (2.7.26)

with some $\nu,\delta>0$. This estimate implies Theorem 2.1.2 since the energy
is conserved along all trajectories. First, we prove similar lower bound for
the energy Hessian.

######  Lemma 2.7.6.

Let conditions of Theorem 2.1.2 hold. Then for each $S\in{\cal S}$

$\langle Y,E^{\prime\prime}(S)Y\rangle>\nu\|Y\|_{\cal W}^{2},\qquad Y\in
N_{S}{\cal S}\cap T_{S}{\cal M},$ (2.7.27)

where $\nu>0$.

###### Proof.

First, we note that $E^{\prime\prime}(S)$ is not complex linear due to the
integral in (2.1.11). Hence, we should express the action of
$E^{\prime\prime}(S)$ in $\varphi_{1}(x):={\rm Re{\hskip
1.42262pt}}\varphi(x)$ and $\varphi_{1}(x):={\rm Im{\hskip
1.42262pt}}\varphi(x)$: formulas (2.7.8) and (2.7.4), (2.7.7) imply that
similarly to (1.6.30),

$E^{\prime\prime}(S)Y=\left(\begin{array}[]{cccl}-\Delta^{\otimes}+4e^{2}\Sigma^{*}(\psi^{\alpha}_{1})G\tilde{\Sigma}(\psi^{\alpha}_{1})&4e^{2}\Sigma^{*}(\psi^{\alpha}_{1})G\tilde{\Sigma}(\psi^{\alpha}_{2})&2L&0\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
4e^{2}\tilde{\Sigma}^{*}(\psi^{\alpha}_{2})G\Sigma(\psi^{\alpha}_{1})&-\Delta^{\otimes}+4e^{2}\Sigma^{*}(\psi^{\alpha}_{2})G\tilde{\Sigma}(\psi^{\alpha}_{2})&0&0\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ 2L^{{\hskip 1.42262pt}*}&0&T&0\\\
0&0&0&M^{-1}\\\ \end{array}\right)Y$ (2.7.28)

for $Y=(\varphi_{1},\varphi_{2},\varkappa,\pi)$. Similarly to (1.6.31), the
operator $L$ corresponds to the matrix

$L(\overline{x},n):=e\Sigma^{*}(\psi^{\alpha})G\nabla\sigma(x-n):~{}~{}~{}x\in{\mathbb
R}^{3},~{}n\in\Gamma,$ (2.7.29)

while $T$ coincides with (1.6.32).

Thus, $E^{\prime\prime}(S)$ is the selfadjoint operator in ${\cal X}$ with the
discrete spectrum, and (2.7.21) implies that the minimal eigenvalue of
$E^{\prime\prime}(S)$ in the invariant space $N_{S}{\cal S}\cap T_{S}{\cal M}$
is positive. Therefore, (2.7.27) follows. ∎

The positivity (2.7.27) implies the lower energy estimate (2.7.26) since the
higher-order terms in (2.7.1) are negligible by the following lemma.

######  Lemma 2.7.7.

Let $\sigma(x)$ satisfy (1.1.1). Then the remainder (1.6.7) admits the bound

$|R(S,Y)|\leq C\|Y\|_{\cal W}^{3}\quad\,\,\,{\rm for}\quad\,\,\,\|Y\|_{\cal
W}\leq 1.$ (2.7.30)

###### Proof.

Due to (2.7.9) it suffices to prove the estimates

$\|\sqrt{G}\rho^{(1)}\|_{L^{2}({\mathbb T})}\leq C_{1}\|Y\|_{\cal
W},\quad\|\sqrt{G}\rho^{(2)}\|_{L^{2}({\mathbb T})}\leq C_{2}\|Y\|_{\cal
W}^{2}\quad\,\,\,{\rm for}\,\,\,\quad\|Y\|_{\cal W}\leq 1.$ (2.7.31)

i) By (2.7.4) we have for $Y=(\varphi,\varkappa,\pi)$

$\sqrt{G}\rho^{(1)}=\sqrt{G}\sigma^{(1)}-2e\sqrt{G}\,\Sigma(\psi^{\alpha})\varphi$
(2.7.32)

in the notation (2.7.6). The operator $\sqrt{G}$ is bounded in $L^{2}({\mathbb
R}^{3})$ by (1.2.1). Hence, (2.7.4) and (1.1.1) imply by the Cauchy-Schwarz
inequality,

$\|\sqrt{G}\sigma^{(1)}\|_{L^{2}({\mathbb T})}\leq C|\varkappa|.$ (2.7.33)

Finally, applying the Cauchy–Schwarz and Hausdorff–Young inequalities to the
second term on the RHS of (2.7.32), we obtain

$\displaystyle\|\sqrt{G}\,\Sigma(\psi^{\alpha})\varphi\|_{L^{2}({\mathbb T})}$
$\displaystyle\leq$ $\displaystyle C\Big{[}\sum_{\xi\in\Gamma^{*}_{N}\setminus
0}\frac{|\tilde{\varphi}(\xi)|^{2}}{|\xi|^{2}}\Big{]}^{1/2}\leq
C\|\tilde{\varphi}\|_{L^{4}(\Gamma^{*}_{N})}\Big{[}\sum_{\xi\in\Gamma^{*}_{N}\setminus
0}|\xi|^{-4}\Big{]}^{1/2}$ (2.7.34) $\displaystyle\leq$ $\displaystyle
C_{1}\|\varphi\|_{L^{4/3}({\mathbb T})}\leq C_{2}\|\varphi\|_{H^{1}({\mathbb
T})}^{2}$

by the Sobolev embedding theorem. Hence, the first inequality (2.7.31) is
proved.
ii) Now we prove the second inequality (2.7.31). According to (2.7.5),

$\sqrt{G}\rho^{(2)}(x)=\sqrt{G}\sigma^{(2)}(x)-e\sqrt{G}\,\Sigma(\varphi)\varphi$
(2.7.35)

Now (1.1.1) and the triangle inequality give

$\|\sqrt{G}\sigma^{(2)}\|_{L^{2}({\mathbb T})}\leq C|\varkappa|^{2}.$ (2.7.36)

At last, we obtain similarly to (2.7.34)

$\displaystyle\|\sqrt{G}\,\Sigma(\varphi)\varphi\|_{L^{2}({\mathbb T})}\leq
C\Big{[}\sum_{\xi\in\Gamma^{*}_{N}\setminus
0}\frac{|\tilde{\beta}(\xi)|^{2}}{|\xi|^{2}}\Big{]}^{1/2}\leq
C_{1}\|\beta\|_{L^{4/3}({\mathbb T})}.$ (2.7.37)

Finally, applying the triangle inequality and the Sobolev embedding theorem,
we obtain

$\displaystyle\|\beta\|_{L^{4/3}({\mathbb T})}$ $\displaystyle\leq$
$\displaystyle\int_{{\mathbb T}^{\overline{N}-1}}[\int_{\mathbb
T}|\varphi(\overline{x})|^{8/3}dx_{j}]^{3/4}\,dx_{1}...\widehat{dx_{j}}...dx_{\overline{N}}$
(2.7.38) $\displaystyle\leq$ $\displaystyle\int_{{\mathbb
T}^{\overline{N}-1}}[\int_{\mathbb
T}|\nabla_{x_{j}}\varphi(\overline{x})|^{2}dx_{j}]\,dx_{1}...\widehat{dx_{j}}...dx_{\overline{N}}\leq
C\|\varphi\|_{H^{1}({\mathbb T})}^{2}.$

Now the lemma is proved. ∎

##  Part II Linear stability of infinite crystals

In the second part of present book, we consider infinite crystals with many
ions per cell of a lattice.

We construct the ground state for 1D, 2D and 3D lattices in 3D space. The main
results are well-posedness of the linearised dynamics in the space of finite-
energy states and its dispersive decay.

The presentation mainly relies on our papers [36, 37, 38, 39, 40, 43] with
suitable extensions.

###  Chapter 3 On existence of ground states

Abstract

A space-periodic ground state is shown to exist for lattices of smeared ions
in ${\mathbb R}^{3}$ coupled to the Schrödinger and scalar fields. The
elementary cell is necessarily neutral.

The 1D, 2D and 3D lattices in ${\mathbb R}^{3}$ are considered, and a ground
state is constructed by minimizing the energy per cell. The case of a 3D
lattice is rather standard, because the elementary cell is compact, and the
spectrum of the Laplacian is discrete.

In the cases of 1D and 2D lattices, the energy functional is differentiable
only on a dense set of variations, due to the presence of the continuous
spectrum of the Laplacian that causes the infrared divergence of the Coulomb
bond. Respectively, the construction of electrostatic potential and the
derivation of the Schrödinger equation for the minimizer in these cases
require an extra argument.

The space-periodic ground states for 1D and 2D lattices give the model of the
nanostructures similar to the carbon nanotubes and graphene respectively.

####  3.1 Introduction

In this chapter we prove the existence of ground states for infinite crystals
in ${\mathbb R}^{3}$ with $d$-dimensional ion lattices

$\Gamma_{d}:=\\{{\bf x}({\bf n})={\bf a}_{1}{\bf n}^{1}+\dots+{\bf a}_{d}{\bf
n}^{d}:{\bf n}\in{\mathbb Z}^{d}\\},$ (3.1.1)

where $d=1,2,3$ and ${\bf a}_{k}\in{\mathbb R}^{3}$ are linearly independent
periods. A 2D (respectively, 1D lattice) is a mathematical model of a
monomolecular film (a wire).

The ions are considered as classical nonrelativistic particles governed by the
Lorentz equations neglecting the magnetic field, while the electrons are
described by the Schrödinger equation neglecting the electron spin. The scalar
potential is the solution to the corresponding Poisson equation.

We consider the crystal with $N$ ions per cell. Let us denote by $\rho_{j}$
the charge density of an ion and by $M_{j}>0$ its mass, $j=1,...,N$. Then the
coupled equations read

$\displaystyle i\hbar\dot{\psi}({\bf x},t)$ $\displaystyle=$
$\displaystyle-\frac{\hbar^{2}}{2{\rm m}}\Delta\psi({\bf x},t)-e\phi({\bf
x},t)\psi({\bf x},t),~{}~{}~{}~{}~{}{\bf x}\in{\mathbb R}^{3},$ (3.1.2)
$\displaystyle\qquad-\Delta\phi({\bf x},t)$ $\displaystyle=$
$\displaystyle\rho({\bf x},t):=\sum_{j=1}^{N}{\hskip 1.42262pt}\sum_{{\bf
n}\in{\mathbb Z}^{d}}\rho_{j}({\bf x}-{\bf x}({\bf n})-{\bf x}_{j}({\bf
n},t))-e|\psi({\bf x},t)|^{2},~{}~{}~{}~{}~{}{\bf x}\in{\mathbb R}^{3},$
(3.1.3) $\displaystyle M_{j}\ddot{\bf x}_{j}({\bf n},t)$ $\displaystyle=$
$\displaystyle-(\nabla\phi({\bf x},t),\rho_{j}({\bf x}-{\bf x}({\bf n})-{\bf
x}_{j}({\bf n},t))),\quad{\bf n}\in{\mathbb Z}^{d},\quad j=1,\dots,N.$ (3.1.4)

Here $e>0$ is the elementary charge, ${\rm m}$ is the electron mass,
$\psi({\bf x},t)$ denotes the wave function of the electron field, and
$\phi({\bf x},t)$ is the electrostatic potential generated by the ions and the
electrons. Further, $(\cdot,\cdot)$ stands for the scalar product in the
Hilbert space $L^{2}({\mathbb R}^{3})$. All derivatives here and below are
understood in the sense of distributions. The system is nonlinear and
translation invariant, i.e., $\psi({\bf x}-{\bf a},t)$, $\phi({\bf x}-{\bf
a},t)$, ${\bf x}_{j}({\bf n},t)+{\bf a}$ is also a solution for any ${\bf
a}\in{\mathbb R}^{3}$ .

The ground state is a $\Gamma_{d}$-periodic stationary solution $\psi^{0}({\bf
x})e^{-i\omega^{0}t}$, $\phi^{0}({\bf x})$, $\overline{\bf x}=({\bf
x}^{0}_{1},~{}\dots,{\bf x}^{0}_{N})$ of the system (3.1.2)–(3.1.4):

$\displaystyle\hbar\omega^{0}\psi^{0}({\bf x})$ $\displaystyle=$
$\displaystyle-\frac{\hbar^{2}}{2{\rm m}}\Delta\psi^{0}({\bf
x})-e\phi^{0}({\bf x})\psi^{0}({\bf x}),~{}~{}~{}~{}{\bf x}\in T_{d},$ (3.1.5)
$\displaystyle-\Delta\phi^{0}({\bf x})$ $\displaystyle=$
$\displaystyle\rho^{0}({\bf x}):=\sigma^{0}({\bf x})-e|\psi^{0}({\bf
x})|^{2},~{}~{}~{}~{}~{}~{}{\bf x}\in T_{d},$ (3.1.6) $\displaystyle 0$
$\displaystyle=$ $\displaystyle-{\hskip 1.42262pt}\langle\nabla\phi^{0}({\bf
x}),\rho^{\rm per}_{j}({\bf x}-{\bf x}^{0}_{j})\rangle,\qquad j=1,\dots,N.$
(3.1.7)

Here, $T_{d}:={\mathbb R}^{3}/\Gamma_{d}$ denotes the ‘elementary cell’ of the
crystal, $\langle\cdot,\cdot\rangle$ stands for the scalar product in the
Hilbert space $L^{2}(T_{d})$ and its different extensions, and

$\sigma^{0}({\bf x}):=\sum_{j=1}^{N}\rho^{\rm per}_{j}({\bf x}-{\bf
x}^{0}_{j}),\quad\rho^{\rm per}_{j}({\bf x}):=\sum_{{\bf n}\in{\mathbb
Z}^{d}}\rho_{j}({\bf x}-{\bf x}({\bf n})),$ (3.1.8)

where we assume that the series converge in an appropriate sense. More
precisely, we will construct a solution to the system (3.1.5)–(3.1.7) with
$\sigma^{0}({\bf x})$ given by the first equation of (3.1.8) where $\rho^{\rm
per}_{j}$ satisfy the following condition:

$\mbox{\bf Condition I.}\qquad\rho^{\rm per}_{j}\in L^{1}(T_{d})\cap
L^{2}(T_{d}),\qquad j=1,...,N.\qquad\qquad$ (3.1.9)

Obviously, $\rho^{\rm per}_{j}\in L^{1}(T_{d})$ if $\rho_{j}\in L^{1}({\mathbb
R}^{3})$. So we consider the case of smeared ions. The case of point ions we
have considered in [39]. In the cases $d=2$ and $d=1$ we will assume
additional conditions (3.3.11) and (3.4.9) respectively.

The elementary cell $T_{d}$ is isomorphic to the 3D torus for $d=3$, to the
direct product of the 2D torus by ${\mathbb R}$ for $d=2$, and to the direct
product of the 1D torus (circle) by ${\mathbb R}^{2}$ for $d=1$.

The system (3.1.5)–(3.1.7) is translation invariant similarly to
(3.1.2)–(3.1.4). Let us note that $\omega^{0}$ should be real since ${\rm
Im{\hskip 1.42262pt}}\omega^{0}\neq 0$ means an instability of the ground
state: the decay as $t\to\infty$ in the case ${\rm Im{\hskip
1.42262pt}}\omega^{0}<0$ and the explosion if ${\rm Im{\hskip
1.42262pt}}\omega^{0}>0$.

Let us denote $\displaystyle Z_{j}:=\int_{T_{d}}\rho^{\rm per}_{j}({\bf
x})d{\bf x}/e$. Then

$\int_{T_{d}}\sigma^{0}({\bf x})d{\bf x}=eZ,\qquad Z:=\sum_{j}Z_{j}.$ (3.1.10)

The total charge per cell vanishes (cf. [6]):

$\int_{T_{d}}\rho^{0}({\bf x})d{\bf x}=\int_{T_{d}}[\sigma^{0}({\bf
x})-e|\psi^{0}({\bf x})|^{2}]d{\bf x}=0.$ (3.1.11)

For $d=3$ this neutrality condition follows directly from equation (3.1.6) by
integration using $\Gamma_{3}$-periodicity of $\phi^{0}({\bf x})$. For $d=1$
and $d=2$ we will deduce it from the finiteness of energy per cell (see
(3.3.8) and (3.4.6)). Equivalently, the neutrality condition can be written as
the normalization

$\int_{T_{d}}|\psi^{0}({\bf x})|^{2}d{\bf x}=Z.$ (3.1.12)

We allow arbitrary $Z_{j}\in{\mathbb R}$, however we assume that $Z>0$:
otherwise the theory is trivial.

Let us comment on our approach. The neutrality condition (3.1.12) defines the
submanifold $\mathcal{M}$ in the space $H^{1}(T_{d})\times T_{d}^{N}$ of
space-periodic configurations $(\psi^{0},\overline{\bf x}^{0})$. We construct
a ground state as a minimizer over $\mathcal{M}$ of the energy per cell
(3.2.3), (3.3.1), (3.4.1).

Our techniques in the case of 3D lattice is rather standard, and we use it as
an ‘Ariadne’s thread’ to manage the more complicated cases of 2D and 1D
lattices, because the corresponding elementary cells are unbounded.

Namely, the derivation of the equations (3.1.5)–(3.1.7) for the minimizer in
the cases of 2D and 1D lattices is not straightforward. The difficulty is that
the energy per cell is finite only on a dense subset of $\mathcal{M}$ due to
the infrared divergence of the Coulomb bond. In these cases we restrict
ourselves by one ion per cell, i.e., by $N=1$. Then $\overline{\bf x}^{0}={\bf
x}^{0}_{1}$ can be chosen arbitrary because of the translation invariance of
the system (3.1.5)–(3.1.7). Respectively, now the energy per cell should be
minimized over $\psi\in M$, where $M$ is the submanifold of $H^{1}(T_{d})$
defined by the neutrality condition (3.1.12).

The main novelties of our approach behind the technical proofs for 2D and 1D
lattices are as follows:

I. The energy per cell consists of two contributions: the kinetic energy, and
the Coulomb bond. Generally, the Coulomb bond for 2D and 1D lattices is
infinite due to the infrared divergence which is caused by the continuous
spectrum of the Laplace operator on the corresponding elementary cells. The
spectrum is continuous since the elementary cells are unbounded in the case of
2D and 1D lattices in $R^{3}$. Let us note that the continuous spectrum and
the infrared singularity also appear in the Schrödinger–Poisson molecular
systems in ${\mathbb R}^{3}$ studied in [5, 33, 65] where the singularity is
summable, contrary to the space-periodic case.

We indicate suitable conditions (3.3.11), (3.4.9) which provide the finiteness
of the Coulomb bond for a dense set of the fields in the case of 2D and 1D
lattice respectively.

Both contributions to the energy per cell (the kinetic energy and the Coulomb