On the Linear Stability of Crystals in the Schrödinger–Poisson Model

We consider the Schrödinger–Poisson–Newton equations for crystals with one ion per cell. We linearize this dynamics at the periodic minimizers of energy per cell and introduce a novel class of the ion charge densities that ensures the stability of the linearized dynamics. Our main result is the energy positivity for the Bloch generators of the linearized dynamics under a Wiener-type condition on the ion charge density. We also adopt an additional ‘Jellium’ condition which cancels the negative contribution caused by the electrostatic instability and provides the ‘Jellium’ periodic minimizers and the optimality of the lattice: the energy per cell of the periodic minimizer attains the global minimum among all possible lattices. We show that the energy positivity can fail if the Jellium condition is violated, while the Wiener condition holds. The proof of the energy positivity relies on a novel factorization of the corresponding Hamilton functional. The Bloch generators are nonselfadjoint (and even nonsymmetric) Hamilton operators. We diagonalize these generators using our theory of spectral resolution of the Hamilton operators with positive definite energy (Komech and Kopylova in, J Stat Phys 154(1–2):503–521, 2014, J Spectral Theory 5(2):331–361, 2015). The stability of the linearized crystal dynamics is established using this spectral resolution.


Introduction
Dyson and Lenard [9,10] were the first to obtain mathematical results on the stability of matter; in their studies a bound from below for the energy was obtained. The thermodynamic limit for the Coulomb systems was first studied by Lebowitz and Lieb [20,21], see the survey and further development in [22]. These results were extended by Catto, Lions, Le Bris to the Thomas-Fermi and Hartree-Fock models [5][6][7]. All these results were concerned either with the thermodynamic limit or the existence of a ground state for infinite particle systems. The dynamical stability of ion-electron dynamics for infinite particle systems with moving ions was never examined before. 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 [2].
In present paper, we analyze for the first time the dynamic stability of a crystal periodic minimizer of energy per cell in linear approximation for the simplest Schrödinger-Poisson model. The periodic minimizer for this model was constructed in [16]. The stability for the nonlinear dynamics will be considered elsewhere.
We consider crystals with one ion per cell. The electron cloud is described by the oneparticle Schrödinger equation; the ions are looked upon as particles that corresponds to the Born and Oppenheimer approximation. The ions interact with the electron cloud via the scalar potential, which is a solution to the corresponding Poisson equation.
This model does not respect the Pauli exclusion principle for electrons. Nevertheless, it provides a convenient framework to introduce suitable functional tools that might be instrumental for physically more realistic models (the Thomas-Fermi, Hartree-Fock, and second quantized models). In particular, we find a novel stability criterion (1.21), (1.23).
We denote by σ (x) ∈ L 1 (R 3 ) the charge density of one ion, R 3 σ (x)dx = eZ > 0, (1.1) where e > 0 is the elementary charge. We assume througout the paper that We consider the cubic lattice = Z 3 for the simplicity of notations. Let ψ(x, t) be the wave function of the electron field, q(n, t) denote the ions displacements, and (x) be the electrostatic potential generated by the ions and electrons. We assume thath = c = m = 1, where c is the speed of light and m is the electron mass. The coupled Schrödinger-Poisson-Newton equations read Mq(n, t) = − ∇ (x, t), σ (x − n − q(n, t)) , n ∈ Z 3 . (1.5) Here the brackets stand for the Hermitian scalar product in the Hilbert space L 2 (R 3 ) and for its various extensions, the series (1.4) converges in a suitable sense, and M > 0. All the derivatives here and below are understood in the sense of distributions. These equations can be written as a Hamilton system with formal Hamilton functional , (1.6) where q := (q(n) : n ∈ Z 3 ), p := ( p(n) : n ∈ Z 3 ), ρ(x) is defined similarly to (1.4), and G := (− ) −1 , i.e., Gρ(x) := 1 4π (1.7) Namely, the system (1.3)-(1.5) can be formally written as iψ(x, t) = ∂ ψ(x) H ,q(n, t) = ∂ p(n) H ,ṗ(n, t) = −∂ q(n) H , (1.8) where ∂ z := 1 2 (∂ z 1 + i∂ z 2 ) with z 1 = Re z and z 2 = Im z. We investigate the stability of periodic minimizers of the energy per cell, which areperiodic stationary solutions of (1.3)-(1.5). We will see that these periodic minimizers can be stable or unstable (then the true ground state of the system might be non-periodic, e.g., quasiperiodic), depending on the choice of the nuclear density σ . However, we only study very special densities σ satisfying some conditions discussed below. A periodic minimizer of a crystal is a -periodic stationary solution ψ 0 (x)e −iω 0 t , 0 (x) , q 0 (n) = q 0 and p 0 (n) = 0 for n ∈ Z 3 (1.9) with a real ω 0 . Such periodic minimizer was constructed in [16] for general lattice with several ions per cell. In our case the ion position q 0 ∈ R 3 can be chosen arbitrarily, and we set q 0 = 0 everywhere below. In present paper, we prove the stability of the formal linearization of the nonlinear system (1.3)-(1.5) at the periodic minimizer (1.9). Namely, substituting into the nonlinear equations (1.3), (1.5) with (x, t) = Gρ(x, t), we formally obtain the linearized equations (see Appendix 1) Here ρ 1 (x, t) is the linearized charge density (1.12) The system (1.11) is linear over R, but it is not complex linear. This is due to the last term in (1.12), which appears from the linearization of the term |ψ| 2 = ψψ in (1.4). However, we need the complex linearity for the application of the spectral theory. That is why we will consider below the complexification of system (1.11) by writing it in the variables The periodic minimizer ψ 0 (x) is a real function up to a phase factor e iφ (see [1] and (1.24) below). This factor can be canceled by multiplying ψ 0 (x) and (x, t) by e −iφ in the first equation (1.11) and in (1.12). Therefore, we will assume that ψ 0 (x) is a real function, and hence, (1.13) Then (1.11) can be written aṡ − e 0 (x) − ω 0 , the operators S and T correspond to matrices (3.3) and (3.4), respectively, and ψ 0 denotes the operators of multiplication by the real function ψ 0 (x). The Hamilton representation (1.8) implies that Our main result is the stability of the linearized system (1.14): for any initial state of finite energy there exists a unique global solution which is bounded in the energy norm. We show that the generator A is densely defined in the Hilbert space and commutes with translations by vectors from . Hence, the equation (1.14) can be reduced with the help of the Fourier-Bloch-Gelfand-Zak transform to equations with the corresponding Bloch generatorsÃ(θ ) = JB(θ ), which depend on the parameter θ from the Brillouin zone * := [0, 2π] 3 . The Bloch energy operatorB(θ ) is given bỹ where * := 2πZ 3 , andH 0 (θ ) : are defined, respectively, by (6.22) and (3.9), (3.12).
for s ∈ R; its spectrum is discrete. However, the operator A is not selfadjoint and even not symmetric in Y 0 -this a typical situation in the linearization of U (1)-invariant nonlinear equations [17,Appendix B]. Respectively, the Bloch generatorsÃ(θ ) are not selfadjoint in The main crux here is that we cannot apply the von Neumann spectral theorem to the nonselfadjoint generators A andÃ(θ ). We solve this problem by applying our spectral theory of abstract Hamilton operators with positive energy [17,18]. This is why we need the positivity of the energy operatorB(θ ): where (θ) > 0 for a.e. θ ∈ * \ * (1.19) and the brackets denote the scalar product in Y 0 (T 3 ). Equivalently, The main result of the present paper is the proof of the positivity (1.20) for the ions charge densities σ satisfying the following two conditions C1 and C2 on the corresponding Fourier transformσ (ξ).
C2. The Jellium Condition: This condition immediately implies that the periodized ions charge density corresponding to the periodic minimizer is a positive constant everywhere in space. In this case the minimum of energy per cell corresponds to the opposite uniform negative electronic charge, so these ion and electronic densities cancel each other, and the potential (x, t) vanishes by (1.4), The energy per cell attains its minimum since the integral (2.8) vanishes (see Lemma 2.1). Thus, the condition (1.23) means that ions can be arranged on an appropriate lattice in a way that their total charge density is constant everywhere in space. This clearly requires that σ has the symmetry of this lattice, which is false for radial densities. The simplest example of such a σ is a constant over the unit cell of a given lattice, which is what physicists usually call Jellium [11]. Here we study this model in the rigorous context of the Schrödinger-Poisson equations. The outstanding role in this Jellium model in our context is provided by the optimality of the lattice : under the condition (1.23) the energy of the periodic minimizer per cell attains the global minimum among all possible lattices (see Lemmas 2.1 and 2.2).
We prove that the stability of this constant-density state under small deformations, is equivalent to the simple condition (1.21). In that case this Jellium periodic minimizer is the crystal ground state, i.e., its small local deformations have a higher energy as well as other periodic arrangements. Also, we use the positivity (1.20) to give a meaning to the associated linearized dynamics, using existing results [17,18].
It is to be noticed that (1.21) is satisfied for the simplest Jellium model, when σ is constant in the unit cell: in this case the Fourier tranformσ is the 'Dirichlet kernel'. Actually, the condition (1.21) holds "generically".
We show that the condition (1.21) is necessary for the positivity (1.20). We expect that the condition (1.23) is also necessary for the positivity (1.20), however, this is still an open challenging problem. This condition cancels the negative energy which is provided by the electrostatic instability ('Earnshaw's Theorem' [29], see Remark 10.2). At least we show in Lemma 10.1 that the positivity (1.20) can break down when condition (1.23) fails. This counterexample relies on a novel small-charge asymptotics of the periodic minimizerψ 0 (x) (Lemma 9.1).
Finally, the positivity (1.20) allows us to construct the spectral resolution ofÃ(θ ), which results in the stability for the linearized dynamics (1.14). The spectral resolution is constructed with application of our spectral theory of abstract Hamilton operators [17,18]. This theory is an infinite-dimensional version of some Gohberg and Krein ideas from the theory of parametric resonance [14,Chap. VI].
In concluzion, all our methods and results extend obviously to equations (1.3)-(1.5) in the case of general lattice where the generators a k ∈ R 3 are linearly independent. In this case the condition (1.23) becomesσ where * denotes the dual lattice, i.e., The condition (1.29) claryfies the relation between the properties of the ions and the resulting crystal geometry. Let us comment on previous results in these directions. The crystal periodic minimizer for the Hartree-Fock equations was constructed by Catto, Le Bris, and Lions [6,7]. For the Thomas-Fermi model similar results were obtained in [5]. The corresponding periodic minimizer in the Schrödinger-Poisson model was constructed in [16]. The stability for the linearized dynamics was not established previously in any model. In [4], Cancès and Stoltz have established the well-posedness for local perturbations of the stationary density matrix in an infinite crystal for the reduced Hartree-Fock model in the random phase approximation with the Coulomb pairwise interaction potential w(x − y) = 1/|x − y|. The space-periodic nuclear potential in the equation (3) of [4] does not depend on time, which corresponds to fixed ion positions. The nonlinear Hartree-Fock dynamics with the Coulomb potential without the random phase approximation was not previously examined, see the discussion in [19] and in the introductions of the papers [3,4]. The paper [3] deals with 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 random potentials which are solutions 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 [23], Lewin and Sabin established the well-posedness for the reduced von Neumann equation with density matrices of infinite trace, describing the Fermi gas with pair-wise interaction potentials w ∈ L 1 (R 3 ). They also proved the asymptotic stability of stationary states for 2D Fermi gas [24]. 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 [27] and the references therein. The scattering theory for short-range and long-range perturbations of such 'periodic operators' was constructed in [12,13].
The paper is organized as follows. In Sect. 2 we recall our result [16] on the existence of a periodic minimizer In Sects. 3-5 we study the Hamiltonian structure of the linearized dynamics and find a bound of the energy from below. In Sect. 6 we calculate the generator of the linearized dynamics in the Fourier-Bloch representation. In Sect. 7 we prove the positivity of the energy. In Sect. 8 we apply this positivity to the stability of the linearized dynamics. Finally, in Sects. 9 and 10 we establish small charge asymptotics of the periodic minimizer and construct examples of negative energy. Some technical calculations are carried out in Appendices.
where we denote the corresponding periodized ion charge density The Poisson equation (2.2) for the -periodic potential 0 implies the neutrality of the periodic cell which is equivalent to the normalization condition by (1.1). We assume that Z > 0, since otherwise the theory is trivial.

The Regularity of the Periodic Minimizer
The existence of the periodic minimizer (1.9) is proved in [16] under the condition which holds by (1.2). The periodic minimizer ψ 0 is constructed as a minimal point of the energy per cell while the operator G per is defined by More precisely, where M denotes the manifold The results [16] imply that there exists a periodic minimizer with ψ 0 , 0 ∈ H 2 (T 3 ). Hence ψ 0 0 ∈ H 2 (T 3 ), and the Eq. (2.1) implies that In other words, (2.14)

The 'Jellium periodic minimizer' and Optimality of the Lattice
The following lemma means that under the condition (1.23) the energy of the periodic minimizer per cell attains at the global minimum among all possible lattices. Proof First we note thatσ by (1.1). Hence, the corresponding periodized ion charge density equals σ 0 (x) := σ (x − n) ≡ eZ, since its Fourier coefficients with nonzero numbers vanish by (1.23): Proof Let ψ 0 1 denote a periodic minimizer for the lattice 1 . There exists at least one point This means that at least one of the Fourier coefficients (2.16), with γ 1 instead of 2πm, does not vanish. Therefore, the corresponding periodized ion charge density This implies that

Linearized Dynamics
Let us calculate the entries of the matrix operator (1.14) under conditions (1.2). (3.1) The conditions (1.2) imply that Let us recall that the periodic minimizer ψ 0 (x) can be taken to be a real function. In this case (1.11)-(1.13) imply that the operator-matrix A is given by (1.14), where S denotes the operator with the 'matrix' Finally, T is the real matrix with entries The operators Gψ 0 : In the next section, we will construct a dense domain for all these operators. On the other hand, the corresponding operators T 1 and T 2 are bounded in view of the following lemma. Denote by the primitive cell (3.5) Let us define the Fourier transform on l 2 aŝ where * = 2π denotes the primitive cell of the lattice * , the series converging in L 2 ( * ). Proof The first operator T 1 reads as the convolution T 1 q(n) = T 1 (n − n )q(n ), where By the Fourier transform (3.6), the convolution operator T 1 becomes the multiplication, for a.e. θ ∈ * \ * . (3.8) By the Bessel-Parseval identity it suffices to check that the 'symbol'T 1 (θ ) is a bounded function. This follows by direct calculation from (3.4). First, we apply the Parseval identitŷ since the last sum over n equals | * | m δ(θ +ξ −2πm) by the Poisson summation formula [15]. Finally, |σ (ξ)| ≤ C ξ −2 by (3.2). Hence, Finally,

The Hamilton Structure and the Domain
In this section we study the domain of the generator A given by (1.14) and (1.15).
Definition 4.1 (i) S + := ∪ ε>0 S ε , where S ε is the space of functions ∈ S (R 3 ) whose Fourier transformsˆ (ξ ) vanish in the ε-neighborhood of the lattice * , (ii) l c is the space of sequences q(n) ∈ R 3 such that q(n) = 0, n > N for some N .
Obviously, D is dense in Y 0 . Proof Formally the matrix (1.15) is symmetric. The following lemma implies that B is defined on D.
(ii) Given a fixed ϕ ∈ S + , we have ϕ ∈ S ε with some ε > 0. First, we note that where F stands for the Fourier transform. Further,ψ 0 (ξ ) = (2π) 3 Moreover, ψ 0 (x) is a bounded function by (2.13). As a result, ψ 0 ∈ L 2 (R 3 ) and ψ 0 * ˜ ∈ L 2 (R 3 ). Hence, belongs to the domain of Gψ 0 and of ψ 0 Gψ 0 . We now consider S * . Applying (3.3), the Parseval identity and (4.2), we get for 2). Hence, integrating by parts twice and taking into account (4.2), we obtain which implies that S * ∈ l 2 . (iii) Let us check that Sq ∈ L 2 (R 3 ) for q ∈ l c . Calculating the Fourier transform of Sq, we obtain that whereq means the Fourier transform (3.6) extended * -periodically to R 3 . Now the Parseval identity gives that It remains to note that the sum over m is finite by (2.14), and since the functionq(ξ ) is bounded for q ∈ l c . Finally, the last integral is finite by (3.2).
This lemma implies that BY ∈ Y 0 for Y ∈ D. The symmetry of B on D is evident from (1.15). Theorem 4.2 is proved.

Corollary 4.5 The proof of Theorem 4.2 shows that AD ⊂ Y 0 , and also A
(4.8)

Factorization of Energy and Bound from Below
The equation (1.14) is formally a Hamiltonian system with the Hamiltonian functional with the notation (3.3)-(3.4), where ψ 0 ∈ C 2 b (R 3 ) by (2.13). Here the first sum is bounded from below, the operator T 2 is bounded in l 2 by Lemma 3.1, while the operator M −1 is positive. Our basic observation is that Indeed, the operators factorize as follows: Now the quadratic form (5.3) becomes the 'perfect square' [26].

Generator in the Fourier-Bloch Transform
We reduce the operators A and B with the help of the Fourier-Bloch-Gelfand-Zak transform [8,25,27].
for any compact subset K ⊂ * + .

Weak Solutions and Linear Stability
We introduce weak solutions and prove the linear stability of the dynamics (1.14) assuming (1.2), (1.21) and (1.23). Then the real periodic minimizer is given by (1.24) with φ = 0, and (1.19) and (1.25) hold by Theorem 7.3.

Weak Solutions
Let us define solutions Y (t) ∈ C(R, Y 1 ) to (1.14) in the sense of vector-valued distributions of t ∈ R. Let us recall that A * V ∈ Y 0 for V ∈ D by Corollary 4.5. We call Y (t) a weak solution to (1.14) if, for every V ∈ D, Equivalently, by the Parseval-Plancherel identity, * Fubini's theorem implies that

Reduction to Selfadjoint Generator
Now we can apply our approach [17] to reduce (8.9) to the dynamics with a selfadjoint generator. By (8.3) for a.e. θ ∈ * . (8.10) Hence, applying˜ (θ ) to the both sides of (8.9), we obtain the equivalent equatioñ (θ, s)ds, t ∈ R for a.e. θ ∈ * , (8.11) whereK for a.e. θ ∈ * (8.12) in the sense of vector-valued distributions. Now the problem is that the domain ofK (θ ) is unknown since the ion density σ (x) generally is not smooth, so we cannot use the PDO techniques. The following lemma plays a key role in our approach (cf. Lemma 2.1 of [17]).
Consider the inverse operator This operator is selfadjoint, since it is bounded and symmetric. Hence, is a densely defined selfadjoint operator by Theorem 13.11, (b) of [28]: where the last inclusion follows by (8.7). (ii) (8.7) implies that˜ −1 (θ ) is a compact operator in Y 0 (T 3 ) by the Sobolev embedding theorem. Hence,K −1 (θ ) is also compact operator in Y 0 (T 3 ) by (8.13).

Small-Charge Asymptotics of the Periodic Minimizer
We will need below the asymptotics as e → 0 of the periodic minimizer (1.9) corresponding to a one-parametric family of ion densities with some fixed function μ ∈ L 2 (R 3 ). We assume that in accordance with (2.7). Now the energy (2.8) reads Denote by ψ 0 e , ω 0 e the family of periodic minimizers with the parameter e ∈ (0, 1]. Formulas (1.24) do not hold in general, since we do not assume (1.23).
In particular, ρ 1 (x, t) is given by (1.12). As a result, we obtain the system (1.11) in the linear approximation.