aa r X i v : . [ m a t h - ph ] O c t On crystal ground state inthe Schr¨odinger–Poisson model
A. I. Komech Faculty of Mathematics of Vienna University andInstitute for Information Transmission Problems RAS
Abstract
A space-periodic ground state is shown to exist for lattices of smeared ions in R coupled to theSchr¨odinger and scalar fields. The elementary cell is necessarily neutral.The 1D, 2D and 3D lattices in R are considered, and a ground state is constructed by mini-mizing the energy per cell. The case of a 3D lattice is rather standard, because the elementary cellis 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 ofvariations, due to the presence of the continuous spectrum of the Laplacian that causes the infrareddivergence of the Coulomb bond. Respectively, the construction of electrostatic potential and thederivation of the Schr¨odinger 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 nanostructuressimilar to the carbon nanotubes and graphene respectively. Supported partly by Alexander von Humboldt Research Award, Austrian Science Fund: P22198-N13, and grants ofDFG and the Russian Foundation for Basic Research. Introduction
We consider d -dimensional ion lattices in R ,(1.1) Γ d := { x ( n ) = a n + · · · + a d n d : n = ( n , ..., n d ) ∈ Z d } , where d = 1 , , a k ∈ R are linearly independent periods. A 2D lattice (respectively, 1D lattice)is a mathematical model of a monomolecular film (a wire).Born and Oppenheimer [6] developed the quantum dynamical approach to the crystal structure,separating the motion of ‘light electrons’ and of ‘heavy ions’. As an extreme form of this separation,the ions could be considered as classical nonrelativistic particles governed by the Coulomb force, whilethe electrons could be described by the Schr¨odinger equation neglecting the electron spin. The scalarpotential is the solution to the corresponding Poisson equation.We consider the crystal with N ions per cell. Let us denote by µ j the charge density of an ion andby M j > j = 1 , ..., N . Then the coupled equations read i ~ ˙ ψ ( x , t ) = − ~
2m ∆ ψ ( x , t ) + eφ ( x , t ) ψ ( x , t ) , x ∈ R , (1.2) h c ∂ t − ∆ i φ ( x , t ) = ρ ( x , t ) := N X j =1 X n ∈ Z d µ j ( x − x ( n ) − x j ( n , t )) + e | ψ ( x , t ) | , x ∈ R , (1.3) M j ¨ x j ( n , t ) = − ( ∇ φ ( x , t ) , µ j ( x − x ( n ) − x j ( n , t ))) , n ∈ Z d , j = 1 , . . . , N. (1.4)Here e < ψ ( x , t ) denotes the wave function of the electron field,and φ ( x , t ) is the electrostatic potential generated by the ions and the electrons. Further, ( · , · ) standsfor the scalar product in the Hilbert space L ( R ). All derivatives here and below are understood inthe sense of distributions. The system is nonlinear and translation invariant, i.e., ψ ( x − a , t ), φ ( x − a , t ), x j ( n , t ) + a is also a solution for any a ∈ R .A dynamical quantum description of the solid state as many-body system is not rigorously estab-lished yet (see Introduction of [25] and Preface of [29]). Up to date rigorous results concern only theground state in different models (see below).The classical ”one-electron” theory of Bethe-Sommerfeld, based on periodic Schr¨odinger equation,does not take into account oscillations of ions. Moreover, the choice of the periodic potential in thistheory is very problematic, and corresponds to a fixation of the ion positions which are unknown.The system (1.2)–(1.4) eliminates these difficulties though it does not respect the electron spinlike the periodic Schr¨odinger equation. To remedy this deficiency we should replace the Schr¨odingerequation by the Hartree–Fock equations as the next step to more realistic model. However, we expectthat the techniques developed for the system (1.2)–(1.4) will be useful also for more realistic dynamicalmodels of crystals. These goals were our main motivation in writing this paper.Here, we make the first step proving the existence of the ground state, which is a Γ d -periodicstationary solution ψ ( x ) e − iω t , φ ( x ), x = ( x , . . . , x N ) to the system (1.2)–(1.4): ~ ω ψ ( x ) = − ~
2m ∆ ψ ( x ) + eφ ( x ) ψ ( x ) , x ∈ T d , (1.5) − ∆ φ ( x ) = ρ ( x ) := σ ( x ) + e | ψ ( x ) | , x ∈ T d , (1.6) 0 = − h∇ φ ( x ) , µ per j ( x − x j ) i , j = 1 , . . . , N. (1.7) 2ere, T d := R / Γ d denotes the ‘elementary cell’ of the crystal, h· , ·i stands for the scalar product inthe Hilbert space L ( T d ) and its different extensions, and(1.8) σ ( x ) := N X j =1 µ per j ( x − x j ) , µ per j ( x ) := X n ∈ Z d µ j ( x − x ( n )) , where we assume that the series converge in an appropriate sense. More precisely, we will constructa solution to the system (1.5)–(1.7) with σ ( x ) given by the first equation of (1.8) where µ per j satisfythe following condition:(1.9) Condition I. µ per j ∈ L ( T d ) ∩ L ( T d ) , j = 1 , ..., N. For instance, µ per j ∈ L ( T d ) if µ j ∈ L ( R ). So we consider the case of smeared ions. The case ofthe point ions will be considered elsewhere. In the cases d = 2 and d = 1 we will assume additionalconditions (3.11) and (4.9) respectively.The elementary cell T d is isomorphic to the 3D torus for d = 3, to the direct product of the 2Dtorus by R for d = 2, and to the direct product of the 1D torus (circle) by R for d = 1.The system (1.5)–(1.7) is translation invariant similarly to (1.2)–(1.4). Let us note that ω shouldbe real since Im ω = 0 means an instability of the ground state: the decay as t → ∞ in the caseIm ω < ω > Z j := Z T d µ per j ( x ) d x / | e | . Then(1.10) Z T d σ ( x ) d x = Z | e | , Z := X j Z j . The total charge per cell should be zero (cf. [3]):(1.11) Z T d ρ ( x ) d x = Z T d [ σ ( x ) + e | ψ ( x ) | ] d x = 0 . For d = 3 this neutrality condition follows directly from equation (1.6) by integration using Γ -periodicity of φ ( x ). For d = 1 and d = 2 it follows from the finiteness of energy per cell. Equivalently,the neutrality condition can be written as the normalization(1.12) Z T d | ψ ( x ) | d x = Z. We allow arbitrary Z j ∈ R , however we assume that Z >
0: otherwise the theory is trivial.Let us comment on our approach. The neutrality condition (1.12) defines the submanifold M in the space H ( T d ) × T Nd of space-periodic configurations ( ψ , x ). We construct a ground state asa minimizer over M of the energy per cell (2.3), (3.1), (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 elementarycells are unbounded.Namely, the derivation of the equations (1.5)–(1.7) for the minimizer in the cases of 2D and 1Dlattices is not straightforward. The difficulty is that the energy per cell is finite only on a dense subsetof M due to the infrared divergence of the Coulomb bond. In these cases we restrict ourselves byone ion per cell, i.e., by N = 1. Then x = x can be chosen arbitrary because of the translationinvariance of the system (1.5)–(1.7). Respectively, now the energy per cell should be minimized over ψ ∈ M , where M is the submanifold of H ( T d ) defined by the neutrality condition (1.12).3he main novelties of our approach behind the technical proofs for 2D and 1D lattices are asfollows: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 iscaused by the continuous spectrum of the Laplace operator on the corresponding elementary cells. Thespectrum is continuous since the elementary cells are unbounded in the case of 2D and 1D latticesin R . Let us note that the continuous spectrum and the infrared singularity also appear in theSchr¨odinger–Poisson molecular systems in R studied in [2, 16, 28] where the singularity is summable,contrary to the space-periodic case.We indicate suitable conditions (3.11), (4.9) which provide the finiteness of the Coulomb bond fora 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 bond) are nonnega-tive. Hence, for any minimizing sequence, both contributions are bounded. The bound for the kineticenergy ensures the compactness in each finite region of a cell by the Sobolev embedding theorem.However, this bound cannot prevent the decay of the electron field, i.e., its escape to infinity. Nev-ertheless, the Coulomb interaction prevents even the partial escape to infinity, as we show in Lemma3.12. Physically this means that the electrostatic potential of the remaining positive charge becomesconfining.II. We construct the solution to the Poisson equation (1.6) as the contour integral, providing thecontinuity and a bound for the electrostatic potential. The main difficulty is a verification of theSchr¨odinger equation (1.5) for the minimizer. Namely, the Lagrange method of multipliers is notapplicable because the energy per cell is infinite outside the submanifold M ⊂ H ( T d ) due to theinfrared divergence of the Coulomb bond. Moreover, the Coulomb bond is infinite for a dense set of ψ ∈ M . Hence, to differentiate the energy functional, we should construct the smooth paths in M lying outside this dense set.III. Finally, the proof that ω is real (which is the stability condition for the ground state) is notstraightforward for 2D and 1D lattices, since the potential φ ( x ) a priori can grow at infinity. Thecorreponding bounds for the potentials are given by (3.15) and (4.12).The minimization strategy ensures the existence of a ground state for any lattice (1.1). One couldexpect that a stable lattice should provide a local minimum of the energy per cell for fixed d , N andfunctions ρ j , but this is still an open problem.Let us comment on related works. For atomic systems in R , a ground state was constructedbyLieb, Simon and P. Lions in the case of the Hartree and Hartree–Fock models [24, 26, 27], and by Nierfor the Schr¨odinger–Poisson model [28]. The Hartree–Fock dynamics for molecular systems in R hasbeen constructed by Canc`es and Le Bris [7].A mathematical theory of the stability of matter started from the pioneering works of Dyson,Lebowitz, Lenard, Lieb and others for the Schr¨odinger many body model [14, 20, 21, 23]; see thesurvey in [17]. Recently, the theory was extended to the quantized Maxwell field [22].These results and methods were developed last two decades by Blanc, Le Bris, Catto, P. Lionsand others to justify the thermodynamic limit for the Thomas–Fermi and Hartree–Fock models withspace-periodic ion arrangement [4, 10, 11, 12] and to construct the corresponding space-periodic groundstates [13], see the survey and further references in [5].Recently, Giuliani, Lebowitz and Lieb have established the periodicity of the thermodynamic limitin 1D local mean field model without the assumption of periodicity of a ion arrangement [15].Canc`es and others studied short-range perturbations of the Hartree–Fock model and proved thatthe density matrices of the perturbed and unperturbed ground states differ by a compact operator,[8, 9]. 4he Hartree–Fock dynamics for infinite particle systems were considered recently by Cances andStoltz [9], and Lewin and Sabin [18]. In [9], the well-posedness is established for local perturbationsof the periodic ground state density matrix in an infinite crystal. However, the space-periodic nuclearpotential in the equation [9, (3)] is fixed that corresponds to the fixed nuclei positions. Thus the backreaction of the electrons onto the nuclei is neglected. In [18], the well-posedness is established forthe von Neumann equation with density matrices of infinite trace for pair-wise interaction potentials w ∈ L ( R ). Moreover, the authors prove the asymptotic stability of the ground state in 2D case [19].Nevertheless, the case of the Coulomb potential for infinite particle systems remains open since thecorresponding generator is infinite.Let us note that 2D and 1D crystals in R were not considered previously. The space-periodicground states for 1D and 2D lattices give the model of the nanostructures similar to the carbonnanotubes and graphene respectively.The plan of our paper is as follows. In Section 2, we consider the 3-dimensional lattice. InSection 3, we construct a ground state, derive equations (1.5)–(1.7) and study smoothness propertiesof a ground state for 2-dimensional lattice. In Section 4, we consider the 1-dimensional lattice. Finally,in Appendix we construct and estimate the potential for 1D lattice. Acknowledgments.
The author thanks H. Spohn for useful remarks and E. Kopylova for helpfuldiscussions.
We consider the system (1.5)–(1.7) for the corresponding functions on the torus T = R / Γ and with x j mod Γ ∈ T . For s ∈ R , we denote by H s the complex Sobolev space on the torus T , and for1 ≤ p ≤ ∞ , we denote by L p the complex Lebesgue space of functions on T . The ground state will be constructed by minimizing the energy in the cell T . To this aim, we willminimize the energy with respect to x := ( x , . . . , x N ) ∈ ( T ) N and ψ ∈ H satisfying the neutralitycondition (1.11):(2.1) Z T ρ ( x ) d x = 0 , ρ ( x ) := σ ( x ) + e | ψ ( x ) | . where we set(2.2) σ ( x ) := X j µ per j ( x − x j ) , similarly to (1.8). Let us note that ρ ∈ L for ψ ∈ H by our condition (1.9) since ψ ∈ L by theSobolev embedding theorem.We define the energy in the periodic cell for ψ ∈ H by E ( ψ, x ) := Z T h ~ |∇ ψ ( x ) | + 12 φ ( x ) ρ ( x ) i d x , φ ( x ) := ( − ∆) − ρ, (2.3)where ( − ∆) − ρ is well-defined by (2.1). Namely, consider the dual lattice(2.4) Γ ∗ = { k ( n ) = b n + b n + b n : n = ( n , n , n ) ∈ Z } , b k a k ′ = 2 πδ kk ′ . Every function ρ ∈ L admits the Fourier representation(2.5) ρ ( x ) = 1 p | T | X k ∈ Γ ∗ ˆ ρ ( k ) e − i kx , ˆ ρ ( k ) = 1 p | T | Z e i kx ρ ( x ) d x . Respectively, we set(2.6) φ ( x ) = ( − ∆) − ρ ( x ) := 1 p | T | X k ∈ Γ ∗ \ ˆ ρ ( k ) k e − i kx . This function φ ∈ H and satisfies the Poisson equation − ∆ φ = ρ , since ˆ ρ (0) = 0 due to the neutralitycondition (2.1). Finally,(2.7) Z T φ ( x ) d x = 0 . Now it is clear that the energy (2.3) is finite for ψ ∈ H . Let us rewrite the energy as(2.8) E ( ψ, x ) = I + I , where I ( ψ ) := ~ Z T |∇ ψ ( x ) | d x ≥ , (2.9) I ( φ ) := 12 Z T ( − ∆) − ρ ( x ) · ρ ( x ) d x = 12 Z T |∇ φ ( x ) | d x ≥ . (2.10)The functional (2.3) is chosen, because(2.11) δEδ x j = −h ( − ∆) − ρ ( x ) , ∇ ρ per j ( x − x j ) i = h∇ φ ( x ) , ρ per j ( x − x j ) i , and the variational derivatives formally reads(2.12) δEδ Ψ( x ) = − ~
2m ∆ ψ + 2 e ( − ∆) − ρ ( x ) ψ ( x ) = − ~
2m ∆ ψ + 2 eφ ( x ) ψ ( x ) . The variation in (2.12) is taken over Ψ( x ) = ( ψ ( x ) , ψ ( x )) ∈ L ( T , R ), where ψ ( x ) = Re ψ ( x ) and ψ ( x ) = Im ψ ( x ). Respectively, all the terms in (2.12) are identified with the corresponding R -valueddistributions. Our purpose here is to minimize the energy with respect to( ψ, x ) ∈ M := M × T N , where M denotes the manifold (cf. (1.12))(2.13) M = { ψ ∈ H : Z T | ψ ( x ) | d x = Z } . The energy is bounded from below since E ( ψ, x ) ≥ ψ n , x n ) ∈ M such that(2.14) E ( ψ n , x n ) → E := inf M E ( ψ, x ) , n → ∞ . Our main result for a 3D lattice is the following: 6 heorem 2.1.
Let condition (1.9) hold. Theni) There exists ( ψ , x ) ∈ M with (2.15) E ( ψ , x ) = E . ii) Moreover, ψ ∈ H and satisfies equations (1.5) – (1.7) with d = 3 , where the potential φ ∈ H isreal, and ω ∈ R . To prove item i), let us denote(2.16) ρ n ( x ) := σ n ( x ) + e | ψ n ( x ) | , σ n ( x ) := X j µ per j ( x − x jn ) . Now the sequence ψ n and the corresponding sequence φ n := ( − ∆) − ρ n are bounded in H by (2.8)-(2.10), (2.7) and (2.13)-(2.14). Hence, both sequences are precompact in L p for any p ∈ [1 ,
6) by theSobolev embedding theorem [1, 30]. Therefore, the sequence ρ n is precompact in L by our assumption(1.9), and respectively, the sequence φ n is precompact in H . As the result, there exist a subsequence n ′ → ∞ for which(2.17) ψ n ′ L p −→ ψ , φ n ′ H −→ φ , x n ′ → x , n ′ → ∞ with any p ∈ [1 , σ n ′ L −→ σ , ρ n ′ L −→ ρ , n ′ → ∞ , where σ ( x ) and ρ ( x ) are defined by (1.8) and (1.6). Hence, the neutrality condition (1.11) holds,( ψ , x ) ∈ M , φ ∈ H , and for these limit functions we have(2.19) − ∆ φ = ρ , Z T φ ( x ) d x = 0 . To prove identity (2.15), we take into account that I ( ψ ) is lower semicontinuous on L , while I ( φ )is continuous on H ; i.e.,(2.20) I ( ψ ) ≤ lim inf n ′ →∞ I ( ψ n ′ ) , I ( φ ) = lim n ′ →∞ I ( φ n ′ ) . These limits, together with (2.14), imply that(2.21) E ( ψ , x ) = I ( ψ ) + I ( φ ) ≤ E . Now (2.15) follows from the definition of E , since ( ψ , x ) ∈ M . Thus Theorem 2.1 i) is proved.We will prove the item ii) in next sections. Theorem 2.1 ii) follows from next proposition.
Proposition 2.2.
The limit functions (2.17) satisfy equations (1.5)–(1.7) with d = 3 and ω ∈ R . E ( ψ ) := E ( ψ, x ). We derive (1.5)in next sections, equating the variation of E ( · ) | M to zero at ψ = ψ . In this section we calculate thecorresponding Gˆateaux variational derivative.We should work directly on M introducing an atlas in a neighborhood of ψ in M . We define theatlas as the stereographic projection from the tangent plane T M ( ψ ) = ( ψ ) ⊥ := { ψ ∈ H : h ψ, ψ i =0 } to the sphere (2.13):(2.22) ψ τ = ψ + τ k ψ + τ k L √ Z, τ ∈ ( ψ ) ⊥ . Obviously,(2.23) ddε (cid:12)(cid:12)(cid:12) ε =0 ψ ετ = τ, τ ∈ ( ψ ) ⊥ , where the derivative exists in H . We define the ‘Gˆateaux derivative’ of E ( · ) | M as(2.24) D τ E ( ψ ) := lim ε → E ( ψ ετ ) − E ( ψ ) ε , if this limit exists. We should restrict the set of allowed tangent vectors τ . Definition 2.3. T is the space of test functions τ ∈ ( ψ ) ⊥ ∩ C ∞ ( T ) . Obviously, T is dense in ( ψ ) ⊥ in the norm of H . Let us rewrite the energy (2.3) as(2.25) E ( ψ ) := Z T h ~ |∇ ψ ( x ) | + 12 | Λ ρ ( x ) | i d x , ρ ( x ) := σ ( x ) + e | ψ ( x ) | , where Λ := ( − ∆) − / is defined similarly to (2.6):(2.26) Λ ρ ( x ) := 1 p | T | X k ∈ Γ ∗ \ ˆ ρ ( k ) | k | e − i kx ∈ L for ρ ∈ L . Lemma 2.4.
Let τ ∈ T . Then the derivative (2.24) exists, and ( cf. (2.12)) , (2.27) D τ E ( ψ ) = Z T h ~
2m ( ∇ τ ∇ ψ + ∇ ψ ∇ τ ) + e Λ ρ Λ( τ ψ + ψ τ ) i d x . Proof.
Let us denote ρ ετ ( x ) := σ ( x ) + e | ψ ετ ( x ) | . Lemma 2.5.
For τ ∈ T we have (2.28) D τ Λ ρ := lim ε → Λ ρ ετ − Λ ρ ε = e Λ( τ ψ + ψ τ ) , where the limit converges in L . Proof.
In the polar coordinates(2.29) ψ ετ = ( ψ + ετ ) cos α, α = α ( ε ) = arctan ε k τ k L k ψ k L . ρ ετ = Λ σ + e cos α Λ | ψ + ετ | = Λ ρ + eε cos α Λ( τ ψ + ψ τ ) + e Λ[ ε | τ | cos α − | ψ | sin α ] . (2.30)Here Λ ρ ∈ L since ρ ∈ L , and similarly Λ[ ψ τ ] ∈ L since ψ τ ∈ L . It remains to estimate thelast term of (2.30),(2.31) R ε := Λ[ ε | τ | cos α − | ψ | sin α ] . Here | ψ | ∈ L since ψ ∈ H ⊂ L . Finally, | τ | ∈ L and sin α ∼ ε . Hence, the convergence(2.28) holds in L .Now (2.27) follows by differentiation in ε of (2.25) with ψ = ψ ετ and ρ = ρ ετ . Since ψ is a minimal point, the Gˆateaux derivative (2.27) vanishes:(2.32) Z T h ~
2m ( ∇ τ ∇ ψ + ∇ ψ ∇ τ ) + e Λ ρ Λ( τ ψ + ψ τ ) i d x = 0 . Substituting iτ instead of τ in this identity and subtracting, we obtain(2.33) − ~ h ∆ ψ , τ i + e h Λ ρ , Λ( ψ τ ) i = 0 . Next step we should evaluate the “nonlinear” term.
Lemma 2.6.
For the limit functions (2.17) – (2.18) we have (2.34) h Λ ρ , Λ( ψ τ ) i = h φ ψ , τ i , τ ∈ T . Proof.
Let us substitute ρ = − ∆ φ . Then, by the Parseval–Plancherel identity,(2.35) h Λ ρ , Λ( ψ τ ) i = X k ∈ Γ ∗ \ k ˆ φ ( k ) | k | · d ψ τ ( k ) | k | = h ˆ φ , d ψ τ i = h φ , ψ τ i = h φ ψ , τ i . which proves (2.34).Using (2.34), we can rewrite (2.33) as the variational identity (cf. (2.12))(2.36) h− ~
2m ∆ ψ + eφ ψ , τ i = 0 , τ ∈ T . Now we prove the Schr¨odinger equation (1.5) with d = 3. Lemma 2.7. ψ is the eigenfunction of the Schr¨odinger operator H = − ~ ∆ + eφ : (2.37) Hψ = λψ , where λ ∈ R . roof. First, Hψ is a well-defined distribution since φ ∈ H ⊂ C ( T ) by (2.17). Second, ψ = 0since ψ ∈ M and Z >
0. Hence, there exists a test function θ ∈ C ∞ ( T ) \ T , i.e.,(2.38) h ψ , θ i 6 = 0 . Then(2.39) h ( H − λ ) ψ , θ i = 0 . for an appropriate λ ∈ C . However, ( H − λ ) ψ also annihilates T by (2.36), hence it annihilatesthe whole space C ∞ ( T ). This implies (2.37) in the sense of distributions with a λ ∈ C . Finally, thepotential is real, and φ ∈ C ( T ). Hence, λ ∈ R .This lemma implies equation (1.5) with ~ ω = λ . Hence, ψ ∈ H since φ ∈ C ( T ). Now Theorem2.1 ii) is proved. We have proved that ψ ∈ H under condition (1.9). Using the Schr¨odinger equation (2.37) we canimprove further the smoothness of ψ strengthening the condition (1.9). Namely, let us assume that(2.40) µ per j ∈ C ∞ ( T ) , j = 1 , ..., N. Then also(2.41) σ ( x ) := N X j =1 µ per j ( x − x j ) ∈ C ∞ ( T ) . For example, (2.40) and (2.41) hold if µ j ∈ S ( R ), where S ( R ) is the Schwartz space of test functions. Lemma 2.8.
Let condition (2.40) hold, and ψ ∈ H , φ ∈ H be a solution to equations (1.5) – (1.7) with d = 3 and some x ∈ T N . Then the functions ψ and φ are smooth. Proof.
First, φ ψ ∈ H since H s is the algebra for s > /
2. Hence, equation (1.5) implies that(2.42) ψ ∈ H ⊂ C ( T ) . Now ρ := σ + e | ψ | ∈ H by (2.40). Then (1.6) implies that φ ∈ H ⊂ C ( T ). Hence, φ ψ ∈ H , ψ ∈ H , ρ ∈ H , etc. For simplicity of notation we will consider the 2D lattice Γ = Z and construct a solution to system(1.5)–(1.7) for the corresponding functions on the ‘cylindrical cell’ T := R / Γ = T × R with thecoordinates x = ( x , x , x ), where ( x , x ) ∈ T and x ∈ R . Now we denote by H s the complexSobolev space on T , and by L p , the complex Lebesgue space of functions on T .We will construct a ground state by minimizing the energy (2.3), where the integral is extendedover T instead of T . The neutrality condition of type (2.1) holds for Γ -periodic states with finiteenergy, as we show below. 10 .1 The energy per cell We restrict ourselves by N = 1, so x = x can be chosen arbitrary because of the translationinvariance of the system (1.5)–(1.7). For example, we can set x = 0.The energy in the cylindrical cell T is defined similarly to (2.3), which we rewrite as (2.25):(3.1) E ( ψ ) := Z T h ~ |∇ ψ ( x ) | + 12 | Λ ρ ( x ) | i d x , ρ ( x ) := σ ( x ) + e | ψ ( x ) | . Here σ ( x ) is defined by (2.2) with N = 1 and x = 0:(3.2) σ = µ per1 ∈ L ∩ L according to our condition (1.9). Hence, we have(3.3) Z T σ ( x ) d x = Z | e | , Z > . Further, Λ is the operator ( − ∆) − / defined by the Fourier transform. Namely, we denote Γ ∗ = 2 π Γ ,and define the Fourier representation for the test functions ϕ ∈ C ∞ ( T ) by(3.4) ϕ ( x ) = 1 √ π X k ∈ Γ ∗ e − i ( k x + k x ) Z R e − iξx ˆ ϕ ( k , ξ ) dξ, x ∈ T , where(3.5) ˆ ϕ ( k , ξ ) = F ϕ ( k , ξ ) = 1 √ π Z T e i ( k x + k x + ξx ) ϕ ( x ) d x , ( k , ξ ) ∈ Σ := Γ ∗ × R . The operator Λ is defined for ϕ ∈ L ∩ L by(3.6) Λ ϕ = F − ˆ ϕ ( k , ξ ) p k + ξ provided the quotient belongs to L (Σ ). In this case(3.7) ˆ ϕ (0 ,
0) = 0 . Let us note that ρ ∈ L ∩ L for ψ ∈ H by our condition (1.9) since ψ ∈ L p with p ∈ [2 ,
6] by theSobolev embedding theorem. For ψ ∈ H with finite energy (3.1) we have Λ ρ ∈ L (Σ ). Therefore,(3.7) with ϕ = ρ implies the neutrality condition (2.1) with T instead of T :(3.8) ˆ ρ (0 ,
0) = Z T ρ ( x ) d x = Z T [ σ ( x ) + e | ψ ( x ) | ] d x = 0 . Now (3.3) gives(3.9) Z T | ψ ( x ) | d x = Z . In other words, the finiteness of the Coulomb energy k Λ ρ k prevents the electron charge from escapingto infinity, as mentioned in Introduction. Definition 3.9. M denotes the set of ψ ∈ H satisfying the neutrality condition (3.9) .
11t is important that the energy be finite for a nonempty set of ψ ∈ H . To find the correspondingcondition, let us rewrite the energy (3.1) using the Parseval-Plancherel identity:(3.10) E ( ψ ) = X k ∈ Γ ∗ ~ Z R ( k + ξ ) | ˆ ψ ( k , ξ ) | dξ + 12 X k ∈ Γ ∗ Z R | ˆ ρ ( k , ξ ) | k + ξ dξ. Here the first term on the right hand side is finite for all ψ ∈ H . The second term is finite up to theinfrared divergence at the point ( k , ξ ) = (0 ,
0) since ρ ∈ L (Σ ) for ψ ∈ H .We note that (3.3) can be written as ˆ µ per1 (0) + eZ = 0. We will assume that moreover,(3.11) Condition II. ˆ µ per1 (0 , ξ ) + eZ | ξ | ∈ L ( − , . For example, this condition holds, provided that(3.12) Z T | x || µ per1 ( x ) | d x < ∞ . Lemma 3.10.
Let conditions (1.9) and (3.11) hold, N = 1 and x ∈ T . Then the energy (3.10) isfinite for a dense set of ψ ∈ H . Proof.
By definition, ˆ ρ (0 , ξ ) = ˆ µ per1 (0 , ξ ) + e ˆ P (0 , ξ ), where P ( x ) := | ψ ( x ) | . Hence, (3.11) impliesthat the energy (3.10) is finite for ψ ∈ M with finite momenta Z T | x | | ψ ( x ) | d x < ∞ . Similarly to the 3D case, the energy is nonnegative, and we choose a minimizing sequence ψ n ∈ M such that(3.13) E ( ψ n ) → E := inf M E ( ψ ) , n → ∞ . The second main result of the present paper is the following.
Theorem 3.11.
Let conditions (1.9) and (3.11) hold, and N = 1 . Theni) There exists ψ ∈ M with (3.14) E ( ψ ) = E . ii) Moreover, ψ ∈ H ( T ) and satisfies equations (1.5) – (1.7) with d = 2 , where the potential φ ∈ H ( T ) is real, x = 0 , and ω ∈ R .iii) The following bound holds (3.15) | φ ( x ) | ≤ C (1 + | x | ) / , x ∈ T . To prove item i), let us note that the sequence ψ n is bounded in H due to (3.1), (3.9) and(3.13). Hence, by the Sobolev embedding theorem [1, 30], the sequence ψ n is bounded in L p witheach p ∈ [2 ,
6) and compact in L pR := L p ( T ( R )) for any R >
0, where T ( R ) = { x ∈ T : | x | < R } .Therefore, there exists a subsequence(3.16) ψ n ′ L pR −→ ψ , ρ n ′ := µ per1 + e | ψ n ′ | L R −→ ρ , n ′ → ∞ , ∀ R > , µ per1 ∈ L ∩ L by (1.9). Hence, ψ ∈ H ∩ L p , and(3.17) ρ ( x ) = µ per1 ( x ) + e | ψ ( x ) | ∈ L ∩ L . Next problem is to check the neutrality condition (3.9) for the limit charge density ρ since theconvergence (3.16) itself is not sufficient. Lemma 3.12.
The limit function ψ ∈ M , and the energy (3.1) for ψ is finite. Proof.
Let us prove that(3.18) E ( ψ ) ≤ E . Indeed, (3.10) with ψ = ψ n ′ reads(3.19) E ( ψ n ′ ) := D ~ | f n ′ ( k , ξ ) | + 12 | g n ′ ( k , ξ ) | E Σ , where h . . . i Σ stands for X k ∈ Γ ∗ Z R . . . dξ and f n ′ ( k , ξ ) := p k + ξ ˆ ψ n ′ ( k , ξ ) , g n ′ ( k , ξ ) := ˆ ρ n ′ ( k , ξ ) p k + ξ . The functions ˆ ψ n ′ and ˆ ρ n ′ are bounded in L (Σ ), and are converging in the sense of distributions dueto (3.16). Hence,(3.20) ˆ ψ n ′ L w − ⇀ ˆ ψ , ˆ ρ n ′ L w − ⇀ ˆ ρ , n ′ → ∞ . Similarly, the functions f n ′ and g n ′ are bounded in L (Σ ) by (3.19), (3.13), and are converging in thesense of distributions due to (3.20). Therefore,(3.21) f n ′ L w − ⇀ f , g n ′ L w − ⇀ g , n ′ → ∞ . Hence, for the limit functions, f ( k , ξ ) = p k + ξ ˆ ψ ( k , ξ ) , g ( k , ξ ) = ˆ ρ ( k , ξ ) p k + ξ , a.a. ( k , ξ ) ∈ Σ . Therefore, (3.18) holds since(3.22) E ( ψ ) = D ~ | f ( k , ξ ) | + 12 | g ( k , ξ ) | E Σ ≤ E by the week convergence (3.21). In particular,(3.23) Λ ρ ∈ L . Therefore, ˆ ρ (0 ,
0) = 0 as in (3.8) since ρ ∈ L by (3.17). Hence, ψ ∈ M .Now (3.18) implies (3.14). Thus Theorem 3.11 i) is proved.13 .3 The Poisson equation Our aim here is to construct the potential which is the solution to the Poisson equation (1.6) with d = 2. It suffices to solve the equation(3.24) ∇ φ ( x ) = G ( x ) , x ∈ T , where G ( x ) := − iF − k ,ξ ) k + ξ ˆ ρ ( k , ξ ) is a real vector field, G ∈ L ⊗ R by (3.23), and rot G ( x ) ≡ Lemma 3.13.
The equation (3.24) admits real solution φ ∈ H ( T ) which is unique up to anadditive constant, and satisfies the bound (3.15) . Proof.
The uniqueness up to constant is obvious. If the solution exists, then φ ∈ H ( T ) by(3.17). Local solutions exist since rot G ( x ) ≡
0. However, the existence of the global solution is notobvious since the cell T is not 1-connected.We will prove the existence using the following arguments. Formally φ ( x ) = F − ρ ( k ,ξ ) k + ξ . However,the last expression is not correctly defined distribution in the neighborhood of the point (0 , ρ = ˆ ρ + ˆ ρ where(3.25) ˆ ρ ( k , ξ ) = (cid:26) ˆ ρ (0 , ξ ) , k = 0 , | ξ | < , , otherwise.Respectively, G = G + G , and the solution φ = φ + φ . Obviously,(3.26) G ( x ) = − iF − (0 , ξ ) ξ ˆ ρ (0 , ξ ) = e g ( x ) , e := (0 , , , and g ( x ) is a smooth function. Moreover, (3.17) implies that g ( x ) is the real function, and g ∈ L ( R ) since G ∈ L ⊗ R . Hence, the solution φ ( x ) = Z x g ( s ) ds is smooth and continuous,and depends on x only. The bound (3.15) for φ follows by the Cauchy-Schwartz inequality.The second solution is given by φ ( x ) = F − ρ ( k ,ξ ) k + ξ , where ˆ ρ ∈ L (Σ ) by (3.17). Moreover,ˆ ρ (0 , ξ ) = 0 for | ξ | <
1, and hence φ ∈ H . Remarks 3.14. i) The function φ ( x ) = (1 + | x | ) / − ε with ε > shows that the bound (3.15) isexact under the condition ∇ φ ∈ L . Note that the potential of uniformly charged plane grows linearlywith the distance.ii) In the Fourier transform, (3.24) implies that (3.27) ( k , ξ ) ˆ φ ( k , ξ ) ∈ L (Σ ) ⊗ C . Theorem 3.11 ii) follows from next proposition.
Proposition 3.15.
The functions ψ , φ satisfy equations (1.5)–(1.7) with d = 2 and ω ∈ R . The equation (1.6) is proved above, and the equation (1.7) follows from (2.11) and (3.14) by thetranslation invariance of the energy. It remains to prove the Schr¨odinger equation (1.5). We are goingto derive (1.5), equating the variation of E ( ψ ) | M to zero at ψ = ψ . In this section we calculate thecorresponding Gˆateaux variational derivative.Similarly to (2.22), we define the atlas in a neighborhood of ψ in M as the stereographic projectionfrom the tangent plane T M ( ψ ) = ( ψ ) ⊥ := { ψ ∈ H : h ψ, ψ i = 0 } to the sphere (3.9):(3.28) ψ τ = ψ + τ k ψ + τ k L p Z , τ ∈ ( ψ ) ⊥ . efinition 3.16. T is the space of test functions τ ∈ ( ψ ) ⊥ ∩ C ∞ ( T ) . Obviously, T is dense in ( ψ ) ⊥ in the norm of H . Lemma 3.17.
Let τ ∈ T . Then i) The energy E ( ψ ετ ) is finite for ε ∈ R . ii) The Gˆateaux derivative (2.24) exists, and similarly to (2.27) , (3.29) D τ E ( ψ ) = Z T h ~
2m ( ∇ τ ∇ ψ + ∇ ψ ∇ τ ) + e Λ ρ Λ( τ ψ + ψ τ ) i d x . Proof. i) We should prove the bound(3.30) E ( ψ ετ ) := ~ Z T |∇ ψ ετ ( x ) | d x + 12 Z T | Λ ρ ετ ( x ) | d x < ∞ , where ρ ετ ( x ) := σ ( x ) + e | ψ ετ ( x ) | . The first integral in (3.30) is finite, since ψ ετ ∈ H . Lemma 3.18. Λ ρ ετ ∈ L for τ ∈ T and ε ∈ R , and (3.31) D τ Λ ρ := lim ε → Λ ρ ετ − Λ ρ ε = e Λ( τ ψ + ψ τ ) , where the limit converges in L . Proof.
We use the polar coordinates (2.29) and the corresponding representation (2.30):(3.32) Λ ρ ετ = Λ ρ + eε cos α Λ( τ ψ + ψ τ ) + e Λ[ ε | τ | cos α − | ψ | sin α ] . Now Λ ρ ∈ L according to (3.23). Further, Λ[ τ ψ ] ∈ L by the following arguments:a) τ ψ ∈ L ,b) d τ ψ is the smooth function on Σ , andc) the orthogonality τ ⊥ ψ implies that(3.33) d τ ψ (0 ,
0) = 0 . It remains to estimate the last term of (3.32), Let us denote T ( x ) := | τ ( x ) | and P ( x ) := | ψ ( x ) | .Then the last term (up to a constant factor) reads(3.34) R ε ( x ) := Λ[ ε T ( x ) cos α − P ( x ) sin α ] . Lemma 3.19. R ε ∈ L for ε ∈ R , and (3.35) k R ε k L = O ( ε ) , ε → . Proof. i) It suffices to check that ε ˆ T (0 , ξ ) cos α − ˆ P (0 , ξ ) sin α | ξ | = ( ε ˆ T (0 , ξ ) − Z tan α ) cos α | ξ | − ( ˆ P (0 , ξ ) − Z ) sin α | ξ | ∈ L ( − , . (3.36) 15et us consider each term of the last line of (3.36) separately.1) The first quotient belongs to L ( − , ε ˆ T (0 , − Z tan α = Z T ε | τ | d x − Z tan α = 0by the definition of α in (2.29) since k ψ k = √ Z .2) The second quotient belongs to L ( − , ρ | ξ | = ˆ µ per1 | ξ | + e ˆ P | ξ | = ˆ µ per1 + eZ | ξ | + e ˆ P − Z | ξ | , where all the functions are taken at the point (0 , ξ ). Here the left-hand side belongs to L ( − , ρ ∈ L , while the first term on the right belongs to L ( − ,
1) by our assumption (3.11). ii)
The bound (3.35) holds for both terms of (3.36) by the arguments above since tan α ∼ sin α ∼ ε as ε → L by (3.35).ii) Lemma 3.18 implies the bound (3.30). Formula (3.29) follows by differentiation of (3.30) in ε . Since ψ is a minimal point, the Gˆateaux derivative (3.29) vanishes:(3.39) Z T h ~
2m ( ∇ τ ∇ ψ + ∇ ψ ∇ τ ) + e Λ ρ Λ( τ ψ + ψ τ ) i d x = 0 . Substituting iτ instead of τ in this identity and subtracting, we obtain(3.40) − ~ h ∆ ψ , τ i + e h Λ ρ , Λ( τ ψ ) i = 0 . Next step we should evaluate the “nonlinear” term.
Lemma 3.20.
For the limit functions (3.16) we have (3.41) h Λ ρ , Λ( τ ψ ) i = h φ ψ , τ i , τ ∈ T , where φ is any potential satisfying (3.24) . Proof.
First we note that Λ ρ ∈ L by 3.23), and Λ( τ ψ ) ∈ L as we have established in the proofof Lemma 3.18. Moreover, ρ = − ∆ φ . Then, by the Parseval–Plancherel identity,(3.42) h Λ ρ , Λ( τ ψ ) i = X k ∈ Γ ∗ \ Z ˆ φ ( k , ξ ) d τ ψ ( k , ξ ) dξ + lim ε → Z | ξ | >ε ˆ φ (0 , ξ ) d τ ψ (0 , ξ ) dξ = h ˆ φ , d τ ψ i , where ˆ φ is the distribution on Σ . The last identity holds (and the right hand side is well defined)by (3.33) since ξ ˆ φ (0 , ξ ) ∈ L ( − ,
1) due to (3.24) with G ∈ L ⊗ R . Finally,(3.43) h ˆ φ , d τ ψ i = h φ , τ ψ i = Z φ ( x ) τ ( x ) ψ ( x ) d x by an obvious extension of the Parseval–Plancherel identity.Using (3.41), we can rewrite (3.40) as the variational identity similar to (2.36):(3.44) h− ~
2m ∆ ψ + eφ ψ , τ i = 0 , τ ∈ T . .6 The Schr¨odinger equation Now we prove the Schr¨odinger equation (1.5) with d = 2. Lemma 3.21. ψ is the eigenfunction of the Schr¨odinger operator: (3.45) Hψ = λψ , where λ ∈ R . Proof.
This equation with λ ∈ C follows as in Lemma 2.7. It remains to verify that λ is real. Ourplan is standard: to multiply (3.45) by ψ and to integrate. Formally , we would obtain(3.46) h Hψ , ψ i = λ h ψ , ψ i . However, it is not clear that the left-hand side is well defined and real since the potential φ ( x ) cangrow by (3.15).To avoid this problem, we multiply by a function ψ ε ∈ H with compact support, where ε > k ψ ε − ψ k H → ε →
0. Then(3.47) h Hψ , ψ ε i = λ h ψ , ψ ε i , and the right-hand side converges to the one of (3.46) as ε →
0. Hence, the left-hand sides alsoconverge. In detail,(3.48) h Hψ , ψ ε i = − ~ h ∆ ψ , ψ ε i + h φ ψ , ψ ε i . For the middle term, the limit exists and is real. Therefore, identity (3.47) implies that the last termis also converging, and hence it remains to make its limit real by a suitable choice of approximations ψ ε . We note that h φ ψ , ψ ε i = lim δ → h φ ψ δ , ψ ε i = lim δ → h φ , ψ δ ψ ε i , (3.49)since φ ∈ H ( T ) ⊂ C ( T ). Hence, we can set(3.50) ψ ε ( x ) = χ ( εx ) ψ ( x ) . where χ is a real function from C ∞ ( R ) with ψ (0) = 1. Now the functions ψ δ ( x ) ψ ε ( x ) are real for all ε, δ >
0. It remains to note that the potential φ ( x ) is also real by Lemma 3.13.This lemma implies equation (1.5). Therefore, ψ ∈ H ( T ) since φ ∈ C ( T ). Theorem 3.11 ii)is proved. We have proved that ψ ∈ H ( T ) under conditions (1.9) and (3.11). Using the Schr¨odinger equation(1.5) we can improve the smoothness of ψ strengthening the condition (1.9). Namely, let us assumethat(3.51) µ per1 ∈ C ∞ ( T ) . For example, (3.51) holds if µ ∈ S ( R ), where S ( R ) is the Schwartz space of test functions. Lemma 3.22.
Let condition (3.51) hold, and ψ ∈ H ( T ) , φ ∈ H ( T ) is a solution to equations (1.5) – (1.7) with d = 2 . Then the functions ψ , φ are smooth. The proof is similar to the one of Lemma 2.8.17
1D lattice
The case of a one dimensional lattice Γ is very similar to the 2D case, though some of our constructionsand arguments require suitable modifications. For d = 1 we can assume Γ = Z without loss ofgenerality and construct a solution to system (1.5)–(1.7) for the corresponding functions on the ‘slab’ T := R / Γ = T × R with coordinates x = ( x , x , x ), where x ∈ T , and ( x , x ∈ R . Now wedenote by H s the complex Sobolev space on T , and by L p , the complex Lebesgue space of functionson T .The existence of the ground state follows by minimizing the energy (2.3), where the integral isextended over T instead of T . The neutrality condition of type (2.1) holds for Γ -periodic stateswith finite energy, as for d = 2.Again we restrict ourselves by N = 1, so x = x can be chosen arbitrary, and we set x = 0.The energy in the slab T is defined by expression similar to (3.1):(4.1) E ( ψ ) := Z T h ~ |∇ ψ ( x ) | + 12 | Λ ρ ( x ) | i d x , ρ ( x ) := σ ( x ) + e | ψ ( x ) | . Here σ = µ per i ∈ L ∩ L as in (3.2). Hence,(4.2) Z T σ ( x ) d x = Z | e | , Z > . Now the Fourier representation for the test functions ϕ ( x ) ∈ C ∞ ( T ) is defined by(4.3) ϕ ( x ) = 12 π X k ∈ Γ ∗ e − i k x Z R e − i ( ξ x + ξ x ) ˆ ϕ ( k , ξ ) dξ, where Γ ∗ = 2 π Γ and(4.4) ˆ ϕ ( k , ξ ) = F ϕ ( k , ξ ) = 12 π Z T e i ( k x + ξ x + ξ x ) ϕ ( x ) d x , ( k , ξ ) ∈ Σ := Γ ∗ × R . The operator Λ = ( − ∆) / is defined for ϕ ∈ L ∩ L by the same formula (3.6) provided the quotientbelongs to L (Σ ). This implies(4.5) ˆ ϕ (0 ,
0) = 0 . For ψ ∈ H with finite energy (4.1) we have Λ ρ ∈ L (Σ ), and hence, (4.5) with ϕ = ρ implies theneutrality condition (3.8) with T instead of T :(4.6) ˆ ρ (0 ,
0) = Z T ρ ( x ) d x = Z T [ σ ( x ) + e | ψ ( x ) | ] d x = 0 . Now (4.2) gives(4.7) Z | ψ ( x ) | d x = Z . Thus, the finiteness of the Coulomb energy k Λ ρ k prevents the electron charge from escaping toinfinity, as in 2D case.Finally, the Fourier transform F : ψ ˆ ψ is a unitary operator from L ( T ) to L (Σ ). Hence,energy (3.1) reads(4.8) E ( ψ ) = X k ∈ Γ ∗ Z R h ~
2m ( k + ξ ) | ˆ ψ ( k , ξ ) | + 12 | ˆ ρ ( k , ξ ) | k + ξ i dξ. efinition 4.1. M denotes the set of ψ ∈ H satisfying the neutrality condition (4.7). We note that (4.2) can be written as ˆ µ per1 (0) + eZ = 0. We assume moreover,(4.9) Condition III. ˆ µ per1 (0 , ξ ) + eZ | ξ | ∈ L ( D ) , D := { ξ ∈ R : | ξ | ≤ } similarly to (3.11). For example, this condition holds, provided that(4.10) Z R (1 + | x | + | x | ) | µ ( x ) | d x < ∞ . The third main result of the present paper is the following.
Theorem 4.2.
Let conditions (1.9) and (4.9) hold, and N = 1 . Theni) There exists ψ ∈ M with (4.11) E ( ψ ) = inf ψ ∈ M E ( ψ ) . ii) Moreover, ψ ∈ H ( T ) and satisfies equations (1.5) – (1.7) with d = 1 , where the potential φ ∈ H ( T ) is real, x = 0 , and ω ∈ R .iii) The following bound holds (4.12) | φ ( x ) | ≤ C (1 + | x | + | x | ) / , x ∈ T . The proof is similar to the one of Theorem 3.11. As in 2D case, we obtain ψ ∈ M as a minimizerfor the energy (4.1). The potential φ can be constructed by a modification of Lemma 3.13, seeAppendix below.Finally, next lemma follows similarly to Lemma 2.8. Lemma 4.3.
The functions ψ , φ are smooth under condition (4.13) µ per1 ∈ C ∞ ( T ) . A The potential of 1D lattice
We start with obvious modifications of the proof of Lemma 3.13. Namely, the potential φ ( x ) for the1D lattice satisfies the equation of type (3.24) with(A. 1) G := − iF − ( k , ξ ) k + ξ ˆ ρ ( k , ξ ) ∈ L ( T ) , rot G ( x ) ≡ . We use the splitting of type (3.25), and respectively, the solution splits as φ = φ + φ . The secondsolution φ ∈ H as in the proof of Lemma 3.13. Hence, φ is bounded continuous function on T bythe Sobolev embedding theorem.On the other hand, the analysis of the first solution needs some modifications. Now G ( x ) = g ( x , x ) ∈ L ( R ) ⊗ R is the real vector field, and supp ˆ g ⊂ { ξ ∈ R : | ξ | ≤ } . Therefore, g is thesmooth function, and(A. 2) ∆ φ = ∇ · g ∈ L ( R ) , rot g ( x ) ≡ . ∇ φ = g is given by the contour integral(A. 3) φ ( x ) = Z x g ( y ) d y + C, x ∈ R , which does not depend on the path in R . This solution is real and smooth.We still should prove the estimate (4.12). We will deduce it from the corresponding estimate ’inthe mean’. Let us denote the circle B := { x ∈ R : | x | < } . Lemma A.4.
For any unit vector e ∈ R (A. 4) k φ k L ( B + e R ) ≤ C (1 + R ) / , R > . Proof.
First, (A. 3) implies that(A. 5) φ ( x + e R ) − φ ( x ) = Z R g ( x + t e ) dt, x ∈ R for any R ∈ R . Now the Cauchy-Schwartz inequality implies that(A. 6) | φ ( x + e R ) | ≤ C + 2 R Z R | g ( x + t e ) | dt, x ∈ B since the function φ is bounded in B . Finally, averaging over x ∈ B , we get(A. 7) Z B | φ ( x + e R ) | d x ≤ C | B | + 2 R Z R Z B | g ( x + t e ) | d x dt ≤ C + C R k g k L ( R ) . Hence, (A. 4) is proved.Now (4.12) follows from the Sobolev embedding theorem:(A. 8) max x ∈ B + e R | φ ( x ) | ≤ C k φ k H ( B + e R ) ≤ C [ k ∆ φ k L ( B + e R ) + k φ k L ( B + e R ) )] ≤ C (1 + R ) / since ∆ φ ∈ L ( R ) by (A. 2). Remark A.5.
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