Periodic Schrödinger operators with local defects and spectral pollution
aa r X i v : . [ m a t h - ph ] N ov Periodic Schr¨odinger operators with local defectsand spectral pollution ∗ Eric Canc`es † Virginie Ehrlacher Yvon Maday ‡ November 8, 2018
Abstract
This article deals with the numerical calculation of eigenvalues of per-turbed periodic Schr¨odinger operators located in spectral gaps. Such op-erators are encountered in the modeling of the electronic structure of crys-tals with local defects, and of photonic crystals. The usual finite elementGalerkin approximation is known to give rise to spectral pollution. In thisarticle, we give a precise description of the corresponding spurious states.We then prove that the supercell model does not produce spectral pollu-tion. Lastly, we extend results by Lewin and S´er´e on some no-pollutioncriteria. In particular, we prove that using approximate spectral projec-tors enables one to eliminate spectral pollution in a given spectral gap ofthe reference periodic Schr¨odinger operator.
Periodic Schr¨odinger operators are encountered in the modeling of the electronicstructure of crystals, as well as the study of photonic crystals. They are self-adjoint operators on L ( R d ) with domain H ( R d ) of the form H = − ∆ + V per , where ∆ is the Laplace operator and V per a R -periodic function of L p loc ( R d ) ( R being a periodic lattice of R d ), with p = 2 if d ≤ p > d = 4 and p = d/ d ≥ ∗ This work was financially supported by the ANR grant MANIF. † Universit´e Paris Est, CERMICS, Projet MICMAC, Ecole des Ponts ParisTech - INRIA, 6& 8 avenue Blaise Pascal, 77455 Marne-la-Vall`ee Cedex 2, France, ( [email protected] , [email protected] ) ‡ Universit´e Pierre et Marie Curie-Paris 6, UMR 7598, Laboratoire J.-L. Lions, Paris, F-75005 France, and Division of Applied Mathematics, Brown University, 182 George Street,Providence, RI 02912, USA, ( [email protected] )
1r extended defects. In solid state physics, local defects are due to impurities,vacancies, or interstitial atoms, while extended defects correspond to disloca-tions or grain boundaries. The properties of the crystal can be dramaticallyaffected by the presence of defects. In this article, we consider the case of a d -dimensional crystal with a single local defect, whose properties are encodedin the perturbed periodic Schr¨odinger operator H = H + W = − ∆ + V per + W, W ∈ L ∞ ( R d ) , W ( x ) → | x |→∞ . (1)Note that we do not assume here that W is compactly supported. This allowsus in particular to handle the mean-field model considered in [6]. In the lattermodel, d = 3 and the self-consistent potential W generated by the defect is ofthe form W = ρ ⋆ | · | − with ρ ∈ L ( R ) ∩ C , C denoting the Coulomb space.Such potentials are continuous and vanish at infinity, but are not compactlysupported in general.Computing the spectrum of the operator H is a key step to understand theproperties of the system. It is well known that the self-adjoint operator H is bounded from below on L ( R d ), and that the spectrum σ ( H ) of H ispurely absolutely continuous, and composed of a finite or countable number ofclosed intervals of R [16]. The open interval laying between two such closed in-tervals is called a spectral gap. The multiplication operator W being a compactperturbation of H , it follows from Weyl’s theorem [16] that H is self-adjointon L ( R d ) with domain H ( R d ), and that H and H have the same essentialspectrum: σ ess ( H ) = σ ess ( H ) = σ ( H ) . Contrarily to H , which has no discrete spectrum, H may possess discreteeigenvalues. While the discrete eigenvalues located below the minimum of σ ess ( H ) are easily obtained by standard variational approximations (in virtue ofthe Rayleigh-Ritz theorem [16]), it is more difficult to compute numerically thediscrete eigenvalues located in spectral gaps, for spectral pollution may occur [5].In Section 2, we recall that the usual finite element Galerkin approximationmay give rise to spectral pollution [5], and give a precise description of thecorresponding spurious states. In Section 3, we show that the supercell modeldoes not produce spectral pollution. Lastly, we extend in Section 4 results byLewin and S´er´e [14] on some no-pollution criteria, which guarantee in particularthat the numerical method introduced in [6], involving approximate spectralprojectors, and is spectral pollution free.2 Galerkin approximation
The discrete eigenvalues of H and the associated eigenvectors can be obtainedby solving the variational problem (cid:26) find ( ψ, λ ) ∈ H ( R d ) × R such that ∀ φ ∈ H ( R d ) , a ( ψ, φ ) = λ h ψ, φ i L , where h· , ·i L is the scalar product of L ( R d ) and a the bilinear form associatedwith H : a ( ψ, φ ) = ˆ R d ∇ ψ · ∇ φ + ˆ R d ( V per + W ) ψφ. A sequence ( X n ) n ∈ N of finite dimensional subspaces of H ( R d ) being given, weconsider for all n ∈ N , the self-adjoint operator H | X n : X n → X n defined by ∀ ( ψ n , φ n ) ∈ X n × X n , h H | X n ψ n , φ n i L = a ( ψ n , φ n ) . The so-called Galerkin method consists in approximating the spectrum of theoperator H by the eigenvalues of the discretized operators H | X n for n largeenough, the latter being obtained by solving the variational problem (cid:26) find ( ψ n , λ n ) ∈ X n × R such that ∀ φ n ∈ X n , a ( ψ n , φ n ) = λ n h ψ n , φ n i L . (2)According to the Rayleigh-Ritz theorem [16], under the natural assumption thatthe sequence ( X n ) n ∈ N satisfies ∀ φ ∈ H ( R d ) , inf φ n ∈ X n k φ − φ n k H −→ n →∞ , (3)the Galerkin method allows to compute the eigenmodes of H associated withthe discrete eigenvalues located below the bottom of the essential spectrum. Itis also known (see e.g. [8] for details) that, as H is bounded below, (3) implies σ ( H ) ⊂ lim inf n →∞ σ ( H | X n ) , (4)where the right-hand side is the limit inferior of the sets σ ( H | X n ), that is theset of the complex numbers λ such that there exists a sequence ( λ n ) n ∈ N , with λ n ∈ σ ( H | X n ) for each n ∈ N , converging toward λ . In particular, any discreteeigenvalue λ of the operator H is well-approximated by a sequence of eigenvaluesof the discretized operators H | X n . On the other hand, (3) is not strong enoughan assumption to prevent spectral pollution. Some sequences of eigenvalues of σ ( H | X n ) may indeed converge to a real number which does not belong to thespectrum of H : lim sup n →∞ σ ( H | X n ) * σ ( H ) in general , (5)where the limit superior of the sets σ ( H | X n ) is the set of the complex numbers λ such that there exists a subsequence ( σ ( H | X nk )) k ∈ N of ( σ ( H | X n )) n ∈ N for which ∀ k ∈ N , ∃ λ n k ∈ σ ( H | X nk ) and lim k →∞ λ n k = λ. the natural approach oftruncating R d to a large compact domain and applying the projection method tothe corresponding Dirichlet problem is prone to spectral pollution ”. Truncating R d indeed seems reasonable since it is known that the bound states of H decayexponentially fast at infinity [15]. The following result provides details on thebehavior of the spurious modes when the approximation space is constructedusing the finite element method. Proposition 2.1.
Let ( T ∞ n ) n ∈ N be a sequence of uniformly regular meshes of R d , invariant with respect to the translations of the lattice R , and such that h n := max K ∈T ∞ n diam( K ) → n →∞ . Let (Ω n ) n ∈ N be an increasing sequence ofclosed convex sets of R d converging to R d , T n := { K ∈ T ∞ n | K ⊂ Ω n } and X n the finite-dimensional approximation space of H (Ω n ) ֒ → H ( R d ) obtained with T n and P m finite elements ( m ∈ N ∗ ). Let λ ∈ lim sup n →∞ σ ( H | X n ) \ σ ( H ) and ( ψ n k , λ n k ) ∈ X n k × R be such that H | X nk ψ n k = λ n k ψ n k , k ψ n k k L = 1 and lim k →∞ λ n k = λ . Then, the sequence ( ψ n k ) k ∈ N , considered as a sequence offunctions of H ( R d ) , converges to weakly in H ( R d ) and strongly in L q loc ( R d ) ,with q = ∞ if d = 1 , q < ∞ if d = 2 and q < d/ ( d − if d ≥ , in the sensethat ∀ K ⊂ R d , K compact , ˆ K | ψ n k | q −→ k →∞ , and it holds ∀ ǫ > , ∃ R > s. t. lim inf k →∞ ˆ ∂ Ω nk + B (0 ,R ) | ψ n k | ≥ − ǫ. (6)The latter result shows that the mass of the spurious states concentrates onthe boundary of the simulation domain Ω n k .This phenomenon is clearly observed on the two dimensional numerical sim-ulations reported below, which have been performed with the finite elementsoftware FreeFem++ [11], with V per ( x, y ) = cos( x ) + 3 sin(2( x + y ) + 1) and W ( x, y ) = − ( x + 2) (2 y − exp( − ( x + y )). We have checked numerically, us-ing the Bloch decomposition method, that there is a gap ( α, β ), with α ≃ − . β ≃ . H = − ∆ + V per . Wehave also checked numerically, using the pollution free supercell method (seeTheorem 3.1 below), that H = H + W has exactly one eigenvalue in the gap( α, β ) approximatively equal to − . P -finite element approximation spaces ( X n ) ≤ n ≤ , wherefor each 40 ≤ n ≤ • Ω n = h − π m n n , π m n n i , with m n = (cid:20) n (cid:18) n − (cid:19)(cid:21) ;4 T ∞ n is a uniform 2 π Z -periodic mesh of R consisting of 2 n isometricalisoceles rectangular triangles per unit cell.The spectra of H | X n in the gap ( α, β ) for 40 ≤ n ≤
100 are displayed on Fig. 1.We clearly see that all these operators have an eigenvalue close to − .
1, whichis an approximation of a true eigenvalue of H . The corresponding eigenfunctionfor n = 88 (blue circle on Fig. 1) is displayed on Fig. 2 (top); as expected,it is localized in the vicinity of the defect. On the other hand, most of thesediscretized operators have several eigenvalues in the range ( α, β ), which cannotbe associated with an eigenvalue of H , and can be interpreted as spurious modes.The eigenfunction of H | X n close to − . n = 88 (blue square onFig. 1), is displayed on Fig. 2 (bottom); in agreement with the analysis carriedout in Proposition 2.1, it is localized in the vicinity of the boundary of thecomputational domain. -0.4-0.3-0.2-0.1 0 0.1 40 50 60 70 80 90 100 E i gen v a l ue s ntrue eigenstatespurious eigenstate Figure 1: Spectrum of H | X n in the gap ( α, β ) for 40 ≤ n ≤ Remark 2.1.
Using the results in [19], it is possible to characterize the spuriousstates generated by finite element discretizations of one-dimensional perturbedSchr¨odinger operators: for R = b Z and Ω n = [ − ( n + t ) b, ( n + t ) b ] , the spuri-ous eigenvalues are the discrete eigenvalues in [min( σ ( H )) , + ∞ ) \ σ ( H ) ofthe operators H + ( t ) and H − ( t ) on L ( R + ) with domains H ( R + ) ∩ H ( R + ) ,respectively defined by H ± ( t ) = − d dx + V per ( x ± tb ) . Besides, the spurious eigenvectors of H | X n converge (in some sense, and up to translation) to thediscrete eigenvectors of H ± ( t ) . As [ t ∈ [0 ,b ) σ ( H ± ( t )) ∩ [min( σ ( H )) , + ∞ ) = [min( σ ( H )) , + ∞ ) , any λ ∈ [min( σ ( H )) , + ∞ ) \ σ ( H ) is a spurious eigenvalue, in the sense thatthere exists an increasing sequence (Ω n ) n ∈ N of closed intervals of R convergingto R such that λ ∈ lim inf n →∞ σ ( H | X n ) . We refer to [10] for a proof and a numerical illustration of this result. The proofof similar results for d ≥ is work in progress. roof of Proposition 2.1. We first notice that, since H = − ∆+ ( − ∆ + 2 V per )+ W , with W bounded in L ∞ ( R d ) and − ∆ + 2 V per bounded below, there exists aconstant C ∈ R + such that ∀ ψ ∈ H ( R d ) , a ( ψ, ψ ) ≥ k∇ ψ k L − C k ψ k L . (7)As ∀ k ∈ N , k ψ n k k L = 1 and a ( ψ n k , ψ n k ) = λ n k −→ k →∞ λ, we infer from (7) that the sequence ( ψ n k ) k ∈ N is bounded in H ( R d ). It thereforeconverges, up to extraction, to some function φ ∈ H ( R d ), weakly in H ( R d ),and strongly in L q loc ( R d ) with q = ∞ if d = 1, q < ∞ if d = 2 and q < d/ ( d −
2) if d ≥
3. It is easy to deduce from (3) and the continuity of a on H ( R d ) × H ( R d )that φ satisfies Hφ = λφ and therefore that φ = 0 since λ / ∈ σ ( H ) by assumption.Consequently, the whole sequence ( ψ n k ) k ∈ N converges to zero weakly in H ( R d )and strongly in L q loc ( R d ).Let us now prove (6) by contradiction. Assume that there exists ǫ > ∀ R > , lim inf k →∞ ˆ ∂ Ω nk + B (0 ,R ) | ψ n k | < − ǫ. As k ψ n k k L = 1 for all k , the above inequality also reads ∀ R > , lim sup k →∞ ˆ Ω Rnk | ψ n k | > ǫ, where Ω Rn k = { x ∈ Ω n k | d ( x, ∂ Ω n k ) ≥ R } . We could then extract from ( ψ n k ) k ∈ N a subsequence, still denoted by ( ψ n k ) k ∈ N , such that there exists an increasingsequence ( R n k ) k ∈ N of real numbers going to infinity such that ∀ k ∈ N , ˆ Ω Rnknk | ψ n k | ≥ ǫ. Let us denote by C ( T ∞ n ) = (cid:8) v ∈ C ( R d ) | ∀ K ∈ T ∞ n , v | K ∈ P m (cid:9) the set of continuous functions built from T ∞ n and P m -finite elements, and by X ∞ n = C ( T ∞ n ) ∩ H ( R d ) . The space X ∞ n is an (infinite dimensional) closed subspace of H ( R d ). Obviously X n ֒ → X ∞ n . We then introduce a sequence ( χ n k ) k ∈ N of functions of C ∞ c ( R d )such that for all k ∈ N ,Supp( χ n k ) ⊂ Ω n k , χ k ≡ R nk n k , and ∀| α | ≤ ( m +1) , k ∂ α χ n k k L ∞ ≤ CR −| α | n k , C ∈ R + independent of k . Let e ψ n k = P n k ( χ n k ψ n k ), where P n k is the interpolation projector on X n k . For all k ∈ N , k e ψ n k k L ≥ ǫ / and for all φ ∞ n k ∈ X ∞ n k ,( a − λ n k )( e ψ n k , φ ∞ n k ) = ( a − λ n k )( χ n k ψ n k , φ ∞ n k ) − ( a − λ n k )( χ n k ψ n k − P n k ( χ n k ψ n k ) , φ ∞ n k )= ( a − λ n k )( ψ n k , χ n k φ ∞ n k ) − ( a − λ n k )( χ n k ψ n k − P n k ( χ n k ψ n k ) , φ ∞ n k ) − ˆ R d (∆ χ n k ψ n k φ ∞ n k + 2 φ ∞ n k ∇ χ n k · ∇ ψ n k )= ( a − λ n k )( ψ n k , χ n k φ ∞ n k − P n k ( χ n k φ ∞ n k )) − ( a − λ n k )( χ n k ψ n k − P n k ( χ n k ψ n k ) , φ ∞ n k ) − ˆ R d (∆ χ n k ψ n k φ ∞ n k + 2 φ ∞ n k ∇ χ n k · ∇ ψ n k ) , where we have used that ( a − λ n k )( ψ n k , P n k ( χ n k φ ∞ n k )) = 0 since P n k ( χ n k φ ∞ n k ) ∈ X n k . Denoting by a ( ψ, φ ) = ˆ R d ∇ ψ · ∇ φ + ˆ R d V per ψφ, we end up with( a − λ n k )( e ψ n k , φ ∞ n k ) = ( a − λ n k )( ψ n k , χ n k φ ∞ n k − P n k ( χ n k φ ∞ n k )) − ( a − λ n k )( χ n k ψ n k − P n k ( χ n k ψ n k ) , φ ∞ n k ) − ˆ R d (∆ χ n k ψ n k φ ∞ n k + 2 φ ∞ n k ∇ χ n k · ∇ ψ n k ) − ˆ R d W e ψ n k φ ∞ n k . (8)Besides, for h n k ≤ ∀ φ ∞ n k ∈ X ∞ n k , k χ n k φ ∞ n k − P n k ( χ n k φ ∞ n k ) k H ≤ Ch n k R − n k k φ ∞ n k k H , (9)for some constant C independent of k and φ ∞ n k . To prove the above inequality,we notice that for all K ∈ T n k , ( χ n k φ ∞ n k ) | K ∈ C ∞ ( K ), and ∂ β φ ∞ n k | K = 0 if8 β | = m + 1, so that k χ n k φ ∞ n k − P n k ( χ n k φ ∞ n k ) k H = X K ∈T nk k ( χ n k φ ∞ n k ) | K − ( P n k ( χ n k φ ∞ n k )) | K k H ( K ) ≤ Ch mn k X K ∈T nk max | α | = m +1 k ∂ α ( χ n k φ ∞ n k ) | K k L ( K ) ≤ Ch mn k X K ∈T nk max | α | = m +1 X β ≤ α k ∂ α − β χ n k k L ∞ k ∂ β φ ∞ n k | K k L ( K ) ≤ Ch mn k R − n k X K ∈T nk max | β |≤ m k ∂ β φ ∞ n k | K k L ( K ) ≤ Ch mn k R − n k X K ∈T nk (1 + h − m − n k ) k φ ∞ n k | K k H ( K ) ≤ Ch n k R − n k k φ ∞ n k k H , where we have used inverse inequalities and the assumption that the sequenceof meshes ( T ∞ n ) n ∈ N is uniformly regular, to obtain the last but one inequality.Using the boundedness of ( ψ n k ) k ∈ N in H ( R d ), the properties of χ n k and W ,and the fact that ( ψ n k ) k ∈ N strongly converges to 0 in L ( R d ), we deduce from(8) and (9) that ∀ φ ∞ n k ∈ X ∞ n k , (cid:12)(cid:12)(cid:12) ( a − λ n k )( e ψ n k , φ ∞ n k ) (cid:12)(cid:12)(cid:12) ≤ η n k k φ ∞ n k k H , where the sequence of positive real numbers ( η n k ) k ∈ N goes to zero when k goesto infinity.We can now use Bloch theory (see e.g. [16]) and expand the functions of X ∞ n k as φ ∞ n k ( x ) = Γ ∗ ( φ ∞ n k ) q ( x ) dq, where Γ ∗ is the first Brillouin zone of the perfect crystal, and where for all q ∈ Γ ∗ , ( φ ∞ n k ) q ( x ) = X R ∈R φ ∞ n k ( x + R ) e − iq · R . For each q ∈ Γ ∗ , the function ( φ ∞ n k ) q belongs to the complex Hilbert space L q (Γ) := (cid:8) v ( x ) e iq · x , v ∈ L ( R d ) , v R -periodic (cid:9) , where Γ denotes the Wigner-Seitz cell of the lattice R (notice that the functions( φ ∞ n k ) q are complex-valued). Recall that if R = b Z d (cubic lattice of parameter b > − b/ , b/ d and Γ ∗ = ( − π/b, π/b ] d . The mesh T ∞ n k beinginvariant with respect to the translations of the lattice R , it holds in fact( φ ∞ n k ) q ∈ C ( T ∞ n k ) ∩ L q (Γ) .
9e thus have for all φ ∞ n k ∈ X ∞ n k ,( a − λ n k )( e ψ n k , φ ∞ n k ) = Γ ∗ ( a q − λ n )(( e ψ n k ) q , ( φ ∞ n k ) q ) dq, where a q ( ψ q , φ q ) = ˆ Γ ∇ ψ ∗ q · ∇ φ q + ˆ Γ V per ψ ∗ q φ q . (10)Let ( ǫ n,l,q , e n,l,q ) ≤ l ≤ N n , ǫ n, ,q ≤ ǫ n, ,q ≤ · · · ≤ ǫ n,N n ,q , be an L q (Γ)-orthonormalbasis of eigenmodes of a q in C ( T ∞ n ) ∩ L q (Γ). Expanding ( e ψ n k ) q in the basis( e n k ,l,q ) ≤ l ≤ N nk , we get ( e ψ n k ) q = N nk X j =1 c n k ,j,q e n k ,j,q . Choosing φ ∞ n k such that( φ ∞ n k ) q = N nk X j =1 c n k ,j,q (1 ǫ nk,j,q − λ nk ≥ − ǫ nk,j,q − λ nk < ) e n k ,j,q , we obtain k φ ∞ n k k L = k e ψ n k k L and( a − λ n k )( e ψ n k , φ ∞ n k ) = Γ ∗ N nk X j =1 | ǫ n k ,j,q − λ n k | | c n k ,j,q | . It is easy to check that lim inf k →∞ max j,q | ǫ n k ,j,q − λ n k | = ζ := dist( λ, σ ( H )) > k →∞ ( a − λ n k )( e ψ n k , φ ∞ n k ) ≥ ζǫ. Besides, k φ ∞ n k k L = k e ψ n k k L and a ( φ ∞ n k , φ ∞ n k ) = a ( e ψ n k , e ψ n k ) , which implies that the sequence ( φ ∞ n k ) k ∈ N is bounded in H ( R d ). Consequently,0 < ζǫ ≤ lim inf k →∞ ( a − λ n k )( e ψ n k , φ ∞ n k ) ≤ lim inf k →∞ η n k k φ ∞ n k k H = 0 . We reach a contradiction.A careful look on the above proof shows that the assumptions in Propo-sition 2.1 can be weakened: in particular, the mesh T n can be refined in theregions where | W | is large, and coarsened in the vicinity of the boundary of Ω n (see [10] for a more precise statement).10 Supercell method
In solid state physics and materials science, the current state-of-the-art tech-nique to compute the discrete eigenvalues of a perturbed periodic Schr¨odingeroperator in spectral gaps is the supercell method. Let R be the periodic latticeof the host crystal and Γ its Wigner-Seitz cell. In the case of a cubic lattice ofparamater b >
0, we have R = b Z d and Γ = ( − b/ , b/ d . The supercell methodconsists in solving the spectral problem (cid:26) find ( ψ L,N , λ
L,N ) ∈ X L,N × R such that ∀ φ L,N ∈ X L,N , a L ( ψ L,N , φ
L,N ) = λ L,N h ψ L,N , φ
L,N i L (Γ L ) , (11)where Γ L = L Γ (with L ∈ N ∗ ) is the supercell, L (Γ L ) = (cid:8) u L ∈ L ( R d ) | u L L R -periodic (cid:9) ,a L ( u L , v L ) = ˆ Γ L ∇ u L ·∇ v L + ˆ Γ L ( V per + W ) u L v L , h u L , v L i L (Γ L ) = ˆ Γ L u L v L , and X L,N is a finite dimensional subspace of H (Γ L ) = n u L ∈ L (Γ L ) | ∇ u L ∈ (cid:0) L (Γ L ) (cid:1) d o . We denote by H L,N = H L | X L,N , where H L is the unique self-adjoint operatoron L (Γ L ) associated with the quadratic form a L . It then holds that D ( H L ) = H (Γ L ), ∀ φ L ∈ H (Γ L ) , H L φ L = − ∆ φ L + ( V per + W L ) φ L , and ∀ φ L,N ∈ X L,N , H
L,N φ L,N = − ∆ φ L,N + Π X L,N (( V per + W L ) φ L,N ) , where W L ∈ L ∞ per (Γ L ) denotes the L R -periodic extension of W | Γ L and Π X L,N isthe orthogonal projector of L (Γ L ) on X L,N for the L (Γ L ) inner product.Again for the sake of clarity, we restrict ourselves to cubic lattices ( R = b Z d )and to the most popular discretization method for supercell model, namely theFourier (also called planewave) method. We therefore consider approximationspaces of the form X L,N = X k ∈ π ( bL ) − Z d | | k |≤ π ( bL ) − N c k e L,k (cid:12)(cid:12) ∀ k, c − k = c ∗ k , where e L,k ( x ) = | Γ L | − / e ik · x .From the classical Jackson inequality for Fourier truncation, we deduce byscaling the following property of the discretization spaces X L,N : for all real11umbers r and s such that 0 ≤ r ≤ s , there exists a constant C > L ∈ N ∗ and all φ L ∈ H s per (Γ L ), k φ L − Π X L,N φ L k H r per (Γ L ) ≤ C (cid:18) LN (cid:19) s − r k φ L k H s per (Γ L ) . (12)Our analysis of the supercell method requires some assumption on the potential V per . We define the functional space M per (Γ) as M per (Γ) = ( v ∈ L (Γ) | k v k M per (Γ) := sup L ∈ N ∗ sup w ∈ H (Γ L ) \{ } k vw k L (Γ L ) k w k H (Γ L ) < ∞ ) . It is quite standard to prove that M per (Γ) is a normed space and that the spaceof the R -periodic functions of class C ∞ is dense in M per (Γ). We denote the R -periodic Lorentz spaces [4] by L p,q per (Γ). Proposition 3.1.
The following embeddings are continuous: for d = 1 , L (Γ) ֒ → M per (Γ) , for d = 2 , L , ∞ per (Γ) ֒ → M per (Γ) , for d = 3 , L , ∞ per (Γ) ֒ → M per (Γ) . Proof.
We only prove the result for d = 3; the other two embeddings are ob-tained by similar arguments. Let us first recall that the Lorentz space L , ∞ (Γ)is a L -multiplier of L , (Γ) (this can be seen by combining results on convolu-tion multiplier spaces [2] and continuity properties of the Fourier transform onLorentz spaces [4]), in the sense that ∃ C ∈ R + | ∀ f ∈ L , ∞ (Γ) , ∀ g ∈ L , (Γ) , k f g k L (Γ) ≤ C k f k L , ∞ (Γ) k g k L , (Γ) . Besides, the embedding of H (Γ) into L , (Γ) is continuous (see [1] for instance) ∃ C ∈ R + | ∀ g ∈ H (Γ) , k g k L , (Γ) ≤ C k g k H (Γ) . (13)Let v ∈ L , ∞ per (Γ). Denoting by I L := R ∩ ( − Lb/ , Lb/ , we have, for all12 ∈ H (Γ L ), k vw k L (Γ L ) = ˆ Γ L | vw | = X R ∈I L ˆ Γ+ R | v ( x ) w ( x ) | dx = X R ∈I L ˆ Γ | v ( x ) w ( x + R ) | dx = X R ∈I L k vw ( . + R ) k L (Γ) ≤ C X R ∈I L k v k L , ∞ (Γ) k w ( . + R ) k L , (Γ) ≤ C k v k L , ∞ (Γ) X R ∈I L k w ( . + R ) k L , (Γ) ≤ C C k v k L , ∞ (Γ) X R ∈I L k w ( . + R ) k H (Γ) ≤ C C k v k L , ∞ (Γ) X R ∈I L ˆ Γ (cid:0) | w ( x + R ) | + |∇ w ( x + R ) | (cid:1) dx ≤ C C k v k L , ∞ (Γ) ˆ Γ L (cid:0) | w ( x ) | + |∇ w ( x ) | (cid:1) dx ≤ C C k v k L , ∞ (Γ) k w k H (Γ L ) . Therefore, v ∈ M per (Γ) and k v k M per (Γ) ≤ C C k v k L , ∞ (Γ) . Remark 3.1.
In dimension 3, the R -periodic Coulomb kernel G defined by − ∆ G = 4 π X R ∈R δ R − | Γ | − ! , min x ∈ R G ( x ) = 0 , is in L , ∞ per (Γ) , hence in M per (Γ) . The functional setting we have introducedtherefore allows us to deal with the electronic structure of crystals containingpoint-like nuclei. Theorem 3.1.
Assume that V per ∈ M per (Γ) . Then lim N,L →∞ |
N/L →∞ σ ( H L,N ) = σ ( H ) . Proof.
Let us first establish that σ ( H ) ⊂ lim inf N,L →∞ |
N/L →∞ σ ( H L,N ) . Let λ ∈ σ ( H ) and ( N L ) L ∈ N ∗ be a sequence of integers such that N L L −→ L →∞ ∞ .Let ǫ > ψ ∈ C ∞ c ( R d ) be such that k ψ k L = 1 and k ( H − λ ) ψ k L ≤ ǫ .13e denote by ψ L the L R -periodic extension of ψ | Γ L . Since ψ is compactlysupported, there exists L ∈ N ∗ such that for all L ≥ L , Supp( ψ ) ⊂ Γ L .Consequently, for all L ≥ L , k ψ L k L (Γ L ) = 1 and k ( H L − λ ) ψ L k L (Γ L ) ≤ ǫ. Let ψ L,N L := Π X L,NL ψ L . We are going to prove that k ( H L − λ ) ψ L − ( H L,N L − λ ) ψ L,N L k L (Γ L ) −→ L →∞ . (14)First, we infer from (12) and the density of H (Ω) in L (Ω) for any boundeddomain Ω of R d , that ∀ φ ∈ L ( R d ) , k (1 − Π X L,NL ) φ L k L (Γ L ) −→ L →∞ , where L ( R d ) denotes the space of the square integrable functions on R d withcompact supports, and where φ L is the L R -periodic extension of φ | Γ L . As ψ ,∆ ψ , V per ψ and W ψ are square integrable, with compact supports, we thereforehave for all L ≥ L , k ψ L − ψ L,N L k L (Γ L ) = (cid:13)(cid:13)(cid:13)(cid:16) − Π X L,NL (cid:17) ψ L (cid:13)(cid:13)(cid:13) L (Γ L ) −→ L →∞ , k − ∆ ψ L + ∆ ψ L,N L k L (Γ L ) = (cid:13)(cid:13)(cid:13)(cid:16) − Π X L,NL (cid:17) ( − ∆ ψ ) L (cid:13)(cid:13)(cid:13) L (Γ L ) −→ L →∞ , k W L ψ L − Π X L,NL ( W L ψ L ) k L (Γ L ) = (cid:13)(cid:13)(cid:13)(cid:16) − Π X L,NL (cid:17) ( W ψ ) L (cid:13)(cid:13)(cid:13) L (Γ L ) −→ L →∞ , k V per ψ L − Π X L,NL ( V per ψ L ) k L (Γ L ) = (cid:13)(cid:13)(cid:13)(cid:16) − Π X L,NL (cid:17) ( V per ψ ) L (cid:13)(cid:13)(cid:13) L (Γ L ) −→ L →∞ . We infer from the last two convergence results that, on the one hand, k W L ψ L − Π X L,NL ( W L ψ L,N L ) k L (Γ L ) ≤ (cid:13)(cid:13)(cid:13) W L ψ L − Π X L,NL ( W L ψ L ) (cid:13)(cid:13)(cid:13) L (Γ L ) + (cid:13)(cid:13)(cid:13) Π X L,NL ( W L ( ψ L − ψ L,N L )) (cid:13)(cid:13)(cid:13) L (Γ L ) ≤ (cid:13)(cid:13)(cid:13) W L ψ L − Π X L,NL ( W L ψ L ) (cid:13)(cid:13)(cid:13) L (Γ L ) + k W k L ∞ k ψ L − ψ L,N L k L (Γ L ) −→ L →∞ , and that, on the other hand, k V per ψ L − Π X L,NL ( V per ψ L,N L ) k L (Γ L ) ≤ (cid:13)(cid:13)(cid:13) V per ψ L − Π X L,NL ( V per ψ L ) (cid:13)(cid:13)(cid:13) L (Γ L ) + (cid:13)(cid:13)(cid:13) Π X L,NL ( V per ( ψ L − ψ L,N L )) (cid:13)(cid:13)(cid:13) L (Γ L ) ≤ (cid:13)(cid:13)(cid:13)(cid:16) − Π X L,NL (cid:17) V per ψ L (cid:13)(cid:13)(cid:13) L (Γ L ) + k V per k M per (Γ) k ψ L − ψ L,N L k H (Γ L ) −→ L →∞ . L large enough, k ( H L,N L − λ ) ψ L,N L k L (Γ L ) ≤ ε. As k ψ L,N L k L (Γ L ) = 1 for all L ≥ L , we infer that for L large enough,dist( λ, σ ( H L,N L )) ≤ ǫ , so that λ ∈ lim inf L →∞ σ ( H L,N L ).Let us now prove that lim sup N,L →∞ |
N/L →∞ σ ( H N,L ) ⊂ σ ( H ) . We argue by contradiction, assuming that there exists λ ∈ R \ σ ( H ) and asequence ( L k , N k ) k ∈ N with L k → k →∞ ∞ , N k → k →∞ ∞ , N k /L k → k →∞ ∞ , such thatfor each k , there exists ( ψ L k ,N k , λ L k ,N k ) ∈ X L k ,N k × R satisfying (cid:26) ∀ φ L k ,N k ∈ X L k ,N k , a L k ( ψ L k ,N k , φ L k ,N k ) = λ L k ,N k h ψ L k ,N k , φ L k ,N k i L (Γ Lk ) k ψ L k ,N k k L (Γ Lk ) = 1 , and lim k →∞ λ L k ,N k = λ . Each function ψ L k ,N k is then solution to the PDE −
12 ∆ ψ L k ,N k + Π X Lk,Nk (( V per + W L k ) ψ L k ,N k ) = λ L k ,N k ψ L k ,N k . (15)Reasoning as in the proof of Proposition 2.1, it can be checked that the sequence( k ψ L k ,N k k H (Γ Lk ) ) k ∈ N is bounded, and that ψ L k ,N k −→ k →∞ L ( R d ) . (16)For all k , we consider a cut-off function χ k ∈ C ∞ c ( R d ) such that 0 ≤ χ k ≤ R d , χ k ≡ L k , Supp( χ k ) ⊂ ( L k + L / k )Γ, k∇ χ k k L ∞ ≤ CL − / k , and k ∆ χ k k L ∞ ≤ CL − k for some constant C ∈ R + independent of k . We then set e ψ k = χ k ψ L k ,N k . It holds e ψ k ∈ H ( R d ), 1 ≤ k e ψ k k L ≤ d/ and −
12 ∆ e ψ k + V per e ψ k − λ e ψ k = χ k (cid:16) V per ψ L k ,N k − Π X Lk,Nk ( V per ψ L k ,N k ) (cid:17) − χ k Π X Lk,Nk ( W L k ψ L k ,N k ) − ∇ χ k · ∇ ψ L k ,N k −
12 ∆ χ k ψ L k ,N k + ( λ L k ,N k − λ ) e ψ k . (17)As ( λ L k ,N k ) k ∈ N converges to λ in R and k e ψ k k L ≤ d/ , we have( λ L k ,N k − λ ) e ψ k −→ k →∞ L ( R d ) . Using the facts that Supp( χ k ) ⊂ L k , k∇ χ k k L ∞ ≤ CL − / k and k ∆ χ k k L ∞ ≤ CL − k for a constant C ∈ R + independent of k , and the boundedness of thesequence ( k ψ L k ,N k k H (Γ Lk ) ) k ∈ N , we get −∇ χ k · ∇ ψ L k ,N k −
12 ∆ χ k ψ L k ,N k −→ k →∞ L ( R d ) .
15t also follows from (16) that the sequence k W L k ψ L k ,N k k L (Γ Lk ) goes to zero,leading to χ k Π X Lk,Nk ( W L k ψ L k ,N k ) −→ k →∞ L ( R d ) . Lastly, χ k (cid:16) V per ψ L k ,N k − Π X Lk,Nk ( V per ψ L k ,N k ) (cid:17) −→ k →∞ L ( R d ) . (18)To show the above convergence result, we consider ǫ > W , ∞ per (Γ) := { W per ∈ L ∞ per (Γ) | ∇ W per ∈ L ∞ per (Γ) } in M per (Γ), we canchoose some e V per ∈ W , ∞ per (Γ) such that k V per − e V per k M per (Γ) ≤ ε . We thendeduce from (12) that, for all k ∈ N , (cid:13)(cid:13)(cid:13) V per ψ L k ,N k − Π X Lk,Nk ( V per ψ L k ,N k ) (cid:13)(cid:13)(cid:13) L (Γ Lk ) ≤ (cid:13)(cid:13)(cid:13) ( V per − e V per ) ψ L k ,N k (cid:13)(cid:13)(cid:13) L (Γ Lk ) + (cid:13)(cid:13)(cid:13) e V per ψ L k ,N k − Π X Lk,Nk (cid:16) e V per ψ L k ,N k (cid:17)(cid:13)(cid:13)(cid:13) L (Γ Lk ) ≤ k V per − e V per k M per (Γ) k ψ L k ,N k k H (Γ Lk ) + L k N k k e V per ψ L k ,n k k H (Γ Lk ) ≤ ε k ψ L k ,N k k H (Γ Lk ) + L k N k k ψ L k ,n k k H (Γ Lk ) ( k e V per k L ∞ + k∇ e V per k L ∞ ) . Since the sequence (cid:16) k ψ L k ,N k k H (Γ Lk ) (cid:17) k ∈ N ∗ is bounded, this yields (cid:13)(cid:13)(cid:13) V per ψ L k ,N k − Π X Lk,Nk ( V per ψ L k ,N k ) (cid:13)(cid:13)(cid:13) L (Γ Lk ) −→ k →∞ , which implies (18).Collecting the above convergence results, we obtain that the right-hand sideof (17) goes to zero strongly in L ( R d ). Therefore, ( e ψ k / k e ψ k k L ) k ∈ N is a Weylsequence for λ , which contradicts the fact that λ / ∈ σ ( H ).A similar result was proved in [18] for compactly supported defects in 2D pho-tonic crystals, with V per ∈ L ∞ ( R ) and N = ∞ . In [7], we prove that the errormade on the eigenvalues and the associated eigenvectors decays exponentiallywith respect to the size of the supercell. We did not consider here the error dueto numerical integration. The numerical analysis of the latter is ongoing workand will be reported in [10].Note that, if instead of supercells of the form Γ L = L Γ, L ∈ N ∗ , we hadused computational domains of the form Γ L + t = ( L + t )Γ, t ∈ (0 , ∂ Γ L + t . Inthe one-dimensional setting ( R = b Z ), and for a fixed value of t , the trans-lated spurious modes φ L,N ( · − ( L + t ) b/
2) strongly converge in H ( R ), when16 goes to infinity, to the normalized eigenmodes of the dislocation operator H ( t ) = − d dx + 1 x< V per ( x + tb/
2) + 1 x> V per ( x − tb/
2) studied in [12]. Werefer to [10] for further details.
Spectral pollution can be avoided by using e.g. the quadratic projection method,introduced in an abstract setting in [17], and applied to the case of perturbedperiodic Schr¨odinger operators in [5]. An alternative way to prevent spec-tral pollution is to impose constraints on the approximation spaces ( X n ) n ∈ N .Consider a gap ( α, β ) ⊂ R \ σ ( H ) in the spectrum of H and denote by P = χ ( −∞ ,γ ] ( H ) where γ = α + β and where χ ( −∞ ,γ ] is the characteristicfunction of the interval ( −∞ , γ ]. Theorem 4.1.
Let ( P n ) n ∈ N be a sequence of linear projectors on L ( R d ) suchthat for all n ∈ N , Ran ( P n ) ⊂ H ( R d ) , and sup n ∈ N k P n k L ( L ) < ∞ , and ( X n ) n ∈ N a sequence of finite dimensional discretization spaces satisfying (3)as well as the following two properties: (A1) ∀ n ∈ N , X n = X + n ⊕ X − n with X − n ⊂ Ran( P n ) and X + n ⊂ Ran(1 − P n ) ; (A2) sup φ n ∈ X n \{ } k ( P − P n ) φ n k H ( R d ) k φ n k H ( R d ) −→ n →∞ .Then, lim n →∞ σ ( H | X n ) ∩ ( α, β ) = σ ( H ) ∩ ( α, β ) . The above result is an extension, for the specific case of perturbed periodicSchr¨odinger operators, to the results in [14, Theorem 2.6] in the sense that (i)the exact spectral projector P is replaced by an approximate projector P n , and(ii) the discretization space X n may consist of functions of H ( R d ) (the formdomain of H ), while in [14], the basis functions are assumed to belong to H ( R d )(the domain of H ). Proof.
From (4), we already know that σ ( H ) ∩ ( α, β ) ⊂ lim inf n →∞ σ ( H | X n ) ∩ ( α, β ). Conversely, let λ ∈ (lim sup n →∞ σ ( H | X n ) ∩ ( α, β )) \ σ ( H ), and ( ψ n k ) k ∈ N be a sequence of functions of H ( R d ) such that for all k ∈ N , ψ n k ∈ X n k , k ψ n k k L ( R d ) = 1 and ( H | X nk − λ ) ψ n k −→ k →∞ L ( R d ). Reasoningas in the proof of Proposition 2.1, we obtain that the sequence ( ψ n k ) k ∈ N con-verges to 0, weakly in H ( R d ), and strongly in L ( R d ). Let us then expand ψ n k as ψ n k = ψ + n k + ψ − n k with ψ + n k := (1 − P n k ) ψ n k ∈ X + n k and ψ − n k := P n k ψ n k ∈ X − n k and notice that( a − λ )( ψ + n k , ψ + n k ) + ( a − λ )( ψ − n k , ψ + n k ) = ( a − λ )( ψ n k , ψ + n k ) − ˆ R d W ψ n k ψ + n k . ψ + n k = (1 − P n ) ψ n k ∈ X n k , (cid:12)(cid:12) ( a − λ )( ψ n k , ψ + n k ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) h ( H | X nk − λ ) ψ n k , (1 − P n ) ψ n k i L (cid:12)(cid:12)(cid:12) ≤ (cid:18) k ∈ N k P n k k L ( L ) (cid:19) k ( H | X nk − λ ) ψ n k k L −→ k →∞ . Besides, as W vanishes at infinity, ( ψ n k ) k ∈ N converges to 0 in L ( R d ) andsup k ∈ N k ψ + n k k L ≤ k ∈ N k P n k k L ( L ) < ∞ , we also have ˆ R d W ψ n k ψ + n k −→ k →∞ . Therefore, ( a − λ )( ψ + n k , ψ + n k ) + ( a − λ )( ψ − n k , ψ + n k ) −→ k →∞ . Likewise,( a − λ )( ψ + n k , ψ − n k )+( a − λ )( ψ − n k , ψ − n k ) = ( a − λ )( ψ n k , ψ − n k ) − ˆ R d W ψ n k ψ − n k −→ k →∞ . Substracting the second equation from the first one, we obtain( a − λ )( ψ + n k , ψ + n k ) − ( a − λ )( ψ − n k , ψ − n k ) −→ k →∞ . Now, we notice that( a − λ )( ψ − n k , ψ − n k ) = ( a − λ )( P n k ψ n k , P n k ψ n k )= ( a − λ )( P ψ n k , P ψ n k ) + 2( a − λ )( P ψ n k , ( P n k − P ) ψ n k )+( a − λ )(( P n k − P ) ψ n k , ( P n k − P ) ψ n k ) , and ( a − λ )( ψ + n k , ψ + n k ) = ( a − λ )((1 − P n k ) ψ n k , (1 − P n k ) ψ n k )= ( a − λ )((1 − P ) ψ n k , (1 − P ) ψ n k )+2( a − λ )((1 − P ) ψ n k , ( P − P n k ) ψ n k )+( a − λ )(( P − P n k ) ψ n k , ( P − P n k ) ψ n k ) . Besides, there exists η + , η − > ψ ∈ H ( R d ),( a − λ )((1 − P ) ψ, (1 − P ) ψ ) ≥ η + k (1 − P ) ψ k L ( R d ) , − ( a − λ )( P ψ, P ψ ) ≥ η − k P ψ k L ( R d ) . Thus,( a − λ )( ψ + n k , ψ + n k ) − ( a − λ )( ψ − n k , ψ − n k ) ≥ min( η + , η − ) k ψ n k k L ( R d ) +2( a − λ )( ψ n k , ( P − P n k ) ψ n k ) . A
2) and the boundedness of ( ψ n k ) k ∈ N in H ( R d ), we deducethat ( a − λ )( ψ n k , ( P − P n k ) ψ n k ) −→ k →∞ , which imply that k ψ n k k L −→ k →∞
0. This contradicts the fact that k ψ n k k L = 1for all k ∈ N .The assumptions made in Theorem 4.1 allow in particular to consider ap-proximation spaces built from approximate spectral projectors of H . Asa matter of illustration, let us consider the case when the approximate spec-tral projectors are constructed by means of the finite element method. Asin Section 2, we consider a sequence ( T ∞ n ) n ∈ N of uniformly regular meshes of R d , invariant with respect to the translations of the lattice R , and such that h n := max K ∈T ∞ n diam( K ) −→ n →∞
0, and denote by X ∞ n the infinite dimen-sional closed vector subspace of H ( R d ) built from ( T ∞ n ) n ∈ N and P m -finite el-ements. Assume that we want to compute the eigenvalues of H = H + W located inside the gap ( α, β ) between the J th and ( J + 1) st bands of H . UsingBloch theory [16], we obtain P = χ ( −∞ ,γ ] ( H ) = Γ ∗ P q dq, where P q is the rank- J orthogonal projector on L q (Γ) defined by P q = J X j =1 | e j,q i h e j,q | , where ( ǫ j,q , e j,q ) j ∈ N ∗ , ǫ ,q ≤ ǫ ,q ≤ · · · , is an L q (Γ)-orthonormal basis of eigen-modes of the quadratic form a q defined by (10). For n large enough, we introduce P n := Γ ∗ J X j =1 | e n,j,q i h e n,j,q | dq, (19)where ( ǫ n,j,q , e n,j,q ) ≤ j ≤ N n , ǫ n, ,q ≤ ǫ n, ,q ≤ · · · ≤ ǫ n,N n ,q , is the L q (Γ)-orthonormalbasis of eigenmodes of a q in C ( T ∞ n ) ∩ L q (Γ) already introduced in the proof ofProposition 2.1.We have seen in Section 2 that using approximation spaces of the form X n = { ψ n ∈ X ∞ n | Supp( ψ n ) ⊂ Ω n } , where (Ω n ) n ∈ N is an increasing sequence of closed convex sets of R d converging to R d , leads, in general, to spectral pollution. We now consider the approximationspaces e X n = X + n ⊕ X − n where X − n = P n X n and X + n = (1 − P n ) X n . (20)Note that e X n = X n + P n X n , so that e X n can be seen as an augmentation of X n .19 orollary 4.1. The sequence of approximation spaces ( e X n ) n ∈ N defined by (20)satisfies (3) and it holds lim n →∞ σ ( H | e X n ) ∩ ( α, β ) = σ ( H ) ∩ ( α, β ) . (21) Proof. As e X n = X n + P n X n with ( X n ) n ∈ N satisfying (3), it is clear that ( e X n ) n ∈ N satisfies (3). The sequence ( P n ) n ∈ N is a sequence of orthogonal projectors of L ( R d ) such that Ran( P n ) ⊂ X ∞ n ⊂ H ( R d ). Besides, k P n k L ( L ) = 1 since theprojector P n is orthogonal. It follows from the minmax principle [16] and usuala priori error estimates for linear elliptic eigenvalue problems [3] thatsup ≤ j ≤ J, q ∈ Γ ∗ ǫ n,j,q −→ n →∞ α and inf j ≥ J +1 , q ∈ Γ ∗ ǫ n,j,q −→ n →∞ β, and that there exists C ∈ R + such that k P n − P k L ( H ) ≤ C sup q ∈ Γ ∗ sup v q ∈ Ran( P q ) k v q k L q (Γ) = 1 inf v nq ∈ C ( T ∞ n ) ∩ L q (Γ) k v q − v nq k H q (Γ) −→ n →∞ . We conclude using Theorem 4.1.Let us finally present some numerical simulations illustrating Corollary 4.1 ina one-dimensional setting, with V per ( x ) = cos( x ) + 3 sin(2 x + 1) and W ( x ) = − ( x + 2) e − x . We focus on the spectral gap ( α, β ) located between the firstand second bands of H = − d dx + V per (corresponding to J = 1). Numericalsimulations done with the pollution-free supercell model show that α ≃ − . β ≃ − .
65, and that H has exactly two discrete eigenvalues λ ≃ − . λ ≃ − .
66 in the gap ( α, β ).The simulations below have been performed with a uniform mesh of R cen-tered on 0, consisting of segments of length h = π/
50, and with Ω = [ − L, L ],for different values of L . The sums over R have been truncated using verylarge cut-offs; likewise, the integrals on the Brillouin zone have been computednumerically on a very fine uniform integration grid, in order to eliminate theso-called k -point discretization errors. The numerical analysis of the approxima-tions resulting from the truncation of the sums over R and from the numericalintegration on Γ ∗ , is work in progress.The spectra of the operators H | X n (standard finite element discretizationspaces) and H | e X n (augmented finite element discretization spaces defined by(20)) are displayed in Figure 4. The variational approximation of H in X n isseen to generate spectral pollution, while, in agreement with Corollary 4.1, nospectral pollution is observed with the discretization spaces e X n .20 Figure 3: The spectra of the variational approximations of H for various sizesof the simulation domain, obtained with standard finite element discretizationspaces X n (top) and with augmented finite element discretization spaces e X n defined by (20) (bottom). Acknowledgements
We thank Fran¸cois Murat for helpful discussions.
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