Absence of binding in a mean-field approximation of quantum electrodynamics
aa r X i v : . [ m a t h - ph ] S e p Absence of binding in a mean-fieldapproximation of quantum electrodynamics
Sok JérémyCeremade, UMR 7534, Université Paris-Dauphine,Place du Maréchal de Lattre de Tassigny,75775 Paris Cedex 16, France.October 8, 2018
Abstract
We study the Bogoliubov-Dirac-Fock model which is a mean-field approximationof QED. It allows to consider relativistic electrons interacting with the Dirac sea. Westudy the system of two electrons in the vacuum: it has been shown in a previouspaper [21] that an electron alone can bind due to the vacuum polarisation, under sometechnical assumptions. Here we prove the absence of binding for the system of twoelectrons:the response of the vacuum is not sufficient to counterbalance the repulsionof the electrons.
The Dirac operator
The theory of relativistic quantum mechanics is based on the Dirac operator D ,that describes the kinetic energy of a relativistic electron. To simplify formulae, wetake relativistic units ~ = c = 4 πε = 1 and set the bare particle mass equal to .In this case, the Dirac operator is defined by [24]: D = − i α · ∇ + β where β, α j ∈M ( C ) are the Dirac matrices: β = (cid:18) Id − Id (cid:19) , α j = (cid:18) σ j σ j (cid:19) , j = 1 , , (1a) σ = (cid:18) (cid:19) , σ = (cid:18) − ii (cid:19) , σ = (cid:18) − (cid:19) . (1b)It acts on the Hilbert space H = L ( R , C ) with domain H ( R , C ) . Its spectrum isnot bounded from below: σ ( D ) = ( −∞ , − ∪ [1 , + ∞ ) , which implies the existenceof states with arbitrarily small negative energy. Dirac postulated that all the negativeenergy states are already occupied by "virtual" electrons forming the so-called Diracsea: by Pauli principle a real electron can only have positive energy.According to this interpretation, the vacuum, filled by the Dirac sea, is a polarizablemedium that reacts to the presence of an electromagnetic field. BDF model
In this paper we study the Bogoliubov-Dirac-Fock (BDF) model which is a no-photon,mean-field approximation of Quantum Electrodynamics (QED) which was introducedby Chaix and Iracane [3]. It enables us to consider a system of relativistic electronsinteracting with the vacuum in the presence of an electrostatic field. This paper is a ontinuation of previous works by Hainzl, Gravejat, Lewin, Séré, Siedentop [10, 7, 8,9, 6] and Sok [21, 20].The derivation of the BDF model from QED is explained in [3] and [7, Appendix]:we refer the reader to these papers for full details.In QED, an electronic system is described by a state in the fermionic Fock space F el [24, Chapter 10] on which (formally) acts the Hamiltonian H QED [7, Appendix]. Themean-field approximation consists to restricting the study to Hartree-Fock type states,called BDF states. They are fully characterized by their one-body density matrix(1pdm) which are orthogonal projectors of H .For instance, the projector P − := χ ( −∞ , ( D ) is the 1pdm of the vacuum state Ω ∈ F el : it must be thought of as the infiniter Slater determinant f ∧ f ∧ · · · where ( f i ) i ≥ is an orthonormal basis (BON) of Ran ( P − ) . A projector P defines a BDF state iff P − P − is Hilbert-Schmidt ( i.e. its integral kernel is square integrable).We take P − as a reference state and define a renormalized Hamiltonian : H QED : bya procedure of normal ordering relative to P − [3, 7]. The energy h Ω P , : H QED : Ω P i of astate Ω P , turns out to be a function of the reduced density matrix (r1pdm) Q := P − P − .Formally this function is e E ν BDF ( Q ) := Tr (cid:0) D Q (cid:1) − αD ( ν, ρ Q ) + α (cid:0) D ( ρ Q , ρ Q ) − k Q k Ex (cid:1) , (2)where α > is the fine structure constant, ν is the external density of charge, ρ Q ( x ) :=Tr C (cid:0) Q ( x, x ) (cid:1) is the density of Q , with Q ( x, y ) the integral kernel of Q , and: D ( ν, ν ) = k ν k C := 4 π Z R | b ν ( k ) | | k | dk and k Q k Ex := ZZ R × R | Q ( x, y ) | | x − y | dxdy. (3)The hat in b ν denotes the Fourier transform and D ( ν, ν ) < + ∞ is the Coulomb energyof ν : it coincides with s ν ( x ) ∗ ν ( y ) | x − y | dxdy whenever this integral makes sense. We alsowrite C := n ν ∈ S ′ ( R ) , b ν measurable and Z | b ν ( k ) | | k | dk < + ∞ o . (4)In (2) we recognize the kinetic energy, the interaction energy with ν , the direct term α D ( ρ Q , ρ Q ) and the exchange term − α k Q k Ex . A priori this formula makes sense onlywhen Q and D Q are trace-class and the variational problem is ill-defined.An ultraviolet cut-off Λ > is necessary. Following [6], we replace D by D := D (cid:0) − ∆Λ (cid:1) with domain H / ( R , C ) , and only consider states Q such that Tr (cid:0) | D | | Q | (cid:1) < + ∞ .By adapting (2), we get a well-defined energy E ν BDF (defined in the next section).
Remark . We use the terms Direct space and Fourier space: a function that dependson position variables (such as a wave function ψ ( x ) or a 1pdm Q ( x, y ) ) is in Directspace, while its Fourier transform that depends on momentum variables is in Fourierspace (such as b ψ ( p ) or b Q ( p, q ) ). Remark . Other choices of cut-off are possible. This one, the smooth cut-off, isconvenient for the study of functions in Direct space. In [7, 8, 9] Hainzl et al. havechosen the sharp cut-off, replacing L ( R , C ) by its subspace H Λ made of square-integrable functions whose Fourier transform vanishes outside the ball B R (0 , Λ) . Remark . We still have χ ( −∞ , ( D ) = P − . We also write P := χ ( −∞ , ( D ) = Id − P − the projector on its positive spectral subspace. Notation . For an operator Q , we define R Q by its integral kernel: R Q ( x, y ) := Q ( x, y ) | x − y | , x, y ∈ R × R , x = y. (5)Moreover for any ρ ∈ C we write v ρ := ρ ∗ |·| . (6) xistence of minimizers For a r1pdm Q = P − P − , the charge of the system is given by its so-called P − -trace Tr P − ( Q ) , defined by Tr P − ( Q ) := Tr (cid:0) P − QP − (cid:1) + Tr (cid:0) P QP (cid:1) . (7)It coincides with the usual trace for trace-class operators and is well-defined for r1pdmbecause of their structure. Indeed as a difference of orthogonal projectors Q satisfies: P ( P − P − ) P − P − ( P − P − ) P − = ( P − P − ) . (8)A minimizer for E ν BDF among states with charge M ∈ N is interpreted as a groundstate of the system with M electrons in the presence of ν . For q ∈ R , the infimum ofthe BDF energy on the charge sector Q Λ ( q ) := { Q : Tr P − ( Q ) = q } is written E ν ( q ) .A sufficient condition for the existence of a minimizer for E ν ( q ) is the validity ofbinding inequalities at level q [9, Theorem 1]. This result is stated for the sharp cut-off,however it is possible to adapt its proof to get this Theorem: Theorem 1.
Let ≤ α < π , Λ > , ν ∈ C and q ∈ R . Then the following assertionsare equivalent:1. the binding inequalities hold: ∀ k ∈ R \{ } , E ν ( q ) < E ν ( q − k ) + E ( k ) ,2. each minimizing sequence ( Q n ) n ≥ for E ν ( q ) is precompact in Q Λ ( q ) and con-verges, up to a subsequence, to a minimizer for E ν ( q ) . If ν = 0 , this result holdsup to translation.If q is an integer, then we can only consider k ∈ Z \{ } in the first assertion. Checking binding inequalities is a difficult task. Hainzl et al. checked them insome cases with non-vanishing ν [9, Theorems 2 and 3]. [9, Theorem 3] states that for ν ∈ L ( R , R + ) ∩ C , there exists a minimizer for E ν ( M ) provided that M − < R ν under technical assumptions on α, Λ .In [21], the existence of a ground state for E (1) is proved, still under technicalassumptions on α, Λ . It is remarkable that an electron can bind alone without anyexternal potential: this is due to the vacuum polarisation. The electron creates a holein the Dirac sea that allows it to bind. This effect causes a charge screening: from faraway the charge of the electron appears smaller as it is surrounded by the hole.Let Q be a minimizer for E (1) , then its density ρ Q is integrable [20], and we havethe charge renormalisation formula : Z ρ Q = 1 × Z ≈ ×
11 + π α log(Λ) = 1 . (9)Here Z is the renormalisation constant . This inadequacy is possible because the min-imizer is not trace-class (hence the mere fact that ρ Q is integrable is non-trivial).We emphasize that these results were proved with the sharp cut-off, but the proofscan be adapted in the present case.Our purpose in this paper is to study the variational problem E (2) , that is twoelectrons in the vacuum. We recall that an electron does not see its own field, but inthe case of two electrons any electron feel the field induced by the other resulting to arepulsive force. If the vacuum polarisation is not strong enough to counterbalance thisrepulsion, then there is no minimizer for E (2) . This constitutes our main Theorem. Theorem 2.
There exist α , Λ , L such that if α ≤ α , Λ ≥ Λ and α log(Λ) ≤ L ,then there is no minimizer for E (2) .Remark . This result is proved in the case of the smooth cut-off, and we expect it tobe true for the sharp one but we were unable to show it. e prove it ad absurdum . Let us give the main ideas.Along this paper we suppose that there exists a minimizer Q for E (2) . Such a min-imizer satisfies a self-consistent equation [9, Proposition 1], [6] and can be decomposedas follows: Q = | ψ ih ψ | + | ψ ih ψ | + γ, (10)where the ψ j ’s are eigenvectors of the so-called mean-field operator: D Q := D + α (cid:0) v ρ Q − R Q (cid:1) , (11)where for a density ρ ∈ C and an operator Q , we define R Q ( x, y ) := Q ( x, y ) | x − y | , x, y ∈ R and v ρ := ρ ∗ |·| . (12)For short we will also write B Q := v ρ Q − R Q . (13)By studying E (2) ≤ E (1) , we get a priori information on the ψ j ’s. In particularwe show that the subspace Span ( ψ , ψ ) splits as followsSpan ( ψ , ψ ) = C h ⊥ ⊕ C h , k h j k L = 1 , where h and h are essentially two bump functions which are some distance R g awayfrom each other. The operator γ is also localised around each h j such that the energy E BDF ( Q ) can be written E BDF ( Q ) = 2 E (1) + θ , where θ > in our range of parameters ( α, Λ) .Roughly speaking the BDF energy should be the sum of the BDF energy of thesetwo parts plus the interaction energy. This interaction energy is too big to ensure E (2) is attained. Remark . Throughout this paper, we work in the regime where α and Λ satisfythese conditions: α ≤ α , α log(Λ) := L ≤ L and Λ ≥ Λ > for small constants α , L , Λ − . K is some constant independent of those numbers while K ( λ ) means aconstant depending on the quantity λ . Symbols o ( · ) , O ( · ) and Θ( · ) are to be understoodin this regime.The paper is organised as follows. In the next section we properly define our modeland give a priori estimates about E (2) and its hypothetical minimizer in Lemma 1.This Lemma is proved in Section 5.Then in Section 3, we study the Pekar-Tomasevitch functional to exploit theseresults (Propositions 3, 4 and 5). These Propositions are proved in Appendix B.Section 4 is devoted to introduce important tools of the proof: the Cauchy expansion(part 4.1) and useful inequalities (part 4.3). We recall in part 4.2 the form of the densityof a minimizer.Section 6 is dedicated to prove Theorem 2. We show how the energy is distributedin Direct space (Proposition 6). This enables us to prove Theorem 2 (part 6.3). Tothis end we first study the localisation of the "real" electrons’ wave functions (Lemma7, proved in Appendix C). We then show how this enables us to get localisation of theenergy of a minimizer (Lemma 8, proved in this Section but using Appendix D). Forthe sake of clarity we explain in Remark 16 how Appendix D is used to prove Lemma8. We have postponed the most technical proofs in the appendices. In Appendix A, weprove Proposition 1 and Lemma 6. This last Lemma shows estimates on a minimizerby bootstrap arguments. Maybe the most difficult results lie in Appendices C and D,dedicated to prove localisation estimates in Direct space. Acknowledgment : The author wishes to thank Éric Séré and Mathieu Lewin for usefuldiscussions and helpful comments. This work was partially supported by the GrantANR-10-BLAN0101 of the French Ministry of research. Presentation of the model
Remark . In this paper, the Fourier transform is defined on L ( R ) by the formula: ∀ f ∈ L ( R ) , b f ( p ) := 1(2 π ) / Z R f ( x ) e − ip · x dx. Notation P ± ) . For an operator Q and e , e ∈ { + , −} we write Q e e := P e QP e . Notation . We recall that for ≤ p ≤ ∞ , the set of compactoperators whose singular values form a sequence in ℓ p is denoted by S p ( H Λ ) [19, 19].The case p = 2 (resp. p = 1 ) corresponds to Hilbert-Schmidt operators (resp. trace-class operators).Those Banach spaces satisfy Hölder-type inequalities [18]. We also recall the Kato-Seiler-Simon inequalities [19]: ∀ ≤ p ≤ ∞ , ∀ f, g ∈ L p ( R ) , k f ( x ) g ( − i ∇ ) k S p ≤ (2 π ) − /p k f k L p k g k L p . (14)Furthermore we write B ( H Λ ) , the set of bounded linear endomorphisms on H Λ . Notation D and D ) . We write s p for d D ( p ) √ | p | the action of sign ( D ) in theFourier space. The function p | p | is also written E ( p ) and E p := p | p | (1 + | p | / Λ ) .Throughout this paper ε [Λ] = ε Λ := 1log(Λ) and a [Λ] := 1 + ε [Λ]2 . (15)We have | D | ε Λ ≤ E (Λ) ε | D | ≤ (1 + e ) | D | , Λ ≥ e = exp (1) . (16) Let ν be an external charge density in C and α, Λ > be given. We want to extend(2): the result is the BDF energy (24) below.Following [6] we define the set: Q Kin := (cid:8) Q ∈ S , | D | / Q, Q | D | / ∈ S , | D | / Q ++ | D | / , | D | / Q −− | D | / ∈ S (cid:9) . (17)The kinetic energy functional is defined on Q Kin by the following formula Tr P − ( D Q ) := Tr( | D | / ( Q ++ − Q −− ) | D | / ) . (18)It coincides with Tr( D Q ) when D Q is trace-class. We will work in the subset of thisspace defined by: K := { Q ∈ Q Kin , − P − ≤ Q ≤ P } ⊂ (cid:8) Q ∈ Q Kin , Q ∗ = Q (cid:9) , (19)the closed convex hull (under that norm) of the difference of two orthogonal projectors: P − P − .We also define Q the Hilbert space of Q ( x, y ) ∈ L ( R × R , C ) such that k Q k Q := ZZ ( E p + E q ) | b Q ( p, q ) | dpdq < + ∞ . (20)The definition of the density ρ Q must coincide with the usual one when Q is (locally)trace-class and ρ Q must be of finite Coulomb norm: k ρ Q k C < + ∞ . For Q in S P − , ρ Q is defined by duality: ∀ V ∈ C ′ , QV ∈ S P − and Tr P − ( QV ) = h V , ρ Q i C ′ ×C . (21)We have the following proposition (proved in Appendix A). roposition 1. The map Q ∈ S P − ρ Q ∈ C is continuous and: k ρ Q k C > k| D | a [Λ] Q ++ | D | a [Λ] k S + k| D | a [Λ] Q −− | D | a [Λ] k S + p log(Λ) k| D | a [Λ] Q k S . (22)Thanks to Kato’s inequality (60), the exchange term is well-defined [1] π ZZ | Q ( x, y ) | | x − y | dxdy ≤ Tr( | D | Q ) = Tr {| D | / Q | D | / } and for Q ∈ K : ≤ Tr {| D | / ( Q ++ − Q −− ) | D | / } ≤ Tr P − ( D Q ) , (23)The BDF energy is defined as follows: E ν BDF ( Q ) := Tr P − ( D Q ) − αD ( ν, ρ Q ) + α (cid:16) D ( ρ Q , ρ Q ) − ZZ | Q ( x, y ) | | x − y | dxdy (cid:17) , Q ∈ K . (24)Any charge sector Q ( q ) := { Q ∈ K , Tr P − ( Q ) = q } leads to a variational problem E ν BDF ( q ) := inf Q ∈Q ( q ) E BDF ( Q ) . (25)By Lieb’s variational principle [9, Proposition 3], a minimizer Q for E ν ( M ) with M ∈ Z is necessarily a difference of two projectors P − P − . To simplify, from this point we assume that ν = 0 . For an integer M ∈ N , let Q bea ground state for E ( M ) , then necessarily Q = P − P − , where P is an orthogonalprojector.The study of the first and second derivative gives more information: we have (cid:2) D Q , P (cid:3) = 0 , and [9, Proposition 1] P = χ ( −∞ ,µ ] (cid:0) D Q (cid:1) , < µ < , (26)where we recall the mean-field operator is defined in (11). We decompose Q withrespect to the positive and negative spectrum: N := χ (0 ,µ ] ( D Q ) and π vac = γ + P − := χ ( −∞ , ( D Q ) , (27)where π vac (resp. n ) is interpreted as the polarized vacuum (resp. as the real electrons).If αM is small enough, then we can show that Tr P − ( γ ) = 0 and thus N has rank M [9, 20]. We will recall the proof below.In the present case, a minimizer for E (2) can be written as in (26)-(27). For smallenough α , we have N = | ψ ih ψ | + | ψ ih ψ | , D Q ψ j = µ j ψ j , < µ ≤ µ = µ < , j ∈ { , } . (28)These equations constitutes the starting point of our proof: they enable us to getestimates on the Sobolev norms of the ψ j ’s. More precisely we will prove Lemma 1.Before stating it, let us recall the Pekar-Tomasevitch functional: E PT ( ψ ) := k∇ ψ k L − ZZ | ψ ( x ) | | ψ ( y ) | | x − y | dxdy, ∀ ψ ∈ H . It describes the energy of a single electron in its own hole. In the case of M electrons,the energy is [5]: ∀ ≤ Γ ≤ , Tr Γ = M, E U PT (Γ) := Tr (cid:0) − ∆ (cid:1) − k ρ Γ k C + U (cid:16) k ρ Γ k C − k Γ k Ex (cid:17) , (29)where U > is some number. By scaling we can assume U = 1 but −k ρ Γ k C has to bereplaced by U − : this last number measures the strength of the polarisation. n this paper, a specific value U = U ( α, Λ) is considered: U − = 1 − Z ( α, Λ) where Z is the renormalisation constant that we have mentionned in the introduction.Its precise expression is given below (57).We write E U PT ( M ) the infimum of the Pekar-Tomasevitch energy on the set { ≤ Γ ≤ , Tr Γ = M } , with U = U . Remark . We assume that U > U c , where U c is the critical value above which, thereis no minimizer for E U PT ( M ) for any integer M ≥ . This important result is proved in[5]. For unitary wave functions φ ⊥ φ , we also write E U PT ( φ ∧ φ ) := E U PT (cid:16) X j =1 | φ j ih φ j | (cid:17) . Lemma 1.
In the regime of Remark 5, let Q = N + γ be a minimizer for E (2) ,decomposed as in (26) - (28) .Let c be (cid:8) α (1 − Z ( α, Λ)) (cid:9) − where Z is defined in (57) . We write ψ j the scalingof ψ j by c : ψ j ( x ) := c / ψ j ( cx ) , x ∈ R , Then we have the following: ( E BDF (1) = 1 + c E PT (1) + O ( αc − ) ,E BDF (2) = E BDF ( Q ) = 2 + c E PT ,U ( ψ ∧ ψ ) + O ( αc − ) . (30) We split each ψ j into an upper spinor ϕ j and a lower one χ j , both in L ( R , C ) . Wewrite n j = | ψ j | (resp n j = | ψ j | ) and n = n + n (resp n = n + n ). Then we have µ j = 1 + k∇ ϕ j k L c − c D ( n j , n ) + O ( αc − ) , (31) in particular: (1 − µ j ) c ? . (32)Estimate (32) follows from (47)-(48). This quantitative error O ( αc − ) gives a priori information about the ψ j ’s thanks to [15, 5] (see the next Section). Notation . Throughout this paper, we will use the following notations. N j = | ψ j ih ψ j | N = N + N ,n j = | ψ j | n = n + n ,γ ′ = Q = = γ + N, ρ ′ γ = ρ γ + n. (33)When we add an underline N j etc. we mean the scaled object by c = ( α (1 − Z )) − .Writing O c : φ ( x ) ∈ L c / φ ( cx ) , we have ψ j = O c ψ j , N j := O c N j O − c , γ = O c γO − c . E U PT (2) Thanks to [15], one knows that there exists but one minimizer for E PT (1) up to aphase and to translation in L ( R , C ) . This minimizer can be chosen positive radi-ally symmetric and decreasing. It is also smooth and with exponential falloff. As R (cid:12)(cid:12) ∇| φ | (cid:12)(cid:12) ≤ R |∇ φ | [16], there holds the same in L ( R , C ) . The set of minimizers isa manifold P ≃ S × R where S is the unit sphere of C . There also holds coercivityinequality [11]: roposition 2. Let φ ∈ H with k φ k L = 1 and let φ ∈ P such that: k φ − φ k H = inf f ∈ P k φ − f k H , then there exists κ > such that (at least in aneighborhood of P ): E PT ( φ ) − E PT (1) ≥ κ k φ − φ k H . Notation . We write P ⊂ P the submanifold of P made of minimizers with center ∈ R : it is isomorphic to S .We are interested in E U PT (2) , with U = U > U c , where U c is the critical valueabove which there is no mminimizers for E PT (2) [5]: in particular E PT (2) = 2 E PT (1) (the proof of [5] also applies for spinor-valued functions). If we choose U > U c : ∀ Ψ ∈ L a ( R × R ) , k Ψ k L = 1 : E PT (Ψ) − E PT (1) ≥ U D ( ρ Ψ , ρ Ψ ) − Tr( γ Ψ R [ γ ψ ])) (34)where we recall ρ Ψ is the density of Ψ and γ Ψ is its one-body density matrix.There holds Lieb’s variational principle: E U PT (2) is also the infimum of E U PT overSlater determinant h ∧ h with h j ∈ H and h h j , h k i = δ jk .Let us consider such a state Ψ = h ∧ h . The plane Span ( h , h ) can be definedwith other orthonormal families: U (2) acts on the set S [Ψ] of those families: (cid:16)(cid:18) a cb d (cid:19) , (cid:18) h h (cid:19)(cid:17) ∈ U (2) × S [Ψ] (cid:18) ah + bh ch + dh (cid:19) ∈ S [Ψ] , (35)The first vector is written ( m · h ) and the second is written ( m · h ) . Characteristic length
For
Ψ = h ∧ h we define the inverse d Ψ of the character-istic length R (Ψ) : d Ψ := inf m ∈ SU (2) D ( | ( m · h ) | , | ( m · h ) | ) = R (Ψ) − . (36)Let φ ∈ P be the radially symmetric and positive function (with φ ( x ) parallelto ( 1 0 0 0 ) ∗ for instance). Let φ x = τ x φ be its translation by x ∈ R . We have: ∀ x , | x | ≥ | x |× D ( | φ | , | φ x | ) ≤ sup | z |≥ | z | sZZ | φ ( x ) | | φ z ( y ) | | x − y | dxdy := Y < + ∞ . (37) Geometric length
For a Slater determinant
Ψ = h ∧ h where h and h satisfy D ( | h | , | h | ) = d Ψ , we define the geometric length R g as follows.Let φ ( j ) ∈ P be the closest function of P to h j in H . Each φ ( j ) is radial withrespect to some vector z j ∈ R , we set R g (Ψ) := | z − z | (or the smallest of such | z − z | ): it should be seen as the interparticle distance . Remark . The geometric length R g does not appear in the energy and R = d − maybe much smaller. Proposition 3.
There exist a > and b = b ( a ) > such that ∀ Ψ = h ∧ h : ∆ E = E U PT (Ψ) − E PT (1) < a ⇒ ∆ E d Ψ ≥ b . (38) Proposition 4.
There exist a ′ > and b ′ > such that: ∀ Ψ = h ∧ h : ∆ E < a ′ ⇒ ZZ | Ψ( x, y ) | | x − y | dxdy ≥ b ′ R g . (39) More precisely:For any < λ let B λj be B ( z j , λ R g ) and B λ := B λ × B λ ∪ B λ ∪ B λ . Then thereexist a λ > , k λ > such that ∀ Ψ = h ∧ h : ∆ E < a λ ⇒ ZZ ( x,y ) ∈B λ | Ψ( x, y ) | | x − y | dxdy ≥ k λ R g (40) emark . It is not possible to replace R − g by d Ψ .To prove Proposition 4, we need to compare R (Ψ) and R g . R (Ψ) and R g Let us consider an almost minimizer for E U PT (2) : Ψ = h ∧ h , E U PT (2) − E U PT (2) > a ≪ , U big enough . (41)We suppose that D ( | h | , | h | ) = d Ψ and write φ j the closest function to h j in P . Wewrite δ j = h j − φ j . By Propositions 2 and 3 we have: d Ψ = R > ε and k δ k H + k δ k H > a . We will here compare R and R g (defined as | z − z | where z j is the center of φ j ).As φ j ( · − z j ) is radial and smooth then: < inf x ∈ R ( | φ j | ∗ |·| )( x ) (cid:0) ( | φ j | ∗ |·| )( x ) (cid:1) / ≤ sup x ∈ R ( | φ j | ∗ |·| )( x ) (cid:0) ( | φ j | ∗ |·| )( x ) (cid:1) / < + ∞ . (42)By Newton’s Theorem [16], writing | φ | = | φ j ( · − z j ) | we have: ∀ x ∈ R , ( | φ | ∗ |·| )( x ) = 1 | x | Z | y |≤| x | | φ ( y ) | dy + Z | y |≥| x | | φ ( y ) | | y | dy ≤ | x | . (43)As a consequence, for sufficiently small a : | D ( Re ( δ ∗ φ ) , | δ | ) | > k δ k L D ( | φ | , | δ | ) , | D ( Re ( δ ∗ φ ) , | φ | ) | > k δ k L R g , (44)where we used Cauchy-Schwarz inequality: Z x | δ ( x ) ∗ φ ( x ) | dx | x − y | ≤ k δ k L { Z x | φ ( x ) | dx | x − y | } / . Thus there holds the following.
Proposition 5.
Let Ψ be as in (41) . We write k δ k = P j k δ j k : there exists κ > suchthat for sufficiently small a > : d Ψ ≥ (1 − κ √ a ) (cid:0) D ( | φ | , | φ | ) + D ( | δ | , | φ | ) + D ( | φ | , | δ | ) (cid:1) + D ( | δ | , | δ | ) , s | h ( x ) | | h ( y ) | | x − y | dxdy > R g + k δ k L k δ k H R g + k δ k L k δ k H , (45) Remark . In particular R = O ( R g ) . Moreover for sufficiently small a , we have ∆ E := X j (cid:0) E PT ( h j ) − E PT (1) (cid:1) = Θ( k δ k H ) . With the help of Proposition 3, we get the following estimates: ZZ | h ( x ) | | h ( y ) | | x − y | dxdy > a . (46) .3 On the decomposition of ψ ∧ ψ In our problem, we consider a couple ( a , b ) described in Lemma 3, and we choose ( α, Λ) such that U ≥ (2 + 1) U c .We consider Ψ = ψ ∧ ψ of Lemma 1. We have: E U PT ( ψ ∧ ψ ) > α and d Ψ > α .This result and the estimate of Remark 10 lead to the following Lemma. Lemma 2.
For ( k, k ′ ) = (1 , or (2 , and ψ k ( x ) = c − / ψ k ( x/c ) , we have k| ψ k ′ | ∗ |·| × ψ k − ( ψ ∗ k ′ ψ k ) ∗ |·| × ψ k ′ k L > c ZZ | h ( x ) | | h ( y ) | | x − y | dxdy > α c . Proof:
Indeed the quantity in the l.h.s. of (2) corresponds to the squared L -normof ( ρ Ψ ∗ |·| ψ k − R [ γ Ψ ] ψ k ) where Ψ := ψ ∧ ψ . Then we decompose ψ k with respect toan orthonormal family ( h , h ) with h ∧ h = Ψ and D ( | h | , | h | ) = d Ψ . We recall that ψ and ψ are eigenvectors of the mean-field operator with eigenvalues µ and µ . In the case µ = µ we cannot choose ψ = h and ψ = h .From the estimation of the µ j ’s (31) we may ask whether the quantity F E ( ψ k ) := E PT ( ψ k ) − D ( | ψ k | , | ψ k ′ | ) (47)is negative and away from or not. As h k = φ k + δ k with φ k ∈ P and k δ k k H = O ( √ ∆ E ) a simple computation shows that: ∀ ( a, b ) ∈ C ∩ S : F E ( ah + bh ) = 32 E PT (1) + O ((∆ E ) / ) . (48) In this part we use the functions s · , E ( · ) and E · and numbers ε λ , a [Λ] defined inNotation 4. We recall Ineq. (16). The results stated here follow from [21, 20].Let e γ be the operator defined by: e γ = χ ( −∞ , ( D + α ( v e ρ − R e Q )) − P − , ( e Q, e ρ ) ∈ Q × C . For instance we can take γ of (27). Provided that k e Q k Kin , k e ρ k C are small enough, byLemma 3 we have | D + α ( v e ρ − R e Q ) | ≥ | D | (cid:0) − α ( k ρ Q k C + k Q k Ex ) (cid:1) = | D | (1 + o (1)) . As a result we can expand e g in power of α , this is the Cauchy expansion [7]: e γ = + ∞ X j =1 α j Q j (cid:2) e Q, e ρ (cid:3) ,Q j (cid:2) e Q, e ρ (cid:3) := − π Z ∞−∞ dω D + iω (cid:16)(cid:0) R e Q − v [ e ρ ] (cid:1) D + iω (cid:17) j . (49)We can further expand each Q j into P jj =0 Q k,j − k (cid:2) e Q, ρ e Q (cid:3) where each Q k,j − k is poly-nomial in R e Q (resp. v [ ρ e Q ] ) of degree k (resp. j − k ).The respective densities of Q k,j − k and Q j are written ρ k,j − k and ρ j . onvergence of the series (49) In [7, 6], Hainzl et al. proved that this seriesis well-defined and in [21, 20] the functions ( Q k,j − k , ρ k,j − k )[ · , · ] are studied in severalnorms.It is possible to adapt the proofs to show that these functions are multilinear con-tinuous in Q × C or more generally in the banach spaces X w = Q w × C w , defined bythe following norms: || Q || Q w := ZZ ( E p + E q ) w ( p − q ) | b Q ( p, q ) | dpdq and k ρ k C w := ZZ w ( k ) | k | | b ρ ( k ) | dk, (50)where √ w : R → [1 , + ∞ ) is a weight function satisfying some sub-additive assump-tions.Furthermore the growth of the norms k ( Q k,j − k , ρ k,j − k ) k B ( X w ) is also polynomial:it follows that there exists some radius A ( α, Λ , w ) such that ( e Q, e ρ ) ∈ B X w (0 , A ) (cid:16)e γ := + ∞ X j =1 α j Q j (cid:2) e Q, e ρ (cid:3) , ρ e γ (cid:17) ∈ B X g (0 , A ) , is well-defined and contractant.The main ingredients of the proof are the following inequalities: k P ± v e ρ P ∓ | D | a [Λ] k S > p log(Λ) k e ρ k C k R e Q |∇| / k S > k e Q k Ex , k v e ρ | D | a [Λ] k S > k e ρ k C k v e ρ |∇| / k B > k e ρ k C (51)In the l.h.s. the first estimate follows from a simple computation in Fourier space[7, 21], and the second one is an application of the KSS inequality (14).In the r.h.s. the first is proved below (Lemma 3) and the last follows from anhomogeneous Sobolev inequality (59). We will say no more about these results andrefer the reader to the cited articles and to [22]. The results of this part are proved in [20].Let Q = γ + N be a minimizer for E ( M ) with M ∈ { , } . It satisfies Eq. (26)-(27)and rank N = M for α sufficiently small. We recall: γ = χ ( −∞ , ( D Q ) − P − . (52)In [7, 21, 20], a fixed-point scheme is used to see γ as a fixed point of some function F (1) (with parameter N ). This scheme enables us to get estimates on γ and N . Bythe Cauchy expansion, Eq. (52) is rewritten as follows: (cid:0) Id − αQ , [ · ] (cid:1)(cid:2) γ ′ (cid:3) = N + αQ , (cid:2) ρ ′ γ (cid:3) + + ∞ X j =2 α j Q j (cid:2) γ ′ , ρ ′ γ (cid:3) . In [20], it is proved that the linear operator (cid:0) Id − αQ , [ · ] (cid:1) is a continuous endomorphismfor Q g and S p ( ≤ p ≤ ) provided that α log(Λ) ≤ L is small enough.Its inverse T is written and it has a uniform bound for all those Banach spaces.This gives γ = α T [ Q , ( N )] + α T [ Q , ( ρ ′ γ )] + + ∞ X j =2 α j T (cid:2) Q j [ γ ′ , ρ ′ γ ] (cid:3) . (53)In [20], the density αρ (cid:2) Q , ( ρ ′ γ ) (cid:3) is computed and we have: αρ (cid:2) Q , ( ρ ′ γ ) (cid:3) = − ˇ f Λ ∗ ρ ′ γ , where ˇ f Λ ∈ L with norm k ˇ f λ k L > L . emark . For the smooth cut-off, the same proof applies for | · | ℓ ˇ f Λ . For any fixedinteger ℓ , there exists K ( ℓ ) > such that, if α ≤ K ( ℓ ) then k | · | ℓ ˇ f Λ k L ≤ n Z | x | ℓ ) (1 + | x | ) | ˇ f Λ ( x ) | dx Z dx | x | (1 + | x | ) o / , > α. (54)The same results hold for ˇ F Λ := F − (cid:16) f Λ f Λ (cid:17) = + ∞ X j =1 ( − j +1 ˇ f ∗ j Λ (55)provided that α ≤ K ′ ( ℓ ) with a smaller bound K ′ ( ℓ ) ≤ K ( ℓ ) .We write τ j [ · ] := ρ (cid:2) T Q j [ · ] (cid:3) and τ k,j − k [ · ] := ρ (cid:2) T Q k,j − k [ · ] (cid:3) . There holds: ρ γ = − ˇ F Λ ∗ n + ( δ − ˇ F Λ ) ∗ (cid:0) ατ , [ N ] + + ∞ X j =2 α j τ j [ γ ′ , ρ ′ γ ] (cid:1) , = − ˇ F Λ ∗ n + ( δ − ˇ F Λ ) ∗ (cid:0) ατ , [ N ] + α e τ [ γ ′ , ρ ′ γ ] (cid:1) . (56)We have ρ γ ∈ L with R ρ γ = − F Λ (0) × M . The renormalisation constant Z is Z := 1 − F Λ (0) = 11 + f Λ (0) ≈
11 + π α log(Λ) and U := 1 F Λ (0) . (57)We also recall [20] ∀ k, k ′ ∈ B R (0 ,
2) : | F Λ ( k ) − F Λ ( k ′ ) | > α | k − k ′ | (58)we will use below with k ′ = 0 . – Let us recall some Sobolev inequalities in R : k f k L > k∇ f k L , k f k L > k|∇| / f k L , k f k L > k|∇| / f k L (59)The last one gives k v e ρ |∇| / k B > k e ρ k C for e ρ ∈ C .– We also recall Kato’s inequality and Hardy’s inequality: Z R | φ ( x ) | | x | dx ≤ π h|∇| φ , φ i , Z R | φ ( x ) | | x | dx ≤ h ( − ∆) φ , φ i . (60)– The following Lemma gives estimates about the operator R Q . Lemma 3.
Let Q ( x, y ) be an operator of finite exchange term and ρ of finite Coulombenergy, then: k |∇| / R Q k S = Tr( R ∗ Q |∇| R Q ) ≤ (cid:0) Z dy | y | | y − e | (cid:1) Tr( Q ∗ R Q ) , ZZ | Q ( x, y ) | | x − y | dxdy = Tr( Q ∗ R Q ) ≤ π π ) ZZ | u || b Q ( u + k/ , u − k/ | dudk, k v ρ |∇| / k B > k ρ k C . In particular k ( v ρ − R Q ) f k L > ( k ρ k C + k Q k Ex ) k|∇| / f k L . roof: The proof for k |∇| / R Q k S is just an application of the Cauchy-Schwarzinequality once we remark that |∇| − is the convolution by Const / | · | [16]. For thelast inequality we write s = x + y and t = x − y and A ( s, t ) := Q ( s + t/ , s − t/ a.e. By Kato’s inequality: ZZ | Q ( x, y ) | | x − y | dxdy = ZZ | A ( s, t ) | | t | dsdt ≤ π Z ds h|∇| A ( s, · ) , A ( s, · ) i≤ π ZZ | u || b Q ( u + k/ , u − k/ | dudk. Those inequalities are true at least for Q ( x, y ) in the Schwartz class S ( R × R ) , weconclude by density. – To end this part we give estimates about D .We have Id − s p s q = s p ( s p − s q ) = ( s p − s q ) s q and | Id − s p s q | ≤ | s p − s q | = (cid:12)(cid:12)(cid:12) c D ( p ) E ( p ) − c D ( p ) E ( q ) + c D ( p ) − c D ( q ) E ( q ) (cid:12)(cid:12)(cid:12) ≤ | p − q | max( E ( p ) , E ( q )) . (61) Notation . The symbol e will always stand for any unitary vector in R . Remark . There holds ( cf [16] for the expression of ( a − ∆) − ): | D | ( x − y ) = 2 π Z + ∞ dω | D | + ω ( x − y )= r π Z + ∞ e − E ω | x − y | | x − y | dω = Cnst K ( | x − y | ) | x − y | where K is the modified Bessel function [25]. A priori estimates on a minimizer for E (2) This part is devoted to prove (63).Let us say γ ′ = γ + N is a minimizer for E (2) written as in (26)-(27).First we prove (28). There holds a priori estimates [20]:
12 Tr (cid:0) − ∆(1 − ∆Λ ) | D | ( γ ′ ) (cid:1) + α k ρ ′ γ k C ≤ E ( γ ′ ) − α γ ′ R [ γ ′ ]) ≤ απ |∇| ( γ ′ ) ) where we have used | D | − ≥ − ∆ (cid:0) − ∆Λ (cid:1) | D | . It follows that: Tr (cid:0) − ∆(1 − ∆Λ ) | D | ( γ ′ ) (cid:1) + α k ρ ′ γ k C ≤ Kα.
As in [20], we can apply a fixed point scheme on ( γ, ρ γ ) with the help of the self-consistent equation (in Q × C for instance). This gives: k γ k Q > √ Lα k ρ ′ γ k C + α k|∇| / γ ′ k S and k ρ γ k C > L k ρ ′ γ k C + √ Lα k|∇| / γ ′ k S . Hence | Tr ( γ ) | ≤ k γ k S < and Tr ( γ ) = 0 as shown in [7]. This proves Tr( N ) =Tr ( N ) = Tr ( γ ′ ) − Tr ( γ ) = 2 . et ( ψ i ) ≤ i ≤ be a basis of orthonormal eigenvectors of χ ,µ ( D γ ′ ) with eigenvalues < µ ≤ µ < . We write N j := | ψ j ih ψ j | and | n j := ψ j | . From the equation satisfiedby ψ j ( D + α ( v [ ρ γ + n ] − R [ γ + N ])) ψ j = µ j ψ j (62)we get the following. Lemma 4.
Let γ ′ and ( ψ j ) j be as above in the regime of Remark 5. Then there holds: π ) Z | p | (cid:0) Λ − (2 + | p | Λ ) + (1 + | p | Λ ) (cid:1) | b ψ j ( p ) | dp ≤ k D ψ j k L − and k D ψ j k L − ≤ α k ρ γ k C k n j k C + α k γ k S k R [ N j ] k S + (cid:0) α k B γ ′ |∇| / k B k|∇| / ψ j k L (cid:1) . As a consequence we also have:
Tr( − ∆(1 − ∆Λ + ∆ Λ ) N ) > c − . (63)It suffices to use the inequalities in the r.h.s. of (51) in Eq. (62). Remark . Compared to the case of E (1) there is an additional term ( v n − R N ) ψ j that has been neglected in − α Re h B N ψ j , ψ j i : this term is non-positive. Notation . From now on, we write v jk = ( ψ ∗ j ψ k ) ∗ |·| and v j := v jj and define a jk := k v k ψ j − v kj ψ k k L . E (1) We compute the energy of a particular test function Q ′ = Q + N , defined as follows[21]. First, we take φ CP = φ a minimizer for E PT (1) in L ( R , C ) ( e.g. real-valuedand positive centered in , cf [15]). Then let ψ be: ψ := t ( φ ∈ L ( R , C ) . (64)Then, we define ψ c := c − / ψ ( c − ( · )) where c − := α F Λ (0) and N := | ψ c ih ψ c | , Q + P − = Π := χ −∞ , (cid:8) D + α (cid:0) ( ρ Q + n ) ∗ |·| − ( R Q + R N ) (cid:1)(cid:9) ,n := | ψ c | , ψ := s −k Π ψ c k L ( ψ c − Π ψ c ) . We have used the fixed point scheme of Section 4.1 to define Q . We also write N := | ψ ih ψ | , Q ′ := Q + N ,B := ( ρ Q + n ) ∗ |·| − α ( R Q + R N ) , D Q := D + αB . The test function Q ′ is the difference between the orthogonal projections Π + N and P − . Following the same method as in [21], the following estimates hold. k Q k Q w > α k n k C w > c − / k Q k Q w > c − k ρ Q k C w > Lc − / k Q k S > αc − / k R N k S > c − (65)where w ( p − q ) = E ( p − q ) and w ( p − q ) = E ( p − q ) .As shown previously in [21, 20] there holds E BDF ( Q ′ ) = h D ψ , ψ i − α Tr ( B [ Q ] Q ) − (Tr( | D + αB | Q ) − Tr( | D | Q ))+ α (cid:0) D ( ρ [ Q ] + n , ρ [ Q ] + n ) − Tr( Q ′ R [ Q ′ ]) (cid:1) (66) stimate of the density ρ Q By Section 4.2, we write ρ Q ′ = ( δ − ˇ F Λ ) ∗ ( n + t [ N ] + α e τ ) , (67) = ( δ − ˇ F Λ ) ∗ n + τ rem . (68)We have k ( δ − ˇ F Λ ) ∗ n ∗ |·| k L ∞ ≤ π k ˇ F Λ k L ) h|∇| ψ c , ψ c i > k∇ ψ c k L = O ( c − ) . We use Ineq. (51) to estimate the norm k τ rem k C of the remainder τ rem . The traces in (66)
By Lemma 3, we can estimate (cid:12)(cid:12) D + αB (cid:12)(cid:12) − (cid:12)(cid:12) D (cid:12)(cid:12) and get thefollowing [21]. Lemma 5.
There holds: | δ Tr | := (cid:12)(cid:12)(cid:12) Tr (cid:8) | D + αB [ Q ′ ] | γ − | D | Q (cid:9)(cid:12)(cid:12)(cid:12) > {k Q k B + α ( k Q k Kin + k τ rem k C ) }k Q k + α {k τ rem k C + k∇ ψ c k L }k Q k S > αc − + αc − × α c − > αc − . (69) h D ψ , ψ i in (66) There holds (1 − Π ) ψ c = − Q ψ c + P ψ c . Then h D ψ c , ψ c i = h D Q ψ c , Q ψ c i − Re h P Q ψ c , P ψ c i + h| D | P ψ c , ψ c ih| D | P ψ c , ψ c i = 1 + k∇ ψ c k L + O ( c − ) . Then thanks to Lemma 3: k| D | / Q ψ c k L ≤ k| D | / Q |∇| / k B k|∇| / ψ c k L and k| D | / Q ψ c k L > αc − . As Q = αQ [ Q ′ , ρ ′ Q ] + α e Q [ Q ′ , ρ ′ Q ] and that Q = Q + − + Q − +1 : P Q ψ c = αQ + − P − ψ c + α P + e Q ψ c . Therefore: α h| D | e Q ψ c , P + ψ c i ≤ α k|∇| / ψ c k L k | D ||∇| / e Q |∇| / k B > α c − × c − = O ( α c − ) α h| D | Q + − P − ψ c , P ψ c i ≤ α k| D | / Q + − |∇| / k B k|∇| / P − ψ c k L k| D | / ψ c k L > αc − / × c − / = O ( αc − ) . Hence: h D (1 − Π ) ψ c , (1 − Π ) ψ c i / (1 − k Π ψ c k L ) = 1 + 12 k∇ ψ c k L + O ( αc − ) . (70) The potential energy in (66)
By the same methods we prove: α (cid:0) D ( ρ [ Q ] , n ) − D ( ρ [ Q ] , n ) − Re (2Tr( Q R [ N ]) − Tr( Q R [ N ])) (cid:1) = − α D ( ˇ F Λ ∗ n , n ) + O ( α c − / ) . (71)For instance by Cauchy-Schwarz inequality followed by Hardy inequality: (cid:12)(cid:12) D (cid:0) ρ [ Q ] , ( P ψ c ) ∗ ( Q ψ c ) (cid:1)(cid:12)(cid:12) ≤ k ρ [ Q ] k C × / k∇ ψ c k / L k Q ψ k L = O ( c − ) . y Ineq. (58), there holds: − α D ( ˇ F Λ ∗ n , n ) = − c D ( n , n ) + O ( α c − + c − k n k L ) = O ( α c − ); indeed: k n k L = k ψ c k L > k|∇| / ψ c k L . As a consequence: E BDF (1) ≤ E
BDF ( Q + N ) = 1 + E PT ( φ )2 c + O ( αc − ) . (72)We have proved the inequality the ≤ part. For the ≥ part, it suffices to take a real minimizer and with the same estimates as above and [21] we prove similar estimates.That there exists a minimizer for E (1) follows from Theorem 1, using the samemethod as in [21]. We have proved E (1) < , then by Lieb’s variational principle weget that for any < q < , E ( q ) > qE (1) , hence the binding inequalities holds for < q < . For q ∈ [0 , c , binding inequalities hold for sufficiently small α . We referto [21] for more details.Similar estimates apply for E (2) , in particular we have E (2) ≤ E (1) ≤ E PT ( φ )2 c + O ( αc − ) . γ ′ for E (2) Bootstrap argument
We write x := Tr( − ∆(1 − ∆Λ + ∆ Λ ) N ) . By Lemma 4, wehave x > c − . This fact enables us to use the method of [4, 21].We scale ψ j by c : ψ j ( x ) = c / ψ j ( cx ) and scale γ accordingly: γ ( x, y ) = c γ ( cx, cy ) .Then writing L A := (1 − ∆ /A ) , the wave function ψ j satisfies: ( c β − ic α · ∇ ) ψ j + αc L − c Λ ( v [ ρ [ γ ] + n ] − R [ g + N ]) ψ j = c µ j L − c Λ ψ j . (73)Splitting ψ j between upper spinor ϕ j and lower spinor χ j both in L ( R , C ) , this gives: k χ k L + k χ k L > c − . Going back to ψ j one gets h D ψ j , ψ j i = 1 + O ( c − ) and it shows that for j = 1 , : < (1 − µ j ) c ≤ K thanks to the equation (28). As ≤ c (1 − L − c Λ ) = − c ∆ c Λ − ∆ ≤ − ∆Λ , then (74) c ( µ j L c Λ − ϕ j = c ( µ j − ϕ j + c ∆ c Λ − ∆ ϕ j = c ( µ j − ϕ j + O L (cid:0) c Λ (cid:1) thanks to Lemma 1 ( O L means in L -norm). We can get another estimate: in thespirit of [21, 20] we can use bootstrap argument with the norms || Q || Q w = ZZ E ( p − q ) k ( E ( p + q )) | b Q ( p, q ) | dpdq and k ρ k C w = Z E ( k ) | b ρ ( k ) | | k | dk, to get the following statement: Lemma 6.
For any fixed k ∈ N ∗ , there exists α ( k ) > such that for α ≤ α ( k ) , ψ j with j = 0 , , is in H k/ with norms O (1) and || γ || Q w , || γ || Q w , k ρ [ γ ] k C w , k ρ [ γ ] k C w > . It is supposed α log(Λ) ≤ L . There also holds: k ∆ ψ k L > min( c − ( c − + Λ − ) , c − / ) , k χ k L > c − and k∇ χ k L > c − , The estimation of E BDF ( γ ′ ) is proven with the help of the estimate k ∆ ψ k L > c − / as shown in the (technical) proof of Lemma 6 in Appendix A.2. emark . By Estimate (63) we can prove that n, γ, ρ γ have estimates of the samekind of those stated in (65) [21, 20]: we have k n k C > c − / , k ρ γ k C > Lc − / , k R N j k S > c − , k| D | / γ k S > c − , k γ k S > αc − / . (75)There also holds k n j k L > c − / .By Lemma 6, we get: k ρ γ k L > Lc − / . Following [20] we can prove ρ γ ∈ L and k ρ γ k L > L . Estimate on c (1 − µ j ) Using estimates on ∇ ϕ j and ∇ χ j (Lemma 6) togetherwith Ineq. (58), we get the following estimate from (28): µ j = 1 + k∇ ϕ j k L c − c D ( n j , n ) + O ( αc − ) . (76)With (47)-(48), we get: (1 − µ j ) c ≤ − E PT (1) + O ( α / ) ? . (77) ψ j ’s It is known ψ ∧ ψ can be split into two almost minimizers of Choquard-Pekar energy h and h : h ∧ h = ψ ∧ ψ . For j ∈ { , } , we write φ j ∈ P the closest Pekarminimizer to h j and its center is written z j . We writeR g := | z − z | . (78)By Section 3, we have: M ( ψ ∧ ψ ) := ZZ | ψ ∧ ψ ( x, y ) | | x − y | dxdy ? R g . (79)Our aim is to show decay estimates far away from z and z . Up to translations,we assume the mean z m = z + z is . Localisation functions
Let ξ ≥ be some radial Schwartz function in S ( R ) satisfying | x | ≤ ⇒ ξ ( x ) = 1 and | x | ≥ ⇒ ξ ( x ) = 0 . We define ξ A ( x ) := ξ ( xA ) for any A > and θ A := p − ξ A . For any x ∈ R we write d ( x ) := min {| x − z | , | x − z |} . (80)Let H be the plane { x : | x − z | = | x − z |} ; the function d ( · ) is differentiable in R \ (cid:0) { z , z } ∪ H (cid:1) . For any A ≫ R g and < λ < we define η λ R g ( x ) := (cid:0) − ξ λ R g ( x − z ) − ξ λ R g ( x − z ) (cid:1) / . (81)We define λ > , defined by the formula λ R g = C L where C ( L, R g ) > is chosen large. (82)The function η λ R g can be seen as the dilation of η λ := p − ξ λ ( · − e ) − ξ λ ( · − e ) byR g where e j := z j − z m R g .At last we define: η ( λ ) c R g ( x ) := q − ξ cλ R g ( x − cz ) − ξ cλ R g ( x − cz ) , (83)we use it in Section D.2.3. emma 7. • For each λ ≤ λ < − , there exists K λ such that: ∀ A > , Z d ( x ) ξ A ( x )( η λ R g ( x )) (cid:16)(cid:12)(cid:12) | D | / ψ ( x ) (cid:12)(cid:12) + (cid:12)(cid:12) | D | / ψ ( x ) (cid:12)(cid:12) (cid:17) dx ≤ K λ (84) Moreover we can choose ( K λ ) λ to be nonincreasing and K λ is (uniformly) bounded inthe regime α, L, Λ − small. • For any λ ≤ λ < − the same holds for d (2) A,λ := d ( x ) ξ A η ( λ ) c R g : Z d ( x ) ξ A ( x )( η ( λ ) c R g ) ( x ) (cid:16)(cid:12)(cid:12) | D | / ψ ( x ) (cid:12)(cid:12) + (cid:12)(cid:12) | D | / ψ ( x ) (cid:12)(cid:12) (cid:17) dx ≤ K ′ λ , (85) where K ′ λ > K λ depends on λ, K λ , ξ . • We can replace | D | / ψ j by ψ j above.Remark . This is a weak estimate due to the presence of v k ψ j − v kj ψ k .This proposition is proved in Appendix C.1. We want to prove that minimizers are localised in space around the centers z , z ofthe electrons. To this end we use localisation operators of [9, 14] with respect to thefunctions ξ cλ R g and η ( λ ) c R g introduced in the previous Section (6.1).By Lemma 7 we know that the wave functions ψ and ψ are localized near z and z . By scaling, it follows that ψ and ψ are localized near cz and cz . We consider: ξ ( λ )1 ( x ) := ξ cλ R g ( x − cz ) and ξ ( λ )2 ( x ) := ξ cλ R g ( x − cz ) ,X ( λ )1 := ( ξ ( λ )1 ) ++ + ( ξ ( λ )1 ) −− and X ( λ )2 := ( ξ ( λ )2 ) ++ + ( ξ ( λ )2 ) −− , and localise γ ′ : ξ ( λ )1 · [ γ ′ ] := X ( λ )1 ( γ ′ ) X ( λ )1 , ξ ( λ )2 · [ γ ′ ] = X ( λ )2 ( γ ′ ) X ( λ )2 . We define the set B λ := (cid:8) B ( cz , cλ R g ) × B ( cz , cλ R g ) (cid:9) ∪ (cid:8) B ( cz , cλ R g ) × B ( cz , cλ R g ) (cid:9) ⊂ R × R . (86)Our aim in this section is to prove: Proposition 6. If γ ′ is a minimizer of E (2) in the regime α, L, Λ − small then: E BDF ( γ ′ ) = E BDF ( ξ − · [ γ ′ ])+ E BDF ( ξ − · [ γ ′ ])+ α ZZ ( x,y ) ∈ B − | ψ ∧ ψ ( x, y ) | | x − y | dxdy + O (cid:16) c R g (cid:17) . (87) Moreover: Tr ( ξ ( 13 ) j · [ γ ′ ]) = 1 + ε j , ε j = o (1) , j = 1 , , Tr ( ξ ( 13 )1 · [ γ ′ ]) + Tr ( ξ ( 13 )2 · [ γ ′ ]) = 2 + O (cid:16) c R g (cid:17) . (88)Assuming this Proposition – proved in Subsection (6.5) – we can prove Theorem 2. By Proposition 4, for sufficiently small α, L , there holds: α ZZ ( x,y ) ∈ B − | ψ ∧ ψ ( x, y ) | | x − y | dxdy ≥ L − c K g R g , or some constant K g > independent of α, Λ in the regime of Remark 5. This gives: E BDF ( γ ′ ) ≥ E BDF (1 + ε ) + E BDF (1 + ε ) + L − K g c R g + O (cid:16) c R g (cid:17) . We know that the function E BDF ( · ) : R R is uniformly Lipschitz with constants and this function is concave on each interval [ M, M + 1] where M ∈ Z [9, Corollary3 mutatis mutandis ]. Furthermore we may assume ε = − ε > up to an error O (cid:16) c R g (cid:17) . The case ε , ε < is easily excluded by concavity of E BDF in [0 , because E BDF (0) = 0 and E BDF (1) ≥ E BDF (2) . Then: E BDF (1 + ε ) + E BDF (1 − ε ) ≥ ε E BDF (2) + (1 − ε ) E BDF (1) + (1 − ε ) E BDF (1) ≥ ε E BDF (2) + (1 − ε )(2 E BDF (1)) ≥ (1 − ε + ε ) E BDF (2) = E BDF (2) . Thus taking F Λ (0) = Θ( α log(Λ)) sufficiently small, the quantity L − is big enough tocompensate the error term O (cid:16) c R g (cid:17) . We get the desired contradiction: E BDF (2) = E BDF ( γ ′ ) ≥ E BDF (2) + 1 c R g K ′ g > E BDF (2) . γ Lemma 8.
For λ ≤ λ < − big enough ( e.g. λ = , , ) there holds: k η ( λ ) c R g ρ γ k C > L p cλ R g and k η ( λ ) c R g | D | / γ k S , k η ( λ ) c R g | D | / γ k S > c p λ R g . (89)This part comes after lots of technicalities: we put together results of Lemma 7,Propositions 7, 8, 9, Remark 18 and the known estimates of Remark 14. We refer thereader to Remark 16 for explanation.Here we assume that L is small enough in such a way that λ R g = O ( L − ) is big enough. Lemma 8 gives that for all λ ≤ λ < − : k η ( λ ) c R g ρ γ k C ≤ ǫ p cλ R g + ǫ k η ( λ/ c R g ρ γ k C , ǫ , ǫ = O ( L ) . (90)We recall that λ R g := C L with C ( L, R g ) > to be chosen. Up to taking a bigger C : C ≤ e C < C we assume λ = 2 − J , J ∈ N . Taking ℓ := c e C L as unity of length, wedefine the sequences ( u m ) , ( v m ) , ( w m ) by the formulae: u = v = w = k η ( λ ) c R g ρ γ k C ,u m := k η (2 m λ ) c R g ρ γ k C , v m = 2 m/ u m ,w m +1 := ǫ r ℓ + ǫ √ w m (91)It is clear from (90) that v m +1 ≤ ǫ r ℓ + ǫ √ v m . Thus we have: ∀ m ∈ N ∗ : v m ≤ w m = w ∞ + (2 / ǫ ) m ( w − w ∞ ) where w ∞ = ǫ (2 /ℓ ) / (1 − ǫ √ − / is well defined provided ǫ < − / . In partic-ular: ∀ m ∈ N ∗ : u m ≤ ǫ √ √ m ℓ + ( √ ǫ ) m √ m n k η ( λ ) c R g ρ γ k C − ǫ √ √ ℓ (1 − ǫ √ o t remains to evaluate at m = J : this gives k η (3 − ) c R g ρ γ k C . Similarly the case m = J − corresponds to − etc. By Hardy-Littlewood-Sobolev inequality [16, Theorem 4.3]: k η ( λ ) c R g ρ γ k C ≤ k| ρ γ |k C > k ρ γ k L / > k ρ γ k L k ρ γ k L > Lc − / . For k η ( λ ) c R g | D | / γ k S it suffices to use this result, Proposition 8 with Lemma 7. Remark . The following holds.1. Lemma 7 states that each ψ j is localized around its center cz j ,2. we give in Remark 14 estimates on the norms of γ , N, ρ γ and n . In particular thedensities have the "correct behaviour" in L , L and Coulomb norms. We callthese estimates: "non-localized estimates".The other cited results are used of as follows. We remark that η ( λ ) c R g = η ( λ ) c R g η ( λ c R g .Proposition 8 gives an estimate of k η ( λ ) c R g | D | / γ k S and k η ( λ ) c R g | D | e a γ k S (where e a ∈ { − , a [Λ] } ) in terms of k η ( λ ) c R g v [ ρ ′ γ ] k L , k η ( λ ) c R g γ k Ex , k η ( λ ) c R g R N k S and k η ( λ ) c R g v [ ρ ′ γ ] k L , and in terms of the non-localized estimates (with the "correct behaviour" with respectto cλ R g , that is as in (89)). Below, we shorten: non. loc. est. w. the c. b.Proposition 9 gives an estimate of k η ( λ ) c R g ∇ v [ ρ γ ] k L in terms of k η ( λ ) c R g | D | / γ k S and k η ( λ ) c R g ρ γ k C = k ρ [ η ( λ ) c R g γ η ( λ c R g ] k C , and in terms of the non. loc. est. w. the c. b.Furthermore, it gives an estimate of k η ( λ ) c R g v ρ γ k L in terms of k η ( λ ) c R g ∇ v ρ γ k L and ofthe non. loc. est. w. the c. b. The term k η ( λ ) c R g γ k Ex is controlled by k η ( λ ) c R g | D | / γ k S and by the non. loc. est. w. the c. b.Thanks to Lemma 7, the term k η ( λ ) c R g R N k Ex (resp. k η ( λ ) c R g n k C ) is proved to be oforder ( c λ R g ) − (resp. ( cλ R g ) − / ).Finally Proposition 7 together with Remark 18 gives an estimate of k ρ [ η ( λ ) c R g γ η ( λ c R g ] k C in terms of k η ( λ c R g P ± γ k S , k η ( λ ) c R g P ± γ k S , and in terms of the non. loc. est. w. the c.b. The presence of P ± is harmless as we can check from the proofs. We consider each term of the BDF energy and write η ( 13 ) c R g ) + ( ξ ( 13 )1 ) + ( ξ ( 13 )2 ) . We use once again Lemma 7, Proposition 8 and Remark 14. We treat one after theother the case of N and γ . We write ( ξ ( λ ) ) := ( ξ ( λ )1 ) + ( ξ ( λ )2 ) . The function ζ refers to ξ ( λ ) or η ( λ ) c R g . γ : Tr (cid:0) ( η ( 13 ) c R g ) | D | / γ | D | / (cid:1) ≤ k ( η ( 13 ) c R g ) | D | / γ k S > c R g Tr (cid:0) ζ ±∓ | D | / γ | D | / (cid:1) ≤ k ζ ±∓ k B k| D | / γ k S > c λ R g inetic energy for N : We recall the following equalities: D ψ j = µ j − αBψ j and ( v n − R N ) ψ = v ψ − v ψ = O L ( α / c − ) . Thus, we have: h η ( 13 ) c R g D ψ j , η ( 13 ) c R g ψ j i = h η ( 13 ) c R g ( µ j − αB ) ψ j , η ( 13 ) c R g ψ j i|h η ( 13 ) c R g D ψ j , η ( 13 ) c R g ψ j i| ≤ (1 + α k v [ ρ ′ γ ] k L ∞ ) k η ( 13 ) c R g ψ j k L + α k η ( 13 ) c R g ψ j k L ( k ( η ( 13 ) c R g ) R γ ψ j k L + α k v kj k L ∞ k η ( 13 ) c R g ψ k k L ) > R g + α k γ k Ex c R g = o ( c − R − g ) . We write : ( ξ ( 13 ) ) = ( P + P − )( ξ ( 13 ) )( P + P − )( ξ ( 13 ) )( P + P − ) , we have to show that h ξ ε ε ξ ε ε D ψ j , ψ j i is O ( c − R − g ) whenever ε = ε or ε = ε .We recall that k P − ψ j k L and α k Bψ j k L are O ( c − ) .The operator ( ξ ( 13 ) ) + − ( ξ ( 13 ) ) − + is O ( c − R − g ) in k·k B -norm. Except for the corre-sponding term, we have ε = − or ε = − , leading to an upper bound: O (cid:0) k ( ξ ( 13 ) ) + − k B ( k P − ψ j k L + α k Bψ j k L ) (cid:1) = O (cid:16) c R g (cid:17) . Similar estimates lead to (88). The estimates ε , ε = o (1) follow from the fact that n = | ψ | + | ψ | = | h | + | h | , where the h j ’s satisfy h ∧ h = ψ ∧ ψ = Ψ and D (cid:0) | h | , | h | (cid:1) = d Ψ . In fact, this o (1) is an O ( α + e − K R g ) . η ( λ ) c R g . By Lemma 7 and Kato’s inequality (Appendix A): k ( η ( λ ) c R g ) n k C > c / λ R g . On the inside: ξ ( 13 ) . We remark the following: ( ξ ( 13 ) ) = ( ξ ( 13 ) ) (cid:0) ( η ( 112 ) c R g ) + ( ξ ( 112 ) ) (cid:1) = ( η ( 112 ) c R g ) − ( η ( 112 ) c R g ) ( η ( 13 ) c R g ) + ( ξ ( 112 ) ) ( ξ ( 13 ) ) = ( η ( 112 ) c R g ) − ( η ( 13 ) c R g ) + ( ξ ( 112 ) ) . (92)Thus: (cid:12)(cid:12)(cid:12) D (cid:0) ( ξ ( 13 ) ) ρ γ , ( η ( 13 ) c R g ) ρ ′ γ (cid:1)(cid:12)(cid:12)(cid:12) ≤ k ( η ( 13 ) c R g ) ρ γ k C ( k ( η ( 13 ) c R g ) ρ γ k C + k ( η ( 112 ) c R g ) ρ ′ γ k C )+ | D (( ξ ( 112 ) ) ρ γ , ( η ( 13 ) c R g ) ρ ′ γ ) | > k ρ γ k L k ρ ′ γ k L c R g + o (cid:0) Lc R g (cid:1) . We treat D (cid:0) ( ξ ( 13 )1 ) ρ γ , ( ξ ( 13 )1 ) ρ ′ γ (cid:1) in a similar way: it is O (cid:0) Lc R g (cid:1) . We have proved sofar: D ( ρ ′ γ , ρ ′ γ ) = D (( ξ ( 13 )1 ) ρ ′ γ , ( ξ ( 13 )1 ) ρ ′ γ ) + D (( ξ ( 13 )2 ) ρ ′ γ , ( ξ ( 13 )2 ) ρ ′ γ )+2 D (( ξ ( 13 )1 ) n, ( ξ ( 13 )2 ) n ) + O (cid:0) Lc R g (cid:1) . In appendix D we prove the following Lemma.
Lemma 9.
For j = 1 , , we have: D (cid:0) ( ξ ( 13 )1 ) ρ ′ γ , ( ξ ( 13 )1 ) ρ ′ γ (cid:1) = D (cid:16) ρ h ξ ( 13 ) j · (cid:2) γ ′ (cid:3)i , ρ h ξ ( 13 ) j · [ γ ′ ] i(cid:17) + O (cid:0) Lc R g (cid:1) . .5.3 Exchange term By Lemma 7 and Kato’s inequality (60): Tr (cid:16) ( η ( λ ) c R g ) NR N (cid:17) > X j k η ( λ ) c R g ψ j k L Tr (cid:0) |∇| N (cid:1) > c ( λ R g ) = o (cid:0) αλ cR g (cid:1) . With the same trick used before, we have: ZZ | γ ′ ( x, y ) | | x − y | dxdy = ZZ (( η ( 13 ) c R g ( x )) +( ξ ( x )) ) | γ ′ ( x, y ) | | x − y | (( η ( 13 ) c R g ( y )) +( ξ ( y )) ) dxdy. We use Kato’s inequality as usual to get: k η ( λ ) c R g γ ′ k Ex > k| D | / η ( λ ) c R g γ ′ k S ≤ k [ | D | / , η ( λ ) c R g ] | D | / k B k| D | / γ ′ k S + k η ( λ ) c R g | D | / γ ′ k S , > c p λ R g . Using trick (92), we get ZZ | γ ′ ( x, y ) | | x − y | dxdy = k ξ γ ′ k Ex + k ξ γ ′ k Ex + 2 ZZ ( ξ ( x )) | γ ′ ( x, y ) | | x − y | ( ξ ( y )) dxdy + O (cid:16) k γ k S c R g + Tr (cid:16) ( η ( 13 ) c R g ) NR N (cid:17) + k η ( 112 ) c R g γ ′ k Ex (cid:17) . Now let us show that for j = 1 , : k ξ j γ ′ k Ex = k ξ j · [ γ ′ ] k Ex + O (cid:0) c R g ) (cid:1) . (93)It suffices to use Kato’s inequality and Eq. (94), we have: k| D | / ξ + − Q k S ≤ π Z + ∞−∞ dω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | D | / D + iω α · ∇ ξ D + iω Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S > k∇ ( ξ cλ R g ) k L ∞ k Q k S Z + ∞−∞ dωE ( ω ) / > k Q k S cλ R g . A Estimates
A.1 [ V, P − ] and proof of Proposition 1 For any smooth complex valued function V , there holds [6]: [ V, P − ] = − i π Z + ∞−∞ D + iη α · ∇ V dηD + iη . (94)Thanks to the KSS inequality as shown in [2], provided smoothness of V ( ∇ V ∈ L p )then this operator is S p ( L ( R , C )) for p > .The integral kernel of its Fourier transform [7] is: F (cid:0) [ V, P − ]; p, q (cid:1) = i π ) / E ( p ) + E ( q ) ( α j d ∂ j V ( p − q ) − s p α j d ∂ j V ( p − q ) s q ) . (95)We prove Proposition 1 by duality, following [6]. Let V be in S ( R ) , Q ∈ S P − (werecall that a [Λ] = 1 + ), then Tr ( QV ) = Tr( P Q ( P + P − ) V P ) + Tr( P − Q ( P + P − ) V P − ) . The operator Q + − | D | a [Λ] 1 | D | a [Λ] [ P − , V ] is in S : indeed thanks to (95) we have ZZ | b V ( p − q ) | | p − q | dpdqE ( p )
1+ 1log(Λ) ( E ( p ) + E ( q )) > log(Λ) k∇ V k L howing k | D | a [Λ] [ P − , V ] k S > p log(Λ) k∇ V k L . This also treats the case Q − + V + − ∈ S . Then we have Q ++ V ++ = Q ++ | D | a [Λ] 1 | D | a [Λ] V ++ ∈ S .Indeed | D | a [Λ] Q ++ | D | a [Λ] ∈ S and | D | a [Λ] V ++ ∈ S with norm O ((log(Λ)) / k∇ V k L ) .Then | D | a [Λ] V ++ 1 | D | a [Λ] ∈ S with norm O ( k∇ V k L ) . So: Tr( Q ++ V ++ ) = Tr (cid:0) | D | a [Λ] | D | a [Λ] Q ++ | D | a [Λ] | D | a [Λ] V ++ (cid:1) = Tr (cid:0) {| D | a [Λ] Q ++ | D | a [Λ] }{ | D | a [Λ] } V ++ 1 | D | a [Λ] (cid:1) = O (cid:0) k| D | a [Λ] Q ++ | D | a [Λ] k S k∇ V k L (cid:1) . The same holds for Q −− V −− . This ends the proof. Remark . In Appendix D we do analogous estimates but with an additional locali-sation operator.We adapt [2, Lemma 5]:
Lemma 10.
Let p be in (3 , + ∞ ] and V a smooth function with ∇ V ∈ L p . Then forany < a < : [ | D | a , V ] | D | a ∈ S p . (96)To prove it we use [17, p. 87] ∀ x > , < a < x a = sin( aπ ) π Z + ∞ dss − a xx + s . (97) A.2 Proof of Lemma 6
Proof:
Let us explain the bootstrap argument.– We show that
Tr(( − ∆) a +1 N ) > . As a consequence: k|∇| a n j k L ≤ P aℓ =0 K ( ℓ, a ) k|∇| ℓ F − ( | b ψ j | ) k L k|∇| a − ℓ F − ( | b ψ j ) |k L > P aℓ =0 K ( ℓ, a ) k|∇| ℓ +3 / F − ( | b ψ j | ) k L k|∇| a − ℓ +3 / F − ( | b ψ j | ) k L > K ( a ) . – As shown in [20], ( γ ′ , ρ ′ γ ) is the fixed point of some function F (1) in a ball of e X a : e X a = { ( Q, ρ ) ∈ S ×S ′ : ZZ E ( p − q ) a E ( p + q ) | b Q ( p, q ) | < + ∞ and Z E ( k ) a | k | | b ρ ( k ) | < + ∞} . – We multiply by | D | ( a +3) / the equation D ψ j = L − ( µ j ψ j − αB γ ′ ψ j ) and we showthat Tr(( − ∆) a +2 N ) > . We have to deal with [ | D | ( a +3) / , v ] ψ j and [ | D | ( a +3) / , R ] ψ j :it suffices to compute in Fourier space and to use Taylor’s formula on the function E ( · ) ( a +3) / . Proof of the estimates
Here as
Tr( − ∆ N ) > , the fixed point method can beapplied on e X a =1 . Indeed k n k L > k|∇| / √ n k L > . We get that ZZ | p − q | E ( p + q ) | b γ ( p, q ) | dpdq > . Let us show the assumption on the H -norm of ψ j .There holds f ( − i ∇ ) D ψ j = f ( − i ∇ )( µ j − αB [ γ ]) ψ j for any f ≥ . Taking the L -normwe have to deal with [ f ( − i ∇ ) , R γ ′ ] and [ f ( − i ∇ ) , v [ ρ ( γ ′ )]] . For f ( − i ∇ ) = |∇| / thereholds k [ |∇| / , v ρ ] ψ k L > ZZ | b ρ ( p − q ) | | p − q | dpdq | q | E ( q ) Z dqE ( q ) | q || b ψ ( q ) | k [ |∇| / , R Q ] ψ k L > ZZ | p − q || b Q ( p, q ) | dpdq k|∇| / ψ k L |∇| / D ψ = µ |∇| / L Λ ψ − α |∇| / L Λ Bψ = O L (1) a priori |∇| / Bψ = [ |∇| / , B ] ψ + B |∇| / |∇| ψ and: |∇| (1 − ∆) ψ , ψ i−h|∇| ψ , ψ i > αc − k v ψ − v ψ k L + c − + α c − = O ( c − + αc − a ) . We get
Tr( | D | N ) > and by the fixed-point Theorem: k γ k Q = ZZ E ( p − q ) E ( p + q ) | b γ ( p, q ) | dpdq > . Notation . The star in k·k ∗ Q means that we replace E ( p − q ) E ( p + q ) by | p − q | | p + q | .Using the methods of [7, 21] we have: k γ k ∗ Q > c − / k ρ ′ γ k L + α ( k γ ′ k ∗ Q ) + α ( k ρ ′ γ k L + k γ ′ k ∗ Q ) + ∞ X k =1 √ k ( αK ( k ρ ′ γ k C + k γ ′ k Q )) k , k [ ∇ , γ ] k S > α ( k ρ ′ γ k L + k N k ∗ Q ) + α ( k ρ ′ γ k L + k γ ′ k ∗ Q ) + ∞ X k =1 √ k ( αK ( k ρ ′ γ k C + k γ ′ k Q )) k , k ρ γ k L > L k n k L + c − / k γ ′ k ∗ Q + α ( k ρ ′ γ k L + k γ ′ k ∗ Q ) + ∞ X k =1 √ k ( αK ( k ρ ′ γ k C + k γ ′ k Q )) k . Therefore k γ ′ k ∗ Q = O ( c − ) , k [ ∇ , γ ] k S = O ( αc − / ) and k ρ γ k L = O ( Lc − / + c − + c − ( √ αa )) . For f ( − i ∇ ) = ∂ k with k = 1 , , we have: ∂ k R Q ψ = [ ∂ k , R [ Q ]] ψ + R Q ∂ k ψ and ∂ k vψ = ( ∂ k v ) ψ + v ( ∂ k ψ ) k [ ∂ k , R Q ] ψ k L = k R ([ ∂ k , Q ]) ψ k L ≤ k [ ∂ k , Q ] k S k∇ ψ k L and k R Q ∂ k ψ k L ≤ k Q k S k ∆ ψ k L k v ρ ( ∂ k ψ ) k L ≤ k v ρ k L k ∂ k ψ k L > k ρ k C k|∇| / ψ k L ≤ k ρ k C p k∇ ψ k L k ∆ ψ k L k ( ∂ k v ρ ) ψ k L > ZZ | b ρ ( k ) | | k | dkdq | q | (1 + | q | ) [ k∇ ψ k L + k ∆ ψ k L ] X k =1 ( k ∂ k D ψ k L ) − k∇ ψ k L ≤ ( µ − k∇ ψ k L + 6 αµ k∇ ψ k L k B [ γ ′ ] ψ k L + α k∇ B [ γ ′ ] ψ k L Tr(∆ (1 − ∆Λ + ∆ Λ ) N ) > αa c − + c − . This gives k ∆ ψ j k L > αc − and in particular: k c (1 − L − c Λ ) ψ j k L = O (cid:0) √ αc Λ (cid:1) . As a consequence we have: k∇ χ j k L = k i σ · ∇ χ j k L = O ( c − ) . (98) Thanks to those estimates, we get: E BDF ( γ + N ) = 2 + E PT ( ψ ∧ ψ )2 c + O ( α c − / + c − ) . (99)We recall that − L − = − ∆Λ − ∆ . Thanks to Section B there holds D ( n , n ) − D ( ψ ∗ ψ , ψ ∗ ψ ) > c − and a > α / c − . rom this point we get better estimate on k ∆ ψ k L > c − but this is still unsatisfactory.Let us be more precise about µ = h ( D + αB ) ψ , ψ i and χ : (1 + µ ) χ = − iσ · ∇ φ − µ ∆Λ − ∆ χ + α L Λ ( v ρ γ χ + ( v χ − v χ ) − ( R γ ψ ) ↓ )= µ ( − iσ · ∇ φ + X ( r )1 ) = − iσ · ∇ φ + O L ( c − / Λ + c − ) h D ψ , ψ i = h D ψ , ψ i − h ∆Λ βψ , ψ i + h ∆Λ − i α · ∇ ψ , ψ i = 1 − k χ k L + 2 Re h− iσ · ∇ ϕ , χ i + O (cid:0) k∇ ψ k L Λ + k ∆ ϕ k L k∇ χ k L Λ (cid:1) = 1 + µ (cid:0) − µ (cid:1) k∇ ϕ k L + Re µ (cid:0) − µ (cid:1) Re h− iσ · ∇ ϕ , X ( r ) i + O (cid:0) k ∆ ϕ k L c Λ (cid:1) = 1 + k∇ ϕ k L + O ( c − + c − Λ − (1 + k ∆ ϕ k L )) . Then: kL − ψ k L = 1 + O ( c − Λ − + k ∆ ψ k L / Λ ) k∇L − ψ k L = k∇ ψ k L + O ( k ∆ ψ k L / ( c Λ ) + k ∆ ψ k L / Λ ) − αµ Re h − ∆ L Λ Bψ , ψ i = − αµ h Bψ , ψ i + O ( α k Bψ k L k ∆ ψ k L / Λ ) k− i α ∇ Bψ k L > k [ ∇ , B ] ψ k L + k B ∇ ψ k L = O ( c − / + k ∆ ψ k / L c − + k ∆ ψ k L c − / ) . and thus: h (1 − ∆) ψ , (1 − ∆) ψ i = µ h − ∆ L ψ , ψ i − αµ Re h − ∆ L Λ Bψ , ψ i + k D L Λ Bψ k L = 1 + 2( µ − − α h Bψ , ψ i ) + k∇ ψ k L + O ( c − ( c − + Λ − ) + k ∆ ψ k L c Λ + k ∆ ψ k L (Λ − + α c − )) . From (62) and the expression of D ψ j , we have k∇ ψ j k L = − α Re h Bψ j , ψ j i . Weconclude k ∆ ψ k L > c − ( c − + Λ − ) and k ∆ ψ k L > min (cid:0) c − , c − ( c − + Λ − (cid:1) . B Proofs of Section 3
B.1 Proof of Proposition 3
Reductio ad absurdum .We assume this is false and take a non-increasing sequence ( a j ) j ≥ tending to suchthat there exists Ψ j that does not satisfy (38) with b = a j : ∆ E < a j and ∆ E d Ψ < a j .In particular (Ψ j ) j is a minimizing sequence for E PT (2) . By geometrical methods [12]we see that Ψ j can be decomposed in two pieces of mass one, each piece tending to aminimizer for E PT (1) . Indeed it is clear that (Tr( − ∆ γ Ψ j )) j is bounded and that thereis no vanishing for ( ρ Ψ j ) j ≥ . If we follow a bubble [13] of ρ Ψ j (one of the biggest) letus show its mass is at the limit.By scaling, for any < λ < we have E PT ( λ ) ≥ λ E PT (1) , where E PT ( λ ) is definedas the infimum of E PT over non-negative one-body density matrix whose trace is λ .Up to following a bubble and extracting a subsequence there holds with Ψ j = h ,j ∧ h ,j : | h ,j ∧ h ,j ih h ,j ∧ h ,j | ⇀ g G ⊕ G ⊕ G , X j =0 Tr( G jj ) = 1 and Tr( G ) < . We recall that each G jj is a density matrix in ( L ) ∧ ( j ) . Following [12, part 5]: G jj = Tr( G jj ) e G jj lim inf j → + ∞ E U PT (Ψ j ) = E U PT (2) ≥ X j =0 ( E U PT ( G jj ) + Tr( G jj ) E U PT (2 − j )) ≥ X j =0 Tr( G jj )( E U PT ( e G jj ) + E U PT (2 − j )) ≥ E U PT (2) . s not all particles are lost (we follow a bubble) either G = 0 or G = 0 . In thecase G , = 0 , [5] enables us to say E U PT ( e G ) > E PT (2) . So G = 0 and G = 0 .Thanks to [15] and Lieb’s variational principle (we may assume G = Tr( G ) | φ ih φ | )there holds E PT ( G ) ≥ (Tr( G )) E PT (1) , then necessarily Tr( G ) = 1 .As a consequence there is exactly two bubbles in ( ρ Ψ j ) j , there exist a decomposition Ψ j = h ,j ∧ h ,j and a sequence ( z ,j ; z ,j ) j of ( R ) such that (up to extraction)1. h h k,j , h ℓ,j i = δ kℓ and | z ,j − z ,j | → j →∞ + ∞ ,2. h k,j ( · − z k,j ) H → j →∞ φ j, ∞ where φ j, ∞ ∈ P is radial.Then it suffices to compute: E U PT (Ψ j ) with this decomposition: E U PT (Ψ j ) = E U PT ( h ,j ) + E U PT ( h ,j ) − D ( | h ,j | , | h ,j | ) + U ZZ | h ,j ∧ h ,j ( x, y ) | dxdy | x − y | = E + E + U W − D ≥ U W + 2 E PT (1) . The last equality holds because we have
U > U c . Let us write ∆ E := E PT ( h ,j ) + E PT ( h ,j ) − E PT (1) . Then: − a j < ∆ E − D < a j and ∆ E ≥ κ X k =1 k h k,j − φ k,j k H where φ k,j ∈ P is the closest function to h k,j in H (Proposition 2). We may assumethat D = d Ψ j because minimizing this quantity corresponds to minimizing ∆ E . Inparticular: | ∆ E − D | < a j = o j → + ∞ ( D ) ⇒ ∆ E ∼ j → + ∞ D ≫ a j . Indeed, let us say that D > d Ψ j , then ( f k,j ( · − z k )) j still converges to φ j, ∞ , inparticular (∆ E ) j converges to . But if ( f ′ ,j , f ′ ,j ) j is a decomposition with D ′ = d Ψ j ,then ∆ ′ E ≤ ∆ E and dist ( f ′ k,j , P ) → j → + ∞ . From now we will drop the subscript j for convenience and suppose D = d Ψ j . Notation . We introduce h k = ( h k − φ k ) + φ k = δ k − φ k in | h k | and in h ∗ h . Weuse the convention k δ k L := k δ k L + k δ k L , k δ k H := k δ k H + k δ k H . We recall that an element of P has an exponential falloff with respect to its center.For some constant ε > , there holds: | h k | = | δ k | + | φ k | + 2 Re ( δ ∗ k φ k ) h ∗ h ∗ = δ ∗ δ + φ ∗ φ + δ ∗ φ + φ ∗ δ k h ∗ h k C = k δ ∗ φ k C + k φ ∗ δ k C + O (cid:0) ( k δ k L k δ k L )( R − g + k δ k L (1 + k∇ δ k L ) + e − ε R g ) (cid:1) D = D ( | φ | , | φ | ) + D ( | δ | , | φ | ) + D ( | φ | , | δ | )+ O (cid:0) k δ k L R g + k δ k L k δ k L ( k δ k L (1 + k∇ δ k L ) + e − ε R g ) (cid:1) Thus: a j U − ? D − k h ∗ h k C ? R g + O j → + ∞ ( k δ k L ) and R g = O j → + ∞ ( a j U − + k δ k L ) . s j → + ∞ , thanks to the coercivity inequality (2) there holds D ∼ ∆ E = Θ( k δ k H + k δ k H ) and R g = o j → + ∞ ( D ) . Studying more precisely M ( h ∧ h ) := s | h ∧ h ( x, y ) | dxdy | x − y | : M ( h ∧ h ) = M ( δ ∧ φ ) + M ( φ ∧ δ ) + O j → + ∞ ( R − g + k δ k L ) = o j → + ∞ ( D ) D = D ( | δ | , | φ | ) + D ( | φ | , | δ | ) + o j → + ∞ ( D ) ? k δ k H + k δ k H . (100)We can easily exclude the case δ , δ = 0 for then it is clear M ( φ ∧ φ ) ? D ( | φ | , | φ | ) thanks to h φ , φ i = 0 . Say then that k δ k H ≥ k δ k H : δ = 0 . The case δ = 0 and δ = 0 is an easy adaptation of what follows, we treat it later. As there holds | φ | ∗ |·| ( x ) ≤ | x − z | where z is the center of φ , Estimate (100) is true only if there lies a mass of δ near z : the quantity k δ ∗ φ k C must compensate D ( | δ | , | φ | ) . Eventually the samephenomena occurs for δ around z the center of φ . Up to extraction: δ k ( · − z k ) k δ k k H ⇀ H ℓ k , and ( ℓ , ℓ ) = (0 , . Indeed up to contraction there is convergence in L loc and if ℓ k = 0 then for all r > and ( i , i ) ∈ (cid:8) (1 , , (2 , (cid:9) lim sup j → + ∞ Z | δ i ( x ) | k δ i k H | φ i | ∗ | · | ( x ) dx ≤ r +lim sup j → + ∞ Z | x − z i |≤ r | δ i ( x ) | k δ i k H | φ | ∗ | · | ( x ) dx = 1 r , this would contradict (100). Then as we have: lim j → + ∞ M (cid:0) δ k δ k H ∧ φ (cid:1) = lim j → + ∞ D M (cid:0) δ ∧ φ (cid:1) = 0 , then necessarily ℓ = ε φ , ∞ with | ε | ≤ . Furthermore, either k δ k H = o j → + ∞ ( k δ k H ) or k δ k H = Θ j → + ∞ ( k δ k H ) .– In the first case then k δ k H = o j → + ∞ ( D ) and ℓ = 0 . We get a contradiction bycomputing: Z h ∗ h = Z φ ∗ φ + Z δ ∗ φ + Z φ ∗ δ + Z δ ∗ δ = O j → + ∞ ( e − ε R g ) + Z δ ∗ φ + O j → + ∞ ( k δ k L (1 + k δ k L ))= Z δ ∗ φ + o j → + ∞ ( k δ k H ) . – In the second case we also get lim j → + ∞ k δ k − H M ( δ ∧ φ ) and ℓ = ε φ , ∞ , | ε | ≤ .Writing for k = k ′ : h k = φ k + ε k k δ k k H φ k ′ + h ( r ) k , up to extraction the following holds: Z h ∗ h = O j → + ∞ ( e − ε R g ) + ε ∗ k δ k H + ε k δ k H + Z ( h ( r )1 ) ∗ h + Z h ∗ h ( r )2 Z ( h ( r )1 ) ∗ h = Z ( h ( r )1 ) ∗ φ + Z ( h ( r )1 ) ∗ ( ε k δ k H φ ) + Z ( h ( r )1 ) ∗ h ( r )2 = o j → + ∞ ( k δ k H ) + O j → + ∞ ( k δ k H ) + O j → + ∞ ( k δ k H k δ k H ) . The o j → + ∞ ( k δ k H ) comes from the L loc -convergence to of h ( r )1 ( ·− z ) k δ k H and the uniformshape of the φ ( · − z ) ’s. In particular: ε ∗ k δ k H = − ε k δ k H + o j → + ∞ ( k δ k H ) . riting ε k δ k H = a and ε k δ k H = b = − a ∗ + ( δa ) : ( h = φ + aφ + h ( r )1 h ( r )1 = δ − aφ h = φ − a ∗ φ + ( δa ) φ + h ( r )2 h ( r )2 = δ − bφ . We apply p − | a | a ∗ − a p − | a | ! with p − | a | =: s (cid:18) g g (cid:19) = φ ( s + | a | − a ( δa )) + φ ( a ( s − sh ( r )1 − ah ( r )2 φ ( s + | a | ) + φ ( a ∗ (1 − s ) + ( δa ) s ) + sh ( r )2 + a ∗ h ( r )1 ! , replacing s = 1 − | a | + O j → + ∞ ( | a | ) and neglecting the term O H ( | a | ) : (cid:18) g g (cid:19) = φ (1 + | a | − a ( δa )) + h ( r )1 − ah ( r )2 + O H ( | a | )(1 + | a | ) φ + φ (( δa )(1 − | a | )) + h ( r )2 + a ∗ h ( r )1 + O H ( | a | ) ! . By L loc -convergence, it is clear that D ( | φ k | , | h ( r ) k ′ | ) = o j → + ∞ ( k δ k ′ k H ) for ( k, k ′ ) equalto (1 , or (2 , . Using δa = o j → + ∞ ( k δ k H ) , at last we have: D ( | g | , | g | ) > D ( | φ | , | φ | ) + o j → + ∞ ( k δ k H ) = o j → + ∞ ( k δ k H ) = o j → + ∞ ( D = d Ψ ) , which gives the desired contradiction.– Let us treat at last the case δ = 0 and δ = 0 . Then as before: D ( | h | , | φ | ) = D ( | δ | , | φ | ) + O (cid:0) k δ k L R g (cid:1) = D ( | δ | , | φ | ) + o j → + ∞ ( D ) . Then necessarily there lies some mass of δ near z and: δ ( · − z ) k δ k H ⇀ H ℓ = 0 . As before necessarily: ℓ = ε φ , ∞ with < | ε | ≤ . But this contradicts: Z h ∗ φ = Z δ ∗ φ + Z φ ∗ φ = Z δ ∗ φ + O j → + ∞ ( e − ε R g ) . B.2 Proof of Proposition 4
The proof is similar to that of Proposition 3: by contradiction we assume the existenceof ( a j ) j decreasing to together with (Ψ j = h ∧ h ) with E U PT (Ψ j ) < a j and M (Ψ j ) : lim j → + ∞ Z B ( z k,j ,A ) | h k,j ( x ) | dx = Z B ( z k,j ,A ) | φ k, ∞ ( x ) | dx. For any − / < λ < let A λ > be the number such that the last integral with A = A λ is equal to λ . We have: ZZ | x − y | < R g +2 A λ | h ∧ h ( x, y ) | | x − y | dxdy ≥ R g + 2 A λ ZZ | x − y | < R g +2 A λ | h ( x ) | | h ( y ) | dxdy − R g + 2 A λ Z dxh ∗ h ( x ) Z y ∈ B ( x, R g +2 A λ ) h ∗ h ( y ) dy lim inf j → + ∞ ZZ | x − y | < R g +2 A λ | h ∧ h ( x, y ) | | x − y | dxdy ≥ R g +2 A λ ( λ − − ) . e used the following trick: if Z h ∗ h = 0 where k h k k L = 1 , then for any Borelianset B : (cid:12)(cid:12)(cid:12) Z B h ∗ h (cid:12)(cid:12)(cid:12) ≤ . The more precise result has the same proof: in the limit there holds similar inequal-ity: for sufficiently small a > , λ R g > A ε where Z | x |≤ A ε | φ ( x ) | dx = ε, ε > − / , φ ∈ P . We conclude with the same argument.
C Localisation in Direct space: the ψ j ’s C.1 Proof of Lemma 7
Notation . For convenience here we write V · ϕ k := v ′ γ ϕ k − R N ϕ k (and a similarexpression for χ k ). The function r k := R γ ψ k is split into its upper part r k, ↑ := ( R γ ψ k ) ↑ and its lower part r k, ↓ both in L ( R , C ) .Moreover we write: P k ( − ∆) := c (1 − µ k L − c Λ ) − ∆ and y c := L − c Λ = c Λ c Λ − ∆ . The operator P k ( − ∆) can be rewritten as follows: with a k := c (1 − µ k ) and b := c Λ then c (1 − µ k y c ) − ∆ = a k (1 + µ k ) − ∆ h µ k c − a k c Λ b b − ∆ + µ k Λ (cid:16) b b − ∆ (cid:17) i = ( a k (1 + µ k ) − ∆) n (cid:0) − a k (1+ µ k ) a k (1+ µ k ) − ∆ (cid:1)h µ k c − a k c Λ b b − ∆ + µ k Λ (cid:16) b b − ∆ (cid:17) io (101) Proof
We remark that n ( x ) = | h ( x ) | + | h ( x ) | = | ψ ( x ) | + | ψ ( y ) | .Thanks to (47)-(48), there holds: ( D + αB ) ψ k = (1 + E PT (1) c + O ( α / c − )) ψ k . (102)Up to applying some m ∈ SU (2) to (cid:18) ψ ψ (cid:19) , we consider ψ k = h k with the following: ( c β − ic α · ∇ h k ) + αcy c ( V · h k − R γ h k ) = ( c − E PT (1)2 ) y c h k + O ( α / y c h ) We write a = − E PT (1)2 and the additional term O ( α / y c h ) = δ k h .– We now rewrite (73) once again: by substitution, we get: ϕ k = αcy c µ k y c P k ( − ∆) ( V · ϕ k − r k, ↑ ) + αy c P k ( − ∆) iσ · ∇ (cid:2) V · χ k − r k, ↓ (cid:3) χ k = α y c P k ( − ∆) iσ · ∇ ( V · ϕ k − r k, ↑ ) + αy c c (1 − µ k y c ) cP k ( − ∆) (cid:2) V · χ k − r k, ↓ (cid:3) (103)There holds similar equation for h k but with additional terms αc ( δ k h ) ↑ with − r k, ↑ and αc ( δ k h ) ↓ with − r k, ↓ .There holds: αc (1 − µ k y c ) = αc (1 − µ k ) + αcµ k (1 − y c ) . For any A ≥ Γ( R g ) R g , we multiply each term by | D | / and then by d A,λ ( · ) definedby d ( · ) ξ A ( · ) η λ R g . e take the L -norm, let us show estimates independent of A (but depending on ξ ): k d A,λ | D | / ψ k k L ≤ K λ + ε ( λ ) k E / A,λ | D | / ψ k k L , with ε ( λ ) < . This will end the proof, the family ( K λ ) λ depending on ( ε ( λ ) ) λ and the latter beingnonincreasing in λ ∈ ( λ , − ) .We prove the estimation of k d (2) A,λ | D | / ψ j k L with j = 1 , by the same method:we need finiteness of k d ( · ) η ( λ/ c R g | D | / ψ k k L with k = 1 , and of k| x − y | γ k S . Werefer to Appendix C for more details.– In Appendix C, we show: | d A,λ ( x ) − d A,λ ( y ) | > | x − y | . (104)Let us first multiply (103) by | D | / : let F j,k := | D | / ∂ j P k ( − ∆) and F ,k := | D | / P k ( − ∆) . Itis clear that they are bounded (convolution) operators, we show in Appendix C that k| · |F j,k k L > , j ∈ { , , } , k ∈ { , } . (105)The function associated to y c is a Yukawa potential Y c [16, Section 6.23]: Y c ( x − y ) = r π c Λ) e − c Λ | x − y | | x − y | , in particular k| · | Y c k L > c Λ . The idea is to take first the commutator [ d A,λ , F j,k ] and [ d A,λ , y c ] . Then we study d A,λ v̟ k ( ̟ k ∈ { ϕ k , χ k } ) and d A,λ r ↑ / ↓ . Estimate of αc k V · ϕ k k L , αc k V · χ k k L We use the same method for both cases.We recall the following: v γ = (cid:0) − ˇ F Λ ∗ n + ( δ − ˇ F Λ ) ∗ ( t N − α e τ ) (cid:1) ∗ | · | = − ˇ F Λ ∗ n ∗ | · | + ρ rem ∗ | · | . By (58): (cid:12)(cid:12) αc ˇ F Λ ∗ n ∗ |·| (cid:12)(cid:12) ( x ) ≤ n ∗ |·| ( x ) + αc (cid:12)(cid:12) ( ˇ F Λ − F Λ (0) δ ) ∗ v n ( x ) (cid:12)(cid:12) ≤ n ∗ |·| ( x ) + O (cid:0) √ c (cid:1) . We used k f k L ∞ > k b f k L , split the integral in Fourier space at level c and usedCauchy-Schwarz inequality. By Appendix A.2 and Proposition 5: (cid:12)(cid:12) αcρ rem ∗ |·| (cid:12)(cid:12) ( x ) > αc ( c / k ρ rem k C + c / k ρ rem k L ) > αc / ( αc − + α c − ) + αc / ( c − + c − ( α ( a + a )) / ) > α √ log(Λ) + √ log(Λ) + α / √ log(Λ) > √ log(Λ) . We recall a jk = k v k ψ k − v kj ψ k k L and by Proposition 5 we know it is O ( c − α / ) . Wedecompose each ψ j in sum of h , h : ψ k = c k h + c k h . Then: v γ ψ k = v γ ( c k h + c k h )( v n − R N ) ψ k = c k ( v | h | h − v h ∗ h h ) + c k ( v | h | h − v h ∗ h h ) . We write h k = δ k + φ k where φ k ∈ P : as in Section B k δ k k H > α . By fast decay ofthe φ k ’s: ( | φ k | ∗ |·| ( x )) = Θ( | φ k | ∗ |·| ( x )) and for | x | ? this is O ( | x − z k | ) .In particular for | x | > λ R g v (cid:2) | h k | (cid:3) ( x ) > k δ k k L | x − z k | + h|∇| δ k , δ k i > λ R g + α, we choose C > such that αcλ R g < − ε where < ε < is fixed (for instance − ).By Cauchy-Schwarz inequality we have v (cid:2) h ∗ h (cid:3) ( x ) , v (cid:2) h ∗ h (cid:3) ( x ) = O ( k δ k L ) . Itfollows that αc k d A,λ V · ϕ k k L > ε ′ ( λ ) k d A,λ ϕ k k L , with < ε ′ ( λ ) < . stimate of αcd A,λ R γψ k | [ d A,λ , R γ ]( x, y ) | > | γ ( x, y ) | so: αc k [ d A,λ , R γ ] ψ k k L > αc k γ k S k ψ k k L > α c / = O ( α √ log(Λ) ) . k R γ d A,λ ψ k k L > Tr( γR γ ) h|∇| d A,λ ψ k , d A,λ ψ k i > c − k| D | / d A,λ ψ k k L . By Lemma (10), [ | D | / , d A,λ ] | D | − / is a bounded operator (with norm O ( k∇ d A,λ k L ∞ ) )and at last we get: αc k d A,λ R γ ψ k k L > αc / (1 + k d A,λ | D | / ψ k k L ) and αc / = O (cid:0) p log(Λ) (cid:1) . We know deal with the case of d (2) A,λ R γψ k , using (108), proved below.The aim is to prove: k d (2) A,λ R γψ k k L > k| x − y | γ k S + k γ k S k d ( · ) η ( λ/ c R g ψ k k L + c / k γ k Ex ( k ψ k k L + k d ( · ) η ( λ/ c R g ψ k k L ) . (106)First of all we use Taylor’s formula (108) to get: k [ d (2)1 ,λ , R γ ] ψ k k L > k| x − y | γ k S + k γ k S k d ( · ) η ( λ/ c R g ψ k k L . Let us prove at the end k| x − y | γ k S = c − k| x − y | γ k S > αc − . There remains k R γd (2) A,λ ψ k k L > k| D | / d (2) A,λ ψ k k L .We commute: using (97), there holds [ | D | / , d (2) A,λ ] = 12 − / π Z + ∞ s / ds − ∆ + s [ − ∆ , d (2) ] 11 − ∆ + s , [ − ∆ , d (2) ] = ( − ∆ d (2) ) − X j =1 ( ∂ j d (2) ) ∂ j . First k ∆ d (2) k L ∞ > . Then thanks to (108): k ( ∂ j d (2) ) ∂ j − ∆+ s ψ k k L > k d ( · ) η ( λ/ c R g ∂ j − ∆+ s ψ k k L > k| x − y | F − ( p j s + | p | ) k L k ψ k k L + k d ( · ) η ( λ/ c R g ψ k k L s >
11 + s ( k ψ k k L + k d ( · ) η ( λ/ c R g ψ k k L ) . To end this section we prove k| x − y | γ k S , k| x − y || D | / γ k S > α . This is almosttrivial: for each j ∈ { , , } we consider ( x j − y j ) γ ( x, y ) and use the Cauchy expansionof γ . For each Q ,k , k ∈ [ | , | ] , we replace at least one P ε v ′ γ P − ε as in (94) ([6]) andwrite: x j − y j = x j − ℓ (1) j + ℓ (1) j − ℓ (2) j + · · · + ℓ ( n ) j − y j . For each convolution operator | D | / D + iη ( x − y ) , P ε D + iη ( x − y ) , D + iω ( x − y ) , multiplying by ( x j − y j ) corresponds to take the derivative ∂ j in Fourier space enabling us to take KSSinequalities (14) under the integral sign. Indeed we have: | ∂ j E / p | > E / p , | ∂ j E ( p )+ iη | > | E p + iη | − , | ∂ j E ( p )+ iω | > E ( ω )+ | p | , | ∂ j P ε ( p ) | > E ( p ) . Then operators of type ρ ∗ |·| or α k ∂ k ( ρ ∗ |·| ) remains unchanged while operators oftype ( x j − y j ) R Q ( x, y ) are trivially Hilbert-Schmidt. This end the proof ; the biggestterm comes from Q , (( x j − y j ) γ ′ ( x, y )) . .2 Proof of (104) and variation for d (2) A,λ
1. We recall that ξ is a radial smooth function with ξ ( x ) = 1 for | x | ≤ and ξ ( x ) = 0 for | x | ≥ . We study d A,λ := d ( · ) ξ A ( · ) η λ R g ( · ) . First remark to be done: H = { x : | x − z | = | x − z |} splits the space into two half-spaces E (set of points closest to z ) and E . Let s H be the orthogonal symmetrywith respect to H : s H ( z ) = z . If x ∈ E and y ∈ E , then | d ( x ) − d ( y ) | = (cid:12)(cid:12) | x − z | − | s H ( y ) − z | (cid:12)(cid:12) ≤ | x − s H ( y ) | ≤ | x − y | . Moreover d A,λ ( y ) = d A,λ ( s H ( y )) and (cid:12)(cid:12) d A,λ ( x ) − d A,λ ( y ) (cid:12)(cid:12) = (cid:12)(cid:12) d A,λ ( x ) − d A,λ ( s H ( y )) (cid:12)(cid:12) . So we may assume that d ( x ) = | x − z | and d ( y ) = | y − z | , and in this case we canwrite: d A,λ ( x ) = F λ ( d ( x )) ξ A ( x ) := d ( x ) q − ξ λ R g ( d ( x )) G A ( | x − z m | ) the same holds for y . We will write F λ ( · ) for x F λ ( d ( x )) for convenience. Thereholds ∇ d A,λ ( x ) = (cid:0) ∇ F λ ( x ) (cid:1) ξ A ( x ) + F λ ( x )( ∇ ξ ( x/A ) A ) , and as we have chosen A ≫ R g we may assume that if ∇ ξ A ( x ) = 0 , then | x − z m | =Θ( d ( x )) . By simple computation: |∇ d A,λ ( x ) | > (cid:0) k| · |∇ ξ k L ∞ + k| · |∇ η λ k L ∞ (cid:1) . (107)2. For x, y ∈ E ε , ε = 1 , (say E ) and A ≫ R g , there holds: d (2) A,λ ( x ) − d (2) A,λ ( y ) = | x − z | ξ A ( x ) η ( λ ) c R g ( x ) − | y − z | ξ A ( y ) η ( λ ) c R g ( y )= | y − z | (cid:0) η ( λ ) c R g ( y ) A ∇ ξ ( yA ) + ξ A ( y ) cλ R g ∇ ( η )( ycλ R g ) (cid:1) · ( x − y )+ ξ A ( y ) η ( λ ) c R g ( y ) h y − z , x − y i + | y − z | + O (cid:0) | x − y | (cid:1) = O (cid:0) d ( y ) η ( λ/ c R g ( y ) | x − y | + | x − y | (cid:1) . (108)Above we used ∇ η ( λ ) c R g = η ( λ/ c R g ∇ η ( λ ) c R g and the O ( · ) depends on ξ , η . This estimateenables us to consider commutators with | D | / σ ·∇ P k ( − ∆) and y c := ( c Λ) ( c Λ) − ∆ , as shown inthe next section. C.3 Proof of (105) and variation for d (2) A,λ
1. For any borelian function F : Z R | x ||F ( x ) | dx ≤ n Z | x | E ( x ) |F ( x ) | dx Z dx | x | E ( x ) o / . To prove | · |F ∈ L it suffices to check all integrals on the right side converge: inFourier space, we have to prove: k ∆ b Fk L + k∇ ∆ b Fk L < + ∞ . Applying this method for F j,k ( x − y ) := | D | / ∂ j P k ( − ∆) ( x − y ) : d F j,k ( p ) = E ( p ) / p j a k + | p | n µ k | p | Λ ( a k + | p | ) 2 b + b | p | ( b + | p | ) o − where we recall b = c Λ , a k = c (1 − µ k ) . From this expression, it is easy to see that for ℓ = 1 , , and m = 1 , we have k ∂ mℓ F j,k k L > . he constant depends on a k but for sufficiently small α, L, Λ − then a k > ε > .2. By the same method we can show that: Z R | x | |F ( x ) | dx ≤ n Z | x | E ( x ) |F ( x ) | dx Z dx | x | E ( x ) o / , enabling us to treat d (2) A,λ . D Localisation in Direct space: γ We recall we explain in Remark 16 how we use the technical results proved here:Propositions 7, 8 and 9.
D.1 Estimates on the localised density
Let Q ∈ K and ≤ ζ ≤ a smooth function ( e.g. ξ λ R g or η λ R g ). Our aim is to give asemi-quantitative estimate of the localisation of the function ζ ρ Q = ρ ζQζ around thesupport of ζ . Proposition 7.
Let Q and ζ be as above, then we have: k ζ ρ Q − ρ [ ζ ++ Qζ ++ + ζ −− Qζ −− ] k C ≤ F est [Λ , ζ, Q ] , (109) with F est [Λ , ζ, Q ] = ( p log(Λ) k∇ ζ k L + k∇ ζ k L ∞ )( k ζP ± | D | a [Λ] Q k S + k∇ ζ k L ∞ k Q k S )+ k∇ ζ k L k| D | a [Λ] Q k S + p log(Λ)( k ζQ ±∓ | D | a [Λ] ζ k S + k ζQ ±∓ k S k∇ ζ k L ∞ )+ p log(Λ) k∇ ζ k L ∞ ( k∇ ζ k L ∞ k Q ±± k S + k ζ | D | a [Λ] Q ±± k S )+(log(Λ)) / k∇ ζ k L ∞ k| D | a [Λ] Q ±± k S . (110) Moreover there holds for ε = ± : k ρ [ ζ εε Qζ εε ] k C ≤ k [ ζ εε , | D | a [Λ] ] k B k Q εε k S + k ζ εε Q εε | D | a [Λ] ζ εε k S > k∇ ζ k L ∞ k Q εε k S + k ζ εε Q εε | D | a [Λ] ζ εε k S . (111) Remark .
1. In the case Q = Π − P − with Π ∗ = Π = Π then ( cf [7]): Q = Q ++ − Q −− ≥ Q ++ . As shown in [20] we can consider an orthonormal familyof eigenvectors of Q that split into those in Ran ( P ) and those in Ran ( P − ) . It is thenclear that: k ζ ++ Q ++ | D | a [Λ] ζ ++ k S ≤ k ζQ ++ | D | a [Λ] ζ k S ≤ k ζ | D | a [Λ] Q k S k ζQ k S
2. There is also an analogous estimate if we choose two different functions ζ , ζ , thatis with ζ ζ ρ ( Q ) = ρ ( ζ Qζ ) . The same proof shows also localisation estimates, butwe have to "polarize" the inequalities just like for a quadratic form and its associatedbilinear form. Proof:
We prove it by duality. Let V be some Schwartz function: we study Tr ( ζQζV ) . By symmetry we just treat ( ζQζV ) ++ . There holds: P ζQζV P = P ζ ( P + P − ) Q ( P + P − ) ζ ( P + P − ) V P = ζ ++ Q ++ ζ ++ V ++ + ζ ++ Q ++ ζ + − V − + + ζ ++ Q + − ζ − + V ++ + ζ ++ Q + − ζ −− V − + + ζ + − Q − + ζ ++ V ++ + ζ + − Q − + ζ + − V − + + ζ + − Q −− ζ − + V ++ + ζ + − Q −− ζ −− V − + . We first show those operators are trace-class and then prove (109). emark . We recall that by Sobolev inequality: k V k L > k∇ V k L .Moreover k| D | − a [Λ] V k S > p log(Λ) k V k L .As shown in Appendix A: ζ − + = i π Z + ∞−∞ D + iη α · ∇ ζ P dηD + iη . (112)It can be rewritten as: ζ − + = i Z + ∞ e − s | D | P − α · ∇ ζP e − s | D | ds, (113)by writing E ( p )+ E ( q ) = R + ∞ e − s ( E ( p )+ E ( q )) in the kernel of its Fourier transform cf Appendix A. ζ ++ Qζ ++ V ++ : ζ ++ Qζ ++ V ++ = ζ ++ ( Q ++ ζ ++ | D | a [Λ] ) 1 | D | a [Λ] V ++ and ( Q ++ ζ ++ | D | a [Λ] ) ∈ S , | D | a [Λ] V ++ ∈ S with norm O ((log(Λ)) / k∇ V k L ) bythe KSS inequality (14). We write k ζQ ++ ζ ++ | D | a [Λ] k S ≤ k ζQ ++ k S k [ ζ ++ , | D | a [Λ] ] k B + k ζQ ++ | D | a [Λ] ζ k S > k ζQ ++ k S k∇ ζ k L ∞ + k ζQ ++ | D | a [Λ] ζ k S . In general whenever there is Q ++ or Q −− we can easily estimate. | Tr( ζ ++ Q ++ ζ + − V − + ) | = | Tr( V − + 1 | D | a [Λ] | D | a [Λ] ζ ++ Q ++ ζ + − ) | > p log(Λ) k∇ V k L k∇ ζ k L ∞ ( k∇ ζ k L ∞ k Q ++ k S + k ζ | D | a [Λ] Q ++ k S ) , | Tr( ζ + − Q −− ζ − + V ++ ) | ≤ k | D | a [Λ] V k S k ζ + − k B k Q −− ζ − + | D | a [Λ] k S > (log(Λ)) / k∇ V k L k∇ ζ k L ∞ k Q −− | D | a [Λ] k S , | Tr( ζ + − Q −− ζ −− V − + ) | > p log(Λ) k∇ V k L k∇ ζ k L ∞ ( k∇ ζ k L ∞ k Q −− k S + k ζ | D | a [Λ] Q −− k S ) . The term ζ + − Q − + ζ + − V − + : k ζ + − Q − + ζ + − V − + k S ≤ k ζ − + k S k Q + − | D | a [Λ] k S k | D | a [Λ] ζ − + V ++ k S k | D | a [Λ] ζ − + V ++ k S > X j =1 π Z + ∞−∞ k | D | a [Λ] ( D + iη ) ∂ j ζ P D + iη V k S dη > X j =1 k ∂ j ζ k L k V k L k E ( · ) / k L Z + ∞−∞ dηE ( η ) / , k ζ − + k S > k∇ ζ k L . The term ζ ++ Q + − ζ −− V − + : | Tr( ζ ++ Q + − ζ −− V − + ) | > p log(Λ) k∇ V k L ( k ζQ + − | D | a [Λ] ζ k S + k ζQ + − k S k∇ ζ k L ∞ ) . The terms ζ + − Q − + ζ ++ V ++ and ζ ++ Q + − ζ − + V ++ These operators are diffi-cult to handle. We use Lemma 10 (Appendix A). First: ζ + − Q − + ζ ++ V ++ = (cid:0) ζ + − | D | ε Λ (cid:1)(cid:0) | D | ε Λ Q − + ζ ++ | D |
12 + ε Λ (cid:1)(cid:0) | D |
12 + ε Λ V ++ (cid:1) ∈ S , with norm O ((log(Λ)) / k∇ ζ k L k V k L k| D | a [Λ] Q k S ) . We used the KSS inequalityand Hölder-type inequality for S p . Similarly we can show that ζ ++ Q + − ζ − + V ++ ∈ S . hen by density of S in S , we approximate (cid:0) | D | ε Λ Q − + ζ ++ | D |
12 + ε Λ (cid:1) by trace-class operators enabling us to say that: Tr( ζ + − Q − + ζ ++ V ++ ) = Tr (cid:16)(cid:0) | D | ε Λ Q − + ζ ++ | D |
12 + ε Λ (cid:1)(cid:0) | D |
12 + ε Λ V ++ (cid:1)(cid:0) ζ + − | D | ε Λ (cid:1)(cid:17) . Let us show that Q − + ζ ++ V ++ ζ + − ∈ S . It suffices to show | D | a [Λ] V ++ η + − ∈ S .We go in Fourier space and used formula (113) to show [ V, P e − sE | D | ] ∈ S . F ([ V, P e − sE | D | ]; p, q ) = 1(2 π ) / b V ( p − q ) (cid:0) P ( q ) e − sE ( q ) − P − ( p ) e − sE ( p ) (cid:1) ; then ( cf Appendix A) P ( q ) e − sE ( q ) − P − ( p ) e − sE ( p ) = ( P ( q ) − P ( p )) e − sE ( q ) + P ( p )( e − sE ( q ) − e − sE ( p ) ) (cid:12)(cid:12) P ( q ) − P ( p ) (cid:12)(cid:12) > | p − q | max( E ( p ) , E ( q )) (cid:12)(cid:12) e − sE ( q ) − e − sE ( p ) (cid:12)(cid:12) = s | E ( p ) − E ( q ) | | e − sE ( q ) − e − sE ( p ) | s | E ( p ) − E ( q ) |≤ s | p − q | min( e − sE ( p ) , e − sE ( q ) ) ≤ s | p − q | ( e − sE ( p ) + e − sE ( q ) ) . By easy computation: k [ V, P e − sE | D | ] k S > s − / e − s/ √ k∇ V k L : Z + ∞ s =0 k [ V, P e − s | D | ] α · ∇ ζe − s | D | k S ds > k∇ ζ k L ∞ k∇ V k L Z + ∞ e − s dss / . At last there remains to show: A [ V, ζ ] = Z + ∞ e − s | D | | D | a [Λ] P (cid:0) V α · ∇ ζ (cid:1) P − e − s | D | ds ∈ S , as in Appendix A it suffices to go in Fourier space and remark k V ∂ j ζ k L ≤ k V k L k ∂ j ζ k L : kA [ V, ζ ] k S > p log(Λ) k V ∇ ζ k L > p log(Λ) k V k L k ∂ j ζ k L . The case of ζ ++ Q + − ζ − + V ++ is similar: first we prove by density that Tr( ζ ++ Q + − ζ − + V ++ ) = Tr( ζ − + V ++ ζ ++ Q + − ) , and we get in fine k ρ [ ζ ++ Q + − ζ − + V ++ ] k C + k ρ [ ζ − + Q + − ζ − + V ++ ] k C > ( p log(Λ) k∇ ζ k L + k∇ ζ k L ∞ )( k ζP | D | a [Λ] Q k S + k∇ ζ k L ∞ k Q k S ) . (114) D.2 Estimates on the localised operator γ Here γ is the vacuum part of a (hypothetical) minimizer of E BDF (2) or a minimizer of E BDF (1) . Our aim is to prove:
Proposition 8.
Let ζ be a smooth function with: ( k∇ ζ k L ∞ , k ∂ j ∂ k ζ k L ∞ < + ∞ , j, k ∈ { , , }k ζv ′ k L , k ζ ∇ v ′ k L , k ζγ k Ex , k ζR N k S < + ∞ . Then there holds: k ζ | D | / γ k S > c − / k ζ ∇ v ′ k L + α ( k ζγ k Ex + k ζR N k S )+ α ( k ζ ∇ v ′ k L + k ζv ′ k L + k ζγ k Ex + k ζR N k S ) + (cid:8) k∇ ζ k L ∞ + P ≤ j,k ≤ k ∂ j ∂ k ζ k L ∞ (cid:9)(cid:8) α ( k ρ ′ γ k C + k|∇| / γ ′ k S ) (cid:9) . (115) The same holds for k ζ | D | e a γ k S with e a ∈ { , a [Λ] } .We can replace k ζγ k Ex + k ζR N k S by k γ ′ k Ex and put P ± γ instead of γ . .2.1 Idea of the proof We will focus on the Cauchy expansion of γ : γ = + ∞ X j =1 α j Q j ( γ ′ , ρ ′ γ ) . As shown in [6, 20, 22], we substitute P ± ( ρ ′ γ ∗ |·| ) P ∓ by its expression (94) wheneverit is necessary (in Q , , Q , , Q , )We multiply γ by | D | e a (or | D | / ) and then by ζ . We consider | D | e a D + iη (or | D | / D + iη ) asa whole operator and we then commute ζ with this operator and maybe some P ε and D + iω (if it was necessary to use (94)) in order to stick ζ with a vρ ′ γ ∗ |·| , a R ′ γ or a ∂ j ρ ′ γ ∗ |·| (if (94) was used). For instance in the case of Q , : Q + − , = Z + ∞−∞ | D | / P D + iη v ′ P − D + iη = i π ZZ R × R | D | / D + iη D + iω α · ∇ v ′ P − D + iω dηdω D + iη . (116)We multiply by ζ and under the integral sign: ζ | D | / D + iη D + iω α · ∇ v ′ = h ζ, | D | / D + iη i D + iω α · ∇ v ′ + | D | / D + iη h ζ, D + iω i α · ∇ v ′ + | D | / D + iη D + iω ζ α · ∇ v ′ . (117)We treat the first two terms in Section D.2.2. For the latter we go in Fourier space andup to a constant the kernel of its Fourier transform is: E / p E p + E q P ( p ) E ( p ) + E ( q ) (cid:0) F ( ζ α · ∇ v ′ ; p − q ) (cid:1) P − ( q ) . In particular its Hilbert-Schmidt norm is O ( p log(Λ) k ζ ∇ v ′ ρ γ k L ) .Doing the same for the other Q k,ℓ , we get terms with commutators treated in D.2.2and other terms with ζv ′ ρ γ , ζ α · ∇ v ′ and ζR γ ′ = R ζγ ′ . In particular taking the k·k S under the integral sign, we get the following estimates on those terms. O (cid:16) c − / k ζ ∇ v ′ k L + α k ζγ ′ k Ex + α ( k ζ ∇ v ′ k L + k ζv ′ k L + k ζγ ′ k Ex ) (cid:17) . (118) Remark . The term k ζγ ′ k Ex is due to Ineq. (51) (l.h.s). Moreover we can dealwith γ and N in γ ′ differently. Indeed as R N ∈ S , k ζγ ′ k Ex can be replaced by K ( k ζγ k Ex + k ζR N k S ) . Remark . The term T [ ζ, v ′ ] := ζ α · ∇ v ′ appears in P − ε v ′ P ε , that equals up to amultiplicative constant to Z + ∞ ω = −∞ dωD + iω T [ ζ, v ′ ] P ε D + iω . Up to a constant its Fourier transform is P − ε ( p ) b T ( p − q ) P ε ( q ) E ( p ) + E ( q ) , and we deal with this term as d P − ε ( p ) b v ′ ( p − q ) c P ε ( q ) in [7, 20, 22]. .2.2 Commutating ζ We recall here that [ ζ, P ε ] is treated in (94), Appendix A.In the same spirit of Lemma 10, we have the following Lemma. Lemma 11.
Let η ∈ R and ζ smooth with k∇ ζ k L ∞ , k ∂ j ∂ k ζ k L ∞ < + ∞ , k, j ∈ { , , } . Then there holds: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)h ζ, | D | / D + iη i | D + iη | / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) B > k∇ ζ k L ∞ + X ≤ j,k ≤ k ∂ j ∂ k ζ k L ∞ . Remark . We can do the same with | D | a [Λ] or | D | / instead of | D | / by usingthe following formula [17, p. 87]: | D | a = sin( aπ ) π Z + ∞ s =0 dss − a | D || D | + s , a = a [Λ] , / . Here we show the proof for | D | / because it enables us to localise the kinetic energy.But we can replace every | D | a [Λ] by | D | / and vice-versa.There is also: Lemma 12.
There exists
K > such that for any η ∈ R and any smooth function ζ with k∇ ζ k L ∞ < + ∞ : (cid:12)(cid:12)(cid:12)h ζ, D + iω i ( x − y ) (cid:12)(cid:12)(cid:12) ≤ K k∇ ζ k L ∞ e − E ( η ) / x − y ) | x − y | . (119) Remark . We recall that up to some constant a − ∆ ( x − y ) = p π e − a | x − y | | x − y | [16].– The interesting fact here is that by taking the commutator of ζ and some functionof − i ∇ we gain some exponent for η or ω . Thus by using KSS inequalities under theintegral sign we get the following estimates for the term with commutators: O (cid:16)(cid:0) k∇ ζ k L ∞ + X ≤ j,k ≤ k ∂ j ∂ k ζ k L ∞ (cid:1)(cid:0) α ( k ρ ′ γ k C + k|∇| / γ k S + k∇ N k S ) (cid:1)(cid:17) (120) Proof of Lemma 11:
We decompose ζ = ζ ++ + ζ + − + ζ − + + ζ −− . We write foreach term ζ εε ′ , ε, ε ′ ∈ { + , −} : h ζ εε ′ , | D | / D + iη i = [ ζ εε ′ , | D | / ] 1 D + iη + | D | / h ζ εε ′ , D + iη i . It follows that: | D | / h ζ εε ′ , D + iη i = | D | / P ε D + iη [ D , ζ ] P ε ′ D + iη . (121) The term | D | / h ζ εε ′ , D + iη i By simple computation we have: [ D , ζ ] = (cid:16) − ∆Λ (cid:17) ( − i α · ∇ ζ ) + ( − ∆ ζ )Λ D + 2 ∇ ζ · ∇ D Λ = ( − i α · ∇ ζ ) − X j =1 (cid:16) ∂ j Λ ( − i α · ∇ ∂ j ζ ) − ∂ j ζ ) D Λ (cid:17) +( − ∆ ζ ) D Λ − X j =1 ∂ j Λ (cid:16) ( − i α · ∇ ζ ) ∂ j Λ − ( ∂ j ζ ) D Λ (cid:17) . (122) hen there holds: k | D | Λ | D | / k B > . (123)Thus substituting in (121), on the right of derivatives of ζ , there is still an operator | D + iη | / available for some KSS inequality. The k·k B − norm of the operator on theirleft is O ( E − / η ) . The k·k B − norm of derivatives of ζ are O ( k∇ ζ k L ∞ + k ∆ ζ k L ∞ ) . The term [ ζ εε ′ , | D | / ] D + iη By symmetry it suffices to study ζ ++ and ζ + − .First: [ ζ ++ , | D | / ] D + iη = π Z + ∞ √ sds P D + s [ D , ζ ] P D + s D + iη . Once again, if we replace [ D , ζ ] by its expression in (122), we see that taking | D + iη | − / from D + iη , there remains | D + iη | / D + iη for some KSS inequality.This enables us to get a finite integral over the s variable: Z + ∞ √ sds (1 + s ) / s ) / < + ∞ . At last: [ ζ + − , | D | / ] 1 D + iη = − π Z + ∞ √ sds P | D | + s ( ζ D + D ζ ) P − | D | + s D + iη = − π Z + ∞ √ sds P | D | + s (2 ζ D + [ D , ζ ]) P − | D | + s D + iη . The term with [ D , ζ ] is dealt with as before. There remains: Z + ∞ √ sds | D | + s ζ + − D | D | + s D + iη . (124)We write ( cf (94)): ζ + − = P [ ζ, P − ] = P π Z + ∞−∞ dω D + iω [ D , ζ ] 1 D + iω , (125)and substitute ζ + − by this expression in (124). We must compensate | D | Λ on the leftside of ζ and | D | D Λ on its right side: we use | D + iω | / on the left side and {| D + iω | / | D + iη | / ( | D | + s ) / } − on the right side: there remains | D + iη | / for someKSS inequality and: Z + ∞ s =0 Z + ∞ ω = −∞ √ sdsdω (1 + s ) / E ( ω ) / < + ∞ . Proof of lemma 12:
This is straightforward because everything is computable: D + iη = D − iηE ( η ) − ∆ . However E ( η ) − ∆ ( x − y ) = e − E ( η ) | x − y | π | x − y | so it is clear that: (cid:12)(cid:12)(cid:12) D + iη ( x − y ) (cid:12)(cid:12)(cid:12) > e − E ( η ) | x − y | / | x − y | . In Direct space we use | ζ ( x ) − ζ ( y ) | ≤ k∇ ζ k L ∞ | x − y | and (cid:12)(cid:12)(cid:12)h ζ, D + iω i ( x − y ) (cid:12)(cid:12)(cid:12) > k∇ ζ k L ∞ e − E ( η ) / x − y ) | x − y | .2.3 Localisation of ∇ v ρ ′ γ and R N We recall that η ( λ ) c R g is the following function: η ( λ ) c R g ( x ) := (cid:8) − ξ cλ R g ( x − cz ) − ξ cλ R g ( x − cz ) (cid:9) − / , λ < λ < − . We will take λ ≤ λ ≤ − ( λ ( L, R g ) is defined in (82)). More generally except for k η ( λ ) c R g ∂v k L , k η ( λ ) c R g v k L , the estimates are true with ζ instead of η ( λ ) c R g in the case where ζ is ζ ( x ) = ζ ( x/A ) with ≤ ζ ≤ fixed . This part gives estimates with respect to ζ and A . Notation . We write θ ( x ) := p − ξ ( x ) , it is clear that k∇ η ( λ ) c R g k L ∞ ≤ k∇ θ k L ∞ cλ R g and so on . Proposition 9.
Let γ + N be a minimizer for E (2) (or E (1) ), ρ ∈ L ∩ L ( e.g. ρ = ρ γ , ρ N ) and λ ≤ λ < − . With the previous notations, there holds: k η ( λ ) c R g R [ N j ] k S > k∇ ψ j k L Z x ( η ( λ ) c R g ) ( x ) | ψ j ( x ) | dx > (cid:8) ( λ R g ) c (cid:9) − , k η ( λ ) c R g γ k Ex > k∇ θ k L ∞ ( cλ R g ) − k| D | / γ k S + k η ( λ ) c R g | D | / γ k S , k η ( λ ) c R g v ρ k L > k ( ∇ η ( λ ) c R g ) v ρ k L + k η ( λ ) c R g ∇ v ρ k L , k ( ∇ η ( λ ) c R g ) v ρ k L > k ρ k L k∇|∇| θ k L ( cλ R g ) − / , k η ( λ ) c R g ∂ j v ρ k L > k η ( λ ) c R g ρ η ( λ/ c R g k C + k ρ k L (cid:16) k∇ θ k / L ∞ ( cλ R g ) / + k∇ θ k L ∞ ( cλ R g ) / (cid:17) + k ρ k / L k ρ k / L k∇ θ k / L ∞ ( cλ R g ) / + k ρ k L (cid:16) k∇ θ k L ∞ ( cλ R g ) / (cid:17) . (126) Moreover if we write γ = αQ , + αQ , + α e Q , ρ N = n we also have: ( k η ( λ ) c R g ρ γ k C > αcλ R g k∇ θ k L ∞ ( k n k C + k αρ , + α e ρ k L / )+ L k η ( λ ) c R g n k C + k η ( λ ) c R g ( αρ , + α e ρ ) k C . (127)We recall that k ρ k L / > k ρ k / L k ρ k / L . Proof:
We will write v ρ = v for convenience. The term k η ( λ ) c R g R N k S k η ( λ ) c R g N j k S = ZZ ( η ( λ ) c R g ) ( x ) | ψ j ( x ) | | ψ j ( y ) | | x − y | dxdy = Z x dx ( η ( λ ) c R g ) ( x ) | ψ j ( x ) | Z y | ψ j ( y ) | | x − y | dy ≤ k∇ ψ j k L Z x ( η ( λ ) c R g ) ( x ) | ψ j ( x ) | dx > λ R g ) c where we have used Lemma 7. The term k η ( λ ) c R g γ k Ex k η ( λ ) c R g γ k Ex ≤ p π k| D | / η ( λ ) c R g γ k S ≤ p π (cid:0) k [ | D | / , η ( λ ) c R g ] γ k S + k η ( λ ) c R g | D | / γ k S (cid:1) > k∇ η ( λ ) c R g k L ∞ k| D | / γ k S + k η ( λ ) c R g | D | / γ k S , and we can treat k η ( λ ) c R g | D | / γ k S as k η ( λ ) c R g | D | a [Λ] γ k S . he term k η ( λ ) c R g v k L We use the Sobolev inequality: k η ( λ ) c R g v k L > k ( ∇ η ( λ ) c R g ) v k L + k η ( λ ) c R g ∇ v k L . We get a term k η ( λ ) c R g ∇ v k L we will treat later.– For the term k ( ∇ η ( λ ) c R g ) v k L , we use the fact that ρ ∗ |·| is L w with weak norm oforder k ρ k L [23] and we use rearrangement inequalities [16]: Z | fg | ≤ Z | f | ∗ | g | ∗ and k∇| f | ∗ k L ≤ k∇| f |k L . k ( ∇ η ( λ ) c R g ) v k L = Z |∇ η ( λ ) c R g | | v | ≤ Z ( |∇ η ( λ ) c R g | ) ∗ ( | v | ) ∗ > Z ( |∇ η ( λ ) c R g | ) ∗ ( x ) k ρ k L | x | dx > k ρ k L k∇ q ( |∇ η ( λ ) c R g | ) ∗ k L = k ρ k L k∇ ( q |∇ η ( λ ) c R g | ) ∗ k L > k ρ k L k∇|∇ η ( λ ) c R g |k L > k ρ k L k∇|∇| θ k L cλ R g . – For the term k η ( λ ) c R g ∂ j v k L , we write: η ( λ ) c R g ∂ j v ( x ) = Z ( y j − x j ) | x − y | ( η ( λ ) c R g ( x ) − η ( y ) c R g ) ρ ( y ) dy + ( η ( λ ) c R g ρ ) ∗ (cid:16) ∂ j | · | (cid:17) . (128)The last term will give k η ( λ ) c R g ρ k C . From this point, due to the particular form of η ( λ ) c R g there holds: η ( λ ) c R g = η ( λ ) c R g η ( λ/ c R g so k η ( λ ) c R g ρ k C = k η ( λ ) c R g ρ η ( λ/ c R g k C . (129)Let us treat the first term of (128). More generally we take ζ ( x ) = ζ ( x/A ) and weuse the properties of η ( λ ) c R g at the very end.Taking the squared norm we have: ZZ ρ ( x ) ρ ( x ) dxdy Z ( ζ ( t ) − ζ ( x ))( ζ ( t ) − ζ ( y ))( t j − x j )( t j − y j ) | t − x | | t − y | dt. We split at level | x − y | = √ A : first if | x − y | ≥ √ A , then ZZ | x − y |≥√ A | ρ ( x ) || ρ ( y ) || x − y | / Z k∇ ζ k / L ∞ dt | t | / | t − e | / ≤ k∇ ζ k / L ∞ √ A k ρ k L > L k∇ ζ k / L ∞ A . If | x − y | ≤ √ A then there holds | x − y |k∇ ζ k L ∞ ≤ k∇ ζ k L ∞ √ A , thus ζ ( x ) = ζ ( y ) + ζ ( x ) − ζ ( y ) and we substitute in the integral over t . We split R in three: | t − x | < | x − y | / , | t − y | < | x − y | / and the remainder domain.a. For the first ball B ( x, | x − y | /
2) = B x : Z B x | ζ ( x ) − ζ ( t ) || ζ ( y ) − ζ ( t ) || t − x | | t − y | dt ≤ k∇ ζ k L ∞ A / Z B x dt | t − x | | t − y | > k∇ ζ k L ∞ A / . b. The same holds for the ball B y . c. For the remainder domain C xy :c.1. we first deal with the term ( ζ ( y ) − ζ ( x )( ζ ( t ) − ζ ( y )) : Z t ∈ C xy dt | ( ζ ( x ) − ζ ( y ))( ζ ( t ) − ζ ( y )) || x − t | | y − t | ≤ ( k∇ ζ k L ∞ ) / A Z dt | x − t | | y − t | / > ( k∇ ζ k L ∞ ) / A | x − y | / nd: ZZ | ρ ( x ) || ρ ( y ) || x − y | / dxdy > k ρ k / L k ρ k / L > L c − / . We used above the Hardy-Littlewood-Sobolev inequality [16, Theorem 4.3].c.2. At last we must handle the term ( ζ ( t ) − ζ ( x )) : ZZ | x − y |≤√ A ρ ( x ) ρ ( y ) dxdy Z t ∈ C xy ( ζ ( t ) − ζ ( y )) ( t j − y j )( t j − x j ) | x − t | | y − t | dt. As t ∈ C xy we can replace | x − t | − by K | y − t | − . Z t ∈ C xy ( ζ ( t ) − ζ ( y )) | t − y | dt ≤ Z t ( ζ ( t ) − ζ ( y )) | t − y | dt. We use now the properties of the function η ( λ ) c R g . It is easy to see that no matter where y ∈ R is, this last integral is O (( cλ R g ) − K ( θ )) . Indeed let Ext be the domain definedby Ext = { y ∈ R : f ( y ) := dist ( y, { η ( λ ) c R g = 1 } ) > cλ R g } .c.2.1. If y ∈ Ext, then it is clear that the previous integral is an O (cid:16) ( cλ R g ) f ( y ) (cid:17) = O (cid:0) cλ R g (cid:1) . c.2.2. Else we split R at level | t − y | = 2 cλ R g : Z | t − y |≤ cλ R g dt ( η ( λ ) c R g ( t ) − η ( λ ) c R g ( y )) | t − y | > k∇ θ k L ∞ ( cλ R g ) ( cλ R g ) = O (cid:0) k∇ θ k L ∞ cλ R g (cid:1) . Z | t − y | > cλ R g dt ( η ( λ ) c R g ( t ) − η ( λ ) c R g ( y )) | t − y | ≤ Z | t − y | > cλ R g dt | t − y | = O (cid:16) cλ R g (cid:17) . Proof of (127)
To begin with we remark that by the Hardy-Littlewood-Sobolevinequality [16]: k ρ k C > k ρ k L / . Then we use formula (56) of ρ γ . We write η ( λ ) c R g ( x ) ˇ F Λ ∗ ρ ( x ) = Z y ( η ( λ ) c R g ( x ) − η ( λ ) c R g ( y )) ˇ F Λ ( x − y ) ρ ( y ) dy + ˇ F Λ ∗ ( η ( λ ) c R g ρ )( x ) . So it suffices to show k|·| ˇ F Λ k L > α to end the proof: this is precisely (54)-(55), appliedwith ℓ = 1 to ˇ F Λ (true if α is less than some K ( ℓ = 1) ). D.2.4 Proof of Lemma 9
We write ξ instead of ξ ( 13 ) j and Q instead of γ ′ for convenience.First remark: for any ε, ε ′ ∈ { + , −} : P ε ξP ε QP ε ′ ξP ε ′ = [ P ε , ξ ] Q ε ε ′ [ ξ, P ε ′ ] + [ P ε , ξ ] Q ε ε ′ ξ + ξQ ε ε ′ [ ξ, P ε ′ ] + ξQ ε ε ′ ξ. (130)This gives the error term between ξQξ and ξ [ Q ] . We estimate their density as in SectionD.1, that is by duality.Second remark: ∂ j ξ ( λ ) = ( ∂ j ξ ) η ( λ/ c R g .As in this section, by using (94), it is clear that k [ P ε , ξ ] Q ε ε ′ [ ξ, P ε ′ ] k C > k| D | a [Λ] Q k S k ξ k L > k| D | a [Λ] Q k S ( c R g ) . We can drop terms involving the density of these operators. e write: ξ + − = i π Z + ∞−∞ D + iω ( η ( λ/ c R g ) ( α · ∇ ξ ) dωD + iω . We commute η ( λ/ c R g with ( D + iω ) − on the right and on the left. As shown beforethere holds: (cid:12)(cid:12)(cid:12) ( η ( λ/ c R g ( x ) − η ( λ/ c R g ( y )) 1 D + iω ( x − y ) (cid:12)(cid:12)(cid:12) > e − E ( ω ) | x − y | / | x − y | k∇ η ( λ/ c R g k L ∞ . So taking KSS inequalities under the integral sign we obtain for instance: Tr (cid:0) P − ξQξ + − V P − (cid:17) = Tr (cid:16) P − ξQ η ( λ/ c R g ξ + − η ( λ/ c R g V P − (cid:17) + O (cid:16) k V k L k Q | D | / k S k∇ ξ k L k∇ η ( λ/ c R g k L ∞ Z R dωE ( ω/ / (cid:17) + O (cid:16) k∇ η ( λ/ c R g k L ∞ k V k L k∇ ξ k L k Q k S Z R dωE ( ω/ (cid:17) . There remains the first trace. First of all, for any V Schwartz, we can show as inSection D.1 that the operator is trace-class with norm controlled by p log(Λ) k∇ ( ξ η ( λ/ c R g V ) k L k| D | / η ( λ/ c R g Q + − k S + k∇ ξ k L ∞ k∇ ( η ( λ/ c R g V ) k L k QP η ( λ/ c R g k S . We have a priori k∇ ( ξ η ( λ/ c R g V ) k L > k η ( λ/ c R g ∇ V k L + k∇ ( ξ η ( λ/ c R g ) k L k V k L . In particular: k [ P ε , ξ ] Q ε ε ′ ξ k C > p log(Λ) k| D | / η ( λ/ c R g Q ε ′ ε k S > L p cλ R g . We use now the fact that we want the trace for a particular V , namely ρ [ ξQξ ] ∗ |·| .So as in Proposition 9, the function ( ξ ρ ′ γ ) ∗ |·| is in L w and k ( ∇ η ( λ/ c R g ) (cid:2) ( ξ ρ ′ γ ) ∗ |·| (cid:3) k L > √ p λc R g k ρ ′ γ k L > √ p λc R g . Then we write ( ξ ρ ′ γ ) ∗ |·| = ρ ′ γ ∗ |·| − (( η ( λ ) c R g ) ρ ′ γ ) ∗ |·| and k η ( λ/ c R g ∇ (cid:2) ( ξ ρ ′ γ ) ∗ |·| (cid:3) k L > k∇ (( η ( λ ) c R g ) ρ ′ γ ) ∗ |·| k L + k η ( λ/ c R g ∇ ( ρ ′ γ ) ∗ |·| k L > k ( η ( λ ) c R g ) ρ ′ γ k C + X j =1 k η ( λ/ c R g ∂ j v ′ ρ γ k L , and those terms are dealt with Propositions 9 and 8.Putting everything together, we get an error term of order: p log(Λ) × c p R g × p c R g = O (cid:16) Lc R g (cid:17) . References [1] V. Bach, J.-M. Barbaroux, B. Helffer, and H. Siedentop. On the stability of therelativistic electron-positron field.
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