The positronium and the dipositronium in a Hartree-Fock approximation of quantum electrodynamics
aa r X i v : . [ m a t h - ph ] S e p The positronium and the dipositronium in aHartee-Fock approximation of quantumelectrodynamics
Sok JérémyCeremade, UMR 7534, Université Paris-Dauphine,Place du Maréchal de Lattre de Tassigny,75775 Paris Cedex 16, France.September 4, 2018
Abstract
The Bogoliubov-Dirac-Fock (BDF) model is a no-photon approximation of quan-tum electrodynamics. It allows to study relativistic electrons in interaction with theDirac sea. A state is fully characterized by its one-body density matrix, an infiniterank nonnegative projector.We prove the existence of the para-positronium, the bound state of an electronand a positron with antiparallel spins, in the BDF model represented by a criticalpoint of the energy functional in the absence of external field.We also prove the existence of the dipositronium, a molecule made of two elec-trons and two positrons that also appears as a critical point. More generally, forany half integer j ∈ + Z + , we prove the existence of a critical point of the energyfunctional made of j + 1 electrons and j + 1 positrons. Contents D . . . . . . . . . . . . . . . . . . . . 9 E j , ± . . . . . . . . . . . . . 17 .3 Strategy of the proof: the para-positronium . . . . . . . . . . . . . . 203.4 Existence of a minimizer for E j , ± . . . . . . . . . . . . . . . . . . . 243.5 Lower bound of E j , ± . . . . . . . . . . . . . . . . . . . . . . . . . . 283.6 Proof of Lemmas 3 and 4 . . . . . . . . . . . . . . . . . . . . . . . . 303.6.1 Proof of Lemma 3 . . . . . . . . . . . . . . . . . . . . . . . . 303.6.2 Proof of Lemma 4 . . . . . . . . . . . . . . . . . . . . . . . . 32 Relativistic quantum mechanics is based on the
Dirac operator D , which is theHamiltonian of the free electron. Its expression is [Tha92]: D := m e c β − i ~ c X j =1 α j ∂ x j (1)where m e is the (bare) mass of the electron, c the speed of light and ~ the reducedPlanck constant and β and the α j ’s are × matrices defined as follows: β := (cid:18) Id C − Id C (cid:19) , α j := (cid:18) σ j σ j (cid:19) , j ∈ { , , } σ := (cid:18) (cid:19) , σ := (cid:18) − ii (cid:19) , σ (cid:18) − (cid:19) . The operator D acts on the Hilbert space H : H := L (cid:0) R , C (cid:1) ; (2)it is self-adjoint on H with domain H ( R , C ) . Its spectrum is σ ( D ) = ( −∞ , m e c ] ∪ [ m e c , + ∞ ) , which leads to the existence of states with arbitrary small energy.Dirac postulated that all the negative energy states are already occupied by"virtual electrons", with one electron in each state: by Pauli’s principle real electronscan only have a positive energy.In this interpretation the Dirac sea, composed by those negatively charged virtualelectrons, constitutes a polarizable medium that reacts to the presence of an externalfield. This phenomenon is called the vacuum polarization .After the transition of an electron of the Dirac sea from a negative energy state toa positive, there is a real electron with positive energy plus the absence of an electronin the Dirac sea. This hole can be interpreted as the addition of a particle with samemass, but opposite charge: the so-called positron. The existence of this particlewas predicted by Dirac in 1931. Although firstly observed in 1929 independentlyby Skobeltsyn and Chung-Yao Chao, it was recognized in an experiment lead byAnderson in 1932. The positronium is the bound state of an electron and a positron. This system wasindependently predicted by Anderson and Mohorovi ˇc ić in 1932 and 1934 and wasexperimentally observed for the first time in 1951 by Martin Deutsch.It is unstable: depending on the relative spin states of the positron and elec-tron, its average lifetime in vacuum is 125 ps (para-positronium) or 142 ns (ortho-positronium) [Kar04]. ere we are interested in positronium states in the Bogoliubov-Dirac-Fock (BDF)model.In a previous paper we have proved the existence of a state that can be interpretedas the ortho-positronium. Our aim in this paper is to find another one that can beinterpreted as the para-positronium and to find another state that can be interpretedas the dipositronium, the bound state of two electrons and two positrons. To findthese states, we use symmetric properties of the Dirac operator. – Following Dirac’s ideas, the free vacuum is described by the negative part of thespectrum σ ( D ) : P − = χ ( −∞ , ( D ) . A correspondence between negative energy states and positron states is given by the charge conjugation C [Tha92]. This is an antiunitary operator that maps Ran P − onto Ran(1 − P − ) . In our convention [Tha92] it is defined by the formula: ∀ ψ ∈ L ( R ) , C ψ ( x ) = iβα ψ ( x ) , (3)where ψ denotes the usual complex conjugation. More precisely: C · ψ ψ ψ ψ = ψ − ψ − ψ ψ . (4)In our convention it is also an involution : C = id. An important property is thefollowing: ∀ ψ ∈ L , ∀ x ∈ R , | C ψ ( x ) | = | ψ ( x ) | . (5)The Dirac operator anti-commutes with D , or equivalently there holds − C D C − = − C D C = D . – There exists another simple symmetry. We define I s := (cid:18) − Id C Id C (cid:19) ∈ C × . (6)This operator is − i the time reversal operator L T [Tha92, 2.5.7] in H , interpretedas a unitary reprsentation of the Poincaré group.It acts on the spinor by simple multiplication, furthermore we have I = − Id and I s : Ran P − ≃ −→ Ran (1 − P − ) ψ ( x ) I s ψ ( x ) Similarly we have − I s D I − = I s D I s = D . – To end this part we recall that SU (2) acts on H [Tha92]. Writing α := ( α j ) j =1 and p := − i ~ ∇ , L := x ∧ p , S := − i α ∧ α = 12 (cid:18) σ σ (cid:19) , (7)we define J := L + S . (8)The operator L is the angular momentum operator and J is the total angular mo-mentum. From a geometrical point of view, − i J gives rise to a unitary representationof SU (2) in H by the following formula: ( e − iθ J · ω ψ ( x ) = e − i S · ω ψ (cid:0) R − ω,θ (cid:1) , ∀ θ ∈ [0 , π ) , ∀ ψ ∈ H , ∀ ω ∈ S , here R ω,θ ∈ SO(3) is the rotation with axis ω and angle θ .As each S j is diagonal by block, it is clear that this group representation canbe decomposed in two representations, the first acting on the upper spinors φ ∈ L ( R , C ) and the second on the lower spinors χ ∈ L ( R , C ) : ψ =: (cid:18) φχ (cid:19) . In [Tha92, pp. 122-129] it is proved that D commutes with the action of SU (2) , thusthe representation can also be decomposed with respect to Ran P − and Ran (1 − P − ) .From an algebraic point of view, there exists a group morphism Φ SU : SU (2) → U ( H Λ ) where U ( H ) is the set of unitary operator of H . We write S := Φ SU (cid:0) SU (2) (cid:1) . (9)The irreducible representations of Φ SU are known and are expressed in terms ofeigenspaces of J , S . The proofs of the following can be found in [Tha92, pp. 122-129].The operators J , J , K all commute with each other, and J , K with D . More-over K commutes with the action Φ SU .We have H Λ ⊂ L ( R ) ≃ L ((0 , ∞ ) , dr ) ⊗ L ( S ) , and J , L only act on the part L ( S ) .Restricted to L ( S ) , we have σ ( J ) = (cid:8) j ( j + 1) , j ∈
12 + Z + (cid:9) , (10)and for each eigenvalue j ( j + 1) ∈ σ J , the eigenspace Ker (cid:0) J − j ( j + 1) (cid:1) maybe decomposed with respect to the eigenspaces of J and S . The correspondingeigenvalues are1. m j = − j, − j + 1 , · · · , j − , j for J ,2. κ j = ± (cid:0) j + (cid:1) for S .The eigenspace k m j ,κ j of a triplet ( j, m j , κ j ) has dimension and is spanned by Φ + m j ,κ j ⊥ Φ − m j ,κ j , which have respectively a zero lower spinor and zero upper spinor. Lemma 1.
For each irreducible subrepresentation Φ ′ SU of Φ SU , there exists ( j, ε, z = [ z : z ] , a ( r ) , a ( r )) ∈ (cid:0)
12 + Z + (cid:1) × { + , −} × C P × (cid:0) S L ((0 , ∞ ) , dr ) (cid:1) , such that the representation Φ ′ SU is spanned by ψ ( x ) defined as follows: ∀ x = rω ∈ R , ψ ( x ) := z ra ( r )Φ + j,ε ( j + 12 ) ( ω ) + z ra ( r )Φ − j,ε ( j + 12 ) . Remark . We recall that for any Hilbert space h and any subspace V ⊂ h , we define S V as the unitary vector in V : S V := { x ∈ V, k x k h = 1 } . We will use this notation throughout this paper.We prove this Lemma in Section 4.
Remark . An irreducible subrepresentation of Φ SU is characterized by the twonumbers ( j, κ j ) . Indeed, the irreducible representations of SU (2) are known: theycan be described by homogeneous polynomials, and for any n ∈ Z + , there is but oneirreducible representation of dimension n + 1 , up to isomorphism.In the case of Φ SU , the two cases κ j = ± ( j + ) are different but isomorphic . Notation . An irreducible subrepresentation of Φ SU spanned by an eigenvector of J and K with respective eigenvalues j ( j + 1) and ε ( j + ) will be refered as beeingof type ( j, ε ) (where ε ∈ { + , −} ). Notation . Throughout this paper we write Proj E to mean the orthonormal pro-jection onto the vector space E . .4 The BDF model This model is a no-photon approximation of quantum electrodynamics (QED) whichwas introduced by Chaix and Iracane in 1989 [CI89], and studied in many papers[BBHS98, HLS05a, HLS05b, HLS07, HLS09, GLS09, Sok14a].It allows to take into account the Dirac vacuum together an electronic system inthe presence of an external field.This is a Hartree-Fock type approximation in which a state of the system "vac-uum plus real electrons" is given by an infinite Slater determinant ψ ∧ ψ ∧ · · · .Such a state is represented by the projector onto the space spanned by the ψ j ’s:its so-called one-body density matrix. For instance P − represents the free Diracvacuum.We do not recall the derivation of the BDF model from QED: we refer the readerto [CI89, HLS05a, HLS07] for full details. Remark . To simplify the notations, we choose relativistic units in which, the massof the electron m e , the speed of light c and ~ are set to .Let us say that there is an external density ν , e.g. that of some nucleus. Wewrite α > the so-called fine structure constant (physically e / (4 πε ~ c ) , where e isthe elementary charge and ε the permittivity of free space).The relative energy of a Hartree-Fock state represented by its 1pdm P withrespect to a state of reference ( P − in [CI89, HLS05a]) turns out to be a function of Q = P − P − , the so-called reduced one-body density matrix.A projector P is the one-body density matrix of a Hartree-Fock state in F elec iff P − P − is Hilbert-Schmidt, that is compact such that its singular values form asequence in ℓ [HLS05a, Appendix].An ultraviolet cut-off Λ > is needed: we only consider electronic states in H Λ := (cid:8) f ∈ H , supp b f ⊂ B (0 , Λ) (cid:9) , where b f is the Fourier transform of f .This procedure gives the BDF energy introduced in [CI89] and studied in [HLS05a,HLS05b]. Notation . Our convention for the Fourier transform F is the following ∀ f ∈ L ( R ) , b f ( p ) := 1(2 π ) / Z f ( x ) e − ixp dx. Let us notice that H Λ is invariant under D and so under P − .We write Π Λ for the orthogonal projection onto H Λ : Π Λ is the Fourier multiplier F − χ B (0 , Λ) F .By means of a thermodynamical limit, Hainzl et al. showed that the formalminimizer and hence the reference state should not be given by Π Λ P − but by anotherprojector P − in H Λ that satisfies the self-consistent equation in H Λ [HLS07]: P − − = − sign (cid:0) D (cid:1) , D = D Π Λ − α P − − )( x − y ) | x − y | (11)We have P − = χ ( −∞ , ( D ) . This operator D was previously introduced by Lieb et al. in [LS00]. In H , the operator D coincides with a bounded, matrix-valuedFourier multiplier whose kernel is H ⊥ Λ ⊂ H . Notation . Throughout this paper we write m = inf σ (cid:0) |D | (cid:1) ≥ , (12)and P := Π Λ − P − = χ (0 , + ∞ ) ( D ) . (13) he resulting BDF energy E ν BDF is defined on Hartree-Fock states represented bytheir one-body density matrix P : N := (cid:8) P ∈ B ( H Λ ) , P ∗ = P = P, P − P − ∈ S ( H Λ ) (cid:9) . We recall that B ( H Λ ) is the set of bounded operators and that for p ≥ , S p ( H Λ ) is the set of compact operators A such that Tr (cid:0) | A | p (cid:1) < + ∞ [RS75, Sim79]. Inparticular S ∞ ( H Λ ) is the set Comp ( H Λ ) of compact operators.This energy depends on three parameters: the fine structure constant α > ,the cut-off Λ > and the external density ν . We assume that ν has finite Coulombenergy , that is b ν measurable and D ( ν, ν ) := 4 π Z R | b ν ( k ) | | k | dk < + ∞ . (14)The above integral coincides with s R × R ν ( x ) ∗ ν ( y ) | x − y | dxdy whenever this last one is well-defined. Remark . The same symmetries holds for P − and P : the charge conjugation C and the operator I s maps Ran P − onto Ran P . Moreover thanks to [Tha92, pp.122-129] we can easily check that D also commutes with the action of SU (2) andwith the operators J and K . For P ∈ N , we have the identity ( P − P − ) = P ( P − P − ) P − P − ( P − P − ) P − ∈ S . (15)The charge of a state P is given by the P − -trace of P − P − , defined by the formula: Tr P − (cid:0) P − P − (cid:1) := Tr (cid:0) P − ( P − P − ) P − + P ( P − P − ) P (cid:1) , (16) = Dim
Ran( P ) ∩ Ran( P ) − Dim
Ran( P − ) ∩ Ran(1 − P ) . (17)A minimizer over states with charge N ∈ N is interpreted as a ground state of asystem with N electrons, in the presence of an external density ν The existence problem was studied in several papers [HLS09, Sok14a, Sok13]: by[HLS09, Theorem 1], it is sufficient to check binding inequalities.The following results hold under technical assumptions on α and Λ (different foreach result).In [HLS09], Hainzl et al. proved existence of minimizers for the system of N electrons with ν ≥ , provided that N − < R ν .In [Sok14a], we proved the existence of a ground state for N = 1 and ν = 0 :an electron can bind alone in the vacuum. This surprising result holds due to thevacuum polarization.In [Sok13], we studied the charge screening effect: due to vacuum polarization,the observed charge of a minimizer P = P − is different from its real charge Tr P − ( P −P − ) . We also proved it is possible to keep track of this effect in the non-relativisticlimit α → : the resulting limit is an altered Hartree-Fock energy.Here we are looking for states with an equal number of electrons and positrons,that is we study E BDF on M := n P ∈ N , Tr P − (cid:0) P − P − (cid:1) = 0 o . (18)From a geometrical point of view M is a Hilbert manifold and E BDF is a differentiablemap on M (Propositions 3 and 4).We thus seek a critical point on M , that is some P ∈ M , P = P − such that ∇E BDF ( P ) = 0 . n [Sok14b], we have found the ortho-positronium by studying the BDF energyrestricted to states with the C -symmetry: P ∈ M s.t. P + C P C = Id H Λ . (19)We write M C the set of such states. We will seek the para-positronium in the set M I of states having the I s -symmetry. Definition 1. M I := { P ∈ M s.t. P + I s P I − = P − I s P I s = Id H Λ } . (20)Equivalently P ∈ M I if and only if Q := P − P − is Hilbert-Schmidt and satisfies − I s Q I − = I s Q I s = Q. We seek a projector P "close" to a state P that can be written as: P = P − + | I s ψ − ih I s ψ − | − | ψ − ih ψ − | , P ψ − = 0 . (21)To deal with the dipositronium, we impose an additional symmetry: we define W ⊂ M C as follows. Definition 2. W := (cid:8) P ∈ M C , ∀ U ∈ S , UP U − = P (cid:9) . (22)Equivalently P ∈ W ⇐⇒ Q := P − P − satisfies − C Q C = Q and UQU − = Q, ∀ U ∈ S . Those sets M C , M I , W have fine properties: they are all submanifolds of M ,invariant under the gradient flow of E BDF (Proposition 5).However while M C has two connected components, M I has only one connectedcomponent and W has countable connected components. So we may find criticalpoints by searching a minimizer of the BDF energy over the different connectedcomponents of W . For the para-positronium, a critical point is found by an argumentof mountain pass. Proposition 1.
There is a one-to-one correspondence between the connected com-ponents of W and the set Z [ X ] of polynomials with coefficients in the ring Z × Z .Let P be in W . The vector space E := Ran P ∩ Ran P has finite dimensionand is invariant under Φ SU . We decompose it into irreducible representations.The projector is associated to P ℓ ℓ =1 t ℓ X ℓ with t ℓ = ( t ℓ, ; t ℓ, − ) if and only if forany j ∈ + Z + :1. The number b j − , of irreducible representations of E of type ( j, +) satisfies b j − , ≡ t j − , [2] .2. The number b j − , − of irreducible representations of E of type ( j, − ) satisfies b j − , − ≡ t j − , − [2] .Notation . The symbols Y and Y denotes respectively C and C or I and I s .Furthermore the different connected components of W are written W p ( X ) with p ( X ) ∈ Z [ X ] .To state our main Theorems, we need to introduce the mean-field operator. Notation . An operator Q ∈ V is Hilbert-Schmidt and wewrite Q ( x, y ) its integral kernel. Its density ρ Q is defined by the formula ∀ x ∈ R , ρ Q ( x ) := Tr C (cid:0) Q ( x, x ) (cid:1) , (23)we prove in the next Section that it is well-defined. The mean-field operator D (Λ) Q associated to Q in the vacuum is : D (Λ) Q := Π Λ (cid:16) D + α (cid:0) ρ Q ∗ | · | − Q ( x, y ) | x − y | (cid:1)(cid:17) . (24) heorem 1. There exist α , L , Λ > such that if α ≤ α ; α log(Λ) := L ≤ L and Λ − ≤ Λ − , then there exists a critical point P = Q + P − of E BDF in M I that satisfies thefollowing equation. ∃ < µ < m, ∃ ψ a ∈ Ker (cid:0) D (Λ) Q − µ (cid:1) , P = χ ( −∞ , (cid:0) D (Λ) Q (cid:1) + | ψ a ih ψ a | − | I s ψ a ih I s ψ a | . (25) As α tends to , the upper spinor of U λ ψ a := λ / ψ a ( λ ( · )) with λ := g ′ (0) αm tendsto a Pekar minimizer.– We recall that the Pekar energy is defined as follows ∀ ψ ∈ H , E PT ( ψ ) := k∇ ψ k L − D (cid:0) | ψ | , | ψ | (cid:1) . The infimum over S L ∩ H is written E PT (1) . Theorem 2.
There exist L , Λ > , and for any j ∈ + Z + , there exists α j suchthat if α ≤ α j ; α log(Λ) := L ≤ L and Λ − ≤ Λ − , then there exists a minimizer P t X ℓ = Q + P − of E BDF over the connected componentof W t X ℓ with t ∈ { (1 , , (0 , } .Moreover there exists < µ ℓ , t < and ψ ∈ Ker (cid:0) D (Λ) Q − µ ℓ , t (cid:1) such that P t X ℓ = χ ( −∞ , ( D (Λ) Q ) + Proj Φ SU ( ψ ) − Proj Φ SU (C ψ ) . Any upper spinor e ϕ of e ψ ∈ Φ SU ( ψ ) can be written as ∀ x = rω x ∈ R , e ϕ =: ra ( r ) j X m = − j c m ( e ϕ )Φ + m,ε ( j + 12 ) , c m ( e ϕ ) ∈ C . Furthermore, as α tends to , the function U λ a ( r ) = λ / a ( λr ) tends to a mini-mizer of the energy E t X ℓ over S L ( R + , r dr ) ∩ H ( R + , r dr ) : E t X ℓ (cid:0) f ( r ) (cid:1) := Tr (cid:0) − ∆ Proj Φ SU ( rf ( r )Φ + j ,ε ( t ) ) (cid:1) − k Proj Φ SU ( rf ( r )Φ + j ,ε ( t ) ) k Ex . (26) In particular, the dipositronium corresponds to the case ℓ = j − = 0 .Notation . The minimum is written E nr t X ℓ for the non-relativistic energy and E j ,ε ( t ) for the BDF energy over W t X j − / . Notation . For t X ℓ ∈ Z [ X ] as in Theorem 2, ε ( t ) ∈ { + , −} denotes + if t = (1 , or − if t = (0 , . Remark . We expect the existence of minimizers over any connected componentsof W (associated to p ( X ) ∈ Z [ X ] ), provided that α is smaller than some α p ( X ) . Remark . The non-relativistic energy can be computed: E t X ℓ (cid:0) f ( r ) (cid:1) := (2 j + 1) + ∞ Z h r | f ′ ( r ) | + ( j + ε )( j + 1 + ε ) | f ( r ) | i dr − x R r r | f ( r ) | | f ( r ) | w j ,ε ( t ) ( r , r ) ,w j ,ε ( t ) ( r , r ) := x ( S ) dn dn | r n − r n | (cid:16) X m ,m ((Φ + m ,ε ( j + 12 ) ) ∗ Φ + m ,ε ( j + 12 ) )( n ) (cid:17) × (cid:16) X m ,m ((Φ + m ,ε ( j + 12 ) ) ∗ Φ + m ,ε ( j + 12 ) )( n ) (cid:17) . (27) t corresponds to the energy E nr (cid:0) Γ (cid:1) := Tr (cid:0) − ∆Γ (cid:1) − k Γ k Ex , ≤ Γ ≤ , Γ ∈ S ( H ( R , C )) restricted to the subspace S ( j ,ε ( t )) := (cid:8) Γ , Γ ∗ = Γ = Γ , Ran (Φ SU ) (cid:12)(cid:12) Γ irreducible of type ( j , ε ( t )) (cid:9) . This subspace is invariant under the action of Φ SU and it is easy to see that it is asubmanifold of (cid:8) Γ , Γ ∗ = Γ = Γ , Tr Γ = 2 j + 1 (cid:9) .The subspace S ( j ,ε ( t )) is invariant under the flow of E nr .The energies can be estimated. Proposition 2.
In the same regime as in Theorem 1, the following holds. Thecritical point P of the BDF functional over M I satisfies E BDF ( P ) = 2 m + α mg ′ (0) E PT (1) + O ( α ) . (28) Furthermore the minimizer P ℓ over W t X ℓ satisfies: E BDF ( P ℓ ) = 2(2 j + 1) + α mg ′ (0) E nr t X ℓ + O ( α K ( j )) . (29) Remark . The Pekar model describes an electron trapped in its own hole in apolarizable medium. Thus it is not surprising to find it here. We recall that there isa unique minimizer of the Pekar energy up to translation and a phase in S (in C ).The asymptotic expansion (28) coincides with that of the ortho-positronium[Sok14b]. In fact, it can be proved that the first difference between the energiesoccurs at order α . Notation . Throughout this paper we write K to mean a constant independent of α, Λ . Its value may differ from one line to the other. When we write K ( a ) , we meana constant that depends solely on a . We also use the symbol > : ≤ a > b meansthere exists K > such that a ≤ Kb .We also recall the reader our use of the notation S V for any subspace V of someHilbert space that denotes the set of unitary vector in V . D D has the following form [HLS07]: D = g ( − i ∇ ) β − i α · ∇|∇| g ( − i ∇ ) (30)where g and g are smooth radial functions on B (0 , Λ) . Moreover we have: ∀ p ∈ B (0 , Λ) , ≤ g ( p ) , and | p | ≤ g ( p ) ≤ | p | g ( p ) . (31) Notation . For α log(Λ) sufficiently small, we have m = g (0) [LL97, Sok14a]. Remark . The smallness of α is needed to get estimates that hold close to thenon-relativistic limit.The smallness of α log(Λ) is needed to get estimates of D : in this case D canbe obtained by a fixed point scheme [HLS07, LL97], and we have [Sok14a, AppendixA]: g ′ (0) = 0 , and k g ′ k L ∞ , k g ′′ k L ∞ ≤ Kα k g ′ − k L ∞ ≤ Kα log(Λ) ≤ and k g ′′ k L ∞ > . (32) Description of the model
Notation . For any ε, ε ′ ∈ { + , −} and A ∈ B ( H Λ ) , we write A ε,ε ′ := P ε A P ε ′ . (33) Notation . For an operator Q ∈ S ( H Λ ) , we write R Q the operator given by theintegral kernel: R Q ( x, y ) := Q ( x, y ) | x − y | . Definition 3 (BDF energy) . Let α > , Λ > and ν ∈ S ′ ( R ) a generalized functionwith D ( ν, ν ) < + ∞ . For P ∈ N we write Q := P − P − and E BDF ( Q ) = Tr P − (cid:0) D Q (cid:1) − αD ( ρ Q , ν ) + α (cid:16) D ( ρ Q , ρ Q ) − k Q k Ex (cid:17) , ∀ x, y ∈ R , ρ Q ( x ) := Tr C (cid:0) Q ( x, x ) (cid:1) , k Q k Ex := x | Q ( x, y ) | | x − y | dxdy, (34)where Q ( x, y ) is the integral kernel of Q . Remark . The term Tr P − (cid:0) D Q (cid:1) is the kinetic energy, − αD ( ρ Q , ν ) is the interactionenergy with ν . The term α D ( ρ Q , ρ Q ) is the so-called diract term and − α k Q k Ex isthe exchange term .Let us see that formula (34) is well-defined whenever Q is P − -trace-class [HLS05a,HLS09]. S P − and the variational set K The set S P − of P − -trace class operator isthe following Banach space: S P − = (cid:8) Q ∈ S ( H Λ ) , Q ++ , Q −− ∈ S ( H Λ ) (cid:9) , (35)with the norm k Q k S P − := k Q + − k S + k Q − + k S + k Q ++ k S + k Q −− k S . (36)We have N ⊂ P − + S P − thanks to (15). The closed convex hull of N − P − under S P − is K := (cid:8) Q ∈ S P − ( H Λ ) , Q ∗ = Q, −P − ≤ Q ≤ P (cid:9) and we have [HLS05a, HLS05b] ∀ Q ∈ K , Q ≤ Q ++ − Q −− . The BDF energy for Q ∈ S P − We have P − ( D Q ) P − = −|D | Q −− ∈ S ( H Λ ) , because |D | ∈ B ( H Λ ) , this proves that the kinetic energy is defined.By the Kato-Seiler-Simon (KSS) inequality [Sim79], Q is locally trace-class: ∀ φ ∈ C ∞ ( R ) , φ Π Λ ∈ S so φQφ = φ Π Λ Qφ ∈ S ( L ( R )) . We recall this inequality states that for all ≤ p ≤ ∞ and d ∈ N , we have ∀ f, g ∈ L p ( R d ) , f ( x ) g ( − i ∇ ) ∈ S p ( H Λ ) and k f ( x ) g ( − i ∇ ) k S p ≤ (2 π ) − d/p k f k L p k g k L p . t follows that the density ρ Q of Q , defined in (34) is well-defined. By the KSSinequality, we can also prove that k ρ Q k C > K (Λ) k Q k S P − [GLS09, Proposition 2].By Kato’s inequality: | · | ≤ π |∇| , (37)the exchange term is well-defined.Moreover the following holds: if α < π , then the BDF energy is bounded frombelow on K [BBHS98, HLS05b, HLS09]. We have ∀ Q ∈ S ( H Λ ) , E BDF ( Q ) ≥ (cid:0) − α π (cid:1) Tr (cid:0) |D || Q | (cid:1) . (38)Here we assume it is the case. This result will be often used throughout thispaper. Minimizers
For Q ∈ K , its charge is its P − -trace: q = Tr P − ( Q ) . We define theCharge sector sets: ∀ q ∈ R , K q := (cid:8) Q ∈ K , Tr( Q ) = q (cid:9) . A minimizer of E ν BDF over K is interpreted as the polarized vacuum in the presenceof ν while a minimizer over charge sector N ∈ N is interpreted as the ground stateof N electrons in the presence of ν , by Lieb’s principle [HLS09, Proposition 3], sucha minimizer is in N − P − .We define the energy functional E ν BDF : ∀ q ∈ R , E ν BDF ( q ) := inf (cid:8) E ν BDF ( Q ) , Q ∈ K q (cid:9) . (39)We also write: K Y := { Q ∈ K , Tr P − ( Q ) = 0 , − Y Q Y − = Q } . (40)Proposition 2 states that this set is sequentially weakly- ∗ closed in S P − ( H Λ ) . We consider V = (cid:8) P − P − , P ∗ = P = P ∈ B ( H Λ ) , Tr P − (cid:0) P − P − (cid:1) = 0 (cid:9) ⊂ S ( H Λ ) . and write: M := P − + V = (cid:8) P, P ∗ = P = P, Tr P − (cid:0) P − P − (cid:1) = 0 (cid:9) . We recall the following proposition, proved in [Sok14b].
Proposition 3.
The set M is a Hilbert manifold and for all P ∈ M , T P M = { [ A, P ] , A ∈ B ( H Λ ) , A ∗ = − A and P A (1 − P ) ∈ S ( H Λ ) } . (41) Writing m P := { A ∈ B ( H Λ ) , A ∗ = − A, P AP = (1 − P ) A (1 − P ) = 0 and P A (1 − P ) ∈ S ( H Λ ) } , (42) any P ∈ M can be written as P = e A P e − A where A ∈ m P . The BDF energy E ν BDF is a differentiable function in S P − ( H Λ ) with: ∀ Q, δQ ∈ S P − ( H Λ ) , d E ν BDF ( Q ) · δQ = Tr P − (cid:0) D Q,ν δQ (cid:1) .D Q,ν := D + α (cid:0) ( ρ Q − ν ) ∗ |·| − R Q (cid:1) . (43)We may rewrite (43) as follows: ∀ Q, δQ ∈ S P − ( H Λ ) , d E ν BDF ( Q ) · δQ = Tr P − (cid:0) Π Λ D Q,ν Π Λ δQ (cid:1) (44)We recall the mean-field operator D (Λ) Q is defined in Notation 24. roposition 4. Let ( P, v ) be in the tangent bundle T M and Q = P − P − . Thenwe have [[Π Λ D Q Π Λ , P ] , P ] ∈ T P M and: d E BDF ( P ) · v = Tr (cid:16)(cid:2)(cid:2) D (Λ) Q , P (cid:3) , P (cid:3) v (cid:17) . (45) In other words: ∀ P ∈ M , ∇E BDF ( P ) = (cid:2)(cid:2) Π Λ D Q Π Λ , P (cid:3) , P (cid:3) . (46) Remark . The operator [[Π Λ D Q Π Λ , P ] , P ] is the "projection" of Π Λ D Q Π Λ ontoT P M .In [Sok14b], we proved that M C is a submanifold of M . We recall that thenotations Y , Y are specified in Notation 5. Proposition 5.
The sets M I and W are submanifolds of M , which are invariant under the flow of E BDF . The following holds: for any P ∈ M Y , writing m Y P = { a ∈ m P , Y a Y − = a } , (47) we have T P M Y = { [ a, P ] , a ∈ m Y P } = { v ∈ T P M , − Y v Y − = v } . (48) Furthermore, for any P ∈ M Y we have ρ P −P − = 0 . For P ∈ W , the same holds with ( m W P := (cid:8) a ∈ m C P , ∀ U ∈ S , UaU − = a (cid:9) , T P W := (cid:8) [ a, P ] , a ∈ m W P (cid:9) . Remark
11 (Lagrangians) . The operator I s induced a symplectic structure on the real Hilbert space ( H Λ , Re h· , ·i H ) : ∀ f, g ∈ H Λ , ω I ( f, g ) := Re h f , I s g i . The manifold M I is constituted by Lagrangians of ω I that are in M .We end this section by stating technical results. The following Theorem is stated in [HLS09, Appendix] and proved in [Sok14b].
Theorem 3 (Form of trial states) . Let P , P be in N and Q = P − P . Thenthere exist M + , M − ∈ Z + such that there exist two orthonormal families ( a , . . . , a M + ) ∪ ( e i ) i ∈ N in Ran P , ( a − , . . . , a − M + ) ∪ ( e − i ) i ∈ N in Ran P − , and a nonincreasing sequence ( λ i ) i ∈ N ∈ ℓ satisfying the following properties:1. The a i ’s are eigenvectors for Q with eigenvalue (resp. − ) if i > (resp. i < ).2. For each i ∈ N the plane Π i := Span ( e i , e − i ) is spanned by two eigenvectors f i and f − i for Q with eigenvalues λ i and − λ i .3. The plane Π i is also spanned by two orthogonal vectors v i in Ran(1 − P ) and v − i in Ran( P ) . Moreover λ i = sin( θ i ) where θ i ∈ (0 , π ) is the angle betweenthe two lines C v i and C e i .4. There holds: Q = M + X i | a i ih a i | − M − X i | a − i ih a − i | + X j ∈ N λ j ( | f j ih f j | − | f − j ih f − j | ) . emark . We have Q ++ = M + X i | a i ih a i | + X j ∈ N sin( θ j ) | e j ih e j | ,Q −− = − M − X i | a − i ih a − i | − X j ∈ N sin( θ j ) | e − j ih e − j | . (49)Thanks to Theorem 3, it is possible to characterize states in M Y and W . Werestate a proposition of [Sok14b] and add the case of I s . Proposition 6.
Let γ = P − P − be in M Y . For − ≤ µ ≤ and X ∈ { γ, γ } , wewrite E Xµ = Ker( X − µ ) . Then for any µ ∈ σ ( γ ) , Y E γµ = E γ − µ . Moreover for | µ | < if we decompose E γµ ⊕ E γ − µ into a sum of planes Π as in Theorem 3, then1. If Y = I s , then we can choose the Π ’s to be I s -invariant.2. If Y = C , then each Π is not C -invariant and Dim E γµ is even.Equivalently Dim E γ µ is divisible by . Moreover there exists a decomposition E γ µ = ⊥ ⊕ ≤ j ≤ N V µ,j and V µ,j = Π aµ,j ⊥ ⊕ CΠ aµ,j where the Π aµ,j ’s and CΠ aµ,j ’s are spectral planes described in Theorem 3. In this part, we introduce a useful trick in the model. The Cauchy expansion (54)is an application of functional calculus: we refer the reader to [HLS05a, Sok14a] forfurther details.We assume Q ∈ S with α k|D | / Q k S ≪ . (50)We recall the following inequality, proved in [Sok14a] ∀ Q ∈ S , k R Q |∇| / k S > k Q k Ex > x | p + q || b Q ( p, q ) | dpdq, (51)From now on, we only deal with Q whose density vanishes: ρ Q = 0 . Themean-field operator D (Λ) Q is away from thanks to (50). Indeed, there holds | Π Λ R Q Π Λ | ≤ |∇| / Π Λ |∇| / R ∗ Q R Q Π Λ |∇| / |∇| / ≤ Π Λ |∇|k |∇| / R Q k B > Π Λ |∇|k Q k Ex > |D | k Q k Ex , thus | D (Λ) Q | ? |D | (cid:0) − αK k Q k Ex (cid:1) . (52)The Cauchy expansion gives an expression of γ := χ ( −∞ , (cid:0) D (Λ) Q (cid:1) − P − := π . We have [HLS05a] χ ( −∞ , (cid:0) D (Λ) Q (cid:1) − P − = 12 π Z + ∞−∞ dω D + iω (cid:0) α Π Λ R Q n Π Λ (cid:1) D Q + iω Π Λ . (53) e also expand in power of Y [ Q ] := − α Π Λ R Q Π Λ : π n − − P − = X j ≥ α j M j [ Y [ Q ]] ,M j [ Y n ] = − π Z + ∞−∞ dω D + iω (cid:16) Y n D + iω (cid:17) j . (54)Each M j [ Y [ Q ]] is polynomial in Π Λ R Q Π Λ of degree j .By using (51), the decomposition (54) is well-defined in several Banach space,provided that α k Q k Ex is small enough.– First, integrating the norm of bounded operator in (53), we obtain k π − P − k B > α k Q k Ex < . – We take the Hilbert-Schmidt norm [HLS05a, Sok14a]: we get k γ k S > α k Q k Ex . (55)– We take the norm k|D | / ( · ) k S we get the rough estimate k|D | / γ k S > min( √ Lα k Q k Ex , α k R Q k S (cid:1) + α k Q k Ex . (56) Remark . The same estimates holds for the differential of Q γ , for sufficientlysmall α . As shown in [Sok14a], the upper bound of each norm is a power series ofkind k γ k ≤ α k M [ Y [ Q ]] k + + ∞ X j =1 p jα j (cid:0) K k Q k Ex (cid:1) j . In the case of the differential, we get an upper bound of kind k d γ k ≤ α k M [ Y [ Q ]] k + + ∞ X j =1 j / α j (cid:0) K k Q k Ex (cid:1) j . The power series converge for sufficiently small α k Q k Ex .– It is also possible to consider other norms, using from the fact that a (scalar)Fourier multiplier F ( p − q ) = F ( − i ∇ x + i ∇ y ) commutes with the operator R [ · ] : Q ( x, y ) Q ( x,y ) | x − y | . We can also consider the norm k Q k w := x w ( p − q )( e E ( p ) + e E ( q )) | b Q ( p, q ) | dpdq, where w ( · ) ≥ is any weight satisfying a subadditive condition [Sok14a]: ∀ p, q ∈ R , p w ( p + q ) ≤ K ( w ) (cid:0)p w ( p ) + p w ( q ) (cid:1) . The existence of a minimizer over W t X ℓ (with t ∈ Z ) is proved with the samemethod used in [Sok14b].We use a lemma of Borwein and Preiss [BP87, HLS09], a smooth generalization ofEkeland’s Lemma [Eke74]: we study the behaviour of a specific minimizing sequence ( P n ) n or equivalently ( P n − P − =: Q n ) n .This sequence satisfies an equation close to the one satisfied by a real minimizerand we show this equation remains in some weak limit. Remark . We recall different topologies over bounded operator, besides the normtopology k·k B [RS75]. . The so-called strong topology , the weakest topology T s such that for any f ∈ H Λ ,the map B ( H Λ ) −→ H Λ A Af is continuous.2. The so-called weak operator topology , the weakest topology T w.o. such that forany f, g ∈ H Λ , the map B ( H Λ ) −→ C A
7→ h
Af , g i is continuous.We can also endow S P − with its weak- ∗ topology, the weakest topology such thatthe following maps are continuous: S P − −→ C Q Tr (cid:0) A ( Q ++ + Q −− ) + A ( Q + − + Q − + ) (cid:1) ∀ ( A , A ) ∈ Comp( H Λ ) × S ( H Λ ) . Lemma 2.
The set K Y , defined in (40) , is weakly- ∗ sequentially closed in S P − ( H Λ ) .Remark . This Lemma was stated for Y = C in [Sok14b]. For Y = I the proofis the same and we refer the reader to this paper. We recall this Theorem as stated in [HLS09]:
Theorem 4.
Let M be a closed subset of a Hilbert space H , and F : M → ( −∞ , + ∞ ] be a lower semi-continuous function that is bounded from below and notidentical to + ∞ . For all ε > and all u ∈ M such that F ( u ) < inf M + ε , thereexist v ∈ M and w ∈ Conv( M ) such that1. F ( v ) < inf M + ε ,2. k u − v k H < √ ε and k v − w k H < √ ε ,3. F ( v ) + ε k v − w k H = min (cid:8) F ( z ) + ε k z − w k H , z ∈ M (cid:9) . – Here we apply this Theorem with H = S ( H Λ ) , M = W p ( X ) − P − and F = E .The BDF energy is continuous in the S P − -norm topology, thus its restrictionover V is continuous in the S ( H Λ ) -norm topology.This subspace H is closed in the Hilbert-Schmidt norm topology because V = M C is closed in S ( H Λ ) and E − − P − is closed in V .Moreover, we have Conv ( W p ( X ) − P − ) S ⊂ K C . – For every η > , we get a projector P η ∈ W p ( X ) and A η ∈ K C such that P thatminimizes the functional F η : P ∈ E −
7→ E BDF ( P − P − ) + ε k P − P − − A η k S . We write Q η := P η − P − , Γ η := Q η − A η , e D Q η := Π Λ (cid:0) D − αR Q η + 2 η Γ η (cid:1) Π Λ . (57)Studying its differential on T P η W , we get: (cid:2) e D Q η , P η (cid:3) = 0 . (58)In particular, by functional calculus, we have: (cid:2) π η − , P η (cid:3) = 0 , π − η := χ ( −∞ , ( e D Q η ) . (59) e also write π + η := χ (0 , + ∞ ) ( e D Q η ) = Π Λ − π − η . (60)We decompose H Λ as follows (here R means Ran ): H Λ = R ( P η ) ∩ R ( π − η ) ⊥ ⊕ R ( P η ) ∩ R ( π + η ) ⊥ ⊕ R (Π Λ − P η ) ∩ R ( π − η ) ⊥ ⊕ R (Π Λ − P η ) ∩ R ( π + η ) . (61)We will prove1. Ran P ∩ Ran π + η has dimension j + 1 and is invariant under Φ SU , spanned bya unitary ψ η ∈ H Λ .2. As η tends to , up to translation and a subsequence, ψ η ⇀ ψ a = 0 , Q η ⇀ Q .There holds P j = Q + P − ∈ W p ( X ) , ψ a is a unitary eigenvector of D (Λ) Q and Q + P − = χ ( −∞ , (cid:0) D (Λ) Q (cid:1) + Proj Φ SU ( ψ a ) − Proj Φ SU (C ψ a ) , (62)where Proj E means the orthonormal projection onto the vector space E .In the following part we write the spectral decomposition of trial states and proveLemma 2. Let ( Q n ) n be any minimizing sequence for E nr t X ( j − / for j ∈ + Z + .Thanks to the upper bound, Dim Ker( Q n −
1) = 1 , as shown in Subsection 3.2.There exist a non-increasing sequence ( λ j ; n ) j ∈ N ∈ ℓ of eigenvalues and an or-thonormal family B n of Ran Q n : B n := ( ψ n , C ψ n ) ∪ ( e aj ; n , e bj ; n , C e aj ; n , C e bj ; n ) , P − ψ n = P − e ⋆j ; n = 0 , ⋆ ∈ { a, b } , (63)such that the following holds. We omit the index n .1. For any j , the vector spaces V ⋆j ; n := Φ SU ( e ⋆j ; n ) are irreducible, and so is V n :=Φ SU ( ψ n ) .2. That last one is of type ( ℓ , ε ( t )) (see Notation 8).3. Moreover for any j ∈ N we write: e a − j := − C e bj and e b − j := C e aj ,V a − j := Φ SU e a − j and V b − j := Φ SU e b − j . (64a) f ⋆j := q − λ j e ⋆ − j + q λ j e ⋆j ,f ⋆ − j := − q λ j e ⋆ − j + q λ j e ⋆j , (64b)and ∀ j ∈ Z ∗ , F ⋆j := Φ SU ( f ⋆j ) . (64c)The trial state Q n has the following form. Q n = Proj V ,n − Proj C V ,n + X j ≥ λ j q j ; n q j ; n = Proj F aj − Proj F a − j + Proj F bj − Proj F b − j . (64d) Remark . Thanks to the cut-off the sequences ( ψ n ) n and ( e j ; n ) n are H -bounded.Up to translation and extraction ( ( n k ) k ∈ N N and ( x n k ) k ∈ ( R ) N ), we can assumethat the weak limit of ( ψ n ) n is non-zero (if it were then there would hold E j ,ε ( t ) =2 m (2 j + 1) ).We can consider the weak limit of each ( e n ) : by means of a diagonal extraction,we assume that all the ( e j,n k ( · − x n k )) k and ( ψ j,n k ( · − x n k )) k , converge along thesame subsequence ( n k ) k . We also assume that ∀ j ∈ N , λ j,n k → µ j , ( µ j ) j ∈ ℓ , ( µ j ) j non-increasing , (65)and that the above convergences also hold in L loc and almost everywhere. .2 Upper bound and rough lower bound of E j , ± We aim to prove the upper bound of Proposition 2. The method will also give arough lower bound of E j , ± . Notation . We write: C ( j ) := j sup − j ≤ m ≤ j k Ψ m,j ± k L ∞ , where the functions Ψ m,j ± are defined in [Tha92, p. 125]: they are the upper orlower spinors of the Φ ± m,κ j ’s.For E j ,ε ( t ) , we only consider t ∈ { (1 , , } and ε ( t ) is defined in Notation8.– We consider trial state of the following form: Q = Proj Φ SU ( ψ ) − Proj Φ SU (C ψ ) , where Φ SU ( ψ ) is of type ( ℓ + , ε ( t )) and P − ψ = 0 . For short, we write N ψ := Proj Φ SU ( ψ ) and N C ψ := Proj Φ SU (C ψ ) . The set of these states is written W t X ℓ . We will prove that the energy of aparticular Q gives the upper bound. The BDF energy of Q ∈ W t X ℓ is: (cid:0) |D | N ψ (cid:1) − α k N ψ k Ex − α Re Tr (cid:0) N ψ R [ N C ψ ] (cid:1) . (66)– We will study the non-relativistic limit α → .– To get an upper bound, we choose a specific trial state in W t X ℓ , the idea is thesame as in [Sok14a, Sok14b]: the trial state is written in (69). Before that, we precisethe structure of elements in W t X ℓ . Minimizer for E nr t X ℓ By an easy scaling argument, there exists a minimizer forthe non-relativistic energy E nr t X ℓ (26). The scaling argument enables us to say thatthis energy is negative. Then it is clear that a minimizing sequence converges to aminimizer Γ , up to extraction. Writing H Γ := − ∆ − R Γ , this minimizer satisfies the self-consistent equation (cid:2) H Γ , Γ (cid:3) = 0 . This comes from Remark 6. In particular, H Γ restricted to Ran Γ is a homothety bysome − e < , so ∀ ψ ∈ Ran Γ , k ψ k L = 1 , k ∆ ψ k L ≤ k R Γ ψ k L > k Γ k Ex k |∇| / ψ k L , and we get k ∆ ψ k / L > k Γ k Ex i.e. k ∆ ψ k L > k Γ k / Ex > (2 j + 1) / . The last estimate comes from a simple study of a minimizer for E nr t X ℓ : we have Tr (cid:0) − ∆Γ (cid:1) − π (cid:0) |∇| Γ (cid:1) ≤ E nr (Γ) < , thus Tr (cid:0) − ∆Γ (cid:1) > j and Tr (cid:0) ( − ∆) Γ (cid:1) > j / . We end this bootstrap argument at k|∇| ψ k L for ψ ∈ Ran ψ : we have |∇| ψ = − ∆ e − ∆ (cid:16) [ |∇| , R Γ ] ψ + R Γ ψ (cid:17) , k |∇| ψ k L > k ∆Γ k S + k∇ Γ k S > j / . rial state We take the following trial state. First, let
Γ =
Proj ra ( r )Ψ j ,j + ε ( t ) 12 be a minimizer for E nr t X ℓ . We form N + := Proj Φ SU P U λ − ( ra ( r )Φ + j ,ε ( t )( j + 12 ) ) (67)where we recall that λ := g ′ (0) αm and U a φ ( x ) := a / φ ( ax ) , a > . This corresponds to dilating Γ by λ − and projecting the range of the dilation onto Ran P . Of course Γ ∈ S ( L ( R , C )) is embedded in S ( L ( R , C × C )) asfollows: Γ (cid:18) Γ 00 0 (cid:19) ∈ S ( L ( R , C × C )) . Then we define N − := C N − C − = C N − C . (68)Our trial state is N := N + − N − . (69) Upper bound for E j , ± We compute E BDF ( N ) .Before that, we study a general projector Proj Φ SU ψ where P − ψ = 0 and Φ SU ψ irreducible of type ( j , ε ( t )) .As an element of Ran P , the wave function ψ can be written ψ = P (cid:18) ϕ (cid:19) . As it spans an irreducible representation of type ( j , ε ( t )) , we can choose ∀ x = rω x ∈ R , ϕ ( x ) := ia ( r )Ψ j j + ε ( t ) 12 ( ω x ) , a ( r ) ∈ L (cid:0) (0 , ∞ ) , r dr (cid:1) , where we used notations of [Tha92, p. 126]. This corresponds to taking ψ := P ra ( r )Φ + j ,ε ( j + 12 ) , ε = ε ( t ) . We recall the following formulae of [Tha92, pp. 125-127] (with ω : x x | x | ) − i α · ∇ = − i ( α · ω ) ∂ r + ir ( α · ω )(2 S · L ) , (cid:8) S · L , α · ω (cid:9) = − α · ω and i σ · ω Ψ m j j ± = Ψ m j j ∓ . (70)This gives P a ( r )Φ + m,ε ( t )( j + 12 ) = 12 i (cid:0) g ( |∇| ) |D | (cid:1) a ( r )Ψ mj + ε g ( |∇| ) |D ||∇| (cid:0) ∂ r ( a ( r )) + ε ( j + ) a ( r ) r (cid:1) Ψ mj − ε , =: ia ↑ ( r )Ψ mj + ε a ↓ ( ε, j ; r )Ψ mj − ε . (71)We write Op := g ( |∇| ) |D ||∇| : the following holds. (cid:12)(cid:12)(cid:12) Tr (cid:0) N ψ R [ N C ψ ] (cid:1)(cid:12)(cid:12)(cid:12) > j sup m k Ψ mj ± k L ∞ k | a ↑ a ↓ ( ε, j , · ) |k C > C ( j ) D (cid:16) | a ↑ | ; | Op · ∂ r ( a ( r )) | + j | Op · r − a ( r ) | (cid:17) , > C ( j ) h|∇| ψ , ψ ik∇ ψ k L =: R em ( j , ψ ) . (72)In fact, we have Tr (cid:0) N ψ R [ N C ψ ] (cid:1) ≥ by direct computation.Let us deal with k N ψ k Ex . otation . We write P ↑ the projection onto the upper part of C × C and P ↓ the projection onto the lower part. That is: P ↑ ψ has no lower spinor and the sameupper spinor as ψ .Similarly, k N ψ k Ex − k P ↑ N ψ P ↑ k Ex = Tr (cid:0) P ↑ N ψ P ↓ R N ψ (cid:1) + Tr (cid:0) P ↓ N ψ P ↑ R N ψ (cid:1) + k P ↓ N ψ P ↓ k Ex , > R em ( j , ψ ) + C ( j ) k∇ ψ k L k |∇| / | D | ψ k L , =: R em ( j , ψ ) . For the trial state (69), this gives: k N + k Ex = k P ↑ N ψ P ↑ k Ex + O (cid:16) C ( j ) (cid:0) α j + α j / (cid:1)(cid:17) = αmg ′ (0) k Γ k Ex (1 + O ( k∇ ψ k L ))+ O (cid:0) k ∆1 − ∆ ψ k L ( k |∇| / − ∆ ψ k L + k∇ ψ k L ) (cid:1) , = αmg ′ (0) k Γ k Ex + O h C ( j ) (cid:16) α j / + inf ≤ s ≤ ( α s j s/ ) (cid:0) α j / + inf − ≤ s ≤ ( α s j s/ ) (cid:1)(cid:17)i . We compute the kinetic energy as in [Sok14a, Sok14b]: we get Tr (cid:0) |D | N + (cid:1) = α mg ′ (0) Tr (cid:0) − ∆Γ (cid:1)(cid:0) Kα (cid:1) + O (cid:0) α Tr (cid:0) (∆) Γ (cid:1)(cid:1) , = α mg ′ (0) Tr (cid:0) − ∆Γ (cid:1) + O (cid:0) α j + α j / (cid:1) . This proves E j ,ε ( t ) ≤ m (2 j + 1) + α mg ′ (0) E nr t X ℓ + O (cid:0) ̺ ( α, j ) (cid:1) ̺ ( α, j ) := α j + α j / + C ( j ) (cid:16) α j / + inf ≤ s ≤ ( α s j s/ ) (cid:0) α j / + inf − ≤ s ≤ ( α s j s/ ) (cid:1)(cid:17) . (73)First, by Kato’s inequality (37), we have k N ψ − N C ψ k Ex ≤ π (cid:0) |∇| ( N ψ + N C ψ ) (cid:1) = π Tr (cid:0) |∇| N ψ (cid:1) . So E BDF ( Q ) ≥ (cid:16) Tr (cid:0) |D | N ψ (cid:1) − α π (cid:0) |∇| N ψ (cid:1)(cid:17) =: 2 (cid:0) (2 j + 1) m + F ( N ψ ) (cid:1) . As α tends to , a minimizer over W t X ℓ should be localized in Fourier space around . Indeed, for α, L sufficiently small, we have ∀ p ∈ B (0 , Λ) , e E ( p ) − m = g ( p ) − m + g ( p ) e E ( p ) + m ≥ p k g k L ∞ | D | , and for any < s ≤ : p k g k L ∞ | D | ≥ s απ | p | ⇐⇒ | p | ≥ αsπ k g k L ∞ p − ( αsπ k g k L ∞ ) =: ϑ s . We get F (cid:0) Π ϑ N ψ Π ϑ (cid:1) ≤ E BDF ( Q ) − j + 1) m. y Cauchy-Schwartz inequality, we get a rough lower bound Tr (cid:0) − ∆Π ϑ N ψ Π ϑ (cid:1) > α (2 j + 1) and E BDF ( Q ) − j + 1) m ? − α (2 j + 1) . For an almost minimizer Q , the same argument shows that Tr (cid:0) − ∆ |D | Q (cid:1) > α (2 j + 1) . (74)A precise lower bound is obtained once we know that there exists a minimizer P j . This state satisfies the self-consistent equation (62): see Subsection 3.5. Remark . The same method can be used to get an upper bound of E nrp ( X ) for any p ( X ) = P ℓ ℓ =0 t ℓ X ℓ . By scaling we have E nrp ( X ) < . The method is more subtle because M I has only one connected component. Wefirst consider the subset M I defined by: M I = (cid:8) P ψ := P − + | ψ ih ψ | − | I s ψ ih I s ψ | , ψ ∈ S Ran P (cid:9) . (75) Lemma 3.
Let F I be the infimum of the BDF energy over M I . Then we have F I ≥ m − α E PT (1) mg ′ (0) + O ( α ) . (76)We will prove the existence of a critical point in the neighbourhood of M I via a mountain pass argument. In this part, we aim to prove the following Proposition. Proposition 7.
1. In the regime of Theorem 1, there exists a bounded sequence in M I − P − of almost critical points: ( Q n = P n − P − ) n such that lim n → + ∞ k∇E BDF ( P n ) k S = 0 with E BDF ( Q n ) = 2 m − α mg ′ (0) E PT (1) + O ( α ) . Furthermore, for sufficiently big n , there exists ψ a ; n such that C ψ a ; n = Ran P n ∩ Ran χ (0 , + ∞ ) (cid:0) D (Λ) Q n − ∇E BDF ( P n ) (cid:1) and P n = χ ( −∞ , (cid:0) D (Λ) Q n − ∇E BDF ( P n ) (cid:1) + | ψ a ; n ih ψ a ; n | − | I s ψ a ; n ih I s ψ a ; n | .
2. Up to a subsequence and up to translation the sequence tends to a critical point Q ∞ of E BDF in M I − P − .Moreover, writing P = Q ∞ + P − , there exists < µ < m and ψ a ∈ S H Λ suchthat P = χ ( −∞ , ( D (Λ) Q ∞ ) + | ψ a ih ψ a | − | I s ψ a ih I s ψ a | , C ψ a = Ker (cid:0) D (Λ) Q ∞ − µ (cid:1) , inf σ ( | D (Λ) Q ∞ | ) = µ. (77) Proof of Proposition 7: first part
For any ψ ∈ S Ran P , we define: c ψ : [0 , −→ M I − P − s
7→ | sin( πs ) ψ + cos( πs )I s ψ ih sin( πs ) ψ + cos( πs )I s ψ | − | I s ψ ih I s ψ | . (78) Remark . The loop c ψ + P − crosses M I at t = where the BDF energy ismaximal: sup s ∈ [0 , E BDF ( c ( s )) . Indeed, there holds E BDF ( c ( s )) = 2 sin( πs ) h|D | ψ , ψ i− α sin( πs ) (cid:2) D (cid:0) | ψ | , | ψ | (cid:1) +cos(2 πs ) D (cid:0) ψ ∗ I s ψ, ψ ∗ I s ψ (cid:1)(cid:3) , nd the derivative with respect to s is: dds E BDF ( c ( s )) = 2 π sin(2 πs ) (cid:16) h|D | ψ , ψ i − α (cid:2) D (cid:0) | ψ | , | ψ | (cid:1) +(sin( πs ) − cos(2 πs )) αD (cid:0) ψ ∗ I s ψ, ψ ∗ I s ψ (cid:1)(cid:3)(cid:17) . For sufficiently small α , this quantity vanishes only at πs ≡ π ] .What happens when we apply the gradient flow Φ BDF ,t of the BDF energy ? Theloop c ψ is transformed into c t := Φ BDF ,t ( c ψ ) and we still have c t ( s = 0) = c t ( s = 1) = 0 . This follows from the fact that P − is the global minimizer of E BDF .We recall that for all s ∈ [0 , , the function c t ( s ) satisfies the equation ∀ t ∈ R + , ddt ( c t ( s )) = −∇E BDF ( c t ( s )) ∈ T c t ( s )+ P − M I . The non-trivial result holds.
Lemma 4.
Let P ψ ∈ M I be a state whose energy is close to the infimum F I : E BDF (cid:0) P ψ (cid:1) < F I + α . Let c ψ be the loop associated to ψ (see (78) ) and c t := Φ BDF ,t ( c ψ ) . Then for all t ∈ R + , the loop c t crosses the set M I at some e s ( t ) ∈ (0 , . Lemma 5.
Let ( c t ) t ≥ be the family of loops defined in Lemma 4 and let ( s ( t )) t ≥ be a family of reals in (0 , such that ∀ t ≥ , E BDF (cid:0) c t ( s ( t )) (cid:1) = sup s ∈ [0 , E BDF ( c t ( s )) . Then there exists an increasing sequence ( t n ) n ∈ N the sequence ( c t n ( s ( t n ))) n ≥ satis-fies the first point of Proposition 7 We prove Lemmas 3 and 4 in Subsection 3.6. We assume they are true to proveLemma 5 and Proposition 7.
Remark . The proof of Lemma 4 uses an index argument. We kept it elementarybut it is possible to rephrase it in terms of the Maslov index [Fur04] once we noticethat I s induces a symplectic structure and that the projectors in M I are Lagrangians(see Remark 11). Spectral decomposition of P n We define F := lim inf t → + ∞ E BDF ( c t ( s ( t ))) = lim inf t → + ∞ sup s ∈ [0 , E BDF ( c t ( s )) . We assume ( t n ) n ≥ is a minimizing sequence for F .We may assume that lim n → + ∞ t n = + ∞ .– First we prove that along the path c t the gradient ∇E BDF (see (46)) is boundedin S . Indeed, for all P = Q + P − ∈ M , we write e Q := P − χ ( −∞ , (cid:0) Π Λ D Q Π Λ (cid:1) , We recall that D (Λ) Q := Π Λ D Q Π Λ : ∇E BDF ( P ) = (cid:2)(cid:2) D Λ Q , P (cid:3) , P (cid:3) = (cid:8) | D (Λ) Q | ; e Q (cid:9) − e QD (Λ) Q e Q, k∇E BDF ( P ) k S > k e Q k S e E (Λ) h (1 + k Q k S )(1 + k e Q k S ) i > K (Λ , F + α ) . (79) e have used the Cauchy expansion (54) to get an expression χ ( −∞ , (cid:0) D (Λ) Q (cid:1) − P − = + ∞ X k =1 α k M k [ Y [ Q ]] where M k [ Y [ Q ]] is a polynomial function of π Λ R Q Π Λ of degree k . We refer thereader to these papers or to (53)-(51) above for more details.From formula (79) and Remark 13 we see that the gradient, as a function of Q is locally Lipschitz , at least in some ball { Q : k|D | / Q k S ≤ C } in which thereholds inf σ (cid:0) | D (Λ) Q | (cid:1) ≥ K ( C ) , where C is some constant. The Lipschitz constant depends on the constant C andin the present case, we can take C > .Let us prove that lim n → + ∞ k∇E BDF ( c t n ( s ( t n ))) k S = 0 . (80)If not, the lim sup is bigger than some η > and then we get a contradiction whenwe consider n large enough such that | F − E BDF ( c t n ( s ( t n ))) | ≪ η and k∇E BDF ( c t n ( s ( t n ))) k S ≥ η , because ∀ τ > , E BDF ( c t n + τ ( s ( t n ))) −E BDF ( c t n ( s ( t n ))) = − Z τ k∇E BDF ( c t n + u )( s t n ) k S du. – We recall that the gradient at P ∈ M is the "projection" of the mean-field operatoronto the tangent plane T P M , in the sens that ∀ v ∈ T P M , P D Q (1 − P ) ∈ S and Tr (cid:0) P D Q (1 − P ) v + (1 − P ) D Q P v (cid:1) = Tr (cid:0) ∇E BDF (cid:1)
Notation . For short, we write Q n := c t n (cid:0) s ( t n ) (cid:1) and P n := Q n and v n := ∇E BDF ( Q n ) . Moreover, we write e D Q n := D Q n − v n and e π − ; n := χ ( −∞ , (cid:0) D (Λ) Q n − v n (cid:1) . We have shown that lim n → + ∞ k v n k S = 0 . But as v n is an element of the tangent plane T P n M , we have (cid:2)(cid:2) v n , P n (cid:3) , P n (cid:3) = P n v n (1 − P n ) + (1 − P n ) v n P n = v n thus (cid:2)(cid:2) D (Λ) Q n − v n , P n (cid:3) , P n (cid:3) = 0 . Equivalently, we have (cid:2) e D (Λ) Q n , P n (cid:3) = (1 − P n ) e D (Λ) Q n P n − P n e D (Λ) Q n (1 − P n ) = 0 . (81)Thus the projector P n commutes with the distorted mean-field operator e D Q n .We recall that lim n k e D (Λ) Q n − D (Λ) Q n k S = 0 , and thus up to taking n big enough, we can neglect the distortion v n : all its Sobolevnorms tend to zero as n tends to infinity thanks to the cut-off .– Thanks to Lemma 3 we have the following energy condition: m + O ( α ) ≤ F ≤ E BDF ( Q n ) ≤ F + α = 2 m + O ( α ) . sing the Cauchy expansion (53)-(51), we have k |D | / ( e π − ; n − P − ) k S > √ Lα k Q n k Ex > √ Lα.
Thus we get (cid:12)(cid:12) k Q n k S − k P n − e π − ; n k S (cid:12)(cid:12) ≤ kP − − e π − ; n k S > √ Lα. As e D Q n and P n commutes, then necessarily k P n − e π − ; n k S is an integer equal totwice the dimension of Ran P n ∩ Ran (1 − e π − ; n ) .But we know that m k Q n k S ≤ Tr (cid:0) |D | Q n (cid:1) ≤ − α π E BDF ( Q n ) ≤ m − α π = 2 m + O ( α ) . Then the above dimension is lesser than and it cannot be because of the energycondition E BDF ( Q n ) ≥ F I ≥ m − Kα ≫ √ Lα.
This proves the first part of Proposition 7. We have
Ran P n ∩ Ran (1 − e π − ; n ) = C ψ a ; n where ψ a ; n is unitary. It is an eigenvector for e D (Λ) Q n with eigenvalue µ n . From theequality: E BDF ( Q n ) = E BDF ( e π − ; n − P − ) + 2 µ n − α x | ψ a ; n ∧ I s ψ a ; n ( x, y ) | | x − y | dxdy, we get < µ n < m . We end the proof as follows. Proof of Proposition 7: second part
We follow the method of [Sok14b].We recall the main steps and refer the reader to this paper for further details.– The idea is simple: we must ensure that there exists a non-vanishing weak-limitand that this weak-limit is in fact a critical point.Let us say that ψ a ; n is associated to the eigenvalue µ n .– The condition of the energy ensures that the sequence ( ψ a ; n ) n does not vanish inthe sense that we do not have the following: ∀ A > , lim sup n sup x ∈ R Z B ( x,A ) | ψ a ; n | = 0 . Up to translation and extraction of a subsequence, we may suppose that ( Q n ) (resp. ( ψ a ; n ) ) converges in the weak topology of H to Q ∞ = 0 (resp. ψ a = 0 ). Inparticular these sequences also converge in L loc and a.e. We recall that thanks tothe cut-off and Kato’s inequality (37) , we have Q n ∈ H ( R × R ) with k| D | Q n k S ≤ e E (Λ) k|D | / Q n k S ≤ e E (Λ)1 − απ/ n E BDF ( Q n ) . A similar estimate hold for ( ψ a ; n ) . We also suppose that lim n µ n = µ ∞ .– As shown in [Sok14b], the operator R Q n converges in the strong operator topologyto R Q ∞ . Thanks to the Cauchy expansion (54), we also haves . lim n h χ ( −∞ , (cid:0) D (Λ) Q n − ∇E BDF ( P n ) (cid:1) − P − i = χ ( −∞ , (cid:0) D (Λ) Q ∞ (cid:1) − P − . By that strong convergence, we also have the weak-convergence of e D (Λ) Q n ψ a ; n to D (Λ) Q ∞ ψ a in L and it follows that: D (Λ) Q ∞ ψ a = µ ∞ ψ a = 0 . – The condition of the energy ensures that for α sufficiently small, the ψ a ; n ’s areclose to a scaled Pekar minimizer: for any n , there exists a Pekar minimizer e φ n suchthat k ψ a ; n − λ − / e φ n ( λ − ( · )) k H ≤ αK where λ := g ′ (0) αm . he constant K depends on the energy estimate of Proposition 7.– Thanks to that, for all n , µ n is an isolated eigenvalue of e D (Λ) Q n , uniformly in n : wehave C ψ a ; n = Ker (cid:0) e D (Λ) Q n − µ n (cid:1) , and dist (cid:16) µ n ; σ (cid:0) e D (Λ) Q n (cid:1) \{ µ n } (cid:17) > Kα . By functional calculus, we finally get the norm convergence of ( ψ a ; n ) n to ψ a in L .– This proves thats . lim n P n = χ ( −∞ , (cid:0) D (Λ) Q ∞ (cid:1) + | ψ a ih ψ a | − | I s ψ a ih I s ψ a | ∈ M I , and ends the proof. E j , ± We consider a family of almost minimizers ( P η n ) n of type (57) where ( η n ) n is anydecreasing sequence. We also consider the spectral decomposition (64) of any Q n := P η n − P − .For short we write P n := P η n and we replace the subscript η n by n (for instance ψ n := ψ η n ). Moreover, we will often write ε instead of ε ( t ) .We study weak limits of ( Q n ) n . We recall that Q n can be written as follows: ( N +; n = P N +; n = Proj Φ SU ψ η n and N − ; n = C N +; n C ,Q n = N +; n − N − ; n + γ n , Ran N ± ; n ∩ Ker γ n = { } . (82)We can suppose ψ n = P a n ( r )Φ + j ,ε t , a n ( r ) ∈ S L ( R + , r dr ) . Remark . The functions ψ ∈ Ran N ± ; n are "almost" radial. We recall (71), giving ∀ x = rω x ∈ R , | ψ ( x ) | ≤ k ψ k L | s n ( r ) |k Φ ± j , ± ( j + 12 ) k L ∞ , | s n ( r ) | := (cid:12)(cid:12) (1 + g ( |∇| ) |D | ) a n (cid:12)(cid:12) ( r ) + (cid:12)(cid:12) g ( |∇| ) |∇||D | ( ∂ r a n + ε a n r ) (cid:12)(cid:12) ( r ) . (83)In particular by Newton’s Theorem for radial function we have: ∀ ψ ∈ Ran N ± ; n , | ψ | ∗ | · | ( x ) ≤ K ( j ) k ψ k L | x | . (84)– We first prove that there is no vanishing, that is ∃ A > , lim sup n sup z ∈ R Z B ( z,A ) | ψ n ( x ) | dx > . Indeed, let assume this is false. Then using (84), it is clear that k N ± ; n k Ex → , and we get lim inf E BDF ≥ j + 1) m + lim inf E BDF ( γ n ) ≥ j + 1) m, an inequality that is false as shown in the previous section. Thus, we have: Q n ⇀ Q ∞ = 0 . – As the BDF energy is sequential weakly lower continuous [HLS05b], we have E j ,ε ≥ E BDF ( Q ∞ ) . Our aim is to prove that Q ∞ + P − ∈ W t X ℓ : in other words that Q ∞ is a minimizerfor E j ,ε . he spectral decomposition (82) is not the relevant one: let us prove we candescribe P n in function of the spectral spaces of the "mean-field operator" e D Q n : thefirst step is to prove (88) below.We recall that Q n satisfies Eq. (58), that we have the decomposition (61).Using (74), we have for all ψ in S Ran N +; n : h e D Q n ψ , ψ i − m = h ( |D | − m ) ψ , ψ i − h ( αR Q n + 2 η n Γ n ) ψ , ψ i , ? − α k Q n k Ex k |∇| / ψ k L − eta n k Γ n k S ? − α (2 j + 1) . Thus
Ran P n ∩ Ran π n + = { } . – Let us prove this subspace has dimension j + 1 : we use the minimizing propertyof Q n . The condition on the first derivative gives (58). The estimation of the energy(from above and below) obtained in the previous section gives this result. Indeed,using the Cauchy expansion and the method of [Sok14a], we have q Tr (cid:0) |D | γ vac ; n (cid:1) > α ( k Q n k Ex + η n k Γ n k S ) > √ Lα √ αj ,γ vac ; n := χ ( −∞ , (cid:0) e D Q n (cid:1) − P − . (85)The Cauchy expansion is explained in (53)-(54) below, we assume the above estimatefor the moment (see (56)).We write Q n = N n + γ n : there holds (cid:12)(cid:12) k N n k S − k Q n k S (cid:12)(cid:12) > L / α (2 j + 1) . As j + 1) ≤ k Q n k S ≤ j + 1) (cid:0) − απ/ (cid:1) − , then necessarily (cid:12)(cid:12) k N n k S − j + 1) (cid:12)(cid:12) > α (2 j + 1) , (86)and for α sufficiently small, the upper bound is smaller than . This provesDim Ran P n ∩ Ran π n + = 2 j + 1 . Remark . There exists a unitary ψ a ; n such that Φ SU ψ a ; n = Ran P n ∩ Ran π n + . We can assume that ψ a ; n ∈ Ker (cid:0) J − j (cid:1) . Then we have N n := Proj Φ SU ψ a ; n − Proj Φ SU C ψ a ; n . (87)Equivalently writing ψ w ; n := C ψ a ; n there holds Φ SU ψ w ; n = Ran (1 − P n ) ∩ Ran π n − .– We have: P n = Proj Φ SU ψ a ; n − Proj Φ SU ψ w ; n + π n − . (88)We thus write Q n = N n + γ vac ; n . (89)As Ran P n is e D Q n invariant and that e D Q n is bounded (with a bound that dependson Λ ), necessarily e D Q n ψ a ; n = µ n ψ a ; n , µ n ∈ R + . As in [Sok14b], studying the Hessian we have m − µ n + 2 η n ≥ . – As for ψ n , there is no vanishing for ( ψ a,n ) n for α sufficiently small: decomposing ψ + ∈ Ran P n : ψ + = aψ a ; n + φ, φ ∈ Ran P n ∩ Ran π n − , e have | a | ≥ µ (cid:0) m + h| e D Q n | φ , φ i − K ( α j + η n k Γ n k S ) (cid:1) . Provided that µ n is close to m , the absence of vanishing for ψ n implies that of ψ a ; n .By Kato’s inequality (37): e D Q n ≥ |D | (cid:0) − α k R Q n |D | − k B − η n k Γ n k B (cid:1) |D |≥ |D | (cid:0) − α k Q n k Ex − η n k Γ n k S (cid:1) Thus (cid:12)(cid:12) e D Q n (cid:12)(cid:12) ≥ |D | (cid:0) − α k Q n k Ex − η n k Γ n k S (cid:1) and µ n ≥ − K ( α j + η n k Γ n k S ) . In the same way we can prove that | µ n − m | > α j + η n k Γ n k S So ψ a,n ⇀ ψ a = 0 . – We decompose γ vac ; n = π n − − P − ∈ W − P − as in (64): using Cauchy’s expansion(53)-(54), we have π n − − P − = 12 π Z + ∞−∞ dω D + iω (cid:0) η n Γ n − α Π Λ R Q n Π Λ + 2 η n Γ n (cid:1) e D Q n + iω Π Λ . (90)To justify this equality, we remark that | e D Q n | is uniformly bounded from below, itfollows that the r.h.s. of (90) is well-defined provided that α ≤ α j : Π Λ R Q n Π > |∇|k Q n k Ex > α (2 j + 1) |∇| ≤ α (2 j + 1) |D | . We must ensure that α p α (2 j + 1) is sufficiently small.Integrating the norm of bounded operator in (90), we obtain k π n − − P − k B > α k Q n k Ex + η n k Γ n k S < . We also expand in power of Y n := − α Π Λ R Q n Π Λ + 2 η n Γ n as in (54) π n − − P − = X j ≥ α j M j [ Y n ] . (91)We have k γ vac ; n k S > α k Q n k Ex + η n k Γ n k S > α . (92)We take the norm k|D | / ( · ) k S : k|D | / γ vac ; n k S > √ Lα k Q N k Ex + η n k Γ n k S > L / αj . (93)– We thus write γ vac ; n = X j ≥ λ j ; n q j ; n , where q j ; n has the same form as the one in (64).Up to a subsequence, we may assume all weak convergence as in Remark (16): thesequence of eigenvalues ( λ j ; n ) n tends to ( µ j ) j ∈ ℓ and each ( e ⋆j ; n ) n (with ⋆ ∈ { a, b } )tends to e ⋆j ; ∞ , ( ψ e ; n ) n tends to ψ e . We can also assume that the sequence ( µ n ) n tends to µ with ≤ µ ≤ m . Notation . For shot we write ψ v := C ψ e .Furthermore, we write P := Q ∞ + P − and π := χ ( −∞ , ( D (Λ) Q ∞ ) .– We will prove that1. (cid:2) D (Λ) Q ∞ , P (cid:3) = 0 , . D (Λ) Q ∞ ψ a = µψ a and so π ψ a = 0 .Moreover D (Λ) Q ∞ C ψ a = − µ C ψ a and h C ψ a , ψ a i = 0 .3. π = P − Proj Φ SU ( ψ a ) + Proj Φ SU (C ψ a ) =: P − N. (94)These results follow from the strong convergences . lim n R Q n = R Q ∞ . (95)This fact enables us to show lim n R Q n ψ a ; n = R Q ∞ ψ a in L , s. op. lim n (cid:0) π n − − P − (cid:1) = π − P − in B ( H Λ ) , w. op. lim n P n = π − P − + Proj Φ SU ψ a − Proj Φ SU ψ w in B ( H Λ ) , lim n ψ a ; n = ψ a in L . (96) Remark . We only write in this paper the proof of R Q n ψ a ; n L −→ n → + ∞ R Q ∞ ψ a and ψ a ; n L −→ n → + ∞ ψ a . The convergence in the weak-topology can be proved using the same method asin [Sok14b]. For the first limit this follows from the convergence of R Q n in thestrong topology. For the proof of this fact and of the strong convergence of γ vac ; n = π n − − P − , we refer the reader to [Sok14b].For R Q n , it suffices to remark that Q n ( x, y ) converges in L loc and a.e. . Toestimate the mass at infinity, we simply use the term | x − y | in Q n ( x,y ) | x − y | .The strong convergence of γ vac ; n follows from that of R Q n and the Cauchy ex-pansion (91).Then, assuming all these convergences, the convergence of Q n resp. (cid:2) e D (Λ) Q n ; P n (cid:3) in the weak operator topology to Q ∞ resp. (cid:2) D (Λ) Q ∞ , P (cid:3) are straightforward.Similarly, using (95), it is clear that e D Q n ψ a ; n ⇀ n → + ∞ D Q ∞ ψ a , and that D (Λ) Q ∞ ψ a = µψ a . To get the existence of minimizer, it suffices to prove that k ψ a k L = 1 or equivalently lim n ψ a ; n = ψ a in L .– To prove the norm convergence of ψ a ; n to ψ a , we need a uniform upper bound of µ n , or precisely, we need the following: lim sup n ( m − µ n ) > . (97)Indeed, we then get ( D − µ n ) ψ a ; n = αR Q n ψ a ; n − η n Γ n ψ a ; n and ψ a ; n = α D − µ n (cid:0) R Q n ψ a ; n − η n Γ n ψ a ; n (cid:1) . (98)Provided that (97) holds and that we have norm convergence of R Q n ψ a ; n we obtainthe norm convergence of ψ a ; n .– To prove the norm convergence of R Q n ψ a ; n to R Q ∞ ψ a , we use the fact that theelement of Φ SU ψ a ; n are "almost radial" (see in Remark 20). We recall (84) holds. n the following, we write δQ n := Q n − Q ∞ and δψ n := ψ a ; n − ψ a and use Cauchy-Schwartz inequality: for any A > there hold Z | x |≥ A (cid:12)(cid:12)(cid:12) Z δQ n ( x, y ) | x − y | ψ a ; n ( y ) dy (cid:12)(cid:12)(cid:12) dx ≤ k δQ n k Ex K ( j ) A , Z | x |≤ A (cid:12)(cid:12)(cid:12) Z δQ n ( x, y ) | x − y | ψ a ; n ( y ) dy (cid:12)(cid:12)(cid:12) dx ≤ π h|∇| ψ a ; n , ψ a ; n i x B (0 ,A ) × B (0 , A ) | δQ n ( x, y ) | | x − y | dxdy + 2 A k δQ n k S k ψ a ; n k L . Thus lim sup n k R [ Q n − Q ∞ ] ψ a ; n k L = 0 . Similarly Z | x |≥ A (cid:12)(cid:12)(cid:12) Q ∞ ( x, y ) | x − y | δψ n ( y ) dy (cid:12)(cid:12)(cid:12) dx ≤ A − A k Q ∞ k S k δψ n k L + 2 k δψ n k L A k Q ∞ k Ex , Z | x |≤ A (cid:12)(cid:12)(cid:12) Q ∞ ( x, y ) | x − y | δψ n ( y ) dy (cid:12)(cid:12)(cid:12) dx ≤ π h|∇| δψ n , δψ n i x B (0 ,A ) × B (0 , A ) | δQ n ( x, y ) | | x − y | dxdy + 2 A k Q ∞ k S k δψ n k L , and lim sup n k R Q ∞ ( ψ a ; n − ψ a ) k L = 0 . This proves that lim n → + ∞ k R Q n ψ a ; n − R Q ∞ ψ a k L = 0 . – Let us prove (97). We have: µ n (2 j + 1) = Tr (cid:16) e D Q n N n (cid:17) , = Tr (cid:16) e D γ vac ; n N n (cid:17) − α k N n k Ex , = E BDF ( Q n ) − E BDF ( γ vac ; n ) − α k N n k Ex ,< m (2 j + 1) − K ( j ) α . (99)This upper bound holds provided that α ≤ α j thanks to the upper bound of E j ,ε obtained in the previous section. E j , ± Our aim is to prove the estimate of Proposition 2. We consider the minimizer Q ∞ = N + γ vac found in the previous subsection. It satisfies Eq. (94) where P = P − + Q ∞ and γ vac = χ ( −∞ , ( D (Λ) Q i nfty ) − P − . (100)– The proof is the same as that in [Sok14a, Sok14b] and relies on estimates on theSobolev norms k |∇| s N + k S where we write N + := Proj Φ SU ψ a = Ker ( D (Λ) Q ∞ − µ ) . (101)Using (101), we get Tr (cid:0) |D | N + (cid:1) = 2(2 j + 1) µ + 2 αµ Tr (cid:0) R Q ∞ N + (cid:1) + α Tr (cid:0) R Q ∞ N + (cid:1) , ≤ j + 1) µ + 4 αµ k Q ∞ k S k∇ N + k S + 4 α k Q ∞ k S k∇ N + k S and provided that α ≤ α j , we get Tr (cid:0) ( − ∆) N + (cid:1) > α (2 j + 1)1 − α (2 j + 1) − k g k L ∞ k g ′′ k L ∞ . (102) e have used Hardy’s inequality: | · | ≤ − ∆ in R . (103)We recall that ≤ k g k L ∞ − > α log(Λ) and k g ′′ k L ∞ > α. See (32) (or [Sok14a, Appendix A] for more details).Thus for sufficiently small α , we have ∀ ψ ∈ S Ran N + , k∇ ψ k L > α − α (2 j + 1) − k g k L ∞ k g ′′ k L ∞ > α . (104)– By bootstrap argument , we can estimate k ∆ N + k S . We have: ∀ ψ ∈ S Ran N + , k |∇| / ψ k L > α p j + 1 and k ∆ ψ k L > α (2 j + 1) / . (105)We prove this result below.Furthermore, using the Cauchy expansion (54) and (51), we get k |D | / γ vac k S > α k∇ N k S + √ Lα k γ vac k Ex + α k Q ∞ k Ex (cid:0) k∇ N k S + k γ vac k Ex (cid:1) , hence k |D | / γ vac k S > α p j + 1 . (106)Now, if we assume (105)-(106), then we getFor α ≤ α j , E BDF (cid:0) Q ∞ (cid:1) = 2 m (2 j + 1) + α mg ′ (0) E nr t X ℓ + O (cid:0) α K ( j ) (cid:1) . We do not prove this fact: the method is the same as in [Sok14a, Sok14b] (in theproof of the lower bound of E BDF (1) resp. E , ).We just recall how we get (105). Proof of (105)
We scale the wave functions of (104) by λ := g ′ (0) αm : ∀ x ∈ R , U λ ψ ( x ) = ψ ( x ) := λ / ψ ( λx ) , and we split ψ (resp. ψ ) into the upper spinor ϕ (resp. ϕ ) and the lower spinor χ (resp. χ ). Thanks to (99), we have α − ( m − µ ) =: α − δm ≥ K ( j ) > provided that α is sufficiently small ( α ≤ α j ) .We write ∀ Q ∈ S , Q := U λ Q U − λ = U λ Q U λ − . For all ψ in S Ran N + we have ( λ δmϕ = iλ σ · ∇ χ + αλ (cid:0) R Q ∞ ψ (cid:1) ↑ ,χ = − iλ σ ·∇ ϕλ ( m + µ ) − αλ (cid:0) R Q ∞ ψ (cid:1) ↓ . (107)– We recall ∀ Q ∈ S , k (cid:2) ∇ , R Q (cid:3) |∇| / k B > x | p − q | | p + q || c Q ( p, q ) | dpdq. (108)This result was previously proved in [Sok14b] and follows from the fact that a (scalar)Fourier multiplier F ( p − q ) = F ( − i ∇ x + i ∇ y ) commutes with the operator R [ · ] : Q ( x, y ) Q ( x,y ) | x − y | . Then it suffices to use Hardy’s inequality (103): k (cid:2) ∇ , R Q ∞ (cid:3) ψ k L > λ x | p − q | | b Q ∞ ( p, q ) | dpdq × k∇ ψ k L . y Hardy’s inequality (103) and (108), the following holds: k χ k S ≤ λ m k∇ ϕ k S + 2 α k R Q ∞ ψ k S > α , k∇ χ k S ≤ λδm ) + 2 α k R Q ∞ ψ k S > ( δm ) α + α (2 j + 1) , k ∆ ϕ k S ≤ λ m k∇ χ k L + 2 α ( k (cid:2) ∇ , R Q ∞ (cid:3) ψ k L + k R Q ∞ k L ∇ ψ ) > ( δm ) α + (2 j + 1) + α (2 j + 1) / , k ∆ χ k S ≤ λ ( δm ) k∇ ϕ k L + 2 α ( k (cid:2) ∇ , R Q ∞ (cid:3) ψ k L + k R Q ∞ k L ∇ ψ ) > ( δm ) α + (2 j + 1) + α (2 j + 1) / . (109)– There remains to estimate x | p − q | | c Q ( p, q ) | dpdq, for Q = N and γ vac . For Q = N , we just have to estimate Tr (cid:0) |∇| N + (cid:1) .The case Q = γ vac is dealt with as in [Sok14a, Sok13]: by a fixed-point argument(valid for α ≤ α j ), we prove that n x | p − q | | d γ vac ( p, q ) | dpdq o / > α min (cid:0) k ∆ N k S , k |∇| / N k S (cid:1) . Now, we can prove that Tr (cid:0) |∇| N + (cid:1) > α / (2 j + 1) / . For a unitary ψ in Ran N + , there holds k |∇| / D ψ k L ≤ µ h|∇| ψ , ψ i + αK k |∇| / ψ k L k R Q ∞ ψ k L + α (cid:0) k [ R Q ∞ , |∇| / ] ψ k L + 2 k Q ∞ k S k |∇| / k L (cid:1) . (110)Similarly, in Fourier space we have: (cid:12)(cid:12)(cid:12) F (cid:0) [ R Q ∞ , |∇| / ]; p, q (cid:1)(cid:12)(cid:12)(cid:12) > | p − q | / | b R Q ∞ ( p, q ) | , and by Hardy’s inequality k [ R Q ∞ , |∇| / ] ψ k L > x | p − q || d Q ∞ ( p, q ) | dpdq k∇ ψ k L > Tr (cid:0) |∇| Q ∞ (cid:1) k∇ ψ k L . Substituting in (110), we get h|∇| ψ , ψ i > α / p j + 1 , hence Tr (cid:0) |∇| N + (cid:1) > α / (2 j + 1) / . We consider a trial state P ψ ∈ M I : Q ψ := P ψ − P − = | ψ ih ψ | − | I s ψ ih I s ψ | , P ψ = ψ ∈ S H Λ . Its BDF energy is E BDF ( Q ψ ) = 2 h|D | ψ , ψ i − α x | ψ ∧ I s ψ ( x, y ) | | x − y | dxdy ≥ m + 2 h (cid:0) |D | − m (cid:1) ψ , ψ i − αD (cid:0) | ψ | , ψ (cid:1) =: 2 m + G I s ( ψ ) . We recall the following |D | − m = 1 |D | + m (cid:0) ( g ( − i ∇ ) − m )( g ( − i ∇ ) + m ) + g ( − i ∇ ) (cid:1) . hanks to Estimates (32) and Kato’s inequality (37), we have G I s ( ψ ) ≤ (1 − Kα ) h − ∆2 |D | ψ , ψ i − α π h|∇| ψ , ψ i We split ψ into two with respect to the frequency cut-off Π αK : we get ψ = Π αK ψ + ψ = ψ + ψ . The constant K is chosen such that α K e E ( αK ) ? απαK . Then we have D (cid:0) | ψ | , | ψ | (cid:1) = D (cid:0) | ψ | , | ψ | (cid:1) + O (cid:0) h|∇| ψ , ψ i + k| ψ | k C k |∇| / ψ k L (cid:1) = D (cid:0) | ψ | , | ψ | (cid:1) + O (cid:0) h|∇| ψ , ψ i + √ α k |∇| / ψ k L (cid:1) , where we recall that k ρ k C = D ( ρ, ρ ) . This gives G I s ( ψ ) = h g ( − i ∇ ) |D | + m ψ , ψ i − α π D (cid:0) | ψ | , | ψ | (cid:1) + K h g ( − i ∇ ) |D | ψ , ψ i + O ( α ) , ≥ α g ′ (0) m k∇ ψ k L − α D (cid:0) | ψ | , | ψ | (cid:1) + O ( α ) , ≥ α m g ′ (0) E PT (1) + O ( α ) . (111)We have obtained a lower bound. Let us prove that it is attained up to an error O ( α ) . That is let us prove there exists a unitary ψ ∈ Ran P such that E BDF ( Q ψ ) − m = G I s ( ψ ) + O ( α )= α mg ′ (0) E PT (1) + O ( α ) . (112)As in [Sok14b], we consider the unique positive radially symetric Pekar minimizer φ PT in L ( R , C ) . We form φ := φ PT ∈ L ( R , C ) , (113)which is a Pekar minimizer in the space of spinors. We scale this wave function by λ − := αmg ′ (0) : ∀ x ∈ R , φ λ − ( x ) := λ − / φ ( λ − x ) . (114)To get a proper ψ ∈ Ran P , we form ψ := 1 kP φ λ − k L P φ λ − . (115)Our trial state is: Q := | ψ ih ψ | − | I s ψ ih I s ψ | . (116)We do not compute its energy: the method is as in [Sok14b] (except that insteadof I s , the operator C is considered in [Sok14b], but that does not change anything).Eventually we refer the reader to the proof of the upper bound of E t X ℓ above inSection 3.2 for the ideas. .6.2 Proof of Lemma 4 We remark the following fact.
Lemma 6.
Let S I s ⊂ H Λ be the set S I s = (cid:8) f ∈ H Λ , k f k L = 1 , h f , I s f i = 0 (cid:9) = (cid:8) f ∈ H Λ , k f k L = 1 , Im hP − f , I s P f i = 0 (cid:9) . There exists a smooth angle operator A : S I s → R /π Z .For two C -colinear wave functions f , f in S I s we have A ( f ) = A ( f ) .Furthermore we have A − (0) = Ran P − and A − ( π ) = Ran P . Proof:
Let f be in S I s : the space Span C ( f, I s f ) is spanned by the eigenvectors g − := f + i I s f k f + i I s f k L and g + := f − i I s f k f − i I s f k L . We haveSpan C ( f, I s f ) = Span ( P − g ± , P g ± ) . It follows that P ± f k P ± g + and P − f k I s P f . As f ∈ S I s , for ε ∈ { + , −} with P ε f = 0 , we have P − ε f ∈ Span R ( P ε f ) . Thus we have with
Span R ( f, I s f ) = Span R ( e − , I s e − ) , e − ∈ Ran P − and k e − k L = 1 . (117)Indeed if P − f = 0 we can choose e − := P − f kP − f k L , else we can choose e − :=I s P f kP f k L .Then we decompose f w.r.t. the basis ( e − , I s e − ) and there exists θ ∈ R / (2 π Z ) with f = cos( θ ) e − + sin( θ )I s e − . In fact the function f ( e − , I s e − ) that maps f toa basis (117) is bi-valued: if ( e − , I s e − ) is a possibility, then ( − e − , − I s e − ) is anotherpossibility. It follows that the angle θ is defined up to π : we thus obtain a function A : S I s → R /π Z . The smoothness of A is straightforward. The end of the proof is also clear. We use the angle operator to get a mountain pass argument: see Lemma 7 below.We use and Theorem 3 and Proposition 6.Let U ⊂ M I be the open subset U ⊂ M I := n P = Q + P − ∈ M I , dim Ker( Q − k Q k B ) = 1 o . For all P = Q + P − ∈ U , the eigenspace Ker( Q − k Q k B ) is spanned by a unitaryvector f . By I s -symmetry, we have I s Ker( Q − k Q k B ) = Ker( Q + k Q k B ) , and we have h f , I s f i = 0 . By Proposition 6, the plane Span C ( f, I s f ) is spannedby f − ∈ Ran P and f + ∈ Ran (1 − P ) .By I s -symmetry, we have I s f − ∈ R f + . In other words: the wave function f − is in S I s . Definition 4.
Let Q + P − ∈ U ⊂ M I and f − as above. We define the smoothfunction A U as follows: A U : Q + P − ∈ U ⊂ M I
7→ A ( f − ) . It is clear it does not depend on the choice of f − but is a function of C f − . Further-more, we have ∀ P ∈ U , ∇A U ( P ) = 0 he following Lemma is an application of classical results in geometry. Lemma 7.
Let M U, I s be the subset M U, I s := (cid:8) Q + P − ∈ U , k Q k B = 1 (cid:9) = A − U (cid:0)(cid:8) π (cid:9)(cid:1) , in other words the set of projectors in U whose range intersects nontrivially Ran P .For any differentiable function c : ( − ε, ε ) → M I such that ε > , c (0) ∈ M U, I s and Tr (cid:0) ∇A U ( c (0)) ∗ dds c (0) (cid:1) = 0 , the following holds: any sufficiently small smooth perturbation c + δc : ( − ε, ε ) → M I , in the norm k e c k := sup s ∈ ( − ε,ε ) k e c ( s ) − P − k S + sup s ∈ ( − ε,ε ) k dds e c ( s ) k S still intersects M U, I s at some s ( δc ) . – Let us now prove Lemma 4. We recall that we have defined a loop c ψ = c thatcrosses M U, I s at s = and we can easily check that Tr (cid:0) A U ( c (2 − )) ∗ dds c (2 − ) (cid:1) =1 = 0 . Furthermore we have defined the family ( c t ) t ≥ by c t := Φ BDF ; t ( c ψ ) where Φ BDF ; t is the gradient flow of the BDF energy.– By Lemma 7, the loop c t still intersects M U, I s for sufficiently small t . We mustensure that this fact holds for all t ≥ to end the proof.We use a continuation principle and set t ∞ := sup n t ≥ , ∀ ≤ τ ≤ t, ∃ s ∈ [0 , c τ crosses M U, I s at s = s o . We also define for all ≤ τ < t ∞ : s − ( τ ) = sup { s ∈ [0 , , ∀ s ′ ≤ s, k c τ ( s ′ ) k B < } > ,s + ( τ ) = inf { s ∈ [0 , , ∀ s ′ ≥ s, k c τ ( s ′ ) k B < } < . – We assume that t ∞ < + ∞ and prove this implies a contradiction.The initial loop c induces L : s ∈ [0 ,
7→ A U ( c ( s )) = πs ∈ T , and we notice that L has a non-trivial homotopy.Thus, at least for τ close to , the following holds.1. There exist < η τ , η ′ τ ≪ such that A U (cid:2) c τ (cid:0) ( s − ( τ ) − η τ , s − ( τ )) (cid:1)(cid:3) ∩ ( π , π + η ′ τ ) = ∅ . (118)2. There exist < η τ , η ′ τ ≪ such that A U (cid:2) c τ (cid:0) ( s + ( τ ) , s + ( τ ) + η τ ) (cid:1)(cid:3) ∩ ( π − η ′ τ , π ) = ∅ . (119)The functions τ ≥ s ± ( τ ) are well-defined and continuous in a neighbourhoodof with s − (0) = s + (0) = . – We prove that by continuity in τ we have ∀ s ∈ [0 , , k c τ ( s ) k B = 1 ⇒ c τ ( s ) + P − ∈ M U, I s (120)and in particular c τ ( s ± ( τ )) ∈ M U, I s − P − . (121) f not, this implies that as τ increases, the second highest eigenvalue of c τ ( s ) alsoincreases to reach where (118) becomes false, at some ( τ , s ) .This cannot occurs because of the energy condition: if this was true, we wouldhave by Kato’s inequality (37) E BDF (cid:0) c τ ( s ) (cid:1) ≥ (1 − α π )Tr (cid:0) |D | c τ ( s ) (cid:1) ≥ m (1 − α π ) > m. Thus (120)-(121) hold for all ≤ τ < t ∞ .– Thanks to this fact, by continuity for all ≤ τ < t ∞ , (118)-(119) hold: if we followthe point s ± ( τ ) from τ = 0 , we see that there cannot exist τ such that (118) or(119) becomes false, because the set { t ≥ , ∀ ≤ τ < t, (118) (resp. (119)) holdsfor τ } is non-empty and open.– Up to an isomorphism of [0 , , we can suppose that for all ≤ τ ≤ t ∞ , ∀ s ∈ [0 , , k ∂ s c τ ( s ) k S > . Remark . In S , the function ∂ s c t ( s ) satsifies the following equation: ddt ∂ s c t ( s ) = ∂ s ∇E BDF ( c t ( s )) ∈ S . These new loops are written e c τ and have the same range as the c τ ’s and definethe same arc length.Studying the limit of e c τ as τ tends to t ∞ , we get that at t = t ∞ , (118)-(119) stillholds for the loop e c t ∞ at some < s − ( t ∞ ) ≤ s + ( t ∞ ) < .Then necessarily, the loop e c t ∞ crosses M U, I s at some s ∈ [ s − ( t ∞ ) , s + ( t ∞ )] . Goingback to c t ∞ , this proves that the same holds for c t ∞ , which contradicts the definitionof t ∞ . Let Φ ′ SU : SU (2) → U ( E ) , E ⊂ H Λ be an irreducible representation of Φ SU . As J and S commutes with the action of SU (2) , then necessarily E is an eigenspace for J and S , associated to j ( j + 1) and κ j = ε ( j + ) where j ∈ + Z + and ε = ± . The eigenspaces are known [Tha92, p.126]: they are spanned by wave functions of type ∀ x = rω x ∈ R , ψ ( x ) := a ( r )Φ ± m,κ j , m = − j, − j + 1 , . . . , j, (122)where a ( r ) ∈ L ( R + , r dr ) , (123a) Φ + m, ± ( j + 12 ) := i Ψ mj ± ! and Φ − m, ± ( j + 12 ) := mj ∓ ! (123b) Ψ mj − = 1 √ j √ j + mY m − j − √ j − mY m + 12 j − and Ψ mj + 12 = 1 √ j + 2 √ j + 1 − mY m − j + 12 −√ j + 1 + mY m + 12 j + 12 . (123c)We recall that the Y mℓ are the spherical harmonics (eigenvectors of L ).Hence E is spanned by a wave function which is a linear combination of that oftype (122).We recall that for any integer n ≥ there is but one irreducible representationof SU (2) of dimension n up to isomorphism. They can be found by the number f eigenvalues of J ′ , the infinitesimal "rotation" around the z axis which induces arepresentation of SO (3) .. Here J ′ corresponds to J .Thus we get that for ε ∈ { + , −} E ε := Φ SU a ( r )Φ εj,κ j is irreducible with respect to Φ SU . By unicity of the irreducible representation ofdimension j + 1 , there exists an isomorphism from E − to E + . As there must bea correspondence between the eigenspace of J ( E − ) and that of J ( E + ) , necessarily C a Φ − m,κ j is sent to C a Φ + m,κ j .In particular as P ↑ E and P ↓ E are also representation of SU (2) with same eigen-values of J , S (or = { } ). If one of them is zero then E is of type E ± . If both arenon-zero, then there exists a ↑ ( r ) , a ↓ ( r ) such that P ↑ E = Φ SU a ↑ ( r )Φ + j,κ j and P ↓ E = Φ SU a ↓ ( r )Φ − j,κ j . Both P ↑ E and P ↓ E are irreducible. We can suppose that there exists f ∈ E with P ↑ f = a ↑ ( r )Φ + j,κ j and P ↓ f = a ↓ ( r )Φ − j,κ j . The isomorphism between the two representations implies that E = Φ SU (cid:0) a ↑ ( r )Φ + j,κ j + a ↓ ( r )Φ − j,κ j (cid:1) . We have to prove that M I and W are submanifold of M . The method is similarto the one used in [Sok14b] to prove that M C is a submanifold of M .Let P = Q + P − ∈ M . We will prove that in a neighbourhood of P in P − + S ,the projectors P in M I (resp. W ) can be written as P = e A P e − A , where A ∈ m I P (resp. m W P ).– If we assume this point, then it is clear that the two sets are submanifolds of M .Indeed e A is a global linear isometry of H Λ , whose restriction to the m · P ’s maps m · P onto m · P .Equivalently it maps the first tangent plane onto the other: { [ a, P ] , a ∈ m · P } → ≃ { [ a, P ] , a ∈ m · P } . – We use Theorem 3 to write Q = + ∞ X j =1 λ j (cid:0) | f j ih f j | − | f − j ih f − j | (cid:1) (124)where ( λ i ) i ∈ ℓ is non-increasing and the f i ’s form an orthonormal basis of Ran Q .Provided that k P − P k S < , then λ < and there is no j such that f j or f − j is in the range of P or P − .We decompose with respect with the eigenvalues µ > µ > · · · > as follows: Q = + ∞ X k =1 µ k (cid:0) Proj
Ker( Q − µ k ) − Proj
Ker( Q + µ k ) (cid:1) . For short we write µ − k := − µ k , and M k := Proj
Ker( Q − µ k ) and E Q µ k := Ker( Q − µ k ) . (125)As any Y ∈ { C , I s } is an isometry (linear or antilinear) and as the eigenvaluesare the sine of the angles between vectors in P and P − , for any k we have Y E Q µ k = E Q − µ k (126)and the eigenspaces E Q µ k ⊕ E Q − µ k = Ker( Q − µ k ) are invariant under Y . ase of W – In the case Y = C and P ∈ W , each eigenspace Ker( Q − µ k ) is also invariant under the action of Φ SU . In other words, Ker( Q − µ k ) is a finite dimensional representation of Φ SU , and we can decompose it intoirreducible representations E ( ℓ ) µ k , where ≤ ℓ ≤ ℓ k .By C -symmetry, we have C E ( ℓ ) µ k = E ( ℓ ′ ) − µ k , there is a one-to-one correspondence between irreducible representations of type E ( ℓ ) µ k and that of type E ( ℓ ) − µ k . Up to changing indices ℓ ′ j , we can suppose that C E ( ℓ ) µ k = E ( ℓ ) − µ k , ≤ ℓ ≤ ℓ k . Decomposing E ( ℓ ) µ k with respect with P − and P , we see that P ± E ( ℓ ) µ k is irreducible , and from the spectral decomposition of Q P − E ( ℓ ) µ k ⊕ P E ( ℓ ) µ k = E ( ℓ ) µ k ⊕ F − µ k , where F − µ k is an irreducible subset of Ker( Q + µ k ) .– Let us show that F − µ k ∩ C E ( ℓ ) µ k = { } . (127)Indeed, from Lemma 1 and the expression of the Φ ± m,κ , we see that CKer (cid:0) J − m (cid:1) = Ker (cid:0) J + m (cid:1) . Thus if the intersection is non-zero, then we have by C -symmetry and Φ SU -symmetry: F − µ k = C E ( ℓ ) µ k . But as shown in [Sok14b], this cannot happen: let us say that E ( ℓ ) µ k is associated tothe eigenvalues j ( j + 1) , κ of J resp. S . We consider: Ker( J − m ) ∩ P ± E ( ℓ ) µ k = C e ± ; m , − j ≤ m ≤ j , k e ± ; m k L = 1 . We would have C e ± ; m = exp iθ ( ± ; m ) e ∓ ; − m . The constant θ ( ± ; m ) does not depend on m by Φ SU -symmetry. Moreover, if Ker( J − m ) ∩ E ( ℓ ) µ k = C f m , then P ± f m k e ± ; m . As in [Sok14b] for M C , the condition C = 1 implies θ + − θ − ≡ π ] while − C Q C = Q implies θ + − θ − ≡ π [2 π ] , which cannot occur.Similarly, we can prove that (127) holds and that in fact F − µ k is orthogonal to C E ( ℓ ) µ k .As a consequence, the number of E ( ℓ ) µ k ’s is even, or equivalently, the number of P − E ( ℓ ) µ k is even.– The fact that P = e A P e − A , with Φ SU A = A, C A C = A, k A k S < + ∞ , (128)follows from Theorem 3 and the different symmetries. he f j ’s in (124) can be written as ( λ j = sin( θ j ) ) f j = r − λ j e − ; j + r λ j e +; j , P ± e ± ; j = e ± ; j . We also have f − j = − r λ j e − ; j + r − λ j e +; j . Then we define A = + ∞ X j =1 θ j (cid:0) | e +; j ih e − ; j | − | e − ; j ih e +; j | (cid:1) . (129)It is easy to check that A satisfies (128). In fact, we can assume that f j spans anirreducible representation of SU (2) , and in this case the same holds for e +; j and e − ; j .As in Section 4.1, the correspondence e − ; j e +; j induces an isomorphism be-tween Φ SU e − ; j and Φ SU e +; j . This fact together with the Φ SU -symmetry impliesthat ∀ U ∈ Ran Φ SU , UAU − = A. The fact that C A C = A was proved in [Sok14b] in the case P , P ∈ M C . Here thisremains true because W ⊂ M C . – We can now determine the connected component of W . Let P , P be in W andlet Q = P − P .We consider E Q := Ker( Q − . If E Q = { } , then we can write P = e A P e − A as in (129). And we see that thepath in ℓ : t ∈ [0 , ( tθ j ) j ∈ ℓ induces a path connecting P and P .If E Q = { } , we count the number of irreducible representation in E Q : let b j,κ j be the number of irr. rep. in Ker (cid:0) J − j ( j + 1) (cid:1) ∩ Ker (cid:0) S − κ j (cid:1) . If all the b j,κ j ’s are even, we can still write P as P = e A P e − A with A as in (129)with the first θ j equal to π . In particular the two projectors can be connected by apath in W .Let us say that b j ,κ ≡ for some j , κ . We have shown that for P ∈ W with k P − P k B < , the number of planes Π j ’s in the decomposition of Theorem 3is even. Precisely, due to the C -symmetry, there exists a sequence ( ℓ µ ( j, κ )) j in N ,with Ker (cid:0) ( P − P ) − µ (cid:1) ∩ Ker (cid:0) J − j ( j + 1) (cid:1) ∩ Ker (cid:0) S − κ (cid:1) = L ≤ ℓ ≤ ℓ µ ( j,κ ) E ( ℓ ) µ , where each E ( ℓ ) µ is irreducible as a representation of Φ SU and ℓ µ ( j, κ ) is even .We show that there cannot exist a continuous path linking P and P by acontradiction argument.Let us say that γ : t ∈ [0 , → W is a continuous path with γ (0) = P and k γ (1) − P k B = 1 .Then by the previous remarks, we have by continuity: ∀ t ∈ [0 , , ∀ j ∈ + Z + , ∀ κ ∈ (cid:8) ± (cid:0) j + (cid:1)(cid:9) ,ℓ ( Q t = γ ( t ) − P ; j, κ ) ≡ . In particular it is not possible to have γ (1) = P . ase of M I For
Y = I s and P ∈ M I , we use (126). For each f ∈ E Qµ , we have I s ∈ E Q − µ where µ ∈ σ ( Q ) . We may assume that µ > .Thus the plane Π :=
Span (cid:0) f, I s f (cid:1) is invariant under Q and I s . We decompose f and I s f with respect to P and − P .By a dimension argument:1. either µ = 1 , P f = 0 and (1 − P )I s f = 0 ,2. or < µ < and C P f = C P I s f and C (1 − P ) f = C (1 − P )I s f. In each case, we write e − a unitary vector in Ran P ∩ Π and e + = I s e − .If we consider the sequence ( µ i ) i of positive eigenvalues of Q (counted withmultiplicities), we get the correspondent sequences ( e − ; j ) j and ( e +; j ) . Moreover byTheorem 3, we know that µ j = sin( θ j ) where θ j ∈ [0 , π ] is the angle between thetwo lines C e − ; j and C f j .Provided that we take − θ j instead of θ j and up to a phase, we can suppose that f j = cos( θ j ) e − ; j + sin( θ j )I s e − ; j . In particular we have P = e A P e − A , with A = X θ j (cid:0) | e +; j ih e − ; j | − | e − ; j ih e +; j | (cid:1) . It is straightforward to check that I s A I − = A . Acknowledgment
The author wishes to thank Éric séré for useful discussions andhelpful comments. This work was partially supported by the Grant ANR-10-BLAN0101 of the French Ministry of research.
References [BBHS98] V. Bach, J.-M. Barbaroux, B. Helffer, and H. Siedentop. On the stabilityof the relativistic electron-positron field.
Comm. Math. Phys , 201:445–460, 1998.[BP87] J. Borwein and D. Preiss. A smooth variational principle with applica-tions to subdifferentiability and to differentiability of convex functions.
Trans. Am. Math. Soc. , 303(2):517–527, 1987.[CI89] P. Chaix and D. Iracane. From quantum electrodynamics to mean-fieldtheory: I. the Bogoliubov-Dirac-Fock formalism.
J. Phys. B: At. Mol.Opt. Phys. , 22:3791–3814, 1989.[Eke74] I. Ekeland. On the variational principle.
J. Math. Anal. Appl. , 47:324–353, 1974.[Fur04] K. Furutani. Fredholm-Lagrangian-Grassmannian and the Maslov index.
Journal of Geometry and Physics , 51(3):269–331, 2004.[GLS09] Ph. Gravejat, M. Lewin, and É. Séré. Ground state and charge renormal-ization in a nonlinear model of relativistic atoms.
Comm. Math. Phys ,286, 2009.[HLS05a] C. Hainzl, M. Lewin, and É. Séré. Existence of a stable polarized vacuumin the Bogoliubov-Dirac-Fock approximation.
Comm. Math. Phys , 257,2005. HLS05b] C. Hainzl, M. Lewin, and É. Séré. Self-consistent solution for the polar-ized vacuum in a no-photon QED model.
J. Phys. A: Math and Gen. ,38(20):4483–4499, 2005.[HLS07] C. Hainzl, M. Lewin, and J. P. Solovej. The mean-field approximation inquantum electrodynamics. the no-photon case.
Comm. Pure Appl. Math. ,60(4):546–596, 2007.[HLS09] C. Hainzl, M. Lewin, and É. Séré. Existence of atoms and moleculesin the mean-field approximation of no-photon quantum electrodynamics.
Arch. Rational Mech. Anal , 192(3):453–499, 2009.[Kar04] S. G. Karshenboim. Precision study of positronium: testing bound sateQED theory.
Int. J. Mod. Phys. A , 19(23):3879–3896, 2004.[LL97] E. H. Lieb and M. Loss.
Analysis . AMS, 1997.[LS00] E. H. Lieb and H. Siedentop. Renormalization of the regularized relativis-tic electron-positron field.
Comm. Math. Phys. , 213(3):673–683, 2000.[RS75] M. Reed and B. Simon.
Methods of Modern Mathematical Physics , volumeI-II. Academic Press Inc., 1975.[Sim79] B. Simon.
Trace Ideals and their Applications , volume 35 of
LondonMathematical Society Lecture Notes Series . Cambridge University Press,1079.[Sok13] J. Sok. Charge renormalisation in a mean-field approximation of QED,2013. preprint, http://arxiv.org/abs/1311.6575.[Sok14a] J. Sok. Existence of ground state of an electron in the BDF approxima-tion.
Rev. Math. Phys. , 26(5), 2014.[Sok14b] J. Sok. The positronium in a mean-field approximation of quantum elec-trodynamics, 2014. http://arxiv.org/abs/1405.3928.[Tha92] B. Thaller.
The Dirac Equation . Springer Verlag, 1992.. Springer Verlag, 1992.