The positronium in a mean-field approximation of quantum electrodynamics.

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The positronium in a mean-field approximation of quantum electrodynamics.

Jérémy Sok

To cite this version:

Jérémy Sok. The positronium in a mean-field approximation of quantum electrodynamics.. 2014.

�hal-00991429�

The positronium in a mean-field

approximation of quantum electrodynamics.

Sok Jérémy

Ceremade, UMR 7534, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny,

75775 Paris Cedex 16, France.

May 15, 2014

Abstract

The Bogoliubov-Dirac-Fock (BDF) model is a no-photon, mean-field approxi- mation of quantum electrodynamics. It describes relativistic electrons in the Dirac sea. In this model, a state is fully characterized by its one-body density matrix, an infinite rank nonnegative operator. We prove the existence of the positronium, the bound state of an electron and a positron, represented by a critical point of the energy functional in the absence of external field. This state is interpreted as the ortho-positronium, where the two particles have parallel spins.

Contents

1 Introduction and main results 2

2 Description of the model 7

2.1 The BDF energy . . . . 7

2.2 Structure of manifold . . . . 8

2.3 Form of trial states . . . . 9

3 Proof of Theorem 1 10

3.1 Strategy and tools of the proof . . . . 10

3.2 Upper and lower bounds of E

. . . . 15

3.3 Existence of a minimizer for E

. . . . 18

3.4 Proof of Theorems 2 and 3 . . . . 25

4 Proofs on results on the variational set 25

4.1 On the manifold

: Theorem 4, Propositions 1, 2 . . . . 25

4.2 On the manifold

: Propositions 3, 4 and 5 . . . . 29

1 Introduction and main results

The Dirac operator

In relativistic quantum mechanics, the kinetic energy of an electron is described by the so-called Dirac operator D

. Its expression is [Tha92]:

D

:= m

c

β − i

c

j=1

α

∂

(1)

where m

is the (bare) mass of the electron, c the speed of light and

the reduced Planck constant, β and the α

’s are 4 × 4 matrices defined as follows:

β :=

Id

0 0 − Id

, α

0 σ

σ

0

, j ∈ { 1, 2, 3 }

σ

:=

0 1 1 0

, σ

:=

0 − i i 0

, σ

1 0

− 1 0

.

The Dirac operator acts on spinors i.e. square-integrable

-valued functions:

H

:= L

,

. (2)

It corresponds to the Hilbert space associated to one electron. The operator D

is self-adjoint on

with domain H

(

,

), but contrary to − ∆/2 in quantum mechanics, it is unbounded from below.

Indeed its spectrum is σ(D

) = ( −∞ , m

c

] ∪ [m

c

, + ∞ ). Dirac postulated that all the negative energy states are already occupied by "virtual electrons", with one electron in each state, and that the uniform filling is unobservable to us. Then, by Pauli’s principle real electrons can only have a positive energy.

It follows that the relativistic vacuum, composed by those negatively charged virtual electrons, is a polarizable medium that reacts to the presence of an external field. This phenomenon is called the vacuum polarization.

If one turns on an external field that gets strong enough, it leads to a transition of an electron of the Dirac sea from a negative energy state to a positive one. The resulting system – an electron with positive energy plus a hole in the Dirac sea – is interpreted as an electron-positron pair. Indeed the absence of an electron in the Dirac sea is equivalent to the addition of a particle with same mass and opposite charge: the positron.

Its existence was predicted by Dirac in 1931. Although firstly observed in 1929 independently by Skobeltsyn and Chung-Yao Chao, it was recognized in an experi- ment lead by Anderson in 1932.

Charge conjugation

Following Dirac’s ideas, the free vacuum is described by the negative part of the spectrum σ(D

):

P

= χ

(D

).

The 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

(

), Cψ(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 the following:

∀ ψ ∈ L

, ∀ x ∈

, | Cψ(x) |

= | ψ(x) |

. (5)

The positronium is the bound state of an electron and a positron. This system was independently predicted by Anderson and Mohoroviˇcić in 1932 and 1934 and was experimentally observed for the first time in 1951 by Martin Deutsch.

It is unstable: depending on the relative spin states of the positron and the electron, its average lifetime in vacuum is 125 ps (para-positronium) or 142 ns (ortho- positronium) (see [Kar04]).

In this paper, we are looking for a positronium state within the Bogoliubov-Dirac- Fock (BDF) model: the state we found can be interpreted as the ortho-positronium where the electron and positron have parallel spins. Our main results are Theorem 1 and 3. In our state, the wave function of the real electron and that of the virtual electron defining the positronium are charge conjugate of each other.

BDF model

The BDF model is a no-photon approximation of quantum electrodynamics (QED) which was introduced by Chaix and Iracane in 1989 [CI89], and studied in many papers [BBHS98, HLS05a, HLS05b, HLS07, HLS09, GLS09, Sok12].

It allows to take into account real electrons together with the Dirac vacuum in the presence of an external field.

This is a Hartree-Fock type approximation in which a state of the system "vac- uum + real electrons" is given by an infinite Slater determinant ψ

∧ ψ

∧ · · · . Equivalently, such a state is represented by the projector onto the space spanned by the ψ

’s: its so-called one-body density matrix. For instance P

represents the free Dirac vacuum.

Here we just give main ideas of the derivation of the BDF model from QED, we refer the reader to [CI89, HLS05a, HLS07] for full details.

Remark 1. To simplify the notations, we choose relativistic units in which, the mass of the electron m

, the speed of light c and

are set to 1.

Let us say that there is an external density ν, e.g. that of some nucleus and let us write α > 0 the so-called fine structure constant (physically e

/(4πε

c), where e is the elementary charge and ε

the permittivity of free space).

The starting point is the (complicated) Hamiltonian of QED

that acts on the Fock space of the electron F

[Tha92]. The (formal) difference between the infinite energy of a Hartree-Fock state Ω

and that of Ω

−

, state of the free vacuum taken as a reference state, gives a function of the reduced one-body density matrix Q := P − P

.

It can be shown that a projector P is the one-body density matrix of a Hartree- Fock state in F

iff P − P

is Hilbert-Schmidt, that is compact such that its singular values form a sequence in ℓ

.

To get a well-defined energy, one has to impose an ultraviolet cut-off Λ > 0: we replace

by its subspace

H_Λ

:=

f ∈

supp f

⊂ B (0, Λ) .

This procedure gives the BDF energy introduced in [CI89] and studied for instance in [HLS05a, HLS05b].

Notation 1. Our convention for the Fourier transform

is the following

∀ f ∈ L

(

), f(p) :=

1 (2π)

Z

f(x)e

dx.

Let us notice that

is invariant under D

and so under P

.

For the sake of clarity, we will emphasize the ultraviolet cut-off and write Π

for the orthogonal projection onto

: Π

is the following Fourier multiplier

Π

:=

χ

. (6)

By means of a thermodynamical limit, Hainzl et al. showed in [HLS07] that the formal minimizer and hence the reference state should not be given by Π

P

but by another projector P

in

that satisfies the self-consistent equation in

:







P

−

= −sign D

, D

= D

− α

2

( P

−

)(x − y)

| x − y |

(7)

We have P

= χ

( D

).

In

, the operator D

coincides with a bounded, matrix-valued Fourier multiplier whose kernel is

⊂

.

The resulting BDF energy E

is defined on Hartree-Fock states represented by their one-body density matrix P :

N

:=

P ∈ B (H

), P

= P

= P, P − P

∈

(H

) .

This energy depends on three parameters: the fine structure constant α > 0, the cut-off Λ > 0 and the external density ν. We assume that ν has finite Coulomb energy, that is

D(ν, ν) := 4π

R3

|b ν(k) |

| k |

dk. (8)

Remark 2. The Coulomb energy coincides with

R3×R3 ν(x)^∗ν(y)

|x−y|

dxdy whenever this integral is well-defined.

Remark 3. The operator D

was previously introduced by Lieb et al. in [LS00] in another context in the case α log(Λ) small.

Notation 2. We recall that B (

) is the set of bounded operators and

(

) the set of compact operators whose singular values form a sequence in ℓ

[RS75, Appendix IX.4 Vol II], [Sim79] (p ≥ 1). In particular

(

) is the set Comp(

) of compact operators.

Notation 3. Throughout this paper we write m = inf σ |D

|

≥ 1, (9)

and

P

:= Π

− P

= χ

( D

). (10) The same symmetry holds for P

and P

: the charge conjugation C maps Ran P

onto Ran P

.

Minimizers and critical points

The charge of a state P ∈

is given by the so-called P

-trace of P − P

Tr

−

P − P

:= Tr P

(P − P

) P

+ P

(P − P

) P

.

This trace is well defined as we can check from the formula [HLS05a]

(P − P

)

= P

(P − P

) P

− P

(P − P

) P

. (11)

Minimizers of the BDF energy with charge constraint N ∈

corresponds to ground

states of a system of N electrons in the presence of an external density ν.

The problem of their existence was studied in several papers [HLS09, Sok12, Sok13]. In [HLS09], Hainzl et al. proved that it was sufficient to check binding inequalities and showed existence of ground states in the presence of an external density ν, provided that N − 1 <

ν, under technical assumptions on α, Λ.

In [Sok12], we proved that, due to the vacuum polarization, there exists a mini- mizer for E

with charge constraint 1: in other words an electron can bind alone in the vacuum without any external charge (still under technical assumptions on α, Λ).

In [Sok13], the effect of charge screening is studied: due to vacuum polarization, the observed charge of a minimizer P 6 = P

is different from its real charge Tr

−

(P − P

).

Here we are looking for a positronium state, that is an electron and a positron in the vacuum without any external density. So we have to study E

on

M

:=

P ∈

, Tr

−

P − P

= 0

. (12)

From a geometrical point of view

is a Hilbert manifold and E

is a differentiable map on

(Propositions 1 and 2).

We thus seek a critical point on

, that is some P ∈

, P 6 = P

such that

∇E

(P ) = 0. We also must ensure that this is a positronium state. A good candidate is a projector P that is obtained from P

by substracting a state ψ

∈ Ran P

and adding a state ψ

∈ Ran P

, that is

P = P

+ | ψ

ih ψ

| − | ψ

ih ψ

| . (13) But there is no reason why such a projector would be a critical point. If it were that would mean that there exists a positronium state in which, apart from the excitation of the virtual electron giving the electron-positron pair, the vacuum is not polarized.

Keeping (13) in mind, we identify a subset

⊂

, made of C-symmetric states.

Definition 1.

The set

of C-symmetric states is defined as:

M_C

= { P ∈

, − C(P − P

)C = P − P

} . (14) Remark 4. Let P ∈

. As − C( P

− P

)C = P

− P

, writing

P − P

=

P − (Π

− P) − P

+ P

, there holds:

P ∈

⇒ P + CP C = Π

, (15)

that is

∀ P ∈

, C : Ran P → Ran(Π

− P ) is an isometry.

The set

has fine properties: this is a submanifold, invariant under the gra- dient flow of E

(Proposition 3). Moreover it has two connected components

and

(Proposition 4). In particular, any extremum of the BDF energy restricted to

is a critical point on

.

So we are lead to seek a minimizer over each of these connected components: the first (

) gives P

, which is the global minimizer over

, but the second gives a non-trivial critical point. It corresponds to the positronium and is a perturbation of a state which can be written as in (13).

Our main Theorems are the following:

Theorem 1.

There exist α

, Λ

, L

> 0 such that if α ≤ α

, Λ

≤ Λ

, and α log(Λ) ≤ L

, then there exists a minimizer of E

over

. Moreover we have

E

:= inf {E

(P ), P ∈

} ≤ 2m + α

m

g

(0)

E

+ O (α

),

where E

< 0 is the Choquard-Pekar energy defined as follows [Lie77]:

E

= inf

k∇ φ k

− D | φ |

, | φ |

, φ ∈ L

(

), k φ k

= 1

. (16)

Under the same assumptions as in Theorem 1, let P be a minimizer for E

. Then there exists an anti-unitary map A ∈

), and P

of form (13) such that

P = e

P

e

,

e

ψ

= ψ

, ε ∈ { +, −} and ψ

= Cψ

, A =

A, P

, P

∈

(

), k A k

α, and CAC = A.

(17)

Moreover, the following holds:

E

= 2m + α

m

g

(0)

E

+ O (α

). (18) We emphasize that ψ

does not represent the electron state.

Theorem 3.

Under the same assumptions as in Theorem 1, let P be a minimizer for E

and Q

= P − P

. Let

be

π

:= χ

Π

D

Π

. (19) Then there holds Ran (Π

−

) ∩ Ran P =

ψ

. The unitary wave function ψ

satisfies the equation

D

ψ

= µ

ψ

, (20)

where µ

is some constant

K

α

≤ m − µ

≤ K

α

, K

, K

> 0.

By C-symmetry ψ

:= Cψ

satisfies D

ψ

= − µ

ψ

, and we have

P =

+ | ψ

ih ψ

| − | ψ

ih ψ

| . (21) Moreover the following holds. We split ψ

into upper spinor ϕ

∈ L

(

,

) and lower spinor χ

∈ L

(

,

) and scale ϕ

by λ :=

:

e

ϕ

(x) := λ

ϕ

λx .

Then in the non-relativistic limit α → 0 (with α log(Λ) kept small), the lower spinor χ

tends to 0 and, up to translation, ϕ

tends to a Pekar minimizer.

Remark 5. As ψ

and ψ

= Cψ

have antiparallel spins, the state P represents one electron in state ψ

and the absence of one electron in state ψ

in the Dirac sea, that is an electron and a positron with parallel spins.

Remark 6. To prove that ϕ

tends to a Pekar minimizer up to translation, it suffices to prove that its Pekar energy tends to E

[Lie77].

Notation 4. Throughout this paper we write K to mean a constant independent of α, Λ. Its value may differ from one line to the other. We also use the symbol

: 0 ≤ a

b means there exists K > 0 such that a ≤ Kb.

Remarks and notations about

D

D

has the following form [HLS07]:

D

= g

( − i ∇ )β − i

· ∇

|∇| g

( − i ∇ ) (22) where g

and g

are smooth radial functions on B(0, Λ) and

= (α

)

. Moreover we have:

∀ p ∈ B(0, Λ), 1 ≤ g

(p), and | p | ≤ g

(p) ≤ | p | g

(p). (23)

Notation 5. For α log(Λ) sufficiently small, we have m = g

(0) [LL97, Sok12].

Remark 7. In general the smallness of α is needed to ensure technical estimates hold. The smallness of α log(Λ) is needed to get estimates of D

: in this case D

can be obtained by a fixed point scheme [HLS07, LL97], and we have [Sok12, Appendix A]:

g

(0) = 0, and k g

k

, k g

k

≤ Kα

k g

− 1 k

≤ Kα log(Λ) ≤

and k g

k

1. (24)

Acknowledgment The author wishes to thank Éric Séré for useful discussions and helpful comments. This work was partially supported by the Grant ANR-10-BLAN 0101 of the French Ministry of Research.

2 Description of the model

2.1 The BDF energy

Definition 2.

Let α > 0, Λ > 0 and ν ∈ S

(

) a generalized function with D(ν, ν) < + ∞ . The BDF energy E

is defined on

as follows: for P ∈

we write Q := P − P

and







E

(Q) = Tr

−

D

Q

− αD(ρ

, ν) + α 2

D(ρ

, ρ

) − k Q k

,

∀ x, y ∈

, ρ

(x) := Tr

Q(x, x)

, k Q k

:=

| Q(x, y) |

| x − y | dxdy, (25) where Q(x, y) is the integral kernel of Q.

Remark 8. The term Tr

−

D

Q

is the kinetic energy, − αD(ρ

, ν) is the interaction energy with ν. The term α

2 D(ρ

, ρ

) is the so-called diract term and − α

2 k Q k

is the exchange term.

1. Let us see that this function is well-defined and more generally that formula (25) is well-defined whenever Q is P

-trace-class [HLS05a, HLS09].

– We start by defining this notion. For any ε, ε

∈ { +, −} and A ∈ B (H

), we write A

:= P

A P

.

The set

0

−

1

of P

-trace class operator is the following Banach space:

S^P

0

−

1

=

Q ∈

(

), Q

, Q

∈

(

) , (26) with the norm

k Q k

S P0

− 1

:= k Q

k

+ k Q

k

+ k Q

k

+ k Q

k

. (27) We have

⊂ P

+

0

−

1

thanks to Eq. (11). The closed convex hull of

− P

in the

0

−

1

-topology gives K :=

Q ∈

0

−

1

(

), Q

= Q, −P

≤ Q ≤ P

and we have [HLS05a, HLS05b]: ∀ Q ∈ K , Q

≤ Q

− Q

. – For Q in

0

−

1

, we show E

(Q) is well defined. We have

P

( D

Q) P

= −|D

| Q

∈

(

), because |D

| ∈ B (

),

this proves that the kinetic energy is defined.

– Thanks to the Kato-Seiler-Simon inequality [Sim79, Chapter 4], the operator Q is locally trace-class:

∀ φ ∈

(

), φΠ

∈

so φQφ = φΠ

Qφ ∈

(L

(

)).

We recall this inequality states that for all 2 ≤ p ≤ ∞ and d ∈

, we have

∀ f, g ∈ L

(

), f(x)g( − i ∇ ) ∈

(H

) and k f(x)g( − i ∇ ) k

≤ (2π)

k f k

k g k

. (28) In particular the density ρ

of Q, given by the formula

∀ x ∈

, ρ

(x) := Tr

Q(x, x)

is well defined. In [HLS05a] Hainzl et al. prove that its Coulomb energy is finite D(ρ

, ρ

) < + ∞ . By Cauchy-Schwartz inequality, D(ν, ρ

) is defined.

– By Kato’s inequality

1

| · | ≤ π

2 |∇| , (29)

the exchange term is well-defined: this implies that k Q k

≤

Tr |∇| Q

Q . – Furthermore the following holds: if α <

, then the BDF energy is bounded from below on K [BBHS98, HLS05b, HLS09]. Here we assume it is the case.

2. For Q ∈ K , its charge is its P

-trace: q = Tr

−

(Q). So we define charge sectors sets:

∀ q ∈

, K

:=

Q ∈ K , Tr(Q) = q .

A minimizer of E

over K is interpreted as the polarized vacuum in the presence of ν while minimizer over charge sector N ∈

is interpretreted as the ground state of N electrons in the presence of ν. We define the energy functional E

:

∀ q ∈

, E

(q) := inf

E

(Q), Q ∈ K

. (30) We also write:

K

:= { Q ∈ K , Tr

−

(Q) = 0, − CQC = Q } . (31) Lemma 1 states that this set is sequentially weakly-∗ closed in

0

− 1

(H

).

Notation 6. For an operator Q ∈

(H

), we write R

the operator given by the integral kernel:

R

(x, y) := Q(x, y)

| x − y | .

2.2 Structure of manifold

We define

=

P − P

, P

= P

= P ∈ B (

), Tr

−

P − P

= 0 ⊂

(

).

Up to adding P

, we deal with

:= P

+

=

P, P

= P

= P, Tr

−

P − P

= 0 .

From a geometrical point of view, we recall that these sets are Hilbert manifolds:

lives in the Hilbert space

(H

) and

lives in the affine space P

+

(H

).

Proposition 1.

The set

is a Hilbert manifold and for all P ∈

,

T

= { [A, P ], A ∈ B (H

), A

= − A and P A(1 − P) ∈

(H

) } . (32) Writing

m_P

:= { A ∈ B (

), A

= − A, P AP = (1 − P)A(1 − P ) = 0 and P A(1 − P) ∈

(

) } ,

(33)

any P

∈

can be written as P

= e

P e

where A ∈

.

The BDF energy E

is a differentiable function in

0

−

1

(H

) with:





∀ Q, δQ ∈

0

−

1

(

), d E

(Q) · δQ = Tr

−

D

δQ . D

:= D

+ α (ρ

− ν) ∗

− R

. (34)

We may rewrite (34) as follows:

∀ Q, δQ ∈

0

−

1

(H

), dE

(Q) · δQ = Tr

−

Π

D

Π

δQ

(35) Notation 7. In the case ν = 0 we write D

:= D

.

Proposition 2.

Let (P, v) be in the tangent bundle T

and Q = P − P

. Then [[Π

D

Π

, P ], P ] is a Hilbert-Schmidt operator in T

and:

d E

(P) · v = Tr

Π

D

Π

, P , P

v

. (36)

Remark 9. The operator [[Π

D

Π

, P ], P ] is the "projection" of Π

D

Π

onto T

. It properly defines a vector in the tangent plane which is exactly the gradient of E

at the point P.

∀ P ∈

, ∇E

(P ) =

Π

D

Π

, P , P

. (37)

We recall

is the set of C-symmetric states (14).

Proposition 3.

The set

is a submanifold of

, which is invariant under the flow of E

. For any P ∈

, writing

m^C_P

= { a ∈

, CaC = a } , (38) we have

T

= { [a, P], a ∈

} = { v ∈ T

, − CvC = v } . (39) Furthermore, for any P ∈

we have

ρ

−

= 0. (40)

Proposition 4.

The set

has two connected components

and

:

∀ P ∈

, P ∈

⇐⇒ Dim Ran P ∩ Ran P

≡ 0[2]. (41) In particular,

contains P

and

contains any P

+ | ψ ih ψ | − | Cψ ih Cψ | where ψ ∈ Ran P

.

We end this section by stating technical results needed to prove Propositions 1,3 and 4.

2.3 Form of trial states

Theorem 4

(Form of trial states). Let P

, P

be in

and Q = P

− P

. Then there exist M

, M

∈

such that there exist two orthonormal families

(a

, . . . , a

) ∪ (e

)

in Ran P

, (a

, . . . , a

) ∪ (e

)

in Ran P

,

and a nonincreasing sequence (λ

)

∈ ℓ

satisfying the following properties.

1. The a

’s are eigenvectors for Q with eigenvalue 1 (resp. − 1) if i > 0 (resp.

i < 0).

2. For each i ∈

the plane Π

:= Span(e

, e

) is spanned by two eigenvectors f

and f

for Q with eigenvalues λ

and − λ

.

3. The plane Π

is also spanned by two orthogonal vectors v

in Ran(1 − P ) and v

in Ran(P ). Moreover λ

= sin(θ

) where θ

∈ (0,

) is the angle between the two lines

v

and

e

.

4. There holds:

Q =

M+

X

i=1

| a

ih a

| −

M−

X

i=1

| a

ih a

| +

j∈N

λ

( | f

ih f

| − | f

ih f

| ).

Remark 10. We have Q

=

M+

X

i=1

| a

ih a

| +

j∈N

sin(θ

)

| e

ih e

| ,

Q

= −

M−

X

i=1

| a

ih a

| −

j∈N

sin(θ

)

| e

ih e

| .

(42)

Thanks to Theorem 4, it is possible to characterize C-symmetric states.

Proposition 5.

Let γ = P − P

be in

. For − 1 ≤ µ ≤ 1 and A = ∈ { γ, γ

} , we write

E

= Ker(A − µ).

Then for any µ ∈ σ(γ), we have CE

= E

. Moreover for | µ | < 1: if we decompose E

⊕ E

into a sum of planes Π as in Theorem 4, then each Π is not C-invariant and dim E

is even. Equivalently dim E

is divisible by 4.

Moreover there exists a decomposition

E

= ⊕

1≤j≤N 2

V

and V

= Π

⊕

CΠ

where the Π

’s and CΠ

’s are spectral planes described in Theorem 4.

3 Proof of Theorem 1

3.1 Strategy and tools of the proof

Topologies

The upper bound in (18) comes from minimization over C-symmetric state of form (13).

We prove the existence of the minimizer over

by using a lemma of Borwein and Preiss [BP87, HLS09], a smooth generalization of Ekeland’s Lemma [Eke74]:

we study the behaviour of a specific minimizing sequence (P

)

or equivalently (P

− P

=: Q

)

.

Each element of the sequence satisfies an equation close to the one satisfied by a real minimizer and we show this equation remains in some weak limit.

Remark 11. We recall different topologies over bounded operators, besides the norm topology k·k

[RS75].

1. The so-called strong topology, the weakest topology T

such that for any f ∈

, the map

B (

) −→

A 7→ Af

is continuous.

2. The so-called weak operator topology, the weakest topology T

such that for any f, g ∈

, the map

B (H

) −→

A 7→ h Af , g i is continuous.

We can also endow

0

−

1

with its weak-∗ topology, the weakest topology such that the following maps are continuous:

S^P

0

−

1

−→

Q 7→ Tr A

(Q

+ Q

) + A

(Q

+ Q

)

∀ (A

, A

) ∈ Comp(H

) ×

(H

).

We emphasize that the weak-∗ topology is different from the weak topology (where Comp(

) must be replaced by B (

)).

The following Lemma is important in our proof.

Lemma 1.

The set K

(defined in (31)) is weakly- ∗ sequentially closed in

0

− 1

(

).

We prove this Lemma at the end of this Subsection.

Borwein and Preiss Lemma

We recall this Theorem as stated in [HLS09]:

Theorem 5.

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 not identical to + ∞ . For all ε > 0 and all u ∈ M such that F (u) < inf

+ε

, there exist v ∈ M and w ∈ Conv( M ) such that

1. F (v) < inf

+ε

,

2. || u − v ||

< √ ε and || v − w ||

< √ ε, 3. F (v) + ε || v − w ||

= min

F (z) + ε || z − w ||

, z ∈ M .

Here we apply this Theorem with H =

(H

), M =

− P

and F = E

. The BDF energy is continuous in the

0

−

1

-norm topology, thus its restriction over

is continuous in the

(H

)-norm topology.

This subspace H is closed in the Hilbert-Schmidt norm topology because

=

is closed in

(H

) and

− P

is closed in

.

Moreover, we have

Conv(

− P

)

⊂ K

.

For every η > 0, we get a projector P

∈

and A

∈ K

such that P

that minimizes the functional

F

: P ∈

7→ E

(P − P

) + ε k P − P

− A

k

. We write

Q

:= P

− P

, Γ

:= Q

− A

, D

:= Π

D

− αR

+ 2ηΓ

Π

. (43) Studying its differential on T

, we get that

e

D

, P

= 0. (44)

In particular, by functional calculus, we get that

π^η

−

, P

= 0,

:= χ

( D

). (45)

We also write

π⁺_η

:= χ

( D

) = Π

−

. (46) We can decompose

as follows (here R means Ran):

H_Λ

= R(P

) ∩ R(

) ⊕R(P

) ∩ R(

) ⊕

R(Π

− P

) ∩ R(

) ⊕R(Π

− P

) ∩ R(

).

(47) We will prove

1. Ran P ∩ Ran

has dimension 1, spanned by a unitary ψ

∈

.

2. As η tends to 0, up to translation and a subsequence, ψ

⇀ ψ

6 = 0, Q

⇀ Q.

There holds Q + P

∈

, ψ

is a unitary eigenvector of Π

D

Π

and Q + P

= χ

Π

D

Π

+ | ψ

ih ψ

| − | Cψ

ih Cψ

| .

In the following part we write the spectral decomposition of trial states and prove Lemma 1.

Spectral decomposition

Let (Q

)

be any minimizing sequence for E

. We consider the spectral decom- position of the trial states Q

: thanks to the upper bound, Dim Ker(Q

− 1) = 1, as shown in Subsection 3.2.

There exist a non-increasing sequence (λ

)

∈ ℓ

of eigenvalues and an or- thonormal basis

of Ran Q

:

Bn

:= (ψ

, Cψ

) ∪ (e

, e

, Ce

, Ce

), P

ψ

= P

e

= 0, ⋆ ∈ { a, b } , (48) such that the following holds. We omit the index n:

∀ j ∈

, e

:= − Ce

, e

:= Ce

, (49a)

f

:=

q1−λj 2

e

+

q1+λj 2

e

, f

:= −

q1+λ_j 2

e

+

q1+λ_j 2

e

.

(49b)





Q

= | ψ

ih ψ

| − | Cψ

ih Cψ

| +

j≥1

λ

q

q

= | f

ih f

| − | f

ih f

| + | f

ih f

| − | f

ih f

| .

(49c) Remark 12. Thanks to the cut-off, the sequences (ψ

)

and (e

)

are H

-bounded.

Up to translation and extraction ((n

)

∈

and (x

)

∈ (

)

), we assume that the weak limit of (ψ

)

is non-zero (if it were then there would hold E

= 2m).

We consider the weak limit of each (e

): by means of a diagonal extraction, we assume that all the (e

( · − x

))

and (ψ

( · − x

))

, converge along the same subsequence (n

)

. We also assume that

∀ j ∈

, λ

→ µ

, (µ

)

∈ ℓ

, (µ

)

non-increasing, (50) and that the above convergences also hold in L

and almost everywhere.

Proof of Lemma 1

Let (Q

)

be a sequence in K

that converges to Q ∈ K in the weak- ∗ topology of

, that is:

∀ (G

, G

) ∈ Comp(H

) ×

(H

) :

Tr(Q

G

) →

n→+∞

Tr(Q

G

) and Tr(Q

G

) →

n→+∞

Tr(Q

G

), Tr(Q

G

) →

n→+∞

Tr(Q

G

) and Tr(Q

G

) →

n→+∞

Tr(Q

G

).

In particular we have S := sup

k Q

k

< + ∞ by the uniform boundedness princi- ple. The C-symmetry is a weak- ∗ condition: for all φ

, φ

∈

:

Tr − CQ

C | φ

ih φ

|

= −h Q

Cφ

, Cφ

i thus − CQC = Q. There remains to prove that Tr

−

(Q) = 0.

We consider the spectral decomposition of P

:= P

+ Q

. We know that this is compact perturbation of P

, thus its essential spectrum is { 0, 1 } and there exist an ONB of

:

(e

)

∪ (f

)

∪ (g

)

, K

∈

and two sequences (r

)

, (s

)

in [0,

) that tend to 0, such that

P

= 1 2

K1

X

k=1

| e

ih e

| +

j∈N

n

r

| f

ih f

| + (1 − s

) | g

ih g

|

. Our aim is to prove we can rewrite P

as follows:

P

= P

+ γ

, γ

=

j

t

| φ

ih φ

| − | Cφ

ih Cφ

| , P

∈

, 2

j

t

≤ Tr Q

− Q

, (φ

)

∪ (Cφ

)

orthonormal family.

(52)

Let us assume this point for the moment. Up to extraction, it is clear that the weak limit γ

of (γ

) has trace 0: the eventual loss of mass of (φ

)

is compensated by that of (Cφ

)

: | φ

(x) |

= | Cφ

(x) |

for all x ∈

. So the weak limit of

t

| φ

ih φ

| − | Cφ

ih Cφ

| has trace 0.

The same goes for Q

:= P

− P

. We write S := lim sup

Tr

−

(Q

) < + ∞ . We decompose each Q

as in (49) and take the same notations. We may have D

:= Dim(Q

− 1) > 2 but the sequence (D

)

is bounded by S . There is at most

S

2

different ψ

in the spectral decomposition of Q

(j = 1, . . . , ⌊

⌋ ).

We study the weak-limit of the ψ

’s and the e

’s: there may be a loss of mass.

However from (42), we see that the loss of mass in ψ

is compensated by that of Cψ

, and that of e

is compensated by that of Ce

.

The subscript ∞ means we take the weak limit. If the sequences of eigenvalues (λ

)

∈ ℓ

weakly converges to (µ

)

∈ ℓ

, then we get that









Q

=

1≤j≤⌊S/2⌋

| ψ

ih ψ

| +

j∈N

µ

| e

ih e

| + | e

ih e

|

Q

= −

1≤j≤⌊S/2⌋

| ψ

ih ψ

| −

j∈N

µ

| e

ih e

| + | e

ih e

|

where | ψ

|

= | ψ

|

resp. | e

|

= | e

|

. Thus Tr Q

+ Q

= 0.

Proof of (52) The condition − CQ

C = Q

is equivalent to CP

C = Π

− P

, so for any µ ∈

we have

CKer P

− µ

= Ker P

− (1 − µ) .

Up to reindexing the sequences, we can assume that r

= s

and up to changing the ONB, we can assume that g

= Cf

. Let us remark that

CB

C = B

where B

:= 1 2

K₀

X

k=1

| e

ih e

| .