Jean-Marie Barbaroux
email: barbarou@univ-tln.fr

Jérémy Faupin
email: jeremy.faupin@univ-lorraine.fr

Jean-Claude Guillot
email: guillot@cmapx.polytechnique.fr

Spectral Theory near Thresholds for Weak Interactions with Massive Particles

Keywords: Mathematical models, quantum field theory, decay of gauge bosons, spectral theory, thresholds. AMS subject classification codes: 47B25, 81Q10, 81T10, 83C47

We consider a Hamiltonian describing the weak decay of the massive vector boson Z 0 into electrons and positrons. We show that the spectrum of the Hamiltonian is composed of a unique isolated ground state and a semiaxis of essential spectrum. Using an infrared regularization and a suitable extension of Mourre's theory, we prove that the essential spectrum below the boson mass is purely absolutely continuous.

Introduction

In this paper, we study a mathematical model for the weak decay of the vector boson Z 0 into electrons and positrons. The model we consider is an example of models of the weak interaction that can be patterned according to the Standard Model of Quantum Field Theory. Another example, describing the weak decay of the intermediate vector bosons W ± into the full family of leptons, has been considered previously in [START_REF] Barbaroux | Spectral theory for a mathematical model of the weak interactions: The decay of the intermediate vector bosons W+[END_REF][START_REF] Aschbacher | Spectral theory for a mathematical model of the weak interaction: The decay of the intermediate vector bosons W+/-, II[END_REF]. Comparable models describing quantum electrodynamics processes can be constructed in a similar manner, see [START_REF] Barbaroux | Quantum electrodynamics of relativistic bound states with cutoffs II[END_REF]. We also mention [START_REF] Georgescu | On the spectral analysis of quantum field Hamiltonians[END_REF][START_REF] Gérard | Spectral and scattering theory for some abstract QFT Hamiltonians[END_REF] where the spectral analysis of some related abstract quantum field theory models have been studied.

Unlike [START_REF] Aschbacher | Spectral theory for a mathematical model of the weak interaction: The decay of the intermediate vector bosons W+/-, II[END_REF], the physical phenomenon considered in the present paper only involves massive particles. In some respects, e.g. as far as the existence of a ground state is concerned, this feature considerably simplifies the spectral analysis of the Hamiltonian associated with the physical system we study. The main drawback is that, due to the positive masses of the particles, an infinite number of thresholds occur in the spectrum of the free Hamiltonian (i.e. the full Hamiltonian where the interaction between the different particles has been turned off). Understanding the nature of the spectrum of the full Hamiltonian near the thresholds as the interaction is turned on then becomes a subtle question. Spectral analysis near thresholds, in particular by means of perturbation theory, is indeed well-known to be a delicate subject. This is the main concern of the present work.

Our main result will provide a complete description of the spectrum of the Hamiltonian below the boson mass. We will show that the spectrum is composed of a unique isolated eigenvalue E (the ground state energy), and the semi-axis of essential spectrum [E + m e , ∞), m e being the electron mass. Moreover, using a version of Mourre's theory allowing for a non self-adjoint conjugate operator and requiring only low regularity of the Hamiltonian with respect to this conjugate operator, we will prove that the essential spectrum below the boson mass is purely absolutely continuous.

In order to prove our main results we use a spectral representation of the selfadjoint Dirac operator generated by the sequence of spherical waves. See [START_REF] Greiner | Relativistic quantum mechanics[END_REF] and Section 2. If we have been using the plane waves, for example the four ones associated with the helicity (see [START_REF] Thaller | The Dirac Equation[END_REF]), the two kernels G (α) (•) of the interaction (see below) would have had to satisfy an infrared regularization with respect to the fermionic variables. By our choice of the sequence of the spherical waves, the kernels of the interaction have to satisfy an infrared regularization for only two values of the discrete parameters characterizing the sequence of spherical waves. For any other value of the discrete parameters, we do not need to introduce an infrared regularization. Thus we have reduced the problem of proving that the spectrum is absolutely continuous in a neighborhood of a threshold to a simpler one, which still remains to be solved.

Before precisely stating our main results in Section 3, we begin with introducing in details the physical model we consider.

2 Description of the model 2.1 The Fock space of electrons, positrons and Z 0 bosons

Free Dirac operator

The energy of a free relativistic electron of mass m e is described by the Dirac Hamiltonian (see [START_REF] Rose | Relativistic Electron Theory[END_REF][START_REF] Thaller | The Dirac Equation[END_REF] and references therein)

H D := α • 1 i ∇ + β m e ,
acting on the Hilbert space H = L 2 (R 3 ; C 4 ), with domain D(H D ) = H 1 (R 3 ; C 4 ). We use a system of units such that = c = 1. Here α = (α 1 , α 2 , α 3 ) and β are the Dirac matrices in the standard form:

β = I 0 0 -I , α i = 0 σ i σ i 0 , i = 1, 2, 3,
where σ i are the usual Pauli matrices. The operator H D is self-adjoint, and spec(H D ) = (-∞, -m e ] ∪ [m e , +∞).

The generalized eigenfunctions associated with the continuous spectrum of the Dirac operator H D are labeled by the total angular momentum quantum numbers

j ∈ 1 2 , 3 2 , 5 2 , .
. . , m j ∈ {-j, -j + 1, . . . , j -1, j},

and by the quantum numbers

κ j ∈ ± (j + 1 2 ) . (2)
In the sequel, we will drop the index j and set γ = (j, m j , κ j ) ,

and a sum over γ will thus denote a sum over j ∈ N + 1 2 , m j ∈ {-j, -j + 1, . . . , j -1, j} and κ j ∈ {±(j + 1 2 )}. We denote by Γ the set {(j, m j , κ j ), j ∈ N + 1 2 , m j ∈ {-j, -j + 1, . . . , j -1, j}, κ j ∈ {±(j + 1 2 )}}. For p ∈ R 3 being the momentum of the electron, and p := |p|, the continuum energy levels are given by ± ω(p), where ω(p) := (m e 2 + p 2 )

1 2 . ( 4
)
We set the notation ξ = (p, γ) ∈ R + × Γ.

(

) 5
The continuum eigenstates of H D are denoted by (see Appendix A for a detailed description)

ψ ± (ξ, x) = ψ ± ((p, γ), x) .

We then have H D ψ ± ((p, γ), x) = ± ω(p) ψ ± ((p, γ), x).

The generalized eigenstates ψ ± are here normalized in such a way that

R 3 ψ † ± ((p, γ), x) ψ ± ((p , γ ), x) dx = δ γγ δ(p -p ), R 3
ψ † ± ((p, γ), x) ψ ∓ ((p , γ ), x) dx = 0 .

Here ψ † ± ((p, γ), x) is the adjoint spinor of ψ ± ((p, γ), x). According to the hole theory [START_REF] Itzykson | Quantum Field theory[END_REF][START_REF] Rose | Relativistic Electron Theory[END_REF][START_REF] Schweber | An Introduction to Relativistic Quantum Field Theory[END_REF][START_REF] Thaller | The Dirac Equation[END_REF][START_REF] Weinberg | The Quantum Theory of Fields[END_REF], the absence in the Dirac theory of an electron with energy E < 0 and charge e is equivalent to the presence of a positron with energy -E > 0 and charge -e.

Let us split the Hilbert space H = L 2 (R 3 ; C 4 ) into H c -= P (-∞,-me] (H D )H and H c + = P [me,+∞) (H D )H.

Here P I (H D ) denotes the spectral projection of H D corresponding to the interval I.

Let Σ := R + × Γ. From now on, we identify the Hilbert spaces H c ± with

H c := L 2 (Σ; C) ⊕ γ L 2 (R + ; C) ,
by using the unitary operators U c ± defined from H c ± to H c as (U c ± φ)(p, γ) = L.i.m ψ † ± ((p, γ) , x) φ(x) dx .

On H c , we define the scalar products (g, h) = g(ξ)h(ξ)dξ = γ∈Γ R + g(p, γ)h(p, γ) dp .

In the sequel, we shall denote the variable (p, γ) by ξ 1 = (p 1 , γ 1 ) in the case of electrons, and ξ 2 = (p 2 , γ 2 ) in the case of positrons, respectively.

The Fock space for electrons and positrons

Let

F a := F a (H c ) = ∞ n=0 ⊗ n a H c ,
be the Fermi-Fock space over H c , and let

F D := F a ⊗ F a
be the Fermi-Fock space for electrons and positrons, with vacuum Ω D (see Appendix C for details).

Creation and annihilation operators for electrons and positrons

We set, for every

g ∈ H c , b γ,+ (g) = b + (P γ g) , b * γ,+ (g) = b * + (P γ g)
, where P γ is the projection of H c onto the γ-th component, and b + (P γ g) and b * + (P γ g) are respectively the annihilation and creation operator for an electron defined in Appendix C.

As above, we set, for every As in [24, Chapter X], we introduce operator-valued distributions b ± (ξ) and b

h ∈ H c , b γ,-(h) = b -(P γ h) , b * γ,-(h) = b * -(P γ h) ,
* ± (ξ) that fulfill for g ∈ H c , b ± (g) = b γ,± (p) (P γ g) (p) dξ b * ± (g) = b * γ,± (p) (P γ g) (p) dξ
where we used the notation of (7).

2.1.4 Fock space for the Z 0 boson.

Let S be any separable Hilbert space. Let ⊗ n s S denote the symmetric n-th tensor power of S. The symmetric Fock space over S, denoted by F s (S), is the direct sum

F s (S) = ∞ n=0 ⊗ n s S , (8)
where ⊗ 0 s S ≡ C. The state Ω s = (1, 0, 0, . . . , 0, . . .) denotes the vacuum state in

F s (S). Let Σ 3 := R 3 × {-1, 0, 1} .
The one-particle Hilbert space for the particle

Z 0 is L 2 (Σ 3 ) with scalar product (f, g) = Σ3 f (ξ 3 )g(ξ 3 )dξ 3 , (9)
with the notations

ξ 3 = (k, λ) and Σ3 dξ 3 = λ=-1,0,1 R 3 dk , (10)
where

ξ 3 = (k, λ) ∈ Σ 3 .
The bosonic Fock space for the vector boson Z 0 , denoted by F Z 0 , is thus

F Z 0 = F s (L 2 (Σ 3 )) . (11)
For f ∈ L 2 (Σ 3 ), we define the annihilation and creation operators, denoted by a(f ) and a * (f ) by

a(f ) = Σ3 f (ξ 3 )a(ξ 3 )dξ 3 (12)
and

a * (f ) = Σ3 f (ξ 3 )a * (ξ 3 )dξ 3 ( 13
)
where the operators a(ξ 3 ) (respectively a * (ξ 3 )) are the bosonic annihilation (respectively bosonic creation) operator for the boson Z 0 (see e.g [START_REF] Kastler | Introduction à l'Electrodynamique Quantique[END_REF][START_REF] Barbaroux | Spectral theory for a mathematical model of the weak interactions: The decay of the intermediate vector bosons W+[END_REF][START_REF] Barbaroux | Spectral theory for a mathematical model of the weak interactions: The decay of the intermediate vector bosons W+[END_REF]).

The Hamiltonian

The free Hamiltonian

The quantization of the Dirac Hamiltonian H D , denoted by H D , and acting on F D , is given by

H D = ω(p 1 ) b * + (ξ 1 ) b + (ξ 1 )dξ 1 + ω(p 2 ) b * -(ξ 2 ) b -(ξ 2 )dξ 2 ,
with ω(p) given in [START_REF] Barbaroux | Spectral theory for a mathematical model of the weak interactions: The decay of the intermediate vector bosons W+[END_REF]. The operator H D is the Hamiltonian of the quantized Dirac field.

Let D D denote the set of vectors Φ ∈ F D for which Φ (r,s) is smooth and has a compact support and Φ (r,s) = 0 for all but finitely many (r, s). Then H D is well-defined on the dense subset D D and it is essentially self-adjoint on D D . The self-adjoint extension will be denoted by the same symbol H D , with domain D(H D ).

The operators number of electrons and number of positrons, denoted respectively by N + and N -, are given by

N + = b * + (ξ 1 ) b + (ξ 1 )dξ 1 and N -= b * -(ξ 2 ) b -(ξ 2 )dξ 2 . ( 14
)
They are essentially self-adjoint on D D . Their self-adjoint extensions will be also denoted by N + and N -.

We have spec(

H D ) = {0} ∪ [m e , ∞).
The set [m e , ∞) is the absolutely continuous spectrum of H D . The Hamiltonian of the bosonic field, denoted by H Z 0 , acting on F Z 0 , is

H Z 0 := ω 3 (k) a * (ξ 3 )a(ξ 3 ) dξ 3
where

ω 3 (k) = |k| 2 + m Z 0 2 . ( 15
)
The operator H Z 0 is essentially self-adjoint on the set of vectors Φ ∈ F Z 0 such that Φ (n) is smooth and has compact support and Φ (n) = 0 for all but finitely many n. Its self-adjoint extension is denoted by the same symbol.

The spectrum of H Z 0 consists of an absolutely continuous spectrum covering [m Z 0 , ∞) and a simple eigenvalue, equal to zero, whose corresponding eigenvector is the vacuum state Ω s ∈ F Z 0 .

The free Hamiltonian is defined on

H := F D ⊗ F Z 0 by H 0 = H D ⊗ 1l + 1l ⊗ H Z 0 . ( 16
)
The operator

H 0 is essentially self-adjoint on D(H D ) ⊗ D(H Z 0 ). Since m e < m Z 0 , the spectrum of H 0 is given by spec(H 0 ) = {0} ∪ [m e , ∞) .
More precisely,

spec pp (H 0 ) = {0}, spec sc (H 0 ) = ∅, spec ac (H 0 ) = [m e , ∞), (17)
where spec pp , spec sc , spec ac denote the pure point, singular continuous and absolutely continuous spectra, respectively. Furthermore, 0 is a non-degenerate eigenvalue associated to the vacuum Ω D ⊗ Ω s .

The Interaction

The interaction between the electrons/positrons and the boson vectors Z 0 , in the Schrödinger representation, is given, up to coupling contant, by (see [18, (4.139)] and [32, (21.3.20)])

I = Ψ e (x)γ α (g V -γ 5 )Ψ e (x)Z α (x) dx + h.c., (18)
where γ α , α = 0, 1, 2, 3, and γ 5 are the Dirac matrices, g V is a real parameter such that g V 0, 074 (see e.g [START_REF] Greiner | Gauge Theory of Weak Interactions[END_REF]), Ψ e (x) and Ψ e (x) are the Dirac fields for the electron e -and the positron e + of mass m e , and Z α is the massive boson field for Z 0 .

With the notations of Subsection 2.1.1, Ψ e (x) is formally defined by

Ψ e (x) = ψ + (ξ, x)b + (ξ) + ψ -(ξ, x)b * -(ξ) dξ, where ψ -(ξ, x) = ψ -((p, γ), x) = ψ -((p, (j, -m j , -κ j )), x) . ( 19
)
The boson field Z α is formally defined by (see e.g. [31, Eq. (5.3.34)]),

Z α (x) = (2π) -3 2 dξ 3 (2(|k| 2 +m Z 0 2 ) 1 2 ) 1 2 α (k, λ)a(ξ 3 )e ik.x + * α (k, λ)a * (ξ 3 )e -ik.
x , with ξ 3 = (k, λ) according to [START_REF] Faupin | Second Order Perturbation Theory for Embedded Eigenvalues[END_REF], and where the vectors α (k, λ) are the polarizations vectors of the massive spin 1 bosons (see [START_REF] Weinberg | The Quantum Theory of Fields[END_REF]Section 5.3]).

If one considers the full interaction I in [START_REF] Greiner | Gauge Theory of Weak Interactions[END_REF] describing the decay of the gauge boson Z 0 into massive leptons and if one formally expands this interaction with respect to products of creation and annihilation operators, we are left with a finite sum of terms with kernels yielding singular operators which cannot be defined as closed operators. Therefore, in order to obtain a well-defined Hamiltonian (see e.g [START_REF] Glimm | Quantum field theory and statistical mechanics[END_REF][START_REF] Barbaroux | Quantum electrodynamics of relativistic bound states with cutoffs II[END_REF][START_REF] Barbaroux | Quantum electrodynamics of relativistic bound states with cutoffs[END_REF][START_REF] Barbaroux | Spectral theory for a mathematical model of the weak interactions: The decay of the intermediate vector bosons W+[END_REF][START_REF] Aschbacher | Spectral theory for a mathematical model of the weak interaction: The decay of the intermediate vector bosons W+/-, II[END_REF]), we replace these kernels by square integrable functions G (α) .

This implies in particular to introduce cutoffs for high momenta of electrons, positrons and Z 0 bosons. Moreover, we confine in space the interaction between the electrons/positrons and the bosons by adding a localization function

f (|x|), with f ∈ C ∞ 0 ([0, ∞))
. The interaction Hamiltonian is thus defined on

H = F D ⊗ F Z 0 by H I = H (1)
I + H (1) I * + H (2)
I + H (2) I * , (20) with H
(1)

I = R 3 f (|x|)ψ + (ξ 1 , x)γ µ (g V -γ 5 ) ψ -(ξ 2 , x) µ (ξ 3 ) 2ω 3 (k) e ik•x dx × G (1) (ξ 1 , ξ 2 , ξ 3 )b * + (ξ 1 )b * -(ξ 2 )a(ξ 3 ) dξ 1 dξ 2 dξ 3 , (21) H (1)
I * = R 3 f (|x|) ψ -(ξ 2 , x)γ µ (g V -γ 5 )ψ + (ξ 1 , x) * µ (ξ 3 ) 2ω 3 (k) e -ik•x dx × G (1) (ξ 1 , ξ 2 , ξ 3 )a * (ξ 3 )b -(ξ 2 )b + (ξ 1 ) dξ 1 dξ 2 dξ 3 , (22)

H

(2)

I = R 3 f (|x|)ψ + (ξ 1 , x)γ µ (g V -γ 5 ) ψ -(ξ 2 , x) * µ (ξ 3 ) 2ω 3 (k) e -ik•x dx × G (2) (ξ 1 , ξ 2 , ξ 3 )b * + (ξ 1 )b * -(ξ 2 )a * (ξ 3 ) dξ 1 dξ 2 dξ 3 , (23)
and

H (2) I * = R 3 f (|x|) ψ -(ξ 2 , x)γ µ (g V -γ 5 )ψ + (ξ 1 , x) µ (ξ 3 ) 2ω 3 (k) e ik•x dx × G (2) (ξ 1 , ξ 2 , ξ 3 )a(ξ 3 )b -(ξ 2 )b + (ξ 1 ) dξ 1 dξ 2 dξ 3 . (24)
Performing the integration with respect to x in the expressions above, we see that H

(1) I and H

(2) I can be written under the form

H (1) I := H (1)
I (F (1) ) := F (1) (ξ 1 , ξ 2 , ξ 3 )b * + (ξ 1 )b * -(ξ 2 )a(ξ 3 ) dξ 1 dξ 2 dξ 3 , (25)

H

(2)

I := H (2)
I (F (2) ) := F (2) (ξ 1 , ξ 2 , ξ 3 )b * + (ξ 1 )b * -(ξ 2 )a * (ξ 3 ) dξ 1 dξ 2 dξ 3 , (26)
where, for α = 1, 2,

F (α) (ξ 1 , ξ 2 , ξ 3 ) := h (α) (ξ 1 , ξ 2 , ξ 3 )G (α) (ξ 1 , ξ 2 , ξ 3 ), (27)
and h (1) (ξ 1 , ξ 2 , ξ 3 ), h (2) (ξ 1 , ξ 2 , ξ 3 ) are given by the integral over x in [START_REF] Kastler | Introduction à l'Electrodynamique Quantique[END_REF] and [START_REF] Mourre | Absence of singular continuous spectrum for certain selfadjoint operators[END_REF], respectively.

Our main result, Theorem 3.9 below, requires the coupling functions F (α) (ξ 1 , ξ 2 , ξ 3 ) to be sufficiently regular near p 1 = 0 and p 2 = 0 (where, recall, ξ l = (p l , γ l ) for l = 1, 2). The behavior of the generalized eigenstates ψ + (ξ, x) and ψ -(ξ, x) near ξ = 0, and therefore the behavior of h (α) (ξ 1 , ξ 2 , ξ 3 ) near p 1 = 0 and p 2 = 0, will be analyzed in Appendix A.

The total Hamiltonian

Definition 2.1. The Hamiltonian of the decay of the boson Z 0 into an electron and a positron is

H := H 0 + gH I .
where g is a real coupling constant.

Main results

For

p ∈ R + , j ∈ { 1 2 , 3 2 , • • • }, γ = (j, m j , κ j
) and γ j = j + 1 2 , we define

A(ξ) = A(p, γ) := (2p) γj Γ(γ j ) ω(p) + m e ω(p) 1 2 ∞ 0 |f (r)|r 2γj (1 + (pr) 2 )dr 1 2 , (28)
where Γ denotes Euler's Gamma function, and f ∈ C ∞ 0 ([0, ∞)) is the localization function appearing in (21)- [START_REF] Reed | Methods of modern mathematical physics[END_REF]. We make the following hypothesis on the kernels G (α) . Hypothesis 3.1. For α = 1, 2,

A(ξ 1 ) 2 A(ξ 2 ) 2 (|k| 2 + m Z 0 2 ) 1 2 G (α) (ξ 1 , ξ 2 , ξ 3 ) 2 dξ 1 dξ 2 dξ 3 < ∞. ( 29
)
Remark 3.2. Note that up to universal constants, the functions A(ξ) in (28) are upper bounds for the integrals with respect to x that occur in [START_REF] Kastler | Introduction à l'Electrodynamique Quantique[END_REF]. These bounds are derived using the inequality (see [START_REF] Weinberg | The Quantum Theory of Fields[END_REF]Eq.(5.3.23)-(5.3.25)])

µ (ξ 3 ) 2ω 3 (k) ≤ C m Z 0 (1 + |k| 2 ) 1 4 . ( 30
)
For C m Z 0 being the constant defined in [START_REF] Triebel | Interpolation theory, function spaces, differential operators[END_REF], and

C Z = 156 C m Z 0 , let us define K 1 (G (α) ) 2 := C Z 2 A(ξ 1 ) 2 A(ξ 2 ) 2 |G (α) (ξ 1 , ξ 2 , ξ 3 )| 2 dξ 1 dξ 2 dξ 3 , K 2 (G (α) ) 2 := C Z 2 A(ξ 1 ) 2 A(ξ 2 ) 2 |G (α) (ξ 1 , ξ 2 , ξ 3 )| 2 (|k| 2 + 1) 1 2 dξ 1 dξ 2 dξ 3 . (31)
Theorem 3.3 (Self-adjointness). Assume that Hypothesis 3.1 holds. Let g 0 > 0 be such that

g 0 2 α=1,2 K 1 (G (α) ) 2 ( 1 m e 2 + 1) < 1 . ( 32
)
Then for any real g such that |g| ≤ g 0 , the operator H = H 0 + gH I is self-adjoint with domain D(H 0 ). Moreover, any core for H 0 is a core for H.

Remark 3.4. 1) Combining [START_REF] Greiner | Relativistic quantum mechanics[END_REF], relative boundedness of H I with respect to H 0 (see Section 4) and standard perturbation theory of isolated eigenvalues (see e.g. [START_REF] Kato | Perturbation Theory for Linear Operators[END_REF]), we deduce that, for |g| m e , inf spec(H) is a non-degenerate eigenvalue of H. In other words, H admits a unique ground state.

2) Let Q be the total charge operator

Q = N + -N -,
where N + and N -are respectively the operator number of electrons and the operator number of positrons given by [START_REF] Georgescu | Spectral theory of massless Pauli-Fierz models[END_REF].

The total Hamiltonian H commutes with Q, and H is decomposed with respect to the spectrum of the total charge operator as

H ⊕ z∈Z H z .
Each H z reduces H and by mimicking the proof given in [START_REF] Takaesu | On the spectral analysis of quantum electrodynamics with spatial cutoffs. I[END_REF] one proves that the ground state of H belongs to H 0 . Theorem 3.3 follows from the Kato-Rellich Theorem together with standard estimates of creation and annihilation operators in Fock space, showing that the interaction Hamiltonian H I is relatively bounded with respect to H 0 . For the convenience of the reader, a sketch of the proof of Theorem 3.3 is recalled in Subsection 4.1.

For a self-adjoint operator A, we denote by spec ess (A) the essential spectrum of A. Theorem 3.5 (Localization of the essential spectrum). Assume that Hypothesis 3.1 holds and let g 0 be as in [START_REF] Weinberg | The Quantum theory of fields[END_REF]. Then, for all |g| ≤ g 0 ,

spec ess (H) = [inf spec(H) + m e , ∞).
Theorem 3.5 is proven in Subsection 4.2. Our proof is based on a method due to Dereziński and Gérard [START_REF] Dereziński | Asymptotic completeness in quantum field theory. Massive Pauli-Fierz Hamiltonians[END_REF] that we adapt to our context.

To establish our next theorems, we need to strengthen the conditions on the kernels G (α) . Given a function f ∈ L 1 ([0, ∞)), we make the convention that the Fourier transform of f is the Fourier transform of the function f ∈ L 1 (R) defined by f (p) = f (p) if p ≥ 0 and f (p) = f (-p) otherwise. Hypothesis 3.6. For α = 1, 2, the kernels

G (α) ∈ L 2 (Σ × Σ × Σ 3 ) satisfy (i) There exists a compact set K ⊂ R + ×R + ×R 3 such that G (α) (p 1 , γ 1 , p 2 , γ 2 , k, λ) = 0 if (p 1 , p 2 , k) / ∈ K.
(ii) There exists ε ≥ 0 such that γ1,γ2,λ

(1 + x 2 1 + x 2 2 ) 1+ε Ĝ(α) (x 1 , γ 1 , x 2 , γ 2 , k, λ) 2 dx 1 dx 2 dk < ∞,
where Ĝ(α) denote the Fourier transform of G (α) with respect to the variables (p 1 , p 2 ), and x j is the variable dual to p j .

(iii) If γ 1j = 1 or γ 2j = 1, where for l = 1, 2, γ lj = |κ j l | (with γ l = (j l , m j l , κ j l )), and if

p 1 = 0 or p 2 = 0, then G (α) (p 1 , γ 1 , p 2 , γ 2 , k, λ) = 0. Remark 3.7. 1)
The assumption that G (α) is compactly supported in the variables (p 1 , p 2 , k) is an "ultraviolet" constraint that is made for convenience. It could be replaced by the weaker assumption that G (α) decays sufficiently fast at infinity.

2) Hypothesis 3.6 (ii) comes from the fact that the coupling functions G (α) must satisfy some "minimal" regularity for our method to be applied. In fact, Hypothesis (ii) could be slightly improved with a refined choice of interpolation spaces in our proof (see Section 5 for more details). In Hypothesis 3.6 (iii), we need in addition an "infrared" regularization. We remark in particular that Hypotheses (ii) and (iii) imply that, if γ 1j = 1 or γ 2j = 1,

G (α) (p 1 , γ 1 , p 2 , γ 2 , k, λ) |p l | 1 2 +ε , l = 1, 2,
for 0 ≤ ε < 1/2. We emphasize, however, that this infrared assumption is required only in the case γ lj = 1, that is, for j = 1/2. For all other j ∈ N + 1 2 , we do not need to impose any infrared regularization on the generalized eigenstates ψ ± (p, γ); They are already regular enough.

3) One verifies that Hypotheses 3.6(i) and 3.6(ii) imply Hypothesis 3.1.

Theorem 3.8 (Localization of the spectrum). Assume that Hypothesis 3.1 holds. There exists g 1 > 0 such that, for all |g| ≤ g 1 ,

spec(H) = {inf spec(H)} ∪ [inf spec(H) + m e , ∞).
In particular, H has no eigenvalue below its essential spectrum except for the ground state energy, inf spec(H), which is a simple eigenvalue. Theorem 3.9 (Absolutely continuous spectrum). Assume that Hypothesis 3.6 holds with ε > 0 in Hypothesis 3.6(ii). For all δ > 0, there exists g δ > 0 such that, for all |g| ≤ g δ , the spectrum of H in the interval

[inf spec(H) + m e , inf spec(H) + m Z 0 -δ]
is purely absolutely continuous. Remark 3.10. In Theorem 5.5 below, we prove a stronger result than Theorem 3.9, which is of independent interest, namely we show that a limiting absorption principle holds for H in the interval

[inf spec(H) + m e , inf spec(H) + m Z 0 -δ].
Another consequence of the limiting absorption principle of Theorem 5.5 is the local decay property (70). Theorems 3.8 and 3.9 are proven in Section 5. Our proofs rely on Mourre's Theory with a non-self adjoint conjugate operator. Such extensions of the usual conjugate operator theory [START_REF] Mourre | Absence of singular continuous spectrum for certain selfadjoint operators[END_REF][START_REF] Amrein | C 0 -groups, commutator methods and spectral theory of N -body Hamiltonians[END_REF] have been considered in [START_REF] Hübner | Spectral properties of the spin-boson Hamiltonian[END_REF], [START_REF] Skibsted | Spectral analysis of N -body systems coupled to a bosonic field[END_REF], and later extended in [START_REF] Georgescu | Commutators, C 0 -semigroups and resolvent estimates[END_REF][START_REF] Georgescu | Spectral theory of massless Pauli-Fierz models[END_REF].

We use in this paper a conjugate operator, A, similar to the ones of [START_REF] Hübner | Spectral properties of the spin-boson Hamiltonian[END_REF] and [START_REF] Georgescu | Commutators, C 0 -semigroups and resolvent estimates[END_REF][START_REF] Georgescu | Spectral theory of massless Pauli-Fierz models[END_REF], and prove regularity of the total Hamiltonian with respect to this conjugate operator. Combined with a Mourre estimate, this regularity property allows us to deduce a virial theorem and a limiting absorption principle, from which we obtain Theorems 3.8 and 3.9.

Our main achievement consists in proving that the regularized physical interaction Hamiltonian H I is regular enough for the Mourre theory to be applied, except for the terms associated to the "first" generalized eigenstates (j = 1/2). For the latter, we need to impose a non-physical infrared condition. To establish the regularity of H I with respect to A, we use in particular real interpolation theory, together with a version of the Mourre theory requiring only low regularity of the Hamiltonian with respect to the conjugate operator.

We remark that if we make the further assumption that the kernels G (α) are sufficiently regular with respect to the Z 0 variable k, similarly to what is assumed in Hypothesis 3.6(ii) for the variables p 1 , p 2 , it is possible to extend the result of Theorem 3.9 to the interval [inf spec(H) + m e , M ), for any M > inf spec(H) + m e . To do that, one would have to add to the conjugate operator A a term acting on F Z 0 , similar to the ones acting on F D (see (40)), which would yield a Mourre estimate on any interval of the form [inf spec(H) + m e , M ), M > inf spec(H) + m e . The regularity of G (α) in p 1 , p 2 and k would insure that H is regular enough with respect to A. For simplicity of exposition, we do not present the details of such an extension of Theorem 3.9 here.

Our paper is organized as follows. As mentioned above, Section 4 is devoted to the proof of Theorems 3.3 and 3.5, and Section 5 is devoted to the proof of Theorems 3.8 and 3.9. In Appendix A, we give the estimates on the generalized eigenfunctions of the Dirac operator that are used in this paper. In Appendix B, we recall the abstract results from Mourre's theory that we need. Finally, for the convenience of the reader, standard definitions and properties of creation and annihilation operators in Fock space are recalled in Appendix C.

Self-adjointness

We sketch the standard proof of Theorem 3.3 relying on the Kato-Rellich Theorem.

Proof of Theorem 3.3. We use the N τ estimates of [START_REF] Glimm | Quantum field theory and statistical mechanics[END_REF] and follow the proof of [7, Theorem 2.6] (see also [START_REF] Barbaroux | Quantum electrodynamics of relativistic bound states with cutoffs II[END_REF]). For

K i (G) 2 := α=1,2 K i (G (α) ) 2 , i = 1, 2 , (33)
and

C 1,β := ( 1 m e 2 + 1 + 2β) 1 2 , C 2,βη := ( η m e 2 (1 + 2β)) 1 2 , B 1,β := (1 + 1 2β ) 1 2 , B 2,βη := (η(1 + 1 2β ) + 1 4η ) 1 2 ,
we obtain, for any ψ ∈ D(H),

H I ψ ≤ (K 1 (G) C 1,β + K 2 (G)C 2,β ) H 0 ψ + (K 1 (G)B 1,β + K 2 (G)B 2,βη ) ψ . (34)
Therefore, with [START_REF] Weinberg | The Quantum theory of fields[END_REF] and for β and η small enough, using the Kato-Rellich Theorem concludes the proof.

If we note that K 2 (G) ≥ K 1 (G), and set

K(G) := K 2 (G) , C βη := C 1,β + C 2,βη , B βη := B 1,β + B 2,βη ,
we obtain from (34) the following relative bound:

Corollary 4.1. For any ψ ∈ D(H),

H I ψ ≤ K(G) (C βη H 0 ψ + B βη ψ ) .
In the sequel, for the sake of simplicity, we shall use this relative bound instead of the stronger result (34).

Localization of the essential spectrum

In this subsection, we prove Theorem 3.5. We use the Dereziński-Gérard partition of unity [START_REF] Dereziński | Asymptotic completeness in quantum field theory. Massive Pauli-Fierz Hamiltonians[END_REF] in a version that accommodates the Fermi-Dirac statistics and the CAR (such a partition of unity was used previously in [START_REF] Ammari | Scattering theory for a class of fermionic models[END_REF]). Let

U a : F a (H c ⊕ H c ) → F a (H c ) ⊗ F a (H c ) = F a ⊗ F a ,
be defined by

U a Ω a = Ω a ⊗ Ω a U a b * (ϕ 1 ⊕ ϕ 2 ) = (b * (ϕ 1 ) ⊗ 1l + (-1) N ⊗ b * (ϕ 2 ))U a ,
where (-1) N denotes the bounded operator on F a defined by its restriction to ⊗ r a h c as (-1) N u = (-1) r u for any u ∈ ⊗ r a h c . Clearly, using the anti-commutation relations, U a extends by linearity to a unitary map on

F a (H c ⊕ H c ). Let j 0 ∈ C ∞ ([0, ∞); [0, 1]
) be such that j 0 ≡ 1 on [0, 1/2] and j 0 ≡ 0 on [1, ∞), and let j ∞ be defined by the relation j 2 0 + j 2 ∞ ≡ 1. Let y := i∇ p account for the position variable of the fermions. Given R > 0, we introduce the bounded operators j R 0 :

= j 0 (|y|/R) and j R ∞ := j ∞ (|y|/R) on F a (H c ). Let j R a : H c → H c ⊕ H c ϕ → (j R 0 ϕ, j R ∞ ϕ).
Lifting the operator j R a to the Fock space F a (H c ) allows one to define a map Γ(j R a ) :

F a (H c ) → F a (H c ⊕H c ). The Dereziński-Gérard partition of unity is defined by Γa (j R a ) : F a → F a ⊗ F a , Γa (j R a ) = U a Γ(j R a )
. Using the relation j 2 0 + j 2 ∞ ≡ 1, one easily verifies that Γa (j R a ) is isometric. We construct a similar partition of unity, Γs (j R s ), acting on the bosonic Fock space

F Z 0 = F s (L 2 (Σ 3 )). It is defined by Γs (j R s ) : F Z 0 → F Z 0 ⊗ F Z 0 , Γs (j R s ) = U s Γ(j R s ),
where

U s : F s (L 2 (Σ 3 ) ⊕ L 2 (Σ 3 )) → F Z 0 ⊗ F Z 0 ,
is the unitary operator defined by

U s Ω s = Ω s ⊗ Ω s U s a * (ϕ 1 ⊕ ϕ 2 ) = (a * (ϕ 1 ) ⊗ 1l + 1l ⊗ a * (ϕ 2 ))U s ,
and j R s is the bounded operator defined by

j R s : L 2 (Σ 3 ) → L 2 (Σ 3 ) ⊕ L 2 (Σ 3 ) ϕ → (j R 0 ϕ, j R ∞ ϕ).
Here we have used similar notations as above, namely j R 0 := j 0 (|y 3 |/R) and j R ∞ := j ∞ (|y 3 |/R), where y 3 := i∇ k accounts for the position variable of the bosons.

Let N denote the number operator, acting either on F a or on F Z 0 . To shorten notations, we define the operators

N 0 := N ⊗ 1l, N ∞ := 1l ⊗ N,
acting on F a ⊗ F a and on F Z 0 ⊗ F Z 0 . We recall the following properties that can be easily proven using the definitions of the operators involved (see [START_REF] Ammari | Scattering theory for a class of fermionic models[END_REF][START_REF] Dereziński | Asymptotic completeness in quantum field theory. Massive Pauli-Fierz Hamiltonians[END_REF]).

i) Let ϕ 1 , . . . , ϕ n ∈ H c . Then Γa (j R a ) n i=1 b * (ϕ i )Ω a = n i=1 b * (j R 0 ϕ i ) ⊗ 1l + (-1) N ⊗ b * (j R ∞ ϕ i ) Ω a ⊗ Ω a . Let ϕ 1 , . . . , ϕ n ∈ L 2 (Σ 3 ). Then Γs (j R s ) n i=1 a * (ϕ i )Ω s = n i=1 a * (j R 0 ϕ i ) ⊗ 1l + 1l ⊗ a * (j R ∞ ϕ i ) Ω s ⊗ Ω s .
(ii) Let ω be an operator on H c such that the commutators [ω, j R # ], defined as quadratic forms on D(ω), extend to bounded operators on H c , where j # stands for j 0 and j ∞ . Then

(N 0 + N ∞ ) -1 2 (dΓ(ω) ⊗ 1l + 1l ⊗ dΓ(ω)) Γa (j R a ) -Γa (j R a )dΓ(ω) N -1 2 P ⊥ Ωa ≤ ǎd ω (j R a ) ,
where P Ωa denotes the orthogonal projection onto the vacuum sector in F a , and

ǎd ω (j R a ) := ([ω, j R 0 ], [ω, j R ∞ ]). The same estimate holds if F a , H c , j R a ,
Γa and Ω a are replaced respectively by F Z 0 , L 2 (Σ 3 ), j R s , Γs and Ω s . Recall that the total Hilbert space can be written as H = F a ⊗ F a ⊗ F Z 0 . As in [START_REF] Ammari | Scattering theory for a class of fermionic models[END_REF][START_REF] Dereziński | Asymptotic completeness in quantum field theory. Massive Pauli-Fierz Hamiltonians[END_REF], it is convenient to introduce an "extended" Hamiltonian, H ext , acting on the "extended" Hilbert space

H ext := 4 i=1 F a ⊗ 2 j=1 F Z 0 .
In our setting, the extended Hamiltonian is given by the expression

H ext := H ext 0 + gH ext I ,
where

H ext 0 :=dΓ(H D ) ⊗ 1l ⊗ 2 Fa ⊗ 1l ⊗ 2 F Z 0 + 1l ⊗ 2 Fa ⊗ dΓ(H D ) ⊗ 1l ⊗ 2 F Z 0 + 1l ⊗ 4 Fa ⊗ dΓ(H Z 0 ) ⊗ 1l F Z 0 + 1l ⊗ 4 Fa ⊗ 1l F Z 0 ⊗ dΓ(H Z 0 ),
and

H ext I
is given by ( 20)-( 24), except that the creation and annihilation operators for the electrons, b

# + = b # ⊗ 1l ⊗ 1l, acting on H = F a ⊗ F a ⊗ F Z 0 , are replaced by b #,0 + := b # ⊗ 1l ⊗ 3 Fa ⊗ 1l ⊗ 2 F Z 0
(acting on H ext ), likewise, the creation and annihilation operators for the positrons, b

# -= (-1) N+ ⊗ b # ⊗ 1l, are replaced by b #,0 -:= (-1) N+,0 ⊗ (-1) N+,∞ ⊗ b # ⊗ 1l Fa ⊗ 1l ⊗ 2 F Z 0 ,
and the creation and annihilation operators for the bosons, a # , are replaced by

a #,0 := 1l ⊗ 4 Fa ⊗ a # ⊗ 1l F Z 0 .
Here we have set

N +,0 := (N ⊗ 1l Fa ) ⊗ 1l ⊗ 2 Fa ⊗ 1l ⊗ 2 F Z 0 , N +,∞ := (1l Fa ⊗ N ) ⊗ 1l ⊗ 2 Fa ⊗ 1l ⊗ 2 F Z 0 ,
on H ext . We define similarly the number operators

N -,0 := 1l ⊗ 2 Fa ⊗ (N ⊗ 1l Fa ) ⊗ 1l ⊗ 2 F Z 0 , N -,∞ := 1l ⊗ 2 Fa ⊗ (1l Fa ⊗ N ) ⊗ 1l ⊗ 2 F Z 0 ,
and

N Z 0 ,0 := 1l ⊗ 4 Fa ⊗ N ⊗ 1l F Z 0 , N Z 0 ,∞ := 1l ⊗ 4 Fa ⊗ 1l F Z 0 ⊗ N ,
and the creation and annihilation operators

b #,∞ + := 1l Fa ⊗ b # ⊗ 1l ⊗ 2 Fa ⊗ 1l ⊗ 2 F Z 0 , b #,∞ - := (-1) N+,0 ⊗ (-1) N+,∞ ⊗ 1l Fa ⊗ b # ⊗ 1l ⊗ 2 F Z 0 , and a #,∞ := 1l ⊗ 4 Fa ⊗ 1l F Z 0 ⊗ a # .
Now, we introduce an isometric map, ΓR : H → H ext , by setting

ΓR := Γa (j R a ) ⊗ Γa (j R a ) ⊗ Γs (j R s ).
Theorem 3.5 will be a consequence of the following lemma.

Lemma 4.3. Assume that Hypothesis 1 holds and let g 0 be as in [START_REF] Weinberg | The Quantum theory of fields[END_REF].

Let χ ∈ C ∞ 0 (R). Then, for all |g| ≤ g 0 , ΓR χ(H) -χ(H ext ) ΓR → 0, as R → ∞.
Proof. Using the Helffer-Sjöstrand functional calculus, we represent χ(H) as the integral

χ(H) = 1 π ∂ χ ∂ z (z)(H -z) -1 d Rez d Imz,
where χ ∈ C ∞ 0 (C) denotes an almost analytic extension of χ satisfying χ| R = χ and |∂ z χ(z)| ≤ C n |Im z| n for any n ∈ N. The same representation holds for χ(H ext ), from which we deduce that

ΓR χ(H) -χ(H ext ) ΓR = 1 π ∂ χ ∂ z (z)(H ext -z) -1 (H ext ΓR -ΓR H)(H -z) -1 d Rez d Imz.
By Lemma 4.2(ii), together with

N 1 2 # (H -z) -1 ≤ C|Im z| -1 , (H ext -z) -1 (N #,0 + N #,∞ ) 1 2 ≤ C|Im z| -1 ,
where N # stands for N + , N -or N Z 0 (and likewise for N #,0 and N #,∞ ), we obtain

(H ext -z) -1 (H ext 0 ΓR -ΓR H 0 )(H -z) -1 ≤ C ǎd ω (j R a ) + ǎd ω3 (j R s ) |Im z| -2 . ( 35
)
Here, ω is given by ( 4) and ω 3 is given by [START_REF] Gérard | Spectral and scattering theory for some abstract QFT Hamiltonians[END_REF]. Using e.g. pseudo-differential calculus, one easily verifies that ǎd ω (j R a ) = O(R -1 ) and ǎd ω3 (j R s ) = O(R -1 ), as R → ∞. Hence, (35) combined with the properties of the almost analytic extension χ show that

∂ χ ∂ z (z)(H ext -z) -1 (H ext 0 ΓR -ΓR H 0 )(H -z) -1 d Rez d Imz = O(R -1 ).
It remains to estimate

∂ χ ∂ z (z)(H ext -z) -1 (H ext I ΓR -ΓR H I )(H -z) -1 d Rez d Imz.
The different interaction terms appearing in the definition ( 20) of H I are treated in the same way. Consider for instance the interaction Hamiltonian H

(1) I

given by ( 21), written under the form given in [START_REF] Rose | Relativistic Electron Theory[END_REF],

H (1) I = F (1) (ξ 1 , ξ 2 , ξ 3 )b * + (ξ 1 )b * -(ξ 2 )a(ξ 3 ) dξ 1 dξ 2 dξ 3 ,
with F (1) ∈ L 2 (dξ 1 dξ 2 dξ 3 ). We let H

I = j 1 (|i∇ p1 |, |i∇ p2 |, |i∇ k |)F (1) (ξ 1 , ξ 2 , ξ 3 )b * ,0 + (ξ 1 )b * ,0 -(ξ 2 )a 0 (ξ 3 ) ΓR dξ 1 dξ 2 dξ 3 , + l>1 j l (|i∇ p1 |, |i∇ p2 |, |i∇ k |)F (1) (ξ 1 , ξ 2 , ξ 3 )b * , + (ξ 1 )b * , -(ξ 2 )a (ξ 3 ) ΓR dξ 1 dξ 2 dξ 3 ,
where we have set

j 1 (|y 1 |, |y 2 |, |y 3 |) = 1 -j 0 (|y 1 |/R)j 0 (|y 2 |/R)j 0 (|y 3 |/R) and, for l = 1, j l (|y 1 |, |y 2 |, |y 3 |) is of the form j l (|y 1 |, |y 2 |, |y 3 |) = j #1 (|y 1 |/R)j #2 (|y 2 |/R)j #3 (|y 3 |/R)
with j #i = j 0 or j #i = j ∞ , and at least one of the j #i 's is equal to j ∞ . Moreover, b * , + stands for b * ,0 + or b * ,∞ + , and likewise for b * , -and a . It follows from the N τ estimates (see [START_REF] Glimm | Quantum field theory and statistical mechanics[END_REF]) that (1) .

(H ext -z) -1 (H (1),ext I ΓR -ΓR H I,(1) )(H -z) -1 ≤ C|Im z| -2 l j l (|i∇ p1 |, |i∇ p2 |, |i∇ k |)F
Therefore, using the fact that

j l (|i∇ p1 |, |i∇ p2 |, |i∇ k |)F (1) → 0,
as R → ∞ and the properties of χ, we deduce that

∂ χ ∂ z (z)(H ext -z) -1 (H (1),ext I ΓR -ΓR H (1)
I )(H -z) -1 d Rez d Imz → 0,
as R → ∞. Since the other interaction terms in [START_REF] Itzykson | Quantum Field theory[END_REF] are treated in the same way, this concludes the proof.

We are now ready to prove Theorem 3.5.

Proof of Theorem 3.5. We prove that

spec ess (H) ⊂ [inf spec(H) + m e , ∞). ( 36
) Let χ ∈ C ∞ 0 ((-∞, inf spec(H) + m e )).
Since ΓR is isometric, we can write

χ(H) = Γ * R ΓR χ(H) = Γ * R χ(H ext ) ΓR + o R (1), ( 37
)
where o R (1) stands for a bounded operator vanishing as R → ∞. The last equality above follows from Lemma 4.3. Observing that N tot,∞ := N +,∞ + N -,∞ + N Z 0 ,∞ commutes with H ext and that

H ext 1l [1,∞) (N tot,∞ ) ≥ (inf spec(H) + m e )1l [1,∞) (N tot,∞ ), we deduce that χ H ext = 1l {0} (N tot,∞ )χ H ext .
Hence (37) yields

χ(H) = Γ * R 1l {0} (N tot,∞ )χ H ext ΓR + o R (1) = Γ * R 1l {0} (N tot,∞ ) ΓR χ(H) + o R (1), (38)
where we used again Lemma 4.3 in the last equality. Inspecting the definition of the operator ΓR , it is easy to see that

Γ * R 1l {0} (N tot,∞ ) ΓR = Γ (j R 0 ) 2 ⊗ Γ (j R 0 ) 2 ⊗ Γ (j R 0 ) 2 . Since Γ (j R 0 ) 2 ⊗ Γ (j R 0 ) 2 ⊗ Γ (j R 0 ) 2 (H 0 + i) -1 is compact, and since (H 0 + i)χ(H) is bounded, we conclude that Γ * R 1l {0} (N tot,∞ ) ΓR χ(H)
is compact. Therefore, by (38), the operator χ(H) is also compact, which proves (36).

To prove the converse inclusion, it suffices to construct, for any λ ∈ (inf spec(H)+ m e , ∞), a Weyl sequence associated to λ. This can be done in the same way as in [START_REF] Dereziński | Asymptotic completeness in quantum field theory. Massive Pauli-Fierz Hamiltonians[END_REF]Theorem 4.1] or [START_REF] Ammari | Scattering theory for a class of fermionic models[END_REF]Theorem 4.3]. We do not give the details.

5 Proofs of Theorems 3.8 and 3.9

In this section, we prove Theorems 3.8 and 3.9 by applying a suitable version of Mourre's theory. We begin with defining the conjugate operator A that we consider in Subsection 5.1; We show that the semi-group generated by A preserves the form domain of the total Hamiltonian H. In Subsection 5.2, we establish regularity of H with respect to A and in Subsection 5.3, we prove a Mourre estimate. Putting all together, we finally deduce in Subsection 5.4 that the statements of Theorems 3.8 and 3.9 hold.

The conjugate operator and its associated semigroup

Let a be the operator on L 2 (R + ) defined by the expression

a = i 2 f (p)∂ p + ∂ p f (p) = if (p)∂ p + i 2 f (p), (39)
where f (p) := p -1 ω(p) = p -1 p 2 + m e 2 and f stands for the derivative of f . The operator a with domain C ∞ 0 ((0, ∞)) is symmetric; its closure is denoted by the same symbol.

We construct the C 0 -semigroup, w t , associated with a. Let

g(p) := p 0 1 f (r) dr = p 2 + m e 2 -m e .
Note that the function g is bijective on [0, ∞), with inverse

g -1 (p) = (p + m e ) 2 -m e 2 .
For all t ≥ 0, let ψ t be defined on [0, ∞) by ψ t (p) := 0 if p < (t + m e ) 2 -m e 2 = g -1 (t) and ψ t (p) := g -1 (-t + g(p)) otherwise. Setting

(w t u)(p) := ∂ p ψ t (p) 1 2 u(ψ t (p)),
one easily verifies that w t is the C 0 -semigroup of isometries associated with a, namely w t+s = w t w s for t, s ≥ 0, and (∂ t w t u)| t=0 (p) = i(au)(p). We observe that a is maximal symmetric with deficiency index n + = dim Ker(a * -i) = 0.

On H c = ⊕ γ L 2 (R + ), the operator ⊕ γ a is still denoted by the symbol a. Our conjugate operator, A, acting on the full Hilbert space

H = F a ⊗ F a ⊗ F Z 0 , is then given by A := dΓ(a) ⊗ 1l ⊗ 1l + 1l ⊗ dΓ(a) ⊗ 1l. ( 40
)
From the properties of a, we deduce that A is maximal symmetric and generates the C 0 -semigroup

W t := Γ(w t ) ⊗ Γ(w t ) ⊗ 1l.
The adjoint semigroup, W * t , with generator -A * , is given as follows: For any p, t ≥ 0, let φ t (p) := g -1 (t + g(p)). One can verify that the adjoint semigroup of w t is the C 0 -semigroup of contractions given by

(w * t u)(p) = ∂ p φ t (p) 1 2 u(ψ t (p)).
We deduce that

W * t = Γ(w * t ) ⊗ Γ(w * t ) ⊗
G := D(|H| 1 2 ) = D(H 1 2 0 ).
Proposition 5.1. For all t ≥ 0, we have that

W t G ⊂ G, W * t G ⊂ G,
and

H 1 2 0 W t (H 1 2 0 + 1l) -1 ≤ 1, H 1 2 0 W * t (H 1 2 0 + 1l) -1 ≤ 1.
In particular, Hypothesis B.1 of Appendix B is satisfied.

Proof. We prove the statement for W * t , the proof for W t is similar. First, we show that w * t D(ω) ⊂ D(ω) and that

ω -1 2 w t ω w * t ω -1 2 ≤ 1, (41)
where, recall, ω is the multiplication operator by ω(p) = p 2 + m e 2 on L 2 (R + ). For any u ∈ C ∞ 0 ((0, ∞)), we have that

ω w * t u 2 = ω(p) 2 ∂ p φ t (p) u(φ t (p)) 2 dp.
Using the definition of φ t , one sees that φ t (p) ≥ p for all t ≥ 0, and hence

ω w * t u 2 ≤ ω(φ t (p)) 2 ∂ p φ t (p) u(φ t (p)) 2 dp = ω u 2 .
Since C ∞ 0 ((0, ∞)) is a core for ω, this implies that w * t D(ω) ⊂ D(ω) and that

ω w * t ω -1 ≤ 1.
Using the fact that w t is isometric and an interpolation argument, we obtain (41). Now, let ϕ ∈ F a,fin (D(ω)) ⊗ F a,fin (D(ω)) ⊗ F Z 0 , where F a,fin (D(ω)) denotes the set of vectors (ϕ 0 , ϕ 1 , . . . ) in ⊕ ∞ n=0 ⊗ n a D(ω) (algebraic tensor product) such that ϕ n = 0 for all but finitely many n's. We compute

H 1 2 0 W * t ϕ 2 = ϕ, W t H 0 W * t ϕ = ϕ, dΓ(w t w * t , w t ωw * t ) ⊗ Γ(w t w * t ) ⊗ 1l + Γ(w t w * t ) ⊗ dΓ(w t w * t , w t ωw * t ) ⊗ 1l + Γ(w t w * t ) ⊗ Γ(w t w * t ) ⊗ dΓ(ω 3 ) ϕ ,
where, for c 1 , c 2 operators on H c , the operator dΓ(c 1 , c 2 ) on F a is defined by (see [START_REF] Ammari | Scattering theory for a class of fermionic models[END_REF][START_REF] Dereziński | Asymptotic completeness in quantum field theory. Massive Pauli-Fierz Hamiltonians[END_REF])

dΓ(c 1 , c 2 )Ω a = 0, dΓ(c 1 , c 2 )| ⊗ n a Hc = n j=1 c 1 ⊗ • • • ⊗ c 1 j-1 ⊗c 2 ⊗ c 1 ⊗ • • • ⊗ c 1 n-j
.

Combining (41), the bound w t w * t ≤ 1, and [1, Lemma 2.3] (see also [9, Lemma 2.8]), we obtain

H 1 2 0 W * t ϕ 2 ≤ (dΓ(ω) 1 2 ⊗ 1l ⊗ 1l ϕ 2 + (1l ⊗ dΓ(ω) 1 2 ⊗ 1l ϕ 2 + (1l ⊗ 1l ⊗ dΓ(ω 3 ) 1 2 ϕ 2 = H 1 2 0 ϕ 2 .
This concludes the proof.

Regularity of the Hamiltonian with respect to the conjugate operator

Recall that the conjugate operator A is defined by the expressions (39) and (40).

In this subsection, we prove the following proposition.

Proposition 5.2. Assume that Hypothesis 3.6 holds. Let |g| m e . Then we have that

H ∈ C 1,1 (A G ; A G * ),
in the sense of Hypothesis B.5 of Appendix B.

To prove Proposition 5.2, we use real interpolation. We have that

[H 0 , iA] = N + ⊗ 1l ⊗ 1l + 1l ⊗ N -⊗ 1l,
H 0 ∈ C 1,1 (A G ; A G * ).
Here we recall that, for all 0 ≤ θ ≤ 1 and 1 ≤ q < ∞,

C θ,q (A G ; A G * ) := T ∈ B(G; G * ), W * t T W t -T ∈ B(G; G * ) for all t ∈ (0, 1), 1 0 t -θq-1 W * t T W t -T q B(G;G * ) dt < ∞ . (42)
In order to prove that H ∈ C 1,1 (A G ; A G * ), it remains to show that the interaction Hamiltonian H I ∈ C 1,1 (A G ; A G * ). Using in particular Proposition B.3, we see that it suffices, in fact, to verify that the commutator [H I , iA] belongs to B(G; G * ) and that [H I , iA] ∈ C 0,1 (A G ; A G * ). This is the purpose of the remainder of this section.

We use the notation [START_REF] Skibsted | Spectral analysis of N -body systems coupled to a bosonic field[END_REF]. Using Hypothesis 3.6 and the estimates of Appendix A (see ( 80)-( 81) and ( 83)-( 86)), we can rewrite

F (α) (ξ 1 , ξ 2 , ξ 3 ) := h(α) (ξ 1 , ξ 2 , ξ 3 ) G(α) (ξ 1 , ξ 2 , ξ 3 ), ( 43
)
where h(α

) (ξ 1 , ξ 2 , ξ 3 ) is of the form h(α) (ξ 1 , ξ 2 , ξ 3 ) = p 1 p 2 s (α) (ξ 1 , ξ 2 , ξ 3 ), (44)
with s (α) satisfying, for all n, m ∈ {0, 1, 2},

∂ n p1 ∂ m p2 s (α) (ξ 1 , ξ 2 , ξ 3 ) p -n 1 p -m 2 , (45)
in a neighborhood of 0. Moreover the kernels G(α) satisfy (a 0 ) There exists a compact set

K ⊂ R + ×R + ×R 3 such that G(α) (p 1 , γ 1 , p 2 , γ 2 , k, λ) = 0 if (p 1 , p 2 , k) / ∈ K.
(b 0 ) There exists ε > 0 such that γ1,γ2,λ

(1 + x 2 1 + x 2 2 ) 1+ε Ĝ(α) (x 1 , γ 1 , x 2 , γ 2 , k, λ) 2 dx 1 dx 2 dk < ∞,
where, recall, Ĝ(α) denote the Fourier transform of G(α) with respect to the variables (p 1 , p 2 ), and x l , l = 1, 2, is the variable dual to p l .

(c 0 ) If p 1 = 0 or p 2 = 0, then G(α) (p 1 , γ 1 , p 2 , γ 2 , k, λ) = 0.

Our strategy consists in working with interaction operators of the form (20) with H

(1)

I , H (2)
I
given by ( 25)- [START_REF] Schweber | An Introduction to Relativistic Quantum Field Theory[END_REF] and F (1) , F (2) satisfying (43)-(45). We then use an interpolation argument for the kernels G(α) . given by (25)-( 26) and F (1) , F (2) satisfying (43)-( 45).

(i) Suppose that G(α) ∈ L 2 (Σ × Σ × Σ 3 ) satisfy the following conditions (i)(a) There exists a compact set K ⊂ R + ×R + ×R 3 such that G(α) (p 1 , γ 1 , p 2 , γ 2 , k, λ) = 0 if (p 1 , p 2 , k) / ∈ K. (i)(b) γ1,γ2,λ (1 + x 2 1 + x 2 2 ) Ĝ(α) (x 1 , γ 1 , x 2 , γ 2 , k, λ) 2 dx 1 dx 2 dk < ∞. (i)(c) If p 1 = 0 or p 2 = 0, then G(α) (p 1 , γ 1 , p 2 , γ 2 , k, λ) = 0. Then H I = [H I , iA] ∈ C 0 (A G ; A G * ) ≡ B(G; G * ).
(ii) Suppose that G(α) ∈ L 2 (Σ × Σ × Σ 3 ) satisfy the following conditions (ii)(a) There exists a compact set

K ⊂ R + ×R + ×R 3 such that G(α) (p 1 , γ 1 , p 2 , γ 2 , k, λ) = 0 if (p 1 , p 2 , k) / ∈ K. (ii)(b) γ1,γ2,λ (1 + x 2 1 + x 2 2 ) 3 Ĝ(α) (x 1 , γ 1 , x 2 , γ 2 , k, λ) 2 dx 1 dx 2 dk < ∞. (ii)(c) If p 1 = 0 or p 2 = 0, then D β G(α) (p 1 , γ 1 , p 2 , γ 2 , k, λ) = 0 for all multi- index β = (β 1 , β 2 ), |β| ≤ 2, with D β = ∂ β1+β2 /∂ x β 1 1 ∂ x β 2 2
.

Then

H I = [H I , iA] ∈ C 1 (A G ; A G * ).
Proof. (i) Recall that the conjugate operator A is defined by Eq. ( 40), with

a = if (p)∂ p + i 2 f (p),
and f (p) = p -1 p 2 + m e 2 . We use the notation

a l = if (p l )∂ p l + i 2 f (p l ), for l = 1, 2. We then have that [H I , iA] = H I (-ia 1 F ) + H I (-ia 2 F ) , (46)
in the sense of quadratic forms on D(H 0 ) ∩ D(A).

Recalling the notations ξ l = (p l , γ l ), we compute

(a 1 F (α) )(ξ 1 , ξ 2 , ξ 3 ) = i 2 p 1 p 2 f (p 1 )s (α) (ξ 1 , ξ 2 , ξ 3 ) + ip 2 f (p 1 )s (α) (ξ 1 , ξ 2 , ξ 3 ) + ip 1 p 2 f (p 1 )(∂ p1 s (α) )(ξ 1 , ξ 2 , ξ 3 ) G(α) (ξ 1 , ξ 2 , ξ 3 ) + ip 1 p 2 f (p 1 )s (α) (ξ 1 , ξ 2 , ξ 3 )(∂ p1 G(α) )(ξ 1 , ξ 2 , ξ 3 ). ( 47
)
Using (45) and the definition of f , we see that the term in brackets satisfy

i 2 p 1 p 2 f (p 1 )s (α) (ξ 1 , ξ 2 , ξ 3 ) + ip 2 f (p 1 )s (α) (ξ 1 , ξ 2 , ξ 3 ) + ip 1 p 2 f (p 1 )(∂ p1 s (α) )(ξ 1 , ξ 2 , ξ 3 ) p -1 1 p 2 , in any compact set. Now, since p 1 → G(α) (p 1 , γ 1 , ξ 2 , ξ 3 ) ∈ H 1 0 (R +
) by the conditions (i)(b) and (i)(c), and since G(α) is compactly supported in the variables (p 1 , p 2 , k) by the condition (i)(a), we deduce that

p -1 1 p 2 G(α) (ξ 1 , ξ 2 , ξ 3 ) ∈ L 2 (dξ 1 dξ 2 dξ 3 ).
Here we used that

p -1 1 p 2 G(α) (ξ 1 , ξ 2 , ξ 3 ) L 2 (dξ1dξ2dξ3) p 2 ∂ p1 G(α) (ξ 1 , ξ 2 , ξ 3 ) L 2 (dξ1dξ2dξ3) ,
by Hardy's inequality at the origin in H 1 0 (R + ). Likewise, we have that

ip 1 p 2 f (p 1 )s (α) (ξ 1 , ξ 2 , ξ 3 ) 1,
in any compact set, and hence, using again that p 1 → G(α) (p 1 , γ 1 , ξ 2 , ξ 3 ) ∈ H 1 0 (R + ) and that G(α) is compactly supported in the variables (p 1 , p 2 , k), it follows that

ip 1 p 2 f (p 1 )s (α) (ξ 1 , ξ 2 , ξ 3 )(∂ p1 G(α) )(ξ 1 , ξ 2 , ξ 3 ) ∈ L 2 (dξ 1 dξ 2 dξ 3 ).
The previous estimates show that

(a 1 F (α) )(ξ 1 , ξ 2 , ξ 3 ) ∈ L 2 (dξ 1 dξ 2 dξ 3 ),
and proceeding in the same way, one verifies that (a 2

F (α) )(ξ 1 , ξ 2 , ξ 3 ) ∈ L 2 (dξ 1 dξ 2 dξ 3 ).
Using the expression (46) of the commutator [H I , iA] and the N τ estimates of [START_REF] Glimm | Quantum field theory and statistical mechanics[END_REF], we immediately deduce that [

H I , iA] ∈ B(G; G * ) = C 0 (A G ; A G * ).
(ii) It suffices to proceed similarly. More precisely, we compute the second commutator

H I , iA , iA = -H I (a 2 1 F ) -H I (a 2 2 F ) -2H I (a 1 a 2 F ). ( 48
)
Computing a 2 1 F , a 2 2 F and a 1 a 2 F yields to several terms that are estimated separately. Each term, however, can be treated in the same way, using Hardy's inequality together with the assumptions (ii)(a), (ii)(b), (ii)(c). We give an example. Consider the first term inside the brackets of ( 47) and apply to it the operator if (p 1 )∂ p1 . This gives in particular a term of the form

- 1 2 p 2 f (p 1 )f (p 1 )s (α) (ξ 1 , ξ 2 , ξ 3 ) G(α) (ξ 1 , ξ 2 , ξ 3 ),
that will appear in the expression of a 2 1 F . From (45) and the definition of f , it follows that

p 2 f (p 1 )f (p 1 )s (α) (ξ 1 , ξ 2 , ξ 3 ) G(α) (ξ 1 , ξ 2 , ξ 3 ) p -3 1 p 2 G(α) (ξ 1 , ξ 2 , ξ 3 ) , in any compact set. Since p 1 → G(α) (p 1 , γ 1 , ξ 2 , ξ 3 ) ∈ H 3 0 (R +
) by the conditions (ii)(b) and (ii)(c), and since G(α) is compactly supported in the variables (p 1 , p 2 , k) by the condition (ii)(a), we obtain as above that

p 2 f (p 1 )f (p 1 )s (α) (ξ 1 , ξ 2 , ξ 3 ) G(α) (ξ 1 , ξ 2 , ξ 3 ) ∈ L 2 (dξ 1 dξ 2 dξ 3 ).
Here we used that

p -3 1 p 2 G(α) (ξ 1 , ξ 2 , ξ 3 ) L 2 (dξ1dξ2dξ3) p 2 ∂ 3 p1 G(α) (ξ 1 , ξ 2 , ξ 3 ) L 2 (dξ1dξ2dξ3) ,
by Hardy's inequality at the origin in H 3 0 (R + ). Treating all the other terms in a similar manner, we deduce that

a 2 1 F + a 2 2 F + 2a 1 a 2 F ∈ L 2 (dξ 1 dξ 2 dξ 3 ),
and therefore that [[H I , iA], iA] ∈ B(G; G * ). Together with Proposition 5.1, this shows (ii).

Proof of Proposition 5.2. By the observation after the statement of Proposition 5.2, we already know that H 0 ∈ C 1,1 (A G ; A G * ). Hence, to conclude the proof of Proposition 5.2, it suffices to verify that

H I ∈ C 1,1 (A G ; A G * ).
Recall that H I is the sum of 4 terms, see [START_REF] Itzykson | Quantum Field theory[END_REF]. We consider for instance the first one, H

I . The other terms can be treated in the same way.

Let 1)

K 0 ⊂ R + × R + × R 3 be a compact set. Let S (i) denote the set of all G(1) ∈ L 2 (Σ × Σ × Σ 3 ) satisfying the conditions (i)(a) (with K = K 0 ), (i)(b) and (i)(c), equipped with the norm G(
S (i) := γ1,γ2,λ (1 + x 2 1 + x 2 2 ) Ĝ(1) (x 1 , γ 1 , x 2 , γ 2 , k, λ) 2 dx 1 dx 2 dk.
Likewise, we denote by S (ii) the set of all G(1) ∈ L 2 (Σ × Σ × Σ 3 ) satisfying the conditions (ii)(a) (with K = K 0 ), (ii)(b) and (ii)(c), equipped with the norm G( 1)

S (ii) := γ1,γ2,λ (1 + x 2 1 + x 2 2 ) 3 Ĝ(1) (x 1 , γ 1 , x 2 , γ 2 , k, λ) 2 dx 1 dx 2 dk.
By Lemma 5.3 and its proof, the map

S (i) G(1) → H (1) I ( h(1) G(1) ) ∈ C 0 (A G ; A G * ) (49)
is linear and continuous, and, likewise, the map

S (ii) G(1) → H (1) I ( h(1) G(1) ) ∈ C 1 (A G ; A G * ) ( 50
)
is linear and continuous. Here we have used the notation

H (1) I ( h(1) G(1) ) := [H (1)
I ( h(1) G(1) ), iA]. By real interpolation, we deduce that

S (i) , S (ii) θ,2 G(1) → H (1) I ( h(1) G(1) ) ∈ C 0 (A G ; A G * ), C 1 (A G ; A G * ) θ,2 , (51)
for all 0 ≤ θ ≤ 1. Now, by [2, Section 5], we have that

C 0 (A G ; A G * ), C 1 (A G ; A G * ) θ,2 = C θ,2 (A G ; A G * ), (52)
for all 0 < θ < 1, and using the definition (42), one easily verifies that

C θ,2 (A G ; A G * ) ⊂ C 0,1 (A G ; A G * ).
On the other hand, from the definition of the interpolated space S (i) , S (ii) θ,2 and mimicking the method allowing one to compute the interpolation of Sobolev spaces (see e.g. [START_REF] Triebel | Interpolation theory, function spaces, differential operators[END_REF]), it is not difficult to verify that, for 0 < ε < 2θ < 1, the set of all kernels G(1) ∈ L 2 (Σ × Σ × Σ 3 ) satisfying the conditions (a 0 ), (b 0 ) and (c 0 ) stated above is included in S (i) , S (ii) θ,2 . This shows, in particular, that

H (1) I ∈ C 0,1 (A G ; A G * )
, and hence that H

(1)

I ∈ C 1,1 (A G ; A G * ). Since the other terms, H (1) I * , H (2)
I
and H

(2) I *

, can be treated in the same way, this concludes the proof.

The Mourre estimate

Given F = ( F (1) , F (2) ) ∈ (H c ⊗ H c ⊗ L 2 (Σ 3 )) 2 ,

and for

H (i) I ( F (i) )
given by ( 25)-( 26), we define

H I ( F ) = H (1) I ( F (1) ) + (H (1) I ( F (1) )) * + H (2) I ( F (2) ) + (H (2) I ( F (2) )) * .
Proposition 5.4. Assume that Hypothesis 3.6 hold and let δ ∈ (0, m e ). There exist g δ > 0, c δ > 0 and C ∈ R such that, for all |g| ≤ g δ , and for

∆ := [δ, m Z 0 -δ],
we have, in the sense of quadratic forms on D(A) ∩ D(H 0 ),

H ≡ [H, iA] ≥ c δ 1l -C1l ⊥ ∆ (H -E) H , ( 53
)
where we have set

E := inf spec(H), 1l ⊥ ∆ (H -E) := 1l -1l ∆ (H -E) and H := (1l + H 2 ) 1/2 .
Proof of Proposition 5.4. As in Subsection 5.2, we have, in the sense of quadratic forms on

D(A) ∩ D(H 0 ), [H 0 , iA] = N + ⊗ 1l ⊗ 1l + 1l ⊗ N -⊗ 1l , (54)
where N + (respectively N -) is the number operator for electrons (respectively positrons) as defined in [START_REF] Georgescu | Spectral theory of massless Pauli-Fierz models[END_REF]. In the sequel, by abuse of notation, we shall omit the identity operators in N + ⊗ 1l ⊗ 1l and 1l ⊗ N -⊗ 1l and denote them respectively again by N + and N -.

Let a 1 = a ⊗ 1l ⊗ 1l be the conjugate operator for electron acting on the p 1 variable in H c ⊗ H c ⊗ L 2 (Σ 3 ) and a 2 = 1l ⊗ a ⊗ 1l be the conjugate operator for positron acting on the p 2 variable. As in (46), we have that

[H I , iA] = H I (-ia 1 F ) + H I (-ia 2 F ) , (55)
in the sense of quadratic forms on D(A) ∩ D(H 0 ). Here we recall that a 1 F and a 2 F belong to L 2 (dξ 1 dξ 2 dξ 3 ) as follows from the estimates of Appendix A and Hypothesis 3.6 (see more precisely the proof of Lemma 5.3 (i)). For P Ωa×Ωa := P Ωa ⊗ P Ωa ⊗ 1l being the projection onto the electron/positron vacuum, we have that

N + + N -+ P Ωa×Ωa ≥ 1l. (56)
Since H = H 0 + gH I , and for E = inf spec(H), we obtain from (54)-( 55) that

[H, iA] = (N + + N -+ P Ωa×Ωa ) -P Ωa×Ωa + g (H I (-ia 1 F ) + H I (-ia 2 F )) ≥ 1l -P Ωa×Ωa + g (H I (-ia 1 F ) + H I (-ia 2 F )) , (57)
where we used the operator inequality (56) in the last inequality. We estimate separately the two remainder terms occuring in the right hand side of (57).

Let us define a function

f ∆ ∈ C ∞ 0 (R) such that 0 ≤ f ∆ ≤ 1 and f ∆ (λ) = 1 if λ ∈ [δ, m Z 0 -δ], 0 if λ < δ/2 or λ > m Z 0 -δ/2. (58)
We observe that

P Ωa×Ωa f ∆ (H 0 ) = 0 . (59)
The last identity holds because P Ωa×Ωa is a projection commuting with H 0 and because supp(f

∆ ) ∩ spec(H 0 P Ωa×Ωa ) = ∅. As in the proof of Lemma 4.3, let f ∈ C ∞ 0 (C) denote an almost analytic extension of f ∆ satisfying f | R = f ∆ and |∂ z f (z)| ≤ C n |Im z| n for any n ∈ N. Thus, for d f (z) := -1 π ∂ f
∂z (z) dRe z dIm z, using Helffer-Sjöstrand functional calculus and the second resolvent equation, we obtain

f ∆ (H -E) -f ∆ (H 0 ) = (H -E -z) -1 (H -E -H 0 )(H 0 -z) -1 d f (z) = (H -E -z) -1 gH I (F )(H 0 -z) -1 d f (z) -E (H -E -z) -1 (H 0 -z) -1 d f (z) . (60)
From Corollary 4.1, since Hypothesis 3.1 holds, there exists a constant C such that

H I (F )(H 0 + 1) -1 ≤ CK(G) , (61)
where h (α) G (α) = F (α) (see [START_REF] Skibsted | Spectral analysis of N -body systems coupled to a bosonic field[END_REF]) and K(G) = K 2 (G) is given by ( 31) and (33). Therefore, with the inequality

(H 0 + 1)(H 0 -z) -1 ≤ 1 + 1 + |z| |Im z| , (62)
and the properties of f , we obtain that there exists a constant C 1 > 0 depending only on f ∆ and K(G) such that

(H -E -z) -1 gH I (F )(H 0 -z) -1 d f (z) ≤ |g| (1 + 1 + |z| |Im z| ) (H -E -z) -1 H I (G)(H 0 + 1) -1 d f (z) ≤ C 1 |g| . ( 63
)
Moreover, using again (61), standard perturbation theory yields that there exists g 1 > 0 such that for all |g| ≤ g 1 , we have

|E| ≤ |g| K(G)B βη 1 -g 1 K(G)C βη , (64)
where B βη and C βη are the positive constants defined in Subsection 4.1. Thus, there exists a constant C 2 depending on f ∆ and K(G) such that

E (H -E -z) -1 (H 0 -z) -1 d f (z) ≤ C 2 |g| . (65)
Inequalities ( 60), ( 63) and (65) give

f ∆ (H -E) -f ∆ (H 0 ) ≤ (C 1 + C 2 ) |g|. (66)
For shortness, let 1l ∆ ≡ 1l ∆ (H -E) and 1l ⊥ ∆ ≡ 1l ⊥ ∆ (H -E). We have that

-P Ωa×Ωa = -1l ∆ P Ωa×Ωa 1l ∆ -1l ∆ P Ωa×Ωa 1l ⊥ ∆ -1l ⊥ ∆ P Ωa×Ωa 1l ∆ -1l ⊥ ∆ P Ωa×Ωa 1l ⊥ ∆ ≥ -1l ∆ P Ωa×Ωa 1l ∆ -1l ∆ P Ωa×Ωa 1l ⊥ ∆ -1l ⊥ ∆ P Ωa×Ωa 1l ∆ -1l ⊥ ∆ . (67)
Using ( 59) and (66), we obtain that

1l ∆ P Ωa×Ωa ≤ f ∆ (H -E)P Ωa×Ωa = f ∆ (H -E) -f ∆ (H 0 ) P Ωa×Ωa ≤ (C 1 + C 2 ) |g|,
from which we deduce that

-1l ∆ P Ωa×Ωa 1l ∆ -1l ∆ P Ωa×Ωa 1l ⊥ ∆ -1l ⊥ ∆ P Ωa×Ωa 1l ∆ ≥ -3(C 1 + C 2 ) |g| 1l.
Together with (67), this shows that

-P Ωa×Ωa ≥ -3(C 1 + C 2 )|g|1l -1l ⊥ ∆ . (68)
To bound the last term in the right hand side of (57), it suffices to use the relative bound in Corollary 4.1 and the fact that Hypothesis 3.6 holds (and hence also Hypothesis 3.1), to obtain that the operators H I (-ia l F ) (l = 1, 2) are norm relatively bounded with respect to H 0 with relative bounds depending on K(G) and K(-ia l G). Therefore, there exists C 3 depending on K(G) and K(-ia l G) such that

g H I (-ia 1 F ) + H I (-ia 2 F ) ≥ -C 3 |g| H = -C 3 |g| H 1l ∆ (H -E) -C 3 |g| H 1l ⊥ ∆ (H -E) ≥ -C 4 |g|1l ∆ (H -E) -C 3 |g| H 1l ⊥ ∆ (H -E) ≥ -C 4 |g|1l -C 5 |g| H 1l ⊥ ∆ (H -E), (69)
for some constants C 4 , C 5 ∈ R.

The estimates (57), ( 68) and ( 69) yield (53), which concludes the proof.

Proofs of the main theorems

Proof of Theorem 3.8. As above, we use the notation E = inf spec(H). The proof of Theorem 3.8 is divided into two main steps.

Step 1. Let 0 < δ < m e . There exists g δ > 0 such that, for all 0 ≤ |g| ≤ g δ , inf spec(H) \ {E}) ≥ δ.

To prove this, we use the min-max principle. Let µ 2 denote the second point above E in the spectrum of H. The min-max principle implies that

µ 2 ≥ inf ψ ∈ D(H), ψ = 1, ψ ∈ [ΩD ⊗ Ωs] ⊥ ψ, Hψ = inf ψ ∈ D(H), ψ = 1, ψ ∈ [ΩD ⊗ Ωs] ⊥ ψ, H 0 ψ + g ψ, H I ψ ,
where [Ω D ⊗ Ω s ] ⊥ denotes the orthogonal complement of the subspace spanned by Ω D ⊗ Ω s in the total Hilbert space H. Since H I is relatively bounded with respect to H 0 , there exists a positive constant C such that ψ, H I ψ ≥ -C ψ, H 0 ψ , and therefore

µ 2 ≥ inf ψ ∈ D(H), ψ = 1, ψ ∈ [ΩD ⊗ Ωs] ⊥ (1 -C|g|) ψ, H 0 ψ ≥ (1 -C|g|)m e ,
the last inequality being a consequence of ( 17). This proves Step 1.

Step 2. Let 0 < δ < m e . There exists g δ > 0 such that, for all 0

≤ |g| ≤ g δ , spec(H) ∩ [δ, m e + E) = ∅.
Observe that E < 0 satisfies E ≥ -C|g| with C a positive constant, as follows from standard perturbation theory (see ( 64)), and therefore, for g δ small enough and |g| ≤ g δ , we have that δ < m e +E. By Theorem 3.5, we know that inf spec ess (H) = m e + E. Thus we only have to show that H do not have discrete eigenvalue in the interval [δ, m e + E): This is a simple, usual consequence of the virial theorem (see Theorem B.4) combined with the Mourre estimate of Proposition 5.4.

We introduce the notation A = (1 + A * A) 1/2 = (1 + |A| 2 ) 1/2 for any closed operator A. As mentioned before, Theorem 3.9 is a consequence of the following stronger result, which itself follows from Propositions 5.1, 5.2, 5.4, and the abstract results of Appendix B. Theorem 5.5 (Limiting absorption principle). Assume that Hypothesis 3.6 holds with ε > 0 in Hypothesis 3.6(ii). For all δ > 0, there exists g δ > 0 such that, for all |g| ≤ g δ and 1/2 < s ≤ 1,

sup z∈ ∆ A -s (H -z) -1 A -s < ∞,
with ∆ := [inf spec(H) + m e , inf spec(H) + m Z 0 -δ] and ∆ := {z ∈ C, Re z ∈ ∆, 0 < |Im z| ≤ 1}, . Moreover, the map z → A -s (H -z) -1 A -s ∈ B(H) is uniformly Hölder continuous of order s -1/2 on ∆ and the limits

A -s (H -λ -i0 ± ) -1 A -s := lim ε→0 ± A -s (H -λ -iε) -1 A -s ,
exist in the norm topology of B(H), uniformly in λ ∈ ∆. Finally, the map λ → A -s (H -λ-i0 ± ) -1 A -s ∈ B(H) is uniformly Hölder continuous of order s-1/2 on ∆ and, for any 1/2 < s ≤ 1, H satisfies the local decay property

A -s e -itH 1l ∆ (H) A -s t -s+ 1 2 , ( 70
)
for all t ∈ R.

A Generalized eigenfunctions of the free Dirac operator

In this section we describe the properties of the generalized eigenfunctions of the Dirac operator H D introduced in subsection 2.1.1. More details can be found in [17, section 9.9, (44), (45), ( 63)].

Recall that the generalized eigenfunctions of H D are labeled by the angular momentum quantum numbers

j ∈ { 1 2 , 3 2 , 5 2
, . . .}, m j ∈ {-j, -j + 1, . . . , j -1, j}, and by the quantum numbers

κ j ∈ {±(j + 1 2 )} .
We define, for γ j := |κ j |,

g κj ,± (p, r) = C ± 1 |ω(p)| 1 2 (2pr) γj r 1 2 √ π Γ(γ j ) Γ(2γ j + 1)
× e -ipr e iηj γ j F (γ j + 1, 2γ j + 1, 2ipr) + e ipr e -iηj γ j F (γ j + 1, 2γ j + 1, -2ipr) We also define

f κj ,± (p, r) = iC ± 2 |ω(p)| 1 2 1 2 √ π (2pr) γj r Γ(γ j ) Γ(2γ j + 1)
× e -ipr e iηj γ j F (γ j + 1, 2γ j + 1, 2ipr) -e ipr e -iηj γ j F (γ j + 1, 2γ j + 1, -2ipr) The functions F that occur in (71) and (72) are the confluent hypergeometric functions. Their integral representations for γ j > 1/2 are

F (γ j + 1, 2γ j + 1, ±2ipr) = Γ(2γ j + 1) Γ(γ j + 1)Γ(γ j ) 1 0 e ±2ipru u γj
) (θ, ϕ) :=   - j-mj +1 2j+2 Y j+ 1 2 ,mj -1 2 (θ, ϕ) j+mj +1 2j+2 Y j+ 1 2 ,mj -1 2 (θ, ϕ)   (75) Φ (2) mj ,(j+ 1 2 ) (θ, ϕ) :=   j+mj 2j Y j-1 2 ,mj -1 2 (θ, ϕ) j-mj 2j Y j-1 2 ,mj + 1 2 (θ, ϕ)   (76)
and

Φ (1) mj ,-(j+ 1 2 ) (θ, ϕ) =Φ (2) mj ,(j+ 1 2 ) (θ, ϕ) Φ (2) mj ,-(j+ 1 2 ) (θ, ϕ) =Φ (1) mj ,(j+ 1 2 ) (θ, ϕ).
-mj ,-κj (θ, ϕ) -f -κj ,-(p, r)Φ

(2) -mj ,-κj (θ, ϕ)

For positive energies ω(p) > m e , we have the following estimates for the functions g κj ,± and f κj ,± ,

|g j+ 1 2 ,+ (p, r)| ≤ ω(p) + m e ω(p) 1 2 p √ π (2pr) γj 1 Γ(γ j ) , |f j+ 1 2 ,+ (p, r)| ≤ ω(p) -m e ω(p) 1 2 2p √ π (2pr) γj -1 1 Γ(γ j ) , |g -(j+ 1 2 ),+ (p, r)| ≤ ω(p) + m e ω(p) 1 2 2p √ π (2pr) γj -1 1 Γ(γ j ) , |f -(j+ 1 2 ),+ (p, r)| ≤ ω(p) -m e ω(p) 1 2 p √ π (2pr) γj 1 Γ(γ j ) , (78)
and for negative energies -ω(p) < -m e , we have

|g j+ 1 2 ,-(p, r)| ≤ ω(p) -m e ω(p) 1 2 p √ π (2pr) γj 1 Γ(γ j ) , |f j+ 1 2 ,-(p, r)| ≤ ω(p) + m e ω(p) 1 2 2p √ π (2pr) γj -1 1 Γ(γ j ) , |g -(j+ 1 2 ),-(p, r)| ≤ ω(p) -m e ω(p) 1 2 2p √ π (2pr) γj -1 1 Γ(γ j ) , |f -(j+ 1 2 ),-(p, r)| ≤ ω(p) + m e ω(p) 1 2 p √ π (2pr) γj 1 Γ(γ j ) . (79)
We also can bound the first and second derivatives. Below, we give such bounds for |p| ≤ 1. For p larger than one, the functions are locally in L q for any value of q.

There exists a constant C such that for |p| ≤ 1, and for positive energies ω(p) > m e we have

∂ ∂p g j+ 1 2 ,+ (p, r) ≤ C Γ(γ j ) (2pr) γj + pr(γ j -1)(2pr) γj -1 + pr(2pr) γj -1 , ∂ ∂p f j+ 1 2 ,+ (p, r) ≤ C Γ(γ j ) p(2pr) γj -1 + p 2 r(γ j -1)(2pr) γj -2 + p 2 r(2pr) γj , ∂ ∂p g -(j+ 1 2 ),+ (p, r) ≤ C Γ(γ j ) (2pr) γj -1 + pr(γ j -1)(2pr) γj -2 + pr(2pr) γj , ∂ ∂p f -(j+ 1 2 ),+ (p, r) ≤ C Γ(γ j ) p(2pr) γj + p 2 r(γ j -1)(2pr) γj -1 + p 2 r(2pr) γj -1 , (80)
and for |p| ≤ 1 and negative energies -ω(p) < -m e , we have

∂ ∂p g j+ 1 2 ,-(p, r) ≤ C Γ(γ j ) p(2pr) γj + p 2 r(γ j -1)(2pr) γj -1 + p 2 r(2pr) γj -1 , ∂ ∂p f j+ 1 2 ,-(p, r) ≤ C Γ(γ j ) (2pr) γj -1 + pr(γ j -1)(2pr) γj -2 + pr(2pr) γj , ∂ ∂p g -(j+ 1 2 ),-(p, r) ≤ C Γ(γ j ) p(2pr) γj -1 + p 2 r(γ j -1)(2pr) γj -2 + p 2 r(2pr) γj , ∂ ∂p f -(j+ 1 2 ),-(p, r) ≤ C Γ(γ j ) (2pr) γj + pr(γ j -1)(2pr) γj -1 + pr(2pr) γj -1 . (81)
The estimates (80) and (81) yield, for a being the operator defined by (39), and for positive energies ω(p) > m e ,

|a g j+ 1 2 ,+ (p, r)| ≤ C Γ(γ j ) ω(p) p (2pr) γj + pr(γ j -1)(2pr) γj -1 + pr(2pr) γj -1 + ω(p)(1 + 1 p 2 )p(2pr) γj ) , |a f j+ 1 2 ,+ (p, r)| ≤ C Γ(γ j ) ω(p) p p(2pr) γj -1 + p 2 r(γ j -1)(2pr) γj -2 + p 2 r(2pr) γj + ω(p)(1 + 1 p 2 )p 2 (2pr) γj ) , |a g -(j+ 1 2 ),+ (p, r)| ≤ C Γ(γ j ) ω(p) p (2pr) γj -1 + pr(γ j -1)(2pr) γj -2 + pr(2pr) γj + ω(p)(1 + 1 p 2 )p(2pr) γj -1 ) , |a f -(j+ 1 2 ),+ (p, r)| ≤ C Γ(γ j ) ω(p) p p(2pr) γj + p 2 (γ j -1)(2pr) γj -1 + p 2 r(2pr) γj -1 + ω(p)(1 + 1 p 2 )p 2 (2pr) γj ) , (82)
And for negatives energies -ω(p) < -m e , we get the same estimates for |a g j+

∂ 2 ∂p 2 g j+ 1 2 ,+ (p, r) ≤ Cγ 2 j Γ(γ j ) p γj -1 r γj , (83)
∂ 2 ∂p 2 f j+ 1 2 ,+ (p, r) ≤ Cγ 2 j Γ(γ j ) p γj -1 r γj -2 , (84)
∂ 2 ∂p 2 g -(j+ 1 2 ),+ (p, r) ≤ C(γ j -1)γ j Γ(γ j ) p γj -2 r γj -1 + Cγ j Γ(γ j ) p γj r γj -1 , (85)
∂ 2 ∂p 2 f -(j+ 1 2 ),+ (p, r) ≤ Cγ 2 j Γ(γ j ) p γj r γj -1 , (86)
and the same estimates for negatives energies hold respectively for

∂ 2 ∂p 2 f -(j+ 1 2 ),-(p, r) , ∂ 2 ∂p 2 g -(j+ 1 2 ),-(p, r) , ∂ 2 ∂p 2 f j+ 1 2 ,-(p, r) and ∂ 2 ∂p 2 g j+ 1 2 ,-(p, r) .

B Mourre theory: abstract framework

In this section, we recall some abstract results from Mourre's theory that were used in Section 5. We work with an extension of the original Mourre theory [START_REF] Mourre | Absence of singular continuous spectrum for certain selfadjoint operators[END_REF] that allows, in particular, the so-called conjugate operator to be maximal symmetric (not necessarily self-adjoint). Such an extension was considered in [START_REF] Hübner | Spectral properties of the spin-boson Hamiltonian[END_REF] and further refined in [START_REF] Georgescu | Commutators, C 0 -semigroups and resolvent estimates[END_REF][START_REF] Georgescu | Spectral theory of massless Pauli-Fierz models[END_REF] (see also [START_REF] Faupin | Second Order Perturbation Theory for Embedded Eigenvalues[END_REF][START_REF] Golénia | Positive commutators, Fermi Golden Rule and the spectrum of 0 temperature Pauli-Fierz Hamiltonians[END_REF]). Here we mainly follow the presentation of [START_REF] Faupin | Second Order Perturbation Theory for Embedded Eigenvalues[END_REF].

Let H be a complex separable Hilbert space. Consider a self-adjoint operator H on H and a symmetric operator H on H such that D(H) ⊂ D(H ). Let

G := D(|H| 1
2 ), equipped with the norm

ϕ 2 G := |H| 1 2 ϕ 2 + ϕ 2 .
We set

ϕ 2 G * := (|H| + 1l) -1 2 ϕ 2 .
The dual space G * of G identifies with the completion of H with respect to the norm • G * , and the operators H, H identify with elements of B(G; G * ), the set of bounded operators from G to G * . Let A be a closed and maximal symmetric operator on H. In particular, the deficiency indices n ∓ = dim Ker(A * ± i) of A obey either n + = 0 or n -= 0. We suppose that n + = 0, so that A generates a C 0 -semigroup of isometries {W t } t≥0 (see e.g. [START_REF] Davies | Linear operators and their spectra[END_REF]Theorem 10.4.4]). Recall that a C 0 -semigroup on [0, ∞) is, by definition, a map t → W t ∈ B(H) such that W 0 = 1l, W t W s = W t+s for t, s ≥ 0, and w-lim t→0 + W t = 1l, where B(H) denotes the set of bounded operators on H and w-lim stands for weak limit. The fact that A is the generator of the C 0 -semigroup {W t } t≥0 means that

D(A) = u ∈ H, lim t→0 + (it) -1 (W t u -u) exists , iAu = lim t→0 + t -1 (W t u -u).
We make the following hypotheses.

Hypothesis B.1. For all t > 0, W t and W * t preserve G and, for all ϕ ∈ G,

sup 0<t<1 W t ϕ G < ∞, sup 0<t<1 W * t ϕ G < ∞. In particular, t → W t | G ∈ B(G) is a C 0 -semigroup,
G; G * ) is of class C 1,1 (A G ; A G * ), i.e. 1 0 [W t , [W t , H]] B(G;G * ) dt t 2 < ∞.
We recall that A = (1 + A * A) 1/2 = (1 + |A| 2 ) 1/2 for any closed operator A. Our last hypothesis is a version of a strict Mourre estimate.

Hypothesis B.6.

There exist an open interval I ⊂ R and constants c 0 > 0, C ∈ R, such that, in the sense of quadratic forms on D(H),

H ≥ c 0 1l -C1l ⊥ I (H) H , (87)
where 1l ⊥ I (H) := 1l -1l I (H).

The following theorem shows that a limiting absorption principle holds for H in any compact interval where a Mourre estimate is satisfied in the sense of Hypothesis B.6. The proof of Theorem B.7 can be found in [START_REF] Georgescu | Commutators, C 0 -semigroups and resolvent estimates[END_REF] (see also [START_REF] Hübner | Spectral properties of the spin-boson Hamiltonian[END_REF] for a similar result under slightly stronger assumptions). For any 1/2 < s ≤ 1, we have that

sup z∈ J A -s (H -z) -1 A -s < ∞,
and the map z → A -s (H -z) -1 A -s ∈ B(H) is uniformly Hölder continuous of order s -1/2 on J. In particular, the limits

A -s (H -λ -i0 ± ) -1 A -s := lim ε→0 ± A -s (H -λ -iε) -1 A -s ,
exist in the norm topology of B(H), uniformly in λ ∈ J. This implies that the spectrum of H in J is purely absolutely continuous. Moreover, the map λ → A -s (H -λ -i0 ± ) -1 A -s ∈ B(H) is uniformly Hölder continuous of order s -1/2 on J.

Remark B.8. 1) Theorem B.7 is established in [START_REF] Georgescu | Commutators, C 0 -semigroups and resolvent estimates[END_REF] in the more general context of singular Mourre theory. More precisely, as shown in [START_REF] Georgescu | Commutators, C 0 -semigroups and resolvent estimates[END_REF], the assumption that the commutator H is relatively bounded with respect to H can be relaxed. This is of fundamental importance for the application to massless quantized fields considered in [START_REF] Georgescu | Spectral theory of massless Pauli-Fierz models[END_REF], but is not needed for the model studied in the present paper. Therefore, we content ourselves with the simpler setting of regular Mourre theory (i.e. we suppose that H is H-bounded).

2) The results in [START_REF] Georgescu | Commutators, C 0 -semigroups and resolvent estimates[END_REF] are formulated under a stronger assumption than Hypothesis B.5, namely that H ∈ C 2 (A G ; A G * ). Nevertheless, as mentioned in [START_REF] Georgescu | Commutators, C 0 -semigroups and resolvent estimates[END_REF], one can verify that Hypothesis B.5 is sufficient for Theorem B.7 to hold.

3) By Fourier transform, Theorem B.7 implies the local decay property A -s e -itH χ(H) A -s = O( t -s+ 1 2 ), for any χ ∈ C ∞ 0 (I; R) and 1/2 < s ≤ 1.

C Creation and annihilation operators in Fermi-Fock space

Let G be any separable Hilbert space. Let ⊗ n a G denotes the antisymmetric n-th tensor power of G, appropriate to Fermi-Dirac statistics. We define the Fermi-Fock space over G, denoted by F a (G), to be the direct sum

F a (G) = ∞ n=0 ⊗ n a G,
where, by definition, we have set ⊗ 0 a G := C. We shall denote by Ω a the vacuum vector in F a (G), i.e., the vector (1, 0, 0, • • • ).

Let F a be the Fermi-Fock space over H c , F a := F a (H c ) .

The Fermi-Fock space for electrons and positrons, denoted by F D , is the following Hilbert space

F D := F a ⊗ F a . (88)
We denote by Ω D := Ω a ⊗ Ω a the vacuum of electrons and positrons. One has Note that with the notation of ( 7), we have b ± (g) = b ± (ξ)g(ξ) dξ .

F D = ∞ ⊕ r,s=0
We now give a representation of b γ,± (p) and b * γ,± (p). Recall that D D denote the set of smooth vectors Φ ∈ F D for which Φ (r,s) has a compact support and Φ (r,s) = 0 for all but finitely many (r, s).

For every ξ 1 = (p, γ), b + (ξ 1 ) maps F

Any other anti-commutators equal zero.

  where b -(P γ g) and b * -(P γ g) are respectively the annihilation and creation operator for a positron defined in Appendix C, according to which b γ,+ (g) and b γ-(g) anticommute (see (89)).

Lemma 4 . 2 .

 42 With the previous notations, we have the following properties.

(

( 1 )

 1 ,ext I be defined by the same expression, except that the creation and annihilation operators b * + , b * -, a are replaced by b * ,0 + , b * ,0 -, a 0 defined above. Using Lemma 4.2(i), a straightforward computation gives H

Lemma 5 . 3 .

 53 Consider the operator H I of the form[START_REF] Itzykson | Quantum Field theory[END_REF] with H

Proof.

  By Propositions 5.1, 5.2 and 5.4, we see that Hypotheses B.1, B.5 and B.6 of Appendix B are satisfied, the open interval I of Hypothesis B.6 being chosen, for instance, as I = (inf spec(H) + m e -δ, inf spec(H) + m Z 0 -δ/2). Therefore we can apply Theorem B.7 with J = ∆, which proves Theorem 5.5.

with C + 1 =

 1 ω(p) + m e when we consider a positive energy ω(p) > m e and C - 1 = ω(p) -m e when we consider a negative energy -ω(p) < -m e .

) with C + 2 =

 2 ω(p) -m e , for energies ω(p) > m e and C - 2 = -ω(p) + m e for energies -ω(p) < -m e .

( 1 -

 1 u) γj du . (73)The generalized eigenfunctionsψ ±,(j,mj ,κj ) (p, x) = ψ ±,γ (p, x) = ψ ± (ξ, x) ,where + refers to positive energies ω(p) > m e and -refers to negative energies -ω(p) < -m e , fulfillH D ψ ± ((p, γ), x) = ± ω(p) ψ ± ((p, γ), x) ,and are defined by ψ ±,(j,mj ,κj ) (p, x) := ig κj ,± (p, r)Φ(1) mj ,κj (θ, ϕ) -f κj ,± (p, r)Φ(2) (mj ,κj ) (θ, ϕ)

  follows from (19) that ψ -,(j,mj ,κj ) (p, x) := ig -κj ,-(p, r)Φ

1 2 ,

 2 -(p, r)|, |a f j+ 1 2 ,-(p, r)|, |a g -(j+ 1 2 ),-(p, r)| and |a f -(j+ 1 2 ),-(p, r)|, respectively for |a f -(j+ 1 2 ),+ (p, r)|, |a g -(j+ 1 2 ),+ (p, r)|, |a f j+ 1 2 ,+ (p, r)| and |a g j+ 1 2 ,+ (p, r)|. Estimates for the second derivatives are given for (p, r) near (0, 0) by

Theorem B. 7 (

 7 Limiting absorption principle). Assume that Hypotheses B.1, B.5 and B.6 hold. Let J ⊂ I be a compact interval, where I is given by Hypothesis B.6, and let J = {z ∈ C, Re z ∈ J, 0 < |Im z| ≤ 1}.

F

  r a H c ) ⊗ (⊗ s a H c ). For every ϕ ∈ H we define in F a (H) the annihilation operator, denoted by b(ϕ) as: b(ϕ)Ω = 0, and, for any n ∈ N, b(ϕ) (A n+1 (ϕ 1 ⊗ . . . ⊗ ϕ n+1 )) = √ n + 1 (n + 1)! σ sgn(σ) (ϕ, ϕ σ(1) ) ϕ σ(2) ⊗ . . . ⊗ ϕ σ(n+1) where ϕ i ∈ H. Note that the operator b(ϕ) maps ⊗ n+1 a H to ⊗ n a H. It extends by linearity to a bounded operator on F a (H). The creation operator, denoted by b * (ϕ), is the adjoint of b(ϕ). The operators b * (ϕ) and b(ϕ) satisfy b(ϕ) = b * (ϕ) = ϕ .We now define the annihilation and creation operators in the Fermi-Fock space F D for electrons and positrons.We first define the creation and annihilation operators for the electrons. For any g ∈ H c , we define in F D = F a ⊗ F a the annihilation operator, denoted by b + (g), asb + (g) := b(g) ⊗ 1l.Observe that b + (g) mapsF (r+1,s) a into F (r,s) a as follows b + (g) (A r+1 (g 1 ⊗ . . . ⊗ g r+1 ) ⊗ A s (h 1 ⊗ . . . ⊗ h s )) = [ b(g)A r+1 (g 1 ⊗ . . . ⊗ g r+1 )] ⊗ A s (h 1 ⊗ . . . ⊗ h s )The creation operator b * + (g) = b * (g) ⊗ 1l is the adjoint of b + (g). The operators b * + (g) and b + (g) are bounded operators in F D . We set, for every g ∈ H c , b γ,+ (g) = b + (P γ g) b * γ,+ (g) = b * + (P γ g) where P γ is the projection of H c onto the γ-th component. We next define the creation and annihilation operators for the positrons. For every h ∈ H c , we define in F D the annihilation operator, denoted by b -(h), as b -(h) := (-1) Ne ⊗ b(h), where (-1) Ne denotes the bounded operator on F a defined by its restriction to ⊗ r a h c as (-1) Ne u = (-1) r u for any u ∈ ⊗ r a h c . In other words, b -(h) maps F (r,s+1) a into F (r,s) a as follows:b-(h)(A r (g 1 ⊗ . . . ⊗ g r ) ⊗ A s+1 (h 1 ⊗ . . . ⊗ h s+1 )) = A r (g 1 ⊗ . . . ⊗ g r ) ⊗ [(-1) r b(h)A s+1 (h 1 ⊗ . . . ⊗ h s+1 )] The creation operator b * -(h) = (-1) Ne ⊗ b * (h) is the adjoint of b -(h); b * -(h) and b -(h) are bounded operators in F D .As above, we set, for everyh ∈ H c , b γ,-(h) = b -(P γ h) b * γ,-(h) = b * -(P γ h) .A simple computation shows that the following anti-commutation relations hold{b γ,± (g 1 ), b * β,± (g 2 )} = δ γ,β (P γ g 1 , P γ g 2 ) L 2 (R+) ,and{b 1 γ,+ (g 1 ), b 2 β,-(g 2 )} = 0 ,where g 1 , g 2 ∈ H c , and i (i = 1, 2) stand either for * or for no symbol.As in[24, chapter X], we introduce operator-valued distributions b γ,± (p) and b * γ,± (p) that fulfills b γ,± (g) = R + b γ,± (p) (P γ g) (p) dp b * γ,± (g) = R + b * γ,± (p) (P γ g) (p) dp where g ∈ H c . We also define for ξ = (p, γ), b ± (ξ) := b γ,± (p) .

∩(- 1 )(- 1 )

 11 D D and we have (b + (ξ 1 )Φ)(r,s) (p 1 , γ 1 , . . . , p r , γ r ; p 1 , γ 1 , . . . , p s , γ s )= √ r + 1Φ (r+1,s) (p, γ, p 1 , γ 1 , . . . , p r , γ r ; p 1 , γ 1 , . . . , p s , γ s ) b * + (ξ 1) is then given by:(b * + (ξ 1 )Φ) (r+1,s) (p 1 , γ 1 , . . . , p r+1 , γ r+1 ; p 1 , γ 1 , . . . , p s , γ s ) i+1 δ γiγ δ(p -p i ) Φ (r,s) (p 1 , γ 1 , . . . , p i , γ i , . . . , p r+1 , γ r+1 ; p 1 , γ 1 , . . . , p s , γ s )where • denotes that the i-th variable has to be omitted. Similarly, forξ 2 = (p , γ ), b -(ξ 2 ) maps F (r,s+1) a ∩ D D into F (r,s) a ∩ D D such that (b -(ξ 2 )Φ) (r,s) (p 1 , γ 1 , . . . , p r , γ r ; p 1 , γ 1 , . . . , p s , γ s ) = ((-1) N+ ⊗ b(ξ 2 )Φ) (r,s) (p 1 , γ 1 , . . . , p r , γ r ; p 1 , γ 1 , . . . , p s , γ s ) = √ s + 1(-1) r Φ (r,s+1) (p 1 , γ 1 , . . . , p r , γ r ; p , γ , p 1 , γ 1 , . . . , p s , γ s ) b * -(ξ 2 ) is then given by (b * -(ξ 2 )Φ) (r,s+1) (p 1 , γ 1 , . . . , p r , γ r ; p 1 , γ 1 , . . . , p s+1 , γ s+1 ) i+1 δ γ ,γ i δ(p -p i ) Φ (r,s) (p 1 , γ 1 , . . . , p r , γ r ; p 1 , γ 1 , . . . , p i , γ i , . . . , p s+1 , γ s+1 )Let us recall that Φ (r,s) is antisymmetric in the electron and the positron variables separately. We have {b γ,+ (p), b * γ ,+ (p )} = {b γ,-(p), b * γ ,-(p )} = δ γ,γ δ(p -p ) .

  1l, and that W * t is a C 0 -semigroup (of contractions) on H. The form domain of H is denoted by

  and the extension of W t to G * (which will be denoted by the same symbol) defines a C 0 -semigroup on B(G * ) (see [10, Remark 1.4.1)]. Their generators are denoted by A G and A G Proposition B.3. Suppose that Hypothesis B.1 holds and that the sesquilinear form [H, iA] defined on D(A) ∩ G by u, [H, iA]v := i u, HAv -i A * u, Hv , extends to a bounded quadratic form on G. Then H is of class C 1 (A G ; A G * ) in the sense of Hypothesis B.2, and the operator H ∈ B(G; G * ) is the operator associated with the quadratic form [H, iA]. The limiting absorption principle stated in Theorem B.7 below requires some more regularity of H with respect to A: Hypothesis B.5. The operator H ∈ B(

	Moreover, for all ϕ ∈ D(H), lim t→0 + 1 t theorem. Theorem B.4 (Virial Theorem). Assume Hypotheses B.1 and B.2. For any eigenstate ϕ of H, we have that ϕ, W Under Hypotheses B.1 and B.2, we have the following version of the virial ϕ, H ϕ = 0.

* , respectively.

Hypothesis B.2. The operator H ∈ B(G; G * ) is of class C 1 (A G ; A G * ), meaning that there exists a positive constant C such that, for all 0 ≤ t ≤ 1, W t H -HW t B(G;G * ) ≤ Ct. t Hϕ -Hϕ, W t ϕ = ϕ, H ϕ .

Self-adjointness and localization of the essential spectrumIn this section we prove Theorems 3.3 and 3.5.

Acknowledgements

The research of J.-M. B. and J. F. is supported by ANR grant ANR-12-JS0-0008-01. We thank Jean-François Bony and Thierry Jecko for helpful discussions.