# Spectral Theory for a Mathematical Model of the Weak Interaction—Part I: The Decay of the Intermediate Vector BosonsW±

Authors: J.-M. Barbaroux; J.-C. Guillot
Journal: Advances in Mathematical Physics
 (2009)

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## Abstract

We consider a Hamiltonian with cutoffs describing the weak decay
of spin 1 massive bosons into the full family of leptons. The Hamiltonian is
a self-adjoint operator in an appropriate Fock space with a unique ground
state. We prove a Mourre estimate and a limiting absorption principle above
the ground state energy and below the first threshold for a sufficiently small
coupling constant. As a corollary, we prove the absence of eigenvalues and absolute
continuity of the energy spectrum in the same spectral interval.


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## Body

## 1. Introduction

In this article, we consider a mathematical model of the weak interaction as patterned according to the Standard Model in Quantum Field Theory (see [1, 2]). We choose the example of the weak decay of the intermediate vector bosons W± into the full family of leptons.The mathematical framework involves fermionic Fock spaces for the leptons and bosonic Fock spaces for the vector bosons. The interaction is described in terms of annihilation and creation operators together with kernels which are square integrable with respect to momenta. The total Hamiltonian, which is the sum of the free energy of the particles and antiparticles and of the interaction, is a self-adjoint operator in the Fock space for the leptons and the vector bosons and it has an unique ground state in the Fock space for a sufficiently small coupling constant.The weak interaction is one of the four fundamental interactions known up to now. But the weak interaction is the only one which does not generate bound states. As it is well known, it is not the case for the strong, electromagnetic, and gravitational interactions. Thus we are expecting that the spectrum of the Hamiltonian associated with every model of weak decays is absolutely continuous above the energy of the ground state, and this article is a first step towards a proof of such a statement. Moreover a scattering theory has to be established for every such Hamiltonian.In this paper we establish a Mourre estimate and a limiting absorption principle for any spectral interval above the energy of the ground state and below the mass of the electron for a small coupling constant.Our study of the spectral analysis of the total Hamiltonian is based on the conjugate operator method with a self-adjoint conjugate operator. The methods used in this article are taken largely from [3, 4] and are based on [5, 6]. Some of the results of this article have been announced in [7].For other applications of the conjugate operator method see [8–19].For related results about models in Quantum Field Theory see [20, 21] in the case of the Quantum Electrodynamics and [22] in the case of the weak interaction.The paper is organized as follows. In Section2, we give a precise definition of the model we consider. In Section 3, we state our main results and in the following sections, together with the appendix, detailed proofs of the results are given.


## 2. The Model

The weak decay of the intermediate bosonsW+ and W- involves the full family of leptons together with the bosons themselves, according to the Standard Model (see [1, formula (4.139)] and [2]).The full family of leptons involves the electrone- and the positron e+, together with the associated neutrino νe and antineutrino ν̅e, the muons μ- and μ+ together with the associated neutrino νμ and antineutrino ν̅μ, and the tau leptons τ- and τ+ together with the associated neutrino ντ and antineutrino ν̅τ.It follows from the Standard Model that neutrinos and antineutrinos are massless particles. Neutrinos are left handed, that is, neutrinos have helicity-1/2 and antineutrinos are right handed, that is, antineutrinos have helicity +1/2.In what follows, the mathematical model for the weak decay of the vector bosonsW+ and W- that we propose is based on the Standard Model, but we adopt a slightly more general point of view because we suppose that neutrinos and antineutrinos are both massless particles with helicity ±1/2. We recover the physical situation as a particular case. We could also consider a model with massive neutrinos and antineutrinos built upon the Standard Model with neutrino mixing [23].Let us sketch how we define a mathematical model for the weak decay of the vector bosonsW± into the full family of leptons.The energy of the free leptons and bosons is a self-adjoint operator in the corresponding Fock space (see below), and the main problem is associated with the interaction between the bosons and the leptons. Let us consider only the interaction between the bosons and the electrons, the positrons, and the corresponding neutrinos and antineutrinos. Other cases are strictly similar. In the Schrödinger representation the interaction is given by (see [1, page 159, equation (4.139)] and [2, page 308, equation (21.3.20)])
(2.1)I=∫d3xΨe¯(x)γα(1-γ5)Ψνe(x)Wα(x)+∫d3xΨνe¯(x)γα(1-γ5)Ψe(x)Wα(x)*,
where γα, α=0,1,2,3 and γ5 are the Dirac matrices and Ψ·(x) and Ψ·¯(x) are the Dirac fields for e-, e+, νe, and ν̅e.We have(2.2)Ψe(x)=(12π)3/2∑s=±1/2∫d3p(be,+(p,s)u(p,s)p0eip·x+be,-*(p,s)v(p,s)p0e-ip·x),Ψe¯(x)=Ψe(x)†γ0.
Here p0=(|p|2+me2)1/2 where me>0 is the mass of the electron, and u(p,s) and v(p,s) are the normalized solutions to the Dirac equation (see [1, Appendix]).The operatorsbe,+(p,s) and be,+*(p,s) (resp., be,-(p,s) and be,-*(p,s)) are the annihilation and creation operators for the electrons (resp., the positrons) satisfying the anticommutation relations (see below).Similarly we defineΨνe(x) and Ψνe¯(x) by substituting the operators cνe,±(p,s) and cνe,±*(p,s) for be,±(p,s) and be,±*(p,s) with p0=|p|. The operators cνe,+(p,s) and cνe,+*(p,s) (resp., cνe,-(p,s) and cνe,-*(p,s)) are the annihilation and creation operators for the neutrinos associated with the electrons (resp., the antineutrinos).For theWα fields we have (see [24, Section  5.3])
(2.3)Wα(x)=(12π)3/2∑λ=-1,0,1∫d3k2k0(ϵα(k,λ)a+(k,λ)eik·x+ϵα*(k,λ)a-*(k,λ)e-ik·x).
Here k0=(|k|2+mW2)1/2 where mW>0 is the mass of the bosons W±. W+ is the antiparticule of W-. The operators a+(k,λ) and a+*(k,λ) (resp.,  a-(k,λ) and a-*(k,λ)) are the annihilation and creation operators for the bosons W- (resp.,  W+) satisfying the canonical commutation relations. The vectors ϵα(k,λ) are the polarizations of the massive spin 1 bosons W± (see [24, Section  5.2]).The interaction (2.1) is a formal operator and, in order to get a well-defined operator in the Fock space, one way is to adapt what Glimm and Jaffe have done in the case of the Yukawa Hamiltonian (see [25]). For that sake, we have to introduce a spatial cutoff g(x) such that g∈L1(ℝ3), together with momentum cutoffs χ(p) and ρ(k) for the Dirac fields and the Wμ fields, respectively.Thus when one develops the interactionI with respect to products of creation and annihilation operators, one gets a finite sum of terms associated with kernels of the form
(2.4)χ(p1)χ(p2)ρ(k)ĝ(p1+p2-k),
where ĝ is the Fourier transform of g. These kernels are square integrable.In what follows, we consider a model involving terms of the above form but with more general square integrable kernels.We follow the convention described in [24, Section  4.1] that we quote: “The state-vector will be taken to be symmetric under interchange of any bosons with each other, or any bosons with any fermions, and antisymmetric with respect to interchange of any two fermions with each other, in all cases, whether the particles are of the same species or not.’’ Thus, as it follows from [24, Section  4.2], fermionic creation and annihilation operators of different species of leptons will always anticommute.Concerning our notations, from now on,ℓ∈{1,2,3} denotes each species of leptons. ℓ=1 denotes the electron e- the positron e+ and the neutrinos νe, ν̅e. ℓ=2 denotes the muons μ-, μ+ and the neutrinos νμ and ν̅μ, and ℓ=3 denotes the tau-leptons and the neutrinos ντ and ν̅τ.Letξ1=(p1,s1) be the quantum variables of a massive lepton, where p1∈ℝ3 and s1∈{-1/2,1/2} is the spin polarization of particles and antiparticles. Let ξ2=(p2,s2) be the quantum variables of a massless lepton where p2∈ℝ3 and s2∈{-1/2,1/2} is the helicity of particles and antiparticles, and, finally, let ξ3=(k,λ) be the quantum variables of the spin 1 bosons W+ and W- where k∈ℝ3 and λ∈{-1,0,1} is the polarization of the vector bosons (see [24, Section  5]). We set Σ1=ℝ3×{-1/2,1/2} for the leptons and Σ2=ℝ3×{-1,0,1} for the bosons. Thus L2(Σ1) is the Hilbert space of each lepton and L2(Σ2) is the Hilbert space of each boson. The scalar product in L2(Σj),  j=1,2 is defined by
(2.5)(f,g)=∫Σjf(ξ)¯g(ξ)dξ,j=1,2.
Here
(2.6)∫Σ1dξ=∑s=+1/2,-1/2∫dp,∫Σ2dξ=∑λ=0,1,-1∫dk,(p,k∈ℝ3).The Hilbert space for the weak decay of the vector bosonsW+ and W- is the Fock space for leptons and bosons that we now describe.Let𝔖 be any separable Hilbert space. Let ⨂an𝔖 (resp., ⨂sn𝔖) denote the antisymmetric (resp., symmetric) nth tensor power of 𝔖. The fermionic (resp., bosonic) Fock space over 𝔖, denoted by 𝔉a(𝔖) (resp., 𝔉s(𝔖)), is the direct sum
(2.7)𝔉a(𝔖)=⨁n=0∞⨂an𝔖(resp.,𝔉s(𝔖)=⨁n=0∞⨂sn𝔖),
where ⨂a0𝔖=⨂s0𝔖≡ℂ. The state Ω=(1,0,0,…,0,…) denotes the vacuum state in 𝔉a(𝔖) and in 𝔉s(𝔖).For everyℓ, 𝔉ℓ is the fermionic Fock space for the corresponding species of leptons including the massive particle and antiparticle together with the associated neutrino and antineutrino, that is,
(2.8)𝔉ℓ=⨂4𝔉a(L2(Σ1))ℓ=1,2,3.
We have
(2.9)𝔉ℓ=⨁qℓ≥0,q̅ℓ≥0,rℓ≥0,r̅ℓ≥0𝔉ℓ(qℓ,q̅ℓ,rℓ,r̅ℓ)
with
(2.10)𝔉ℓ(qℓ,q̅ℓ,rℓ,r̅ℓ)=(⨂aqℓL2(Σ1))⊗(⨂aq̅ℓL2(Σ1))⊗(⨂arℓL2(Σ1))⊗(⨂ar̅ℓL2(Σ1)).
Here qℓ (resp., q̅ℓ) is the number of massive fermionic particle (resp., antiparticles) and rℓ (resp., r̅ℓ) is the number of neutrinos (resp., antineutrinos). The vector Ωℓ is the associated vacuum state. The fermionic Fock space denoted by 𝔉L for the leptons is then
(2.11)𝔉L=⨂ℓ=13𝔉ℓ,
and ΩL=⨂ℓ=13Ωℓ is the vacuum state.The bosonic Fock space for the vector bosonsW+ and W-, denoted by 𝔉W, is then
(2.12)𝔉W=𝔉s(L2(Σ2))⊗𝔉s(L2(Σ2))≃𝔉s(L2(Σ2)⊕L2(Σ2)).
We have
(2.13)𝔉W=⨁t≥0,t̅≥0𝔉W(t,t̅),
where 𝔉W(t,t̅)=(⨂stL2(Σ2))⊗(⨂st̅L2(Σ2)). Here t (resp., t̅) is the number of bosons W- (resp., W+). The vector ΩW is the corresponding vacuum.The Fock space for the weak decay of the vector bosonsW+ and W-, denoted by 𝔉, is thus
(2.14)𝔉=𝔉L⊗𝔉W,
and Ω=ΩL⊗ΩW is the vacuum state.For everyℓ∈{1,2,3} let 𝔇ℓ denote the set of smooth vectors ψℓ∈𝔉ℓ for which ψℓ(qℓ,q̅ℓ,rℓ,r̅ℓ) has a compact support and ψℓ(qℓ,q̅ℓ,rℓ,r̅ℓ)=0 for all but finitely many (qℓ,q̅ℓ,rℓ,r̅ℓ). Let
(2.15)𝔇L=⨂̂ℓ=13𝔇ℓ.
Here ⨂̂ is the algebraic tensor product.Let𝔇W denote the set of smooth vectors ϕ∈𝔉W for which ϕ(t,t̅) has a compact support and ϕ(t,t̅)=0 for all but finitely many (t,t̅).Let(2.16)𝔇=𝔇L⊗̂𝔇W.
The set 𝔇 is dense in 𝔉.LetAℓ be a self-adjoint operator in 𝔉ℓ such that 𝔇ℓ is a core for Aℓ. Its extension to 𝔉L is, by definition, the closure in 𝔉L of the operator A1⊗12⊗13 with domain 𝔇L when ℓ=1, of the operator 11⊗A2⊗13 with domain 𝔇L when ℓ=2, and of the operator 11⊗12⊗A3 with domain 𝔇L when ℓ=3. Here 1ℓ is the operator identity on 𝔉ℓ.The extension ofAℓ to 𝔉L is a self-adjoint operator for which 𝔇L is a core and it can be extended to 𝔉. The extension of Aℓ to 𝔉 is, by definition, the closure in 𝔉 of the operator Ãℓ⊗1W with domain 𝔇, where Ãℓ is the extension of Aℓ to 𝔉L. The extension of Aℓ to 𝔉 is a self-adjoint operator for which 𝔇 is a core.LetB be a self-adjoint operator in 𝔉W for which 𝔇W is a core. The extension of the self-adjoint operator Aℓ⊗B is, by definition, the closure in 𝔉 of the operator A1⊗12⊗13⊗B with domain 𝔇 when ℓ=1, of the operator 11⊗A2⊗13⊗B with domain 𝔇 when ℓ=2, and of the operator 11⊗12⊗A3⊗B with domain 𝔇 when ℓ=3. The extension of Aℓ⊗B to 𝔉 is a self-adjoint operator for which 𝔇 is a core.We now define the creation and annihilation operators.For eachℓ=1,2,3, bℓ,ϵ(ξ1) (resp.,  bℓ,ϵ*(ξ1)) is the fermionic annihilation (resp.,  fermionic creation) operator for the corresponding species of massive particle when ϵ=+ and for the corresponding species of massive antiparticle when ϵ=-. The operators bℓ,ϵ(ξ1) and bℓ,ϵ*(ξ1) are defined as usually (see, e.g., [20, 26]; see also the detailed definitions in [27]).Similarly, for eachℓ=1,2,3, cℓ,ϵ(ξ2) (resp.,  cℓ,ϵ*(ξ2)) is the fermionic annihilation (resp.,  fermionic creation) operator for the corresponding species of neutrino when ϵ=+ and for the corresponding species of antineutrino when ϵ=-. The operators cℓ,ϵ(ξ2) and cℓ,ϵ*(ξ2) are defined in a standard way, but with the additional requirements that for any ℓ, ℓ′, ϵ and ϵ′, the operators bℓ,ϵ♯(ξ1) and cℓ′,ϵ′♯(ξ2) anticommutes, where ♯ stands either for a * or for no symbol (see the detailed definitions in [27]).The operatoraϵ(ξ3) (resp.,  aϵ*(ξ3)) is the bosonic annihilation (resp.,  bosonic creation) operator for the boson W- when ϵ=+ and for the boson W+ when ϵ=- (see, e.g., [20, 26], or [27]). Note that a♯(ξ3) commutes with bℓ,ϵ♯(ξ1) and cℓ′,ϵ′♯(ξ2).The following canonical anticommutation and commutation relations hold:(2.17){bℓ,ϵ(ξ1),bℓ′,ϵ′*(ξ1′)}=δℓℓ′δϵϵ′δ(ξ1-ξ1′),{cℓ,ϵ(ξ2),cℓ′,ϵ′*(ξ2′)}=δℓℓ′δϵϵ′δ(ξ2-ξ2′),[aϵ(ξ3),aϵ′*(ξ3′)]=δϵϵ′δ(ξ3-ξ3′),{bℓ,ϵ(ξ1),bℓ′,ϵ′(ξ1′)}={cℓ,ϵ(ξ2),cℓ′,ϵ′(ξ2′)}=0,[aϵ(ξ3),aϵ′(ξ3′)]=0,{bℓ,ϵ(ξ1),cℓ′,ϵ′(ξ2)}={bℓ,ϵ(ξ1),cℓ′,ϵ′*(ξ2)}=0,[bℓ,ϵ(ξ1),aϵ′(ξ3)]=[bℓ,ϵ(ξ1),aϵ′*(ξ3)]=[cℓ,ϵ(ξ2),aϵ′(ξ3)]=[cℓ,ϵ(ξ2),aϵ′*(ξ3)]=0,
where we used the notation δ(ξj-ξj′)=δλλ′δ(k-k′).We recall that the following operators, withφ∈L2(Σ1),
(2.18)bℓ,ϵ(φ)=∫Σ1bℓ,ϵ(ξ)φ(ξ)¯dξ,cℓ,ϵ(φ)=∫Σ1cℓ,ϵ(ξ)φ(ξ)¯dξ,bℓ,ϵ*(φ)=∫Σ1bℓ,ϵ*(ξ)φ(ξ)dξ,cℓ,ϵ*(φ)=∫Σ1cℓ,ϵ*(ξ)φ(ξ)dξ
are bounded operators in 𝔉 such that
(2.19)∥bℓ,ϵ♯(φ)∥=∥cℓ,ϵ♯(φ)∥=∥φ∥L2,
where b♯ (resp., c♯) is b (resp., c) or b* (resp., c*).The operatorsbℓ,ϵ♯(φ) and cℓ,ϵ♯(φ) satisfy similar anticommutaion relations (see, e.g., [28]).The free HamiltonianH0 is given by
(2.20)H0=H0(1)+H0(2)+H0(3)=∑ℓ=13‍∑ϵ=±∫wℓ(1)(ξ1)bℓ,ϵ*(ξ1)bℓ,ϵ(ξ1)dξ1+∑ℓ=13‍∑ϵ=±∫wℓ(2)(ξ2)cℓ,ϵ*(ξ2)cℓ,ϵ(ξ2)dξ2+∑ϵ=±∫w(3)(ξ3)aϵ*(ξ3)aϵ(ξ3)dξ3,
where
(2.21)wℓ(1)(ξ1)=(|p1|2+mℓ2)1/2,with0<m1<m2<m3,wℓ(2)(ξ2)=|p2|,w(3)(ξ3)=(|k|2+mW2)1/2,
where mW is the mass of the bosons W+ and W- such that mW>m3.The spectrum ofH0 is [0,∞) and 0 is a simple eigenvalue with Ω as eigenvector. The set of thresholds of H0, denoted by T, is given by
(2.22)T={pm1+qm2+rm3+smW;(p,q,r,s)∈ℕ4,p+q+r+s≥1},
and each set [t,∞), t∈T, is a branch of absolutely continuous spectrum for H0.The interaction, denoted byHI, is given by
(2.23)HI=∑α=12HI(α),
where
(2.24)HI(1)=∑ℓ=13‍∑ϵ≠ϵ′∫Gℓ,ϵ,ϵ′(1)(ξ1,ξ2,ξ3)bℓ,ϵ*(ξ1)cℓ,ϵ′*(ξ2)aϵ(ξ3)dξ1dξ2dξ3+∑ℓ=13‍∑ϵ≠ϵ′∫Gℓ,ϵ,ϵ′(1)(ξ1,ξ2,ξ3)¯aϵ*(ξ3)cℓ,ϵ′(ξ2)bℓ,ϵ(ξ1)dξ1dξ2dξ3,HI(2)=∑ℓ=13∑ϵ≠ϵ′∫Gℓ,ϵ,ϵ′(2)(ξ1,ξ2,ξ3)bℓ,ϵ*(ξ1)cℓ,ϵ′*(ξ2)aϵ*(ξ3)dξ1dξ2dξ3+∑ℓ=13∑ϵ≠ϵ′∫Gℓ,ϵ,ϵ′(2)(ξ1,ξ2,ξ3)¯aϵ(ξ3)cℓ,ϵ′(ξ2)bℓ,ϵ(ξ1)dξ1dξ2dξ3.
The kernels Gℓ,ϵ,ϵ′(2)(·,·,·), α=1,2, are supposed to be functions.The total Hamiltonian is then(2.25)H=H0+gHI,g>0,
where g is a coupling constant.The operatorHI(1) describes the decay of the bosons W+ and W- into leptons. Because of HI(2) the bare vacuum will not be an eigenvector of the total Hamiltonian for every g>0 as we expect from the physics.Every kernelGℓ,ϵ,ϵ′(ξ1,ξ2,ξ3), computed in theoretical physics, contains a δ-distribution because of the conservation of the momentum (see [1] and [24, Section  4.4]). In what follows, we approximate the singular kernels by square integrable functions.Thus, from now on, the kernelsGℓ,ϵ,ϵ′(α) are supposed to satisfy the following hypothesis.Hypothesis 2.1.
Forα=1,2,  ℓ=1,2,3, ϵ,ϵ′=±, we assume
(2.26)Gℓ,ϵ,ϵ′(α)(ξ1,ξ2,ξ3)∈L2(Σ1×Σ1×Σ2).Remark 2.2.
A similar model can be written down for the weak decay of pionsπ- and π+ (see [1, Section  6.2]).Remark 2.3.
The total Hamiltonian is more general than the one involved in the theory of weak interactions because, in the Standard Model, neutrinos have helicity-1/2 and antineutrinos have helicity 1/2.
In the physical case, the Fock space, denoted by𝔉′, is isomorphic to 𝔉L′⊗𝔉W, with
(2.27)𝔉L′=⨂ℓ=13𝔉ℓ′,𝔉ℓ′=(⨂a2L2(Σ1))⊗(⨂a2L2(ℝ3)).
The free Hamiltonian, now denoted by H0′, is then given by
(2.28)H0′=∑ℓ=13∑ϵ=±∫wℓ(1)(ξ1)bℓ,ϵ*(ξ1)bℓ,ϵ(ξ1)dξ1+∑ℓ=13∑ϵ=±∫ℝ3|p2|cℓ,ϵ*(p2)cℓ,ϵ(p2)dp2+∑ϵ=±∫w(3)(ξ3)aϵ*(ξ3)aϵ(ξ3)dξ3,
and the interaction, now denoted by HI′, is the one obtained from HI by supposing that G(α)(ξ1,(p2,s2),ξ3)=0 if s2=ϵ(1/2). The total Hamiltonian, denoted by H′, is then given by H′=H0′+gHI′. The results obtained in this paper for H hold true for H′ with obvious modifications.Under Hypothesis2.1 a well-defined operator on 𝔇 corresponds to the formal interaction HI as it follows.The formal operator(2.29)∫Gℓ,ϵ,ϵ′(1)(ξ1,ξ2,ξ3)bℓ,ϵ*(ξ1)cℓ,ϵ′*(ξ2)aϵ(ξ3)dξ1dξ2dξ3
is defined as a quadratic form on (𝔇ℓ⊗𝔇W)×(𝔇ℓ⊗𝔇W) as
(2.30)∫(cℓ,ϵ′(ξ2)bℓ,ϵ(ξ1)ψ,Gℓ,ϵ,ϵ′(1)aϵ(ξ3)ϕ)dξ1dξ2dξ3,
where ψ, ϕ∈𝔇ℓ⊗𝔇W.By mimicking the proof of [29, Theorem X.44], we get a closed operator, denoted by HI,ℓ,ϵ,ϵ′(1), associated with the quadratic form such that it is the unique operator in 𝔉ℓ⊗𝔉W such that 𝔇ℓ⊗𝔇W⊂𝒟(HI,ℓ,ϵ,ϵ′(1)) is a core for HI,ℓ,ϵ,ϵ′(1) and
(2.31)HI,ℓ,ϵ,ϵ′(1)=∫Gℓ,ϵ,ϵ′(1)(ξ1,ξ2,ξ3)bℓ,ϵ*(ξ1)cℓ,ϵ′*(ξ2)aϵ(ξ3)dξ1dξ2dξ3
as quadratic forms on (𝔇ℓ⊗𝔇W)×(𝔇ℓ⊗𝔇W).Similarly for the operator(HI,ℓ,ϵ,ϵ′(1))*, we have the equality as quadratic forms
(2.32)(HI,ℓ,ϵ,ϵ′(1))*=∫Gℓ,ϵ,ϵ′(1)(ξ1,ξ2,ξ3)¯aϵ*(ξ3)cℓ,ϵ′(ξ2)bℓ,ϵ(ξ1)dξ1dξ2dξ3.Again, there exists two closed operatorsHI,ℓ,ϵ,ϵ′(2) and (HI,ℓ,ϵ,ϵ′(2))* such that 𝔇ℓ⊗𝔇W⊂𝒟(HI,ℓ,ϵ,ϵ′(2)), 𝔇ℓ⊗𝔇W⊂𝒟((HI,ℓ,ϵ,ϵ′(2))*), and 𝔇ℓ⊗𝔇W is a core for HI,ℓ,ϵ,ϵ′(2) and (HI,ℓ,ϵ,ϵ′(2))* and such that
(2.33)HI,ℓ,ϵ,ϵ′(2)=∫Gℓ,ϵ,ϵ′(2)(ξ1,ξ2,ξ3)bℓ,ϵ*(ξ1)cℓ,ϵ′*(ξ2)aϵ*(ξ3)dξ1dξ2dξ3,(HI,ℓ,ϵ,ϵ′(2))*=∫Gℓ,ϵ,ϵ′(2)(ξ1,ξ2,ξ3)aϵ(ξ3)cℓ,ϵ′(ξ2)bℓ,ϵ(ξ1)dξ1dξ2dξ3
as quadratic forms on (𝔇ℓ⊗𝔇W)×(𝔇ℓ⊗𝔇W).We will still denote byHI,ℓ,ϵ,ϵ′(α) and (HI,ℓ,ϵ,ϵ′(α))*  (α=1,2) their extensions to 𝔉. The set 𝔇 is then a core for HI,ℓ,ϵ,ϵ′(α) and (HI,ℓ,ϵ,ϵ′(α))*.Thus(2.34)H=H0+g∑α=1,2∑ℓ=13∑ϵ≠ϵ′(HI,ℓ,ϵ,ϵ′(α)+(HI,ℓ,ϵ,ϵ′(2))*)
is a symmetric operator defined on 𝔇.We now want to prove thatH is essentially self-adjoint on 𝔇 by showing that HI,ℓ,ϵ,ϵ′(α) and (HI,ℓ,ϵ,ϵ′(α))* are relatively H0-bounded.Once again, as above, for almost everyξ3∈Σ2, there exists closed operators in 𝔉L, denoted by Bℓ,ϵ,ϵ′(α)(ξ3) and (Bℓ,ϵ,ϵ′(α)(ξ3))* such that
(2.35)Bℓ,ϵ,ϵ′(1)(ξ3)=-∫Gℓ,ϵ,ϵ′(1)(ξ1,ξ2,ξ3)¯bℓ,ϵ(ξ1)cℓ,ϵ′(ξ2)dξ1dξ2,(Bℓ,ϵ,ϵ′(1)(ξ3))*=∫Gℓ,ϵ,ϵ′(1)(ξ1,ξ2,ξ3)bℓ,ϵ*(ξ1)cℓ,ϵ′*(ξ2)dξ1dξ2,Bℓ,ϵ,ϵ′(2)(ξ3)=∫Gℓ,ϵ,ϵ′(2)(ξ1,ξ2,ξ3)bℓ,ϵ*(ξ1)cℓ,ϵ′*(ξ2)dξ1dξ2,(Bℓ,ϵ,ϵ′(2)(ξ3))*=-∫Gℓ,ϵ,ϵ′(2)(ξ1,ξ2,ξ3)¯bℓ,ϵ(ξ1)cℓ,ϵ′(ξ2)dξ1dξ2
as quadratic forms on 𝔇ℓ×𝔇ℓ.We have that𝔇ℓ⊂𝒟(Bℓ,ϵ,ϵ′(α)(ξ3)) (resp., 𝔇ℓ⊂𝒟((Bℓ,ϵ,ϵ′(α)(ξ3))*) is a core for Bℓ,ϵ,ϵ′(α)(ξ3) (resp., for (Bℓ,ϵ,ϵ′(α)(ξ3))*). We still denote by Bℓ,ϵ,ϵ′(α)(ξ3)) and (Bℓ,ϵ,ϵ′(α)(ξ3))*) their extensions to 𝔉L.It then follows that the operatorHI with domain 𝔇 is symmetric and can be written in the following form:
(2.36)HI=∑α=1,2∑ℓ=13∑ϵ≠ϵ′(HI,ℓ,ϵ,ϵ′(α)+(HI,ℓ,ϵ,ϵ′(α))*)=∑α=1,2∑ℓ=13∑ϵ≠ϵ′∫Bℓ,ϵ,ϵ′(α)(ξ3)⊗aϵ*(ξ3)dξ3+∑α=1,2∑ℓ=13∑ϵ≠ϵ′∫(Bℓ,ϵ,ϵ′(α)(ξ3))*⊗aϵ(ξ3)dξ3.
Let Nℓ denote the operator number of massive leptons ℓ in 𝔉ℓ, that is,
(2.37)Nℓ=∑ϵ∫bℓ,ϵ*(ξ1)bℓ,ϵ(ξ1)dξ1.
The operator Nℓ is a positive self-adjoint operator in 𝔉ℓ. We still denote by Nℓ its extension to 𝔉L. The set 𝔇L is a core for Nℓ.We then have the following.Proposition 2.4.
For almost everyξ3∈Σ2, 𝒟(Bℓ,ϵ,ϵ′(α)(ξ3)), 𝒟((Bℓ,ϵ,ϵ′(α)(ξ3))*)⊃𝒟(Nℓ1/2), and for Φ∈𝒟(Nℓ1/2)⊂𝔉L one has
(2.38)∥Bℓ,ϵ,ϵ′(α)(ξ3)Φ∥𝔉L≤∥Gℓ,ϵ,ϵ′(α)(·,·,ξ3)∥L2(Σ1×Σ1)∥Nℓ1/2Φ∥𝔉L,∥(Bℓ,ϵ,ϵ′(α)(ξ3))*Φ∥𝔉L≤∥Gℓ,ϵ,ϵ′(α)(·,·,ξ3)∥L2(Σ1×Σ1)∥Nℓ1/2Φ∥𝔉L.Proof.
The estimates (2.38) are examples of Nτ estimates (see [25]). The proof is quite similar to the proof of [20, Proposition  3.7]. Details can be found in [27] but are omitted here.Let(2.39)H0,ϵ(3)=∫w(3)(ξ3)aϵ*(ξ3)aϵ(ξ3)dξ3.
Then H0,ϵ(3) is a self-adjoint operator in 𝔉W, and 𝔇W is a core for H0,ϵ(3).We get  the following.Proposition 2.5.
One has(2.40)∥∫(Bℓ,ϵ,ϵ′(α)(ξ3))*⊗aϵ(ξ3)dξ3Ψ∥2≤(∫Σ1×Σ1×Σ2|Gℓ,ϵ,ϵ′(α)(ξ1,ξ2,ξ3)|2w(3)(ξ3)dξ1dξ2dξ3)∥(Nℓ+1)1/2⊗(H0,ϵ(3))1/2Ψ∥2,(2.41)∥∫Bℓ,ϵ,ϵ′(α)(ξ3)⊗aϵ*(ξ3)dξ3Ψ∥2≤(∫Σ1×Σ1×Σ2|Gℓ,ϵ,ϵ′(α)(ξ1,ξ2,ξ3)|2w(3)(ξ3)dξ1dξ2dξ3)∥(Nℓ+1)1/2⊗(H0,ϵ(3))1/2Ψ∥2+(∫Σ1×Σ1×Σ2|Gℓ,ϵ,ϵ′(α)(ξ1,ξ2,ξ3)|2dξ1dξ2dξ3)(η∥(Nℓ+1)1/2⊗1Ψ∥2+14η∥Ψ∥2)
for every Ψ∈𝒟(H0) and every η>0.Proof.
Suppose thatΨ∈𝒟(Nℓ1/2)⊗̂𝒟((H0,ϵ(3))1/2). Let
(2.42)Ψϵ(ξ3)=w(3)(ξ3)1/2((Nℓ+1)1/2⊗aϵ(ξ3))Φ.
We have
(2.43)∫(Bℓ,ϵ,ϵ′(α)(ξ3))*⊗aϵ(ξ3)dξ3Ψ=∫Σ21(w(3)(ξ3))1/2((Bℓ,ϵ,ϵ′(α)(ξ3))*(Nℓ+1)-1/2⊗1)Ψϵ(ξ3)dξ3.
Therefore, for Ψ∈𝒟(Nℓ1/2)⊗̂𝒟((H0,ϵ(3))1/2), (2.40) follows from Proposition 2.4.
We now have(2.44)∥∫Bℓ,ϵ,ϵ′(α)(ξ3)⊗aϵ*(ξ3)Ψdξ3∥𝔉2=∫(Bℓ,ϵ,ϵ′(α)(ξ3)⊗aϵ(ξ3′)Ψ,Bℓ,ϵ,ϵ′(α)(ξ3′)⊗aϵ(ξ3)Ψ)dξ3dξ3′+∫∥(Bℓ,ϵ,ϵ′(α)(ξ3)⊗1)Ψ∥2dξ3,(2.45)∫Σ2×Σ2(Bℓ,ϵ,ϵ′(α)(ξ3)⊗aϵ(ξ3′)Ψ,Bℓ,ϵ,ϵ′(α)(ξ3′)⊗aϵ(ξ3)Ψ)dξ3dξ3′=∫Σ2×Σ21w(3)(ξ3)1/2w(3)(ξ3′)1/2×((Bℓ,ϵ,ϵ′(α)(ξ3)(Nℓ+1)-1/2⊗1)Ψϵ(ξ3′),(Bℓ,ϵ,ϵ′(α)(ξ3′)(Nℓ+1)-1/2⊗1)Ψϵ(ξ3))dξ3dξ3′≤(∫Σ21w(3)(ξ3)1/2∥Bℓ,ϵ,ϵ′(α)(ξ3)(Nℓ+1)-1/2∥𝔉L∥Ψϵ(ξ3)∥dξ3)2≤(∫Σ1×Σ1×Σ2|G(α)(ξ1,ξ2,ξ3)|2w(3)(ξ3)dξ1dξ2dξ3)∥(Nℓ+1)1/2⊗(H0,ϵ(3))1/2Ψ∥2.
Furthermore
(2.46)∫Σ2∥(Bℓ,ϵ,ϵ′(α)(ξ3)⊗1)Ψ∥2dξ3=∫Σ2∥(Bℓ,ϵ,ϵ′(α)(ξ3)(Nℓ+1)-1/2⊗1)((Nℓ+1)1/2⊗1)Ψ∥2dξ3≤(∫Σ1×Σ1×Σ2|Gℓ,ϵ,ϵ′(α)(ξ1,ξ2,ξ3)|2dξ1dξ2dξ3)(η∥(Nℓ+1)Ψ∥2+14η∥Ψ∥2)
for every η>0.
By (2.40), (2.45), and (2.46), we finally get (2.41) for every Ψ∈𝒟(Nℓ1/2)⊗̂𝒟(H0,ϵ(3)). It then follows that (2.40) and (2.41) are verified for every Ψ∈𝒟(H0).We now prove thatH is a self-adjoint operator in 𝔉 for g sufficiently small.Theorem 2.6.
Letg1>0 be such that
(2.47)3g12mW(1m12+1)∑α=1,2∑ℓ=13∑ϵ≠ϵ′∥Gℓ,ϵ,ϵ′(α)∥L2(Σ1×Σ1×Σ2)2<1.
Then for every g satisfying g≤g1, H is a self-adjoint operator in 𝔉 with domain 𝒟(H)=𝒟(H0), and 𝔇 is a core for H.Proof.
LetΨ be in 𝔇. We have
(2.48)∥HIΨ∥2≤12∑α=1,2∑ℓ=13∑ϵ≠ϵ′{∥∫(Bℓ,ϵ,ϵ′(α)(ξ3))*⊗aϵ(ξ3)Ψdξ3∥2+∥∫(Bℓ,ϵ,ϵ′(α)(ξ3))⊗aϵ*(ξ3)Ψdξ3∥2}.
Note that
(2.49)∥H0,ϵ(3)Ψ∥≤∥H0(3)Ψ∥≤∥H0Ψ∥,∥NℓΨ∥≤1mℓ∥H0,ℓΨ∥≤1m1∥H0,ℓΨ∥≤1m1∥H0Ψ∥,
where
(2.50)H0,ℓ=∑ϵ∫wℓ(1)(ξ1)bℓ,ϵ*(ξ1)bℓ,ϵ(ξ1)dξ1+∑ϵ∫wℓ(2)(ξ2)cℓ,ϵ*(ξ2)cℓ,ϵ(ξ2)dξ2.
We further note that
(2.51)∥(Nℓ+1)1/2⊗(H0,ϵ(3))1/2Ψ∥2≤12(1m12+1)∥H0Ψ∥2+β2m12∥H0Ψ∥2+(12+18β)∥Ψ∥2
for β>0, and
(2.52)η∥((Nℓ+1)⊗1)Ψ∥2+14η∥Ψ∥2≤ηm12∥H0Ψ∥2+ηβm12∥H0Ψ∥2+η(1+14β)∥Ψ∥2+14η∥Ψ∥2.
Combining (2.48) with (2.40), (2.41), (2.51), and (2.52) we get for η>0, β>0(2.53)∥HIΨ∥2≤6(∑α=1,2∑ℓ=13∑ϵ≠ϵ′∥Gℓ,ϵ,ϵ′(α)∥2)×(12mW(1m12+1)∥H0Ψ∥2+β2mWm12∥H0Ψ∥2+12mW(1+14β)∥Ψ∥2)+12(∑α=1,2∑ℓ=13∑ϵ≠ϵ′∥Gℓ,ϵ,ϵ′(α)∥2)(ηm12(1+β)∥H0Ψ∥2+(η(1+14β)+14η)∥Ψ∥2),
by noting
(2.54)∫Σ1×Σ1×Σ2|Gℓ,ϵ,ϵ′(ξ1,ξ2,ξ3)|2w(3)(ξ3)dξ1dξ2dξ3≤1mW∥Gℓ,ϵ,ϵ′(α)∥2.
By (2.53) the theorem follows from the Kato-Rellich theorem.


## 3. Main Results

In the sequel, we will make the following additional assumptions on the kernelsGℓ,ϵ,ϵ′(α).Hypothesis 3.1.
(i) Forα=1,2, ℓ=1,2,3, ϵ,ϵ′=±,
(3.1)∫Σ1×Σ1×Σ2|Gℓ,ϵ,ϵ′(α)(ξ1,ξ2,ξ3)|2|p2|2dξ1dξ2dξ3<∞.
(ii) There existsC>0 such that for α=1,2, ℓ=1,2,3, ϵ,ϵ′=±,
(3.2)(∫Σ1×{|p2|≤σ}×Σ2|Gℓ,ϵ,ϵ′(α)(ξ1,ξ2,ξ3)|2dξ1dξ2dξ3)1/2≤Cσ2.
(iii) Forα=1,2, ℓ=1,2,3, ϵ,ϵ′=±, and i,j=1,2,3(3.3)(iii.a)∫Σ1×Σ1×Σ2|[(p2·∇p2)Gℓ,ϵ,ϵ′(α)](ξ1,ξ2,ξ3)|2dξ1dξ2dξ3<∞,(iii.b)∫Σ1×Σ1×Σ2p2,i2p2,j2|∂2Gℓ,ϵ,ϵ′(α)∂p2,i∂p2,j(ξ1,ξ2,ξ3)|2dξ1dξ2dξ3<∞.
(iv) There existsΛ>m1 such that for α=1,2, ℓ=1,2,3, ϵ,ϵ′=±,
(3.4)Gℓ,ϵ,ϵ′(α)(ξ1,ξ2,ξ3)=0if|p2|≥Λ.Remark 3.2.
Hypothesis3.1(ii) is nothing but an infrared regularization of the kernels Gℓ,ϵ,ϵ′(α). In order to satisfy this hypothesis it is, for example, sufficient to suppose that
(3.5)Gℓ,ϵ,ϵ′(α)(ξ1,ξ2,ξ3)=|p2|1/2G̃ℓ,ϵ,ϵ′(α)(ξ1,ξ2,ξ3),
where G̃ℓ,ϵ,ϵ′(α) is a smooth function of (p1,p2,p3) in the Schwartz space.
Hypothesis3.1(iv), which is a sharp ultraviolet cutoff, is actually not necessary, and can be removed at the expense of some additional technicalities. However, in order to simplify the proof of Proposition 3.5, we will leave it.Our first result is devoted to the existence of a ground state forH together with the location of the spectrum of H and of its absolutely continuous spectrum when g is sufficiently small.Theorem 3.3.
Suppose that the kernelsGℓ,ϵ,ϵ′(α) satisfy Hypotheses 2.1 and 3.1(i). Then there exists 0<g2≤g1 such that H has a unique ground state for g≤g2. Moreover
(3.6)σ(H)=σac(H)=[infσ(H),∞)
with infσ(H)≤0.According to Theorem3.3 the ground state energy E=infσ(H) is a simple eigenvalue of H, and our main results are concerned with a careful study of the spectrum of H above the ground state energy. The spectral theory developed in this work is based on the conjugated operator method as described in [5, 6, 30]. Our choice of the conjugate operator denoted by A is the second quantized dilation generator for the neutrinos.Leta denote the following operator in L2(Σ1):
(3.7)a=12(p2·i∇p2+i∇p2·p2).
The operator a is essentially self-adjoint on C0∞(ℝ3,ℂ2). Its second quantized version dΓ(a) is a self-adjoint operator in 𝔉a(L2(Σ1)). From the definition (2.8) of the space 𝔉ℓ, the following operator in 𝔉ℓ(3.8)Aℓ=1⊗1⊗dΓ(a)⊗1+1⊗1⊗1⊗dΓ(a)
is essentially self-adjoint on 𝔇L.Let nowA be the following operator in 𝔉L:
(3.9)A=A1⊗12⊗13+11⊗A2⊗13+11⊗12⊗A3.
Then A is essentially self-adjoint on 𝔇L.We will denote again byA its extension to 𝔉. Thus A is essentially self-adjoint on 𝔇 and we still denote by A its closure.We also set(3.10)〈A〉=(1+A2)1/2.We then have the following.Theorem 3.4.
Suppose that the kernelsGℓ,ϵ,ϵ′(α) satisfy Hypotheses 2.1 and 3.1. For any δ>0 satisfying 0<δ<m1 there exists 0<gδ≤g2 such that, for 0<g≤gδ, the following points are satisfied.(i)
The spectrum ofH in (infσ(H),m1-δ] is purely absolutely continuous.(ii)
Limiting absorption principle.
For everys>1/2 and φ, ψ in 𝔉, the limits
(3.11)limε→0(φ,〈A〉-s(H-λ±iε)〈A〉-sψ)
exist uniformly for λ in any compact subset of (infσ(H),m1-δ].(iii)
Pointwise decay in time.
Supposes∈(1/2,1) and f∈C0∞(ℝ) with suppf⊂(infσ(H),m1-δ). Then
(3.12)∥〈A〉-se-itHf(H)〈A〉-s∥=𝒪(t1/2-s)
as t→∞.The proof of Theorem3.4 is based on a positive commutator estimate, called the Mourre estimate, and on a regularity property of H with respect to A (see [5, 6, 30]). According to [4], the main ingredient of the proof is auxiliary operators associated with infrared cutoff Hamiltonians with respect to the momenta of the neutrinos that we now introduce.Letχ0(·), χ∞(·)∈C∞(ℝ,[0,1]) with χ0=1 on (-∞,1], χ∞=1 on [2,∞) and χ02+χ∞2=1.Forσ>0 we set
(3.13)χσ(p)=χ0(|p|σ),χσ(p)=χ∞(|p|σ),χ̃σ(p)=1-χσ(p),
where p∈ℝ3.The operatorHI,σ is the interaction given by (2.23) and (2.24) and associated with the kernels χ̃σ(p2)Gℓ,ϵ,ϵ′(α)(ξ1,ξ2,ξ3). We then set
(3.14)Hσ:=H0+gHI,σ.Let(3.15)Σ1,σ=Σ1∩{(p2,s2);|p2|<σ},Σ1σ=Σ1∩{(p2,s2);|p2|≥σ}𝔉ℓ,2,σ=𝔉a(L2(Σ1,σ))⊗𝔉a(L2(Σ1,σ)),𝔉ℓ,2σ=𝔉a(L2(Σ1σ))⊗𝔉a(L2(Σ1σ)),𝔉ℓ,2=𝔉ℓ,2,σ⊗𝔉ℓ,2σ,𝔉ℓ,1=⨂2𝔉a(L2(Σ1)).
The space 𝔉ℓ,1 is the Fock space for the massive leptons ℓ, and 𝔉ℓ,2 is the Fock space for the neutrinos and antineutrinos ℓ.Set(3.16)𝔉ℓσ=𝔉ℓ,1⊗𝔉ℓ,2σ𝔉ℓ,σ=𝔉ℓ,2,σ.
We have
(3.17)𝔉ℓ≃𝔉ℓσ⊗𝔉ℓ,σ.
Set
(3.18)𝔉Lσ=⨂ℓ=13𝔉ℓσ𝔉L,σ=⨂ℓ=13𝔉ℓ,σ.
We have
(3.19)𝔉L≃𝔉Lσ⊗𝔉L,σ.
Set
(3.20)𝔉σ=𝔉Lσ⊗𝔉W.
We have
(3.21)𝔉≃𝔉L,σ⊗𝔉σ.
Set
(3.22)H0(1)=∑ℓ=13∑ϵ=±∫wℓ(1)(ξ1)bℓ,ϵ*(ξ1)bℓ,ϵ(ξ1)dξ1,H0(2)=∑ℓ=13∑ϵ=±∫wℓ(2)(ξ2)cℓ,ϵ*(ξ2)cℓ,ϵ(ξ2)dξ2,H0(3)=∑ϵ=±∫w(3)(ξ3)aϵ*(ξ3)aϵ(ξ3)dξ3,H0(2)σ=∑ℓ=13∑ϵ=±∫|p2|>σwℓ(2)(ξ2)cℓ,ϵ*(ξ2)cℓ,ϵ(ξ2)dξ2,H0,σ(2)=∑ℓ=13∑ϵ=±∫|p2|≤σwℓ(2)(ξ2)cℓ,ϵ*(ξ2)cℓ,ϵ(ξ2)dξ2.
We have on 𝔉σ⊗𝔉σ(3.23)H0(2)=H0(2)σ⊗1σ+1σ⊗H0,σ(2).
Here, 1σ (resp., 1σ) is the identity operator on 𝔉σ (resp., 𝔉σ).Define(3.24)Hσ=Hσ|𝔉σ,H0σ=H0|𝔉σ.We get(3.25)Hσ=H0(1)+H0(2)σ+H0(3)+gHI,σon𝔉σ,Hσ=Hσ⊗1σ+1σ⊗H0,σ(2)on𝔉σ⊗𝔉σ.
In order to implement the conjugate operator theory, we have to show that Hσ has a gap in its spectrum above its ground state.We now set, forβ>0 and η>0,
(3.26)Cβη=(3mW(1+1m12)+3βmWm12+12ηm12(1+β))1/2,Bβη=(3mW(1+14β)+12(η(1+14β)+14η))1/2.
Let
(3.27)G=(Gℓ,ϵ,ϵ′(α)(·,·,·))α=1,2;ℓ=1,2,3;ϵ,ϵ′=±,ϵ≠ϵ′,
and set
(3.28)K(G)=(∑α=1,2∑ℓ=13∑ϵ≠ϵ′∥Gℓ,ϵ,ϵ′(α)∥L2(Σ1×Σ1×Σ2)2)1/2.
Let
(3.29)C̃βη=Cβη(1+g1K(G)Cβη1-g1K(G)Cβη),B̃βη=(1+g1K(G)Cβη1-g1K(G)Cβη(2+g1K(G)BβηCβη1-g1K(G)Cβη))Bβη.Let(3.30)K̃(G)=(∑α=1,2∑ℓ=13∑ϵ≠ϵ′∫Σ1×Σ1×Σ2|Gℓ,ϵ,ϵ′(α)(ξ1,ξ2,ξ3)|2|p2|2dξ1dξ2dξ3)1/2.
Let δ∈ℝ be such that
(3.31)0<δ<m1.We set(3.32)D̃=sup(4Λγ2m1-δ,1)K̃(G)(2m1C̃βη+B̃βη),
where Λ>m1 has been introduced in Hypothesis 3.1(iv).Let us define the sequence(σn)n≥0 by
(3.33)σ0=Λ,σ1=m1-δ2,σ2=m1-δ=γσ1,σn+1=γσn,n≥1,
where γ=1-δ/(2m1-δ).Letgδ(1) be such that
(3.34)0<gδ(1)<inf(1,g1,γ-γ23D̃).
For 0<g≤gδ(1) we have
(3.35)0<γ<(1-3gD̃γ),(3.36)0<σn+1<(1-3gD̃γ)σn,n≥1.
Set
(3.37)Hn=Hσn;H0n=H0σn,n≥0,En=infσ(Hn),n≥0.
We then get the following.Proposition 3.5.
Suppose that the kernelsGℓ,ϵ,ϵ′(α) satisfy Hypotheses 2.1, 3.1(i), and 3.1(iv). Then there exists 0<g̃δ≤gδ(1) such that, for g≤g̃δ and n≥1, En is a simple eigenvalue of Hn and Hn does not have spectrum in (En,En+(1-3gD̃/γ)σn).The proof of Proposition3.5 is given in the appendix.We now introduce the positive commutator estimates and the regularity property ofH with respect to A in order to prove Theorem 3.4.The operatorA has to be split into two pieces depending on σ.Let(3.38)ησ(p2)=χ2σ(p2),ησ(p2)=χ2σ(p2),aσ=ησ(p2)aησ(p2),aσ=ησ(p2)aησ(p2).
Since ησ2+(ησ)2=1, and [ησ,[ησ,a]]=0=[ησ,[ησ,a]], we obtain (see [4])
(3.39)a=aσ+aσ.Note that we also have(3.40)aσ=12(ησ(p2)2p2·i∇p2+i∇p2·ησ(p2)2p2),aσ=12(ησ(p2)2p2·i∇p2+i∇p2·ησ(p2)2p2).The operatorsa, aσ, and aσ are essentially self-adjoint on C0∞(ℝ3,ℂ2) (see [5, Proposition  4.2.3]). We still denote by a, aσ, and aσ their closures. If ã denotes any of the operator a, aσ, and aσ, we have
(3.41)𝒟(ã)={u∈L2(Σ1);ãu∈L2(Σ1)}.The operatorsdΓ(a), dΓ(aσ), and dΓ(aσ) are self-adjoint operators in 𝔉a(L2(Σ1)), and we have
(3.42)dΓ(a)=dΓ(aσ)+dΓ(aσ).By (2.8), the following operators in 𝔉ℓ, denoted by Aℓσ and Aσℓ, respectively,
(3.43)Aℓσ=1⊗1⊗dΓ(aσ)⊗1+1⊗1⊗1⊗dΓ(aσ),Aσℓ=1⊗1⊗dΓ(aσ)⊗1+1⊗1⊗1⊗dΓ(aσ)
are essentially self-adjoint on 𝔇ℓ.LetAσ and Aσ be the following two operators in 𝔉L:
(3.44)Aσ=A1σ⊗12⊗13+11⊗A2σ⊗13+11⊗12⊗A3σ,Aσ=Aσ1⊗12⊗13+11⊗Aσ2⊗13+11⊗12⊗Aσ3.
The operators Aσ and Aσ are essentially self-adjoint on 𝔇L. Still denoting by Aσ and Aσ their extensions to 𝔉, Aσ and Aσ are essentially self-adjoint on 𝔇 and we still denote by Aσ and Aσ their closures.We have(3.45)A=Aσ+Aσ.The operatorsa, aσ, and aσ are associated to the following C∞-vector fields in ℝ3, respectively:
(3.46)v(p2)=p2,vσ(p2)=ησ(p2)2p2,vσ(p2)=ησ(p2)2p2.
Let 𝒱(p) be any of these vector fields. We have
(3.47)|𝒱(p)|≤Γ|p|
for some Γ>0, and we also have
(3.48)𝒱(p)=ṽ(|p|)p,
where the ṽ's are defined by (3.46) and (3.48) and fulfil |p|α(dα/d|p|α)ṽ(|p|) bounded for α=0,1,2.Letψt(·):ℝ3→ℝ3 be the corresponding flow generated by 𝒱:
(3.49)ddtψt(p)=𝒱(ψt(p)),ψ0(p)=p.ψt(p) is a C∞-flow and we have
(3.50)e-Γ|t||p|≤|ψt(p)|≤eΓ|t||p|.ψt(p) induces a one-parameter group of unitary operators U(t) in L2(Σ1)≃L2(ℝ3,ℂ2) defined by
(3.51)(U(t)f)(p)=f(ψt(p))(det∇ψt(p))1/2.
Let ϕt(·), ϕtσ(·), and ϕσt(·) be the flows associated with the vector fields v(·), vσ(·), and vσ(·), respectively.LetU(t), Uσ(t), and Uσ(t) be the corresponding one-parameter groups of unitary operators in L2(Σ1). The operators a, aσ, and aσ are the generators of U(t), Uσ(t), and Uσ(t), respectively, that is,
(3.52)U(t)=e-iat,Uσ(t)=e-iaσt,Uσ(t)=e-iaσt.Let(3.53)w(2)(ξ2)=(wℓ(2)(ξ2))ℓ=1,2,3,dΓ(w(2))=∑ℓ=13∑ϵ∫wℓ(2)(ξ2)cℓ,ϵ*(ξ2)cℓϵ(ξ2)dξ2.
Let V(t) be any of the one-parameter groups U(t), Uσ(t), and Uσ(t). We set
(3.54)V(t)w(2)V(t)*=(V(t)wℓ(2)V(t)*)ℓ=1,2,3,
and we have
(3.55)V(t)w(2)V(t)*=w(2)(ψt).
Here ψt is the flow associated to V(t).This yields, for anyφ∈𝔇, (see [11, Lemma  2.8])
(3.56)e-iAtH0eiAtφ-H0φ=(dΓ(e-iatw(2)eiat)-dΓ(w(2)))φ=(dΓ(w(2)∘ϕt-w(2)))φ,(3.57)e-iAσtH0eiAσtφ-H0φ=(dΓ(e-iaσtw(2)eiaσt)-dΓ(w(2)))φ=(dΓ(w(2)∘ϕtσ-w(2)))φ,(3.58)e-iAσtH0eiAσtφ-H0φ=(dΓ(e-iaσtw(2)eiaσt)-dΓ(w(2)))φ=(dΓ(w(2)∘ϕσt-w(2)))φ.Proposition 3.6.
Suppose that the kernelsGℓ,ϵ,ϵ′(α) satisfy Hypothesis 2.1.
For everyt∈ℝ one has, for g≤g1,
(3.59)(i)eitA𝒟(H0)=eitA𝒟(H)⊂𝒟(H0)=𝒟(H),(ii)eitAσ𝒟(H0)=eitAσ𝒟(H)⊂𝒟(H0)=𝒟(H),(iii)eitAσ𝒟(H0)=eitAσ𝒟(H)⊂𝒟(H0)=𝒟(H).Proof.
We only prove (i), since (ii) and (iii) can be proved similarly. By (3.56) we have, for φ∈𝔇,
(3.60)e-itAH0eitAφ=(H0(1)+H0(3)+dΓ(w(2)∘ϕt))φ.
It follows from (3.50) and (3.60) that
(3.61)∥H0eitAφ∥≤eΓ|t|∥H0φ∥.
This yields (i) because 𝔇 is a core for H0. Moreover we get
(3.62)∥H0eitA(H0+1)-1∥≤eΓ|t|.
In view of 𝔇(H0)=𝔇(H), the operators H0(H+i)-1 and H(H0+i)-1 are bounded, and there exists a constant C>0 such that
(3.63)∥HeitA(H+i)-1∥≤CeΓ|t|.
Similarly, we also get
(3.64)∥H0eitAσ(H0+1)-1∥≤eΓ|t|,∥H0eitAσ(H0+1)-1∥≤eΓ|t|,∥HeitAσ(H+i)-1∥≤CeΓ|t|,∥HeitAσ(H+i)-1∥≤CeΓ|t|.LetHI(G) be the interaction associated with the kernels G=(Gℓ,ϵ,ϵ′(α))α=1,2;ℓ=1,2,3;ϵ≠ϵ′=±, where the kernels (Gℓ,ϵ,ϵ′(α)) satisfy Hypothesis 2.1.We set(3.65)V(t)G=(V(t)Gℓ,ϵ,ϵ′(α))α=1,2;ℓ=1,2,3;ϵ≠ϵ′=±.
We have for φ∈𝔇 (see [11, Lemma  2.7]),
(3.66)e-iAtHI(G)eiAtφ=HI(e-iatG)φ,e-iAσtHI(G)eiAσtφ=HI(e-iaσtG)φ,e-iAσtHI(G)eiAσtφ=HI(e-iaσtG)φ.According to [5, 6], in order to prove Theorem 3.4 we must prove that H is locally of class C2(Aσ), C2(Aσ), and C2(A) in (-∞,m1-δ/2) and that A and Aσ are locally strictly conjugate to H in (E,m1-δ/2).Recall thatH is locally of class C2(A) in (-∞,m1-δ/2) if, for any φ∈C0∞((-∞,m1-δ/2)), φ(H) is of class C2(A); that is, t→e-iAtφ(H)eitAψ is twice continuously differentiable for all φ∈C0∞((-∞,m1-δ/2)) and all ψ∈𝔉.Thus, one of our main results is the following one.Theorem 3.7.
Suppose that the kernelsGℓ,ϵ,ϵ′(α) satisfy Hypotheses 2.1, 3.1(i) and 3.1(iii).  (a)
H is locally of class C2(A), C2(Aσ), and C2(Aσ) in (-∞,m1-δ/2).(b)
Hσ is locally of class C2(Aσ) in (-∞,m1-δ/2).It follows from Theorem3.7 that [H,iA], [H,iAσ], [H,iAσ], and [Hσ,iAσ] are defined as sesquilinear forms on ⋃KEK(H)𝔉, where the union is taken over all the compact subsets K of (-∞,m1-δ/2).Furthermore, by Proposition3.6, Theorem 3.7 and [4, Lemma  29], we get for all φ∈C0∞((E,m1-δ/2)) and all ψ∈𝔉,
(3.67)φ(H)[H,iA]φ(H)ψ=limt→0φ(H)[H,eitA-1t]φ(H)ψ,φ(H)[H,iAσ]φ(H)ψ=limt→0φ(H)[H,eitAσ-1t]φ(H)ψ,φ(H)[H,iAσ]φ(H)ψ=limt→0φ(H)[H,eitAσ-1t]φ(H)ψ,φ(Hσ)[Hσ,iAσ]φ(Hσ)ψ=limt→0φ(Hσ)[Hσ,eitAσ-1t]φ(Hσ)ψ.The following proposition allows us to compute[H,iA], [H,iAσ], [H,iAσ], and [Hσ,iAσ] as sesquilinear forms. By Hypotheses 2.1 and 3.1(iii.a), the kernels Gℓ,ϵ,ϵ′(α)(ξ1,·,ξ3) belong to the domains of a, aσ, and aσ.Proposition 3.8.
Suppose that the kernelsGℓ,ϵ,ϵ′(α) satisfy Hypotheses 2.1 and 3.1(iii.a). Then  (a)
for allψ∈𝒟(H) one has
(i)
limt→0[H,(eitA-1)/t]ψ=(dΓ(w(2))+gHI(-iaG))ψ,(ii)
limt→0[H,(eitAσ-1)/t]ψ=(dΓ((ησ)2w(2))+gHI(-iaσG))ψ,(iii)
limt→0[H,(eitAσ-1)/t]ψ=(dΓ((ησ)2w(2))+gHI(-iaσG))ψ,(iv)
limt→0[Hσ,(eitAσ-1)/t]ψ=(dΓ((ησ)2w(2))+gHI(-iaσ(χ̃σ(p2)G)))ψ;(b)
and
(i)
sup0<|t|≤1∥[H,(eitA-1)/t](H+i)-1∥<∞,(ii)
sup0<|t|≤1∥[H,(eitAσ-1)/t](H+i)-1∥<∞,(iii)
sup0<|t|≤1∥[H,(eitAσ-1)/t](H+i)-1∥<∞,(iv)
sup0<|t|≤1∥[Hσ,(eitAσ-1)/t](H+i)-1∥<∞.Proof.
Part(b) follows from part (a) by the uniform boundedness principle. For part (a), we only prove (a)(i), since other statements can be proved similarly.
By (3.50), we obtain
(3.68)1|t||wℓ(2)(ϕt(p2))-wℓ(2)(p2)|≤1|t|(eΓ|t|-1)wℓ(2)(p2)
for ℓ=1,2,3.
By (3.56)–(3.58) and Lebesgue's Theorem we then get for all ψ∈𝒟(H0)(3.69)limt→0[H0,eitA-1t]ψ=limt→01t[e-itAH0eitA-H0]ψ=dΓ(w(2))ψ,limt→0[H0,eitAσ-1t]ψ=limt→01t[e-itAσH0eitAσ-H0]ψ=dΓ((ησ)2w(2))ψ,limt→0[H0,eitAσ-1t]ψ=limt→01t[e-itAσH0eitAσ-H0]ψ=dΓ((ησ)2w(2))ψ.
By (3.66), we obtain, for all ψ∈𝒟(H),
(3.70)limt→0[HI(G),eitA-1t]ψ=limt→01t[e-itAHI(G)eitA-HI(G)]ψ=HI(-i(aG))ψ,limt→0[HI(G),eitAσ-1t]ψ=limt→01t[e-itAσHI(G)eitAσ-HI(G)]ψ=HI(-i(aσG))ψ,limt→0[HI(G),eitAσ-1t]ψ=limt→01t[e-itAσHI(G)eitAσ-HI(G)]ψ=HI(-i(aσG))ψ,limt→0[HI(χ̃σ(p2)G),eitAσ-1t]ψ=limt→01t[e-itAσHI(χ̃σ(p2)G)eitAσ-HI(χ̃σ(p2)G)]ψ=HI(-i(aσ(χ̃σ(p2)G)))ψ.
This concludes the proof of Proposition 3.8.Combining (3.67) with Proposition 3.8, we finally get for every φ∈C0∞((-∞,m1-δ/2)) and every ψ∈𝔉(3.71)φ(H)[H,iA]φ(H)ψ=φ(H)[dΓ(w(2))+gHI(-i(aG))]φ(H)ψ,(3.72)φ(H)[H,iAσ]φ(H)ψ=φ(H)[dΓ((ησ)2w(2))+gHI(-i(aσG))]φ(H)ψ,(3.73)φ(H)[H,iAσ]φ(H)ψ=φ(H)[dΓ((ησ)2w(2))+gHI(-i(aσG))]φ(H)ψ,(3.74)φ(Hσ)[Hσ,iAσ]φ(Hσ)ψ=φ(Hσ)[dΓ((ησ)2w(2))+gHI(-i(aσ(χ̃σG)))]φ(Hσ)ψ.We now introduce the Mourre inequality.LetN be the smallest integer such that
(3.75)Nγ≥1.We have, forg≤gδ(1),
(3.76)γ<γ+1N(1-3gD̃γ-γ)<1-3gD̃γ,γN≤γ-1N(1-3gD̃γ-γ)<γ.
Let
(3.77)ϵγ=12N(1-3gδ(1)D̃γ-γ).
We choose f∈C0∞(ℝ) such that 1≥f≥0 and
(3.78)f(λ)={1ifλ∈[(γ-ϵγ)2,γ+ϵγ],0ifλ>γ+1N(1-3gδ(1)D̃γ-γ)=γ+2ϵγ,0ifλ<(γ-1N(1-3gδ(1)D̃γ-γ))2=(γ-2ϵγ)2.
Note that γ+2ϵγ<1-3gD̃/γ for g≤gδ(1) and γ-ϵγ>γ/N.We set, forn≥1,
(3.79)fn(λ)=f(λσn).Let(3.80)Hn=Hσn,En=infσ(Hn),H0n(2)=H0σn(2).
Let Pn denote the ground state projection of Hn. It follows from Proposition 3.5 that, for n≥1 and g≤g̃δ≤gδ(1),
(3.81)fn(Hn-En)=Pn⊗fn(H0,n(2)).
Note that
(3.82)En=En=infσ(Hn).
Set
(3.83)an=aσn,an=aσn,An=Aσn,An=Aσn,𝔉n=𝔉σn,𝔉n=𝔉σn.
We have
(3.84)𝔉≃𝔉n⊗𝔉n,A=An+An.
We further note that
(3.85)anχ̃σn(p2)=an.
By (3.72), (3.74), and (3.85), we obtain
(3.86)[H,iAn]=[Hn,iAn]⊗1
as sesquilinear forms with respect to 𝔉=𝔉n⊗𝔉n.Furthermore, it follows from the virial theorem (see [6, Proposition  3.2] and Proposition 6.1) that
(3.87)Pn[Hn,iAn]Pn=0.
By (3.81) and (3.87) we then get, for g≤g̃δ≤gδ(1),
(3.88)fn(Hn-En)[H,iAn]fn(Hn-En)=0.We then have the following.Proposition 3.9.
Suppose that the kernelsGℓ,ϵ,ϵ′(α) satisfy Hypotheses 2.1 and 3.1. Then there exists C̃δ>0 and g̃δ(1)>0 such that g̃δ(1)≤g̃δ and
(3.89)fn(Hn-En)[H,iAn]fn(Hn-En)≥C̃δγ2N2σnfn(Hn-En)2
for n≥1 and g≤g̃δ(1).LetEΔ(H-E) be the spectral projection for the operator H-E associated with the interval Δ, and let
(3.90)Δn=[(γ-ϵγ)2σn,(γ+ϵγ)σn],n≥1.
Note that
(3.91)[σn+2,σn+1]⊂((γ-ϵγ)2σn,(γ+ϵγ)σn),n≥1.Theorem 3.10.
Suppose that the kernelsGℓ,ϵ,ϵ′(α) satisfy Hypotheses 2.1 and 3.1. Then there exists Cδ>0 and g̃δ(2)>0 such that g̃δ(2)≤g̃δ(1) and
(3.92)EΔn(H-E)[H,iA]EΔn(H-E)≥Cδγ2N2σnEΔn(H-E)
for n≥1 and g≤g̃δ(2).


## 4. Existence of a Ground State and Location of the Absolutely  Continuous Spectrum

We now prove Theorem3.3. The scheme of the proof is quite well known (see [9, 31]). It follows from Proposition 3.5 that Hn has a unique ground state, denoted by ϕn, in 𝔉n:
(4.1)Hnϕn=Enϕn,ϕn∈𝒟(Hn),∥ϕn∥=1,n≥1.
Therefore Hn has an unique normalized ground state in 𝔉, given by ϕ̃n=ϕn⊗Ωn, where Ωn is the vacuum state in 𝔉n:
(4.2)Hnϕ̃n=Enϕ̃n,ϕ̃n∈𝒟(Hn),∥ϕ̃n∥=1,n≥1.
Since ∥ϕ̃n∥=1, there exists a subsequence (nk)k≥1, converging to ∞ such that (ϕ̃nk)k≥1 converges weakly to a state ϕ̃∈𝔉. We have to prove that ϕ̃≠0. By adapting the proof of Theorem 4.1 in [22] (see also [20]), the key point is to estimate ∥cℓ,ϵ(ξ2)Φ̃n∥𝔉 in order to show that
(4.3)∑ℓ=13∑ϵ∫∥cℓ,ϵ(ξ2)ϕ̃n∥2dξ2=𝒪(g2),
uniformly with respect to n.The estimate (4.3) is a consequence of the so-called “pull-through’’ formula as it follows.LetHI,n denote the interaction HI associated with the kernels 1{|p2|≥σn}(p2)Gℓ,ϵ,ϵ′(α). We thus have
(4.4)H0cℓ,ϵ(ξ2)ϕ̃n=cℓ,ϵ(ξ2)H0ϕ̃n-wℓ(2)(ξ2)cℓ,ϵ(ξ2)ϕ̃n,gHI,ncℓ,ϵ(ξ2)ϕ̃n=cℓ,ϵ(ξ2)gHI,nϕ̃n+gVℓ,ϵ,ϵ′(ξ2)ϕ̃n
with
(4.5)Vℓ,ϵ,ϵ′(ξ2)=g∫Gℓ,ϵ′ϵ(1)(ξ1,ξ2,ξ3)bℓ,ϵ′*(ξ1)aϵ(ξ3)dξ1dξ3+g∫Gℓ,ϵ′ϵ(2)(ξ1,ξ2,ξ3)bℓ,ϵ′*(ξ1)aϵ*(ξ3)dξ1dξ3.
This yields
(4.6)(Hn-En+wℓ(2)(ξ2))cℓ,ϵ(ξ2)ϕ̃n=Vℓ,ϵ,ϵ′(ξ2)ϕ̃n.
By adapting the proof of Propositions 2.4 and 2.5 we easily get
(4.7)∥Vℓ,ϵ,ϵ′ψ∥𝔉≤gmW1/2(∑α=1,2∥Gℓ,ϵ,ϵ′(α)(·,ξ2,·)∥L2(Σ1×Σ2))∥H01/2ψ∥+g∥Gℓ,ϵ,ϵ′(2)(·,ξ2,·)∥L2(Σ1×Σ2)∥ψ∥,
where ψ∈𝒟(H0).Let us estimate∥H0ϕ̃n∥. By (2.53), (2.54), (3.26), and (3.28) we have
(4.8)g∥HI,nϕ̃n∥≤gK(G)(Cβη∥H0ϕ̃n∥+Bβη),∥H0ϕ̃n∥≤|En|+g∥HI,nϕ̃n∥.
Therefore
(4.9)∥H0ϕ̃n∥≤|En|1-g1K(G)Cβη+gK(G)Bβη1-g1K(G)Cβη.
By (3.82), (A.11), and (4.9), there exists C>0 such that
(4.10)∥H0ϕ̃n∥≤C,
uniformly in n and g≤g1.By (4.6), (4.7), and (4.10) we get
(4.11)∥cℓ,ϵϕ̃n∥≤g|p2|(C1/2(∑α=12∥Gℓ,ϵ,ϵ′(α)(·,ξ2,·)∥L2(Σ1×Σ2))+∥Gℓ,ϵ,ϵ′(2)(·,ξ2,·)∥L2(Σ1×Σ2)).
By Hypothesis 3.1(i), there exists a constant C(G)>0 depending on the kernels   G=(Gℓ,ϵ,ϵ′(α))ℓ=1,2,3;α=1,2;ϵ≠ϵ′=± and such that
(4.12)∑ℓ=13∑ϵ∫∥cℓ,ϵ(ξ2)ϕ̃n∥2dξ2≤C(G)2g2.
The existence of a ground state ϕ̃ for H follows by choosing g sufficiently small, that is, g≤g2, as in [20, 22]. By adapting the method developed in [32] (see [32, Corollary  3.4]), one proves that the ground state of H is unique. We omit here the details.Statements aboutσ(H) are consequences of the existence of a ground state and follows from the existence of asymptotic Fock representations for the CAR associated with the cℓ,ϵ♯(ξ2)'s. For f∈L2(ℝ3,ℂ2), we define on 𝒟(H0) the operators
(4.13)cℓ,ϵ♯t(f)=eitHe-itH0cℓ,ϵ♯(f)eitH0e-itH.
By mimicking the proof given in [21, 31] one proves, under the hypothesis of Theorem 3.3 and for f∈C0∞(ℝ3,ℂ2), that the strong limits of cℓ,ϵ♯t(f) when t→±∞ exist for ψ∈𝒟(H0):
(4.14)limt→±∞cℓ,ϵ♯t(f)ψ:=cℓ,ϵ♯±(f)ψ.
The operators cℓ,ϵ♯±(f) satisfy the CAR and we have
(4.15)cℓ,ϵ±(f)ϕ̃=0,f∈C0∞(ℝ3,ℂ2),
where ϕ̃ is the ground state of H.It then follows from (4.14) and (4.15) that the absolutely continuous spectrum of H equals to [infσ(H),∞). We omit the details (see [21, 31]).


## 5. Proof of the Mourre Inequality

We first prove Proposition3.9. In view of Proposition 3.8(a)(iii) and (3.73), we have, as sesquilinear forms,
(5.1)[H,iAσ]=(1-g)dΓ((ησ)2w(2))+g(dΓ((ησ)2w(2)))+gHI(-i(aσG)).
Let 𝔉ℓ(1) (resp., 𝔉ℓ(2)) be the Fock space for the massive leptons ℓ (resp., the neutrinos and antineutrinos ℓ).We have(5.2)𝔉ℓ≃𝔉ℓ(1)⊗𝔉ℓ(2).Let(5.3)𝔉(1)=𝔉W⊗(⨂ℓ=13𝔉ℓ(1)),𝔉(2)=⨂ℓ=13𝔉ℓ(2).We have(5.4)𝔉≃𝔉(1)⊗𝔉(2),𝔉(1) is the Fock space for the massive leptons and the bosons W±, and 𝔉(2) is the Fock space for the neutrinos and antineutrinos.We have, as sesquilinear forms and with respect to (5.4),
(5.5)dΓ((ησ)2w(2))+HI(-i(aσG))=∑ℓ=13∑ϵ∫ησ(p2)2|p2|cℓ,ϵ*(ξ2)cℓ,ϵ(ξ2)dξ2+∑ℓ=13∑ϵ≠ϵ′∫|p2|(11⊗ησ(p2)cℓ,ϵ*(ξ2)+∑α=1,2ℳℓ,ϵ,ϵ′,σ(α)*(ξ2)|p2|⊗12)×(11⊗ησ(p2)cℓ,ϵ(ξ2)+∑α=1,2ℳℓ,ϵ,ϵ′,σ(α)(ξ2)|p2|⊗12)dξ2-∑ℓ=13∑ϵ≠ϵ′∫(∑α=1,2ℳℓ,ϵ,ϵ′,σ(α)*(ξ2)|p2|1/2⊗12)(∑α=1,2ℳℓ,ϵ,ϵ′,σ(α)(ξ2)|p2|1/2⊗12)dξ2,
where
(5.6)ℳℓ,ϵ,ϵ′,σ(α)(ξ2)=i∫(∑α=1,2(aησ(p2)Gℓ,ϵ,ϵ′(α)(ξ1,ξ2,ξ3)))bℓ,ϵ′*(ξ1)aϵ′(ξ3)dξ1dξ3,
and where 1j is the identity operator in 𝔉(j).By mimicking the proofs of Propositions2.4 and 2.5, we get, for every ψ∈𝔇,
(5.7)∑ℓ=13∑ϵ≠ϵ′(ψ,∫(∑α=1,2ℳℓ,ϵ,ϵ′,σ(α)*(ξ2)|p2|1/2⊗12)(∑α=1,2ℳℓ,ϵ,ϵ′,σ(α)(ξ2)|p2|1/2⊗12)ψdξ2)=∑ℓ=13∑ϵ≠ϵ′∥∫(∑α=1,2ℳℓ,ϵ,ϵ′,σα(ξ2)|p2|1/2⊗12)ψdξ2∥2≤(∫|∑α=1,2(aησ(p2)Gℓ,ϵ,ϵ′(α))(ξ1,ξ2,ξ3)|2w(3)(ξ3)|p2|dξ1dξ2dξ3)∥(H0(3))1/2ψ∥.
Noting that |(aησ)(p2)|≤C uniformly with respect to σ, it follows from Hypotheses 2.1 and 3.1 that there exists a constant C(G)>0 such that
(5.8)∫|∑α=1,2(aησ(p2)Gℓ,ϵ,ϵ′(α))(ξ1,ξ2,ξ3)|2w(3)(ξ3)|p2|dξ1dξ2dξ3≤C(G)σ.
This yields
(5.9)-∫(∑α=1,2ℳℓ,ϵ,ϵ′,σ(α)*(ξ2)|p2|1/2⊗12)(∑α=1,2ℳℓ,ϵ,ϵ′,σ(α)(ξ2)|p2|1/2⊗12)dξ2≥-C(G)σ.Combining (5.1), (5.5) with (5.9), we obtain
(5.10)[H,iAn]≥(1-g)dΓ((ησn)2w(2))-gC(G)σn.We have(5.11)dΓ((ησn)2w(2))≥H0n(2).By (3.76), (3.81), and (5.11) we get
(5.12)fn(Hn-En)dΓ(ησn2w(2))fn(Hn-En)≥Pn⊗fn(H0n(2))H0n(2)fn(H0n(2))≥γ2N2σnfn(Hn-En)2
for g≤gδ(1).This, together with (5.10), yields for g≤gδ(1)(5.13)fn(Hn-En)[H,iAn]fn(Hn-En)≥(1-gδ(1))γ2N2σnfn(Hn-En)2-gC(G)σnfn(Hn-En)2.
Setting
(5.14)gδ(2)=inf(gδ(1),1-gδ(1)2C(G)γ2N2),
we get
(5.15)fn(Hn-En)[H,iAn]fn(Hn-En)≥1-gδ(1)2γ2N2σnfn(Hn-En)2
for g≤gδ(2).Proposition3.9 is proved by setting g̃δ(1)=gδ(2) and C̃δ=(1-gδ(1))/2.The proof of Theorem3.10 is the consequence of the following two lemmas.Lemma 5.1.
Assume that the kernelsGℓ,ϵ,ϵ′(α) satisfy Hypotheses 2.1 and 3.1(ii). Then there exists a constant D>0 such that
(5.16)|E-En|≤gDσn2
for n≥1 and g≤g(2).Proof.
Letϕ (resp., ϕ̃n) be the unique normalized ground state of H (resp., Hn). We have
(5.17)E-En≤(ϕ̃n,(H-Hn)ϕ̃n),En-E≤(ϕ,(Hn-H)ϕ)
with
(5.18)H-Hn=gHI(χσn(p2)G).
Combining (2.53) and (2.54) with (3.26)–(3.28) and (5.18), we get
(5.19)∥(H-Hn)ϕ̃n∥≤gK(χσn(p2)G)(Cβη∥H0ϕ̃n∥+Bβη),∥(H-Hn)ϕ∥≤gK(χσn(p2)G)(Cβη∥H0ϕ∥+Bβη).
It follows from Hypothesis 3.1(ii), (4.10), and (5.19) that there exists a constant D>0 such that
(5.20)max(∥(H-Hn)ϕ̃n∥,∥(H-Hn)ϕ∥)≤gDσn2
for n≥1 and g≤g(2).
By (5.17), this proves Lemma 5.1.Lemma 5.2.
Suppose that the kernelsGℓ,ϵ,ϵ′(α) satisfy Hypotheses 2.1 and 3.1(ii). Then there exists a constant C>0 such that
(5.21)∥fn(H-E)-fn(Hn-En)∥≤gCσn
for n≥1 and g≤g(2).Proof.
Letf̃(·) be an almost analytic extension of f(·) given by (3.78) satisfying
(5.22)|∂z̅f̃(x+iy)|≤Cy2.
Note that f̃(x+iy)∈C0∞(ℝ2). We thus have
(5.23)f(s)=∫df̃(z)z-s,df̃(z)=-1π∂f̃∂z̅dxdy.
Using the functional calculus based on this representation of f(s), we get
(5.24)fn(H-E)-fn(Hn-En)=σn∫1H-E-zσn(H-Hn+En-E)1Hn-En-zσndf̃(z).
Combining (2.53) and (2.54) with (3.26)–(3.28) and Hypothesis 3.1(ii), we get, for every ψ∈𝒟(H0) and for g≤g(2),
(5.25)g∥HI(χσnG)ψ∥≤2gCσn2K(G)(Cβη∥(H0+1)ψ∥+(Cβη+Bβη)∥ψ∥).
This yields
(5.26)g∥HI(χσn(p2)G)(H0+1)-1∥≤gC1σn2
for some constant C1>0 and for g≤g(2).
By mimicking the proof of (A.21) we show that there exists a constant C2>0 such that
(5.27)∥(H0+1)(Hn-En-zσn)-1∥≤C2(1+1|Imz|σn)
or g≤g(1).
Combining Lemma5.1 and (5.24) with (5.25)–(5.27) we obtain
(5.28)∥fn(H-E)-fn(Hn-En)∥≤gCσn∫|(∂f̃/∂z̅)(x+iy)|y2dxdy
for some constant C>0 and for g≤g(2).
Using (5.22) and f̃(x+iy)∈C0∞(ℝ2) one concludes the proof of Lemma 5.2.We now prove Theorem3.10.Proof.
It follows from Proposition3.9 that
(5.29)fn(Hn-En)[H,iA]fn(Hn-En)=fn(Hn-En)[H,iAn]fn(Hn-En)≥C̃δγ2N2σnfn(Hn-En)2
for n≥1 and g≤g̃δ(1).
This yields(5.30)fn(H-E)[H,iAn]fn(H-E)≥C̃δγ2N2σnfn(H-E)2-fn(H-E)[H,iA](fn(Hn-En)-fn(H-E))-(fn(Hn-En)-fn(H-E))[H,iA]fn(Hn-En)+C̃δγ2N2σn(fn(Hn-En)-fn(H-E))2+C̃δγ2N2σnfn(H-E)(fn(Hn-En)-fn(H-E))+C̃δγ2N2σn(fn(Hn-En)-fn(H-E))fn(H-E).
Combining Proposition 3.8(i) and (5.23) with (5.26) and (5.27) we show that [H,iA]fn(Hn-En) and fn(H-E)[H,iA] are bounded operators uniformly with respect to n. This, together with Lemma 5.2, yields
(5.31)fn(H-E)[H,iA]fn(H-E)≥C̃δγ2N2σnfn(H-E)2-C̃gσn
for some constant C̃>0 and for g≤inf(g(2),g̃δ(1)).
Multiplying both sides of (5.31) with EΔn(H-E) we then get
(5.32)EΔn(H-E)[H,iA]EΔn(H-E)≥C̃δγ2N2σnEΔn(H-E)-C̃gσnEΔn(H-E).
Setting
(5.33)g̃δ(2)<inf(C̃δC̃γ2N2,g(2),g̃δ(1)),
Theorem 3.10 is proved with Cδ=C̃δ-C̃(N2/γ2)g̃δ(2)>0.


## 6. Proof of Theorem3.7

We set(6.1)At=eitA-1t,adAt·=[At,·],Atσ=eitAσ-1t,Aσt=eitAσ-1t.
The fact that H is of class C1(A), C1(Aσ), and C1(Aσ) in (-∞,m1-δ/2) is the consequence of the following proposition.Proposition 6.1.
Suppose that the kernelsGℓ,ϵ,ϵ′(α) satisfy Hypotheses 2.1 and 3.1(iii.a). For every φ∈C0∞((-∞,m1-δ/2)) and g≤g1, one then has
(6.2)sup0<|t|≤1∥[φ(H),At]∥<∞,sup0<|t|≤1∥[φ(H),Atσ]∥<∞,sup0<|t|≤1∥[φ(H),Aσt]∥<∞,sup0<|t|≤1∥[φ(Hσ),Atσ]∥<∞.Proof.
We use the representation(6.3)φ(H)=∫dϕ(z)(z-H)-1,
where ϕ(z) is an almost analytic extension of φ with
(6.4)|∂z̅ϕ(x+iy)|≤C|y|2,dϕ(z)=-1π∂∂z̅ϕ(z)dxdy.
Note that ϕ(x+iy)∈C0∞(ℝ2).
We get(6.5)adAtφ(H)=∫dϕ(z)(z-H)-1[At,H](z-H)-1.
This yields
(6.6)∥adAtφ(H)∥≤sup0<|t|≤1∥[At,H](i-H)-1∥∫|dϕ(z)|∥(z-H)-1∥∥(i-H)(z-H)-1∥.
It is easy to prove that
(6.7)∫|dϕ(z)|∥(z-H)-1∥∥(i-H)(z-H)-1∥≤C∫|dϕ(z)||Imz|2<∞.
By Proposition3.8(b)(i) and (6.7) we finally get, for g≤g1,
(6.8)sup0<|t|≤1∥adAtφ(H)∥<∞.
In a similar way we obtain, for g≤g1,
(6.9)sup0<|t|≤1∥[Atσ,φ(H)]∥<∞,sup0<|t|≤1∥[Aσt,φ(H)]∥<∞,sup0<|t|≤1∥[Atσ,φ(Hσ)]∥<∞.The proof of Theorem3.7 is the consequence of the following proposition.Proposition 6.2.
Suppose that the kernelsGℓ,ϵ,ϵ′(α) satisfy Hypotheses 2.1, 3.1(i) and 3.1(iii). One then has, for g≤g1,
(6.10)sup0<|t|≤1∥[At,[At,H]](H+i)-1∥<∞,sup0<|t|≤1∥[Atσ,[Atσ,H]](H+i)-1∥<∞,sup0<|t|≤1∥[Aσt,[Aσt,H]](H+i)-1∥<∞,sup0<|t|≤1∥[Atσ,[Atσ,Hσ]](Hσ+i)-1∥<∞.Proof.
We have, for everyψ∈𝒟(H),
(6.11)[At,[At,H]]ψ=1t2e2itA(e-2itAHe2itA-2e-itAHeitA+H)ψ.
By (3.56) we get
(6.12)[At,[At,H0]]ψ=1t2e2itA(dΓ(w(2)∘ϕ2t-2w(2)∘ϕt+w(2)))ψ,
where, for ℓ=1,2,3,
(6.13)(wℓ(2)∘ϕ2t)(p2)-2(wℓ(2)∘ϕt)(p2)+wℓ(2)(p2)=|ϕ2t(p2)|-2|ϕt(p2)|+|p2|.
We further note that
(6.14)1t2||ϕ2t(p2)|-2|ϕt(p2)|+|p2||≤sup|s|≤2|t||∂2∂s2|ϕs(p2)||,∂2∂s2|ϕs(p2)|=|ϕs(p2)|≤eΓ|s||p2|.
Combining (6.12) with (6.13) and (6.14) we get
(6.15)∥[At,[At,H0]](H0+1)-1∥≤e2Γ|t|,sup0<|t|≤1∥[At,[At,H0]](H0+1)-1∥≤e2Γ.
In a similar way we obtain
(6.16)sup0<|t|≤1∥[Atσ,[Atσ,H0]](H0+1)-1∥≤Ce2Γ,sup0<|t|≤1∥[Aσt,[Aσt,H0]](H0+1)-1∥≤Ce2Γ.
Here C is a positive constant.
Let us now prove that(6.17)sup0<|t|≤1∥[At,[At,HI(G)]](H+i)-1∥<∞.
By (3.66) and (6.11) we get, for every ψ∈𝒟(H),
(6.18)[At,[At,HI(G)]]ψ=∑α=1,2∑ℓ=1,2,3∑ϵ≠ϵ′e2itAt2(e-2itAHI(Gℓ,ϵ,ϵ′(α))e2itA-2e-itAHI(Gℓ,ϵ,ϵ′(α))eitA+HI(Gℓ,ϵ,ϵ′(α)))ψ=∑α=1,2∑ℓ=1,2,3∑ϵ≠ϵ′e2itAt2(HI(Gℓ,ϵ,ϵ′;2t(α))-2HI(Gℓ,ϵ,ϵ′;t(α))+HI(Gℓ,ϵ,ϵ′;0(α)))ψ,
where
(6.19)Gℓ,ϵ,ϵ′;t(α)(ξ1,ξ2,ξ3)=(Dϕt(p2))1/2Gℓ,ϵ,ϵ′(α)(ξ1;ϕt(p2),s2;ξ3)=(e-itaGℓ,ϵ,ϵ′(α))(ξ1,ξ2,ξ3).
Combining (2.53) and (2.54) with (3.26)–(3.28) and (6.18) we get
(6.20)∥[At,[At,HI(G)]]ψ∥≤gK(Gt)(Cβη∥(H0+I)ψ∥+(Cβη+Bβη)∥ψ∥).
Here K(Gt)>0 and
(6.21)K(Gt)2=∑α=1,2∑ℓ=1,2,3∑ϵ≠ϵ′1t2∥Gℓ,ϵ,ϵ′;2t(α)-2Gℓ,ϵ,ϵ′;t(α)+Gℓ,ϵ,ϵ′(α)∥L2(Σ1×Σ1×Σ2)2.
We further note that, for 0<|t|≤1,
(6.22)K(Gt)≤sup0<|s|≤2(∑α=1,2∑ℓ=1,2,3∑ϵ≠ϵ′∥∂2∂s2Gℓ,ϵ,ϵ′;s(α)∥L2(Σ1×Σ1×Σ2)2)1/2.
We get
(6.23)(∂∂tGℓ,ϵ,ϵ′;t(α))=32(e-itaGℓ,ϵ,ϵ′(α))+(e-ita(p2·∇p2Gℓ,ϵ,ϵ′(α))),(∂2∂t2Gℓ,ϵ,ϵ′;t(α))=94(e-itaGℓ,ϵ,ϵ′(α))+72(e-ita(p2·∇p2Gℓ,ϵ,ϵ′(α)))+∑i,j=1,2,3e-ita(p2,ip2,j∂p2,ip2,j2Gℓ,ϵ,ϵ′(α)).
Recall that e-ita is an one parameter group of unitary operators in L2(Σ1×Σ1×Σ2).
Combining Hypothesis3.1(iii.a) and (iii.b) with (6.20)–(6.23) we finally get
(6.24)sup0<|t|≤1∥[At,[At,HI(G)]](H0+1)-1∥<∞.
In view of 𝒟(H)=𝒟(H0) the operators H0(H+i)-1 and H(H0-1)-1 are bounded and we obtain
(6.25)sup0<|t|≤1∥[At,[At,H0]](H+i)-1∥<∞,sup0<|t|≤1∥[At,[At,HI(G)]](H+i)-1∥<∞.
This yields
(6.26)sup0<|t|≤1∥[At,[At,H]](H+i)-1∥<∞
for g≤g1.
LetV(p2) denote any of the two C∞-vector fields vσ(p2) and vσ(p2) and let ã denote the corresponding aσ and aσ operators. We get
(6.27)(∂2∂t2(e-iãtGℓ,ϵ,ϵ′(α)))(ξ1,ξ2,ξ3)=14(e-iãt((divV(p2))2Gℓ,ϵ,ϵ′(α)))(ξ1,ξ2,ξ3)+12(e-iãt((divV(p2))V(p2)·∇p2Gℓ,ϵ,ϵ′(α)))(ξ1,ξ2,ξ3)+12(e-iãt(∑i,j=13(Vi(p2)(∂p2,ip2,j2Vj(p2)))Gℓ,ϵ,ϵ′(α)))(ξ1,ξ2,ξ3)+12(e-iãt(∑i,j=13Vi(p2)∂Vj∂p2,i(p2)∂∂p2,jGℓ,ϵ,ϵ′(α)))(ξ1,ξ2,ξ3)+12(e-iãt(∑i,j=13Vi(p2)Vj(p2)∂2∂p2,i∂p2,jGℓ,ϵ,ϵ′(α)))(ξ1,ξ2,ξ3).
Combining the properties of the C∞ fields vσ(p2) and vσ(p2) together with Hypotheses 2.1,  3.1(i) and 3.1(iii) we get, from (6.25) and by mimicking the proof of (6.26),
(6.28)sup0<|t|≤1∥[Atσ,[Atσ,H]](H+i)-1∥<∞,sup0<|t|≤1∥[Aσt,[Aσt,H]](H+i)-1∥<∞
for g≤g1.
Similarly, by mimicking the proof of (6.28), we easily get, for g≤g1,
(6.29)sup0<|t|≤1∥[Atσ,[Atσ,Hσ]](Hσ+i)-1∥<∞.
This concludes the proof of Proposition 6.2.We now prove Theorem3.7.Proof of Theorem3.7.
In view of [5, Lemma  6.2.3] (see also [4, Proposition  28]), the proof of Theorem 3.7 will follow from Proposition 6.1 and the following estimates:
(6.30)sup0<|t|≤1∥[At,[At,φ(H)]]∥<∞,(6.31)sup0<|t|≤1∥[Atσ,[Atσ,φ(H)]]∥<∞,sup0<|t|≤1∥[Aσt,[Aσt,φ(H)]]∥<∞,sup0<|t|≤1∥[Atσ,[Atσ,φ(Hσ)]]∥<∞
for every φ∈C0∞((-∞,m1-δ/2)) and for g≤g1.
Let us prove (6.30). The estimates (6.31) can be proved similarly.
To this end, letϕ be an almost analytic extension of φ satisfying
(6.32)|∂z̅ϕ(x+iy)|≤C|y|3,φ(H)=∫(z-H)-1dϕ(z),dϕ(z)=-1π∂∂z̅ϕ(z)dxdy.
It follows that
(6.33)[At[At,φ(H)]]=∫((z-H)-1[At[At,H]](z-H)-1+2(z-H)-1[At,H](z-H)-1[At,H](z-H)-1)dϕ(z).
We note that(6.34)∥(H+i)(H-z)-1∥≤C|Imz|,forz∈suppϕ.
We also have(6.35)sup0<|t|≤1∥∫(z-H)-1[At[At,H]](z-H)-1dϕ(z)∥≤sup0<|t|≤1∫∥[At[At,H]](H+i)-1∥∥(H+i)(z-H)-1∥|dϕ(z)||Imz|≤Csup0<|t|≤1∥[At,[At,H]](H+i)-1∥∫|dϕ(z)||Imz|2.
Therefore, combining Proposition3.8(b)(i) and (6.34) we obtain
(6.36)sup0<|t|≤1∥∫dϕ(z)(H-z)-1[At,H](H-z)-1[At,H](H-z)-1∥=sup0<|t|≤1∥∫(H-z)-1[At,H](H+i)-1(H+i)(H-z)-1[At,H](H+i)-1(H+i)(H-z)-1∥dϕ(z)≤C(∫|dϕ(z)||y|3)sup0<|t|≤1∥[At,H](H+i)-1∥2<∞.
Inequality (6.36) together with (6.35) yields (6.30), and H is locally of class C2(A) on (-∞,m1-δ/2) for g≤g1.
In a similar way it follows from Propositions3.8(b), 6.1, and 6.2 that H is locally of class C2(Aσ) and C2(Aσ) in (-∞,m1-δ/2) and that Hσ is locally of class C2(Aσ) in (-∞,m1-δ/2), for g≤g1. This ends the proof of Theorem 3.7.


## 7. Proof of Theorem3.4

By (3.91), ⋃n≥1((γ-ϵγ)2σn,(γ+ϵγ)σn)) is a covering by open sets of any compact subset of (E,m1-δ] and of the interval (E,m1-δ] itself. Theorem 3.4(i) and (ii) follow from [6, Theorems  0.1 and  0.2] and Theorems 3.7 and 3.10 above with gδ=g̃δ(2), where g̃δ(2) is given in Theorem 3.10. Theorem 3.4(iii) follows from [30, Theorem  25].



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