Spectral Theory for a Mathematical Model of the Weak Interaction—Part I: The Decay of the Intermediate Vector Bosons 𝑊^±

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.

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 Section 2, 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 bosons 𝑊⁺ 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 electron 𝑒⁻ and the positron 𝑒⁺, together with the associated neutrino 𝜈_𝑒 and antineutrino 𝜈_𝑒, the muons 𝜇⁻ and 𝜇⁺ together with the associated neutrino 𝜈_𝜇 and antineutrino 𝜈_𝜇, and the tau leptons 𝜏⁻ and 𝜏⁺ together with the associated neutrino 𝜈_𝜏 and antineutrino 𝜈_𝜏.

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 bosons 𝑊⁺ 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 bosons 𝑊^± 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)]) 𝐼=d³𝑥Ψ_𝑒(𝑥)𝛾^𝛼1−𝛾₅Ψ_𝜈𝑒(𝑥)𝑊_𝛼(𝑥)+d³𝑥Ψ_𝜈𝑒(𝑥)𝛾^𝛼1−𝛾₅Ψ_𝑒(𝑥)𝑊_𝛼(𝑥)^∗,(2.1) where 𝛾^𝛼, 𝛼=0,1,2,3 and 𝛾₅ are the Dirac matrices and Ψ_⋅(𝑥) and Ψ_⋅(𝑥) are the Dirac fields for 𝑒₋, 𝑒₊, 𝜈_𝑒, and 𝜈_𝑒.

We have Ψ_𝑒(𝑥)=12𝜋^3/2_𝑠=±1/2d³𝑝𝑏_𝑒,+(𝑝,𝑠)𝑢(𝑝,𝑠)√𝑝₀e^{𝑖𝑝⋅𝑥}+𝑏^∗𝑒,−(𝑝,𝑠)𝑣(𝑝,𝑠)√𝑝₀e^{−𝑖𝑝⋅𝑥},Ψ_𝑒(𝑥)=Ψ_𝑒(𝑥)^†𝛾⁰.(2.2) Here 𝑝₀=(|𝑝|²+𝑚^2𝑒)^1/2 where 𝑚_𝑒>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 operators 𝑏_𝑒,+(𝑝,𝑠) and 𝑏^∗𝑒,+(𝑝,𝑠) (resp., 𝑏_𝑒,−(𝑝,𝑠) and 𝑏^∗𝑒,−(𝑝,𝑠)) are the annihilation and creation operators for the electrons (resp., the positrons) satisfying the anticommutation relations (see below).

Similarly we define Ψ_𝜈𝑒(𝑥) and Ψ_𝜈𝑒(𝑥) by substituting the operators 𝑐_𝜈𝑒,±(𝑝,𝑠) and 𝑐^∗𝜈_𝑒,±(𝑝,𝑠) for 𝑏_𝑒,±(𝑝,𝑠) and 𝑏^∗𝑒,±(𝑝,𝑠) with 𝑝₀=|𝑝|. The operators 𝑐_𝜈𝑒,+(𝑝,𝑠) and 𝑐^∗𝜈_𝑒,+(𝑝,𝑠) (resp., 𝑐_𝜈𝑒,−(𝑝,𝑠) and 𝑐^∗𝜈_𝑒,−(𝑝,𝑠)) are the annihilation and creation operators for the neutrinos associated with the electrons (resp., the antineutrinos).

For the 𝑊_𝛼 fields we have (see [24, Section  5.3]) 𝑊_𝛼(𝑥)=12𝜋^3/2_{𝜆=−1,0,1}d³𝑘√2𝑘₀𝜖_𝛼(𝑘,𝜆)𝑎₊(𝑘,𝜆)e^{𝑖𝑘⋅𝑥}+𝜖^∗𝛼(𝑘,𝜆)𝑎^∗−(𝑘,𝜆)e^{−𝑖𝑘⋅𝑥}.(2.3) Here 𝑘₀=(|𝑘|²+𝑚^2𝑊)^1/2 where 𝑚_𝑊>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 𝜖_𝛼(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 𝑊_𝜇 fields, respectively.

Thus when one develops the interaction 𝐼 with respect to products of creation and annihilation operators, one gets a finite sum of terms associated with kernels of the form 𝜒𝑝₁𝜒𝑝₂𝜌(𝑘)𝑔̂𝑝₁+𝑝₂−𝑘,(2.4) 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 𝜈_𝑒, 𝜈_𝑒. ℓ=2 denotes the muons 𝜇⁻, 𝜇⁺ and the neutrinos 𝜈_𝜇 and 𝜈_𝜇, and ℓ=3 denotes the tau-leptons and the neutrinos 𝜈_𝜏 and 𝜈_𝜏.

Let 𝜉₁=(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(Σ_𝑗), 𝑗=1,2 is defined by (𝑓,𝑔)=_Σ𝑗𝑓(𝜉)𝑔(𝜉)d𝜉,𝑗=1,2.(2.5) Here _Σ1d𝜉=_{𝑠=+1/2,−1/2}d𝑝,_Σ2d𝜉=_{𝜆=0,1,−1}d𝑘,𝑝,𝑘∈ℝ³.(2.6)

The Hilbert space for the weak decay of the vector bosons 𝑊⁺ and 𝑊⁻ is the Fock space for leptons and bosons that we now describe.

Let 𝔖 be any separable Hilbert space. Let ⨂^𝑛𝑎𝔖 (resp., ⨂^𝑛𝑠𝔖) denote the antisymmetric (resp., symmetric) 𝑛th tensor power of 𝔖. The fermionic (resp., bosonic) Fock space over 𝔖, denoted by 𝔉_𝑎(𝔖) (resp., 𝔉_𝑠(𝔖)), is the direct sum 𝔉_𝑎(𝔖)=^∞_𝑛𝑛=0_𝑎𝔖resp.,𝔉_𝑠(𝔖)=^∞_𝑛𝑛=0_𝑠𝔖,(2.7) where ⨂^0𝑎𝔖=⨂^0𝑠𝔖≡ℂ. The state Ω=(1,0,0,…,0,…) denotes the vacuum state in 𝔉_𝑎(𝔖) and in 𝔉_𝑠(𝔖).

For every ℓ, 𝔉_ℓ 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, 𝔉_ℓ=⁴𝔉^𝑎𝐿²Σ₁ℓ=1,2,3.(2.8) We have 𝔉_ℓ=_{𝑞ℓ≥0,𝑞ℓ≥0,𝑟ℓ≥0,𝑟ℓ≥0}𝔉^{(𝑞ℓ,𝑞ℓ,𝑟ℓ,𝑟ℓ)ℓ}(2.9) with 𝔉^{(𝑞ℓ,𝑞ℓ,𝑟ℓ,𝑟ℓ)ℓ}=^𝑞ℓ_𝑎𝐿²Σ₁⊗⎛⎜⎜⎝^𝑞ℓ_𝑎𝐿²Σ₁⎞⎟⎟⎠⊗^𝑟ℓ_𝑎𝐿²Σ₁⊗⎛⎜⎜⎝^𝑟ℓ_𝑎𝐿²Σ₁⎞⎟⎟⎠.(2.10) Here 𝑞_ℓ (resp., 𝑞_ℓ) is the number of massive fermionic particle (resp., antiparticles) and 𝑟_ℓ (resp., 𝑟_ℓ) is the number of neutrinos (resp., antineutrinos). The vector Ω_ℓ is the associated vacuum state. The fermionic Fock space denoted by 𝔉_𝐿 for the leptons is then 𝔉_𝐿=³_ℓ=1𝔉_ℓ,(2.11) and Ω_𝐿=⨂^3ℓ=1Ω_ℓ is the vacuum state.

The bosonic Fock space for the vector bosons 𝑊⁺ and 𝑊⁻, denoted by 𝔉_𝑊, is then 𝔉_𝑊=𝔉_𝑠𝐿²Σ₂⊗𝔉_𝑠𝐿²Σ₂≃𝔉_𝑠𝐿²Σ₂⊕𝐿²Σ₂.(2.12) We have 𝔉_𝑊=_{𝑡≥0,𝑡≥0}𝔉^{(𝑡,𝑊𝑡)},(2.13) where 𝔉^{(𝑡,𝑊𝑡)}=(⨂^𝑡𝑠𝐿²(Σ₂))⊗(⨂^𝑡𝑠𝐿²(Σ₂)). Here 𝑡 (resp., 𝑡) is the number of bosons 𝑊⁻ (resp., 𝑊⁺). The vector Ω_𝑊 is the corresponding vacuum.

The Fock space for the weak decay of the vector bosons 𝑊⁺ and 𝑊⁻, denoted by 𝔉, is thus 𝔉=𝔉_𝐿⊗𝔉_𝑊,(2.14) and Ω=Ω_𝐿⊗Ω_𝑊 is the vacuum state.

For every ℓ∈{1,2,3} let 𝔇_ℓ denote the set of smooth vectors 𝜓_ℓ∈𝔉_ℓ for which 𝜓^{(𝑞ℓ,𝑞ℓ,𝑟ℓ,𝑟ℓ)ℓ} has a compact support and 𝜓^{(𝑞ℓ,𝑞ℓ,𝑟ℓ,𝑟ℓ)ℓ}=0 for all but finitely many (𝑞_ℓ,𝑞_ℓ,𝑟_ℓ,𝑟_ℓ). Let 𝔇_𝐿=⨂^3ℓ=1𝔇_ℓ.(2.15) Here ⨂ is the algebraic tensor product.

Let 𝔇_𝑊 denote the set of smooth vectors 𝜙∈𝔉_𝑊 for which 𝜙^(𝑡,𝑡) has a compact support and 𝜙^(𝑡,𝑡)=0 for all but finitely many (𝑡,𝑡).

Let 𝔇=𝔇_𝐿⊗𝔇_𝑊.(2.16) The set 𝔇 is dense in 𝔉.

Let 𝐴_ℓ be a self-adjoint operator in 𝔉_ℓ such that 𝔇_ℓ is a core for 𝐴_ℓ. Its extension to 𝔉_𝐿 is, by definition, the closure in 𝔉_𝐿 of the operator 𝐴₁⊗1₂⊗1₃ with domain 𝔇_𝐿 when ℓ=1, of the operator 1₁⊗𝐴₂⊗1₃ with domain 𝔇_𝐿 when ℓ=2, and of the operator 1₁⊗1₂⊗𝐴₃ with domain 𝔇_𝐿 when ℓ=3. Here 1_ℓ is the operator identity on 𝔉_ℓ.

The extension of 𝐴_ℓ to 𝔉_𝐿 is a self-adjoint operator for which 𝔇_𝐿 is a core and it can be extended to 𝔉. The extension of 𝐴_ℓ to 𝔉 is, by definition, the closure in 𝔉 of the operator 𝐴_ℓ⊗1_𝑊 with domain 𝔇, where 𝐴_ℓ is the extension of 𝐴_ℓ to 𝔉_𝐿. The extension of 𝐴_ℓ to 𝔉 is a self-adjoint operator for which 𝔇 is a core.

Let 𝐵 be a self-adjoint operator in 𝔉_𝑊 for which 𝔇_𝑊 is a core. The extension of the self-adjoint operator 𝐴_ℓ⊗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 𝐴_ℓ⊗𝐵 to 𝔉 is a self-adjoint operator for which 𝔇 is a core.

We now define the creation and annihilation operators.

For each ℓ=1,2,3, 𝑏_ℓ,𝜖(𝜉₁) (resp., 𝑏^∗ℓ,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 𝑏_ℓ,𝜖(𝜉₁) and 𝑏^∗ℓ,𝜖(𝜉₁) are defined as usually (see, e.g., [20, 26]; see also the detailed definitions in [27]).

Similarly, for each ℓ=1,2,3, 𝑐_ℓ,𝜖(𝜉₂) (resp., 𝑐^∗ℓ,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 𝑐_ℓ,𝜖(𝜉₂) and 𝑐^∗ℓ,𝜖(𝜉₂) are defined in a standard way, but with the additional requirements that for any ℓ, ℓ^, 𝜖 and 𝜖^, the operators 𝑏^♯ℓ,𝜖(𝜉₁) and 𝑐^♯ℓ_′,𝜖′(𝜉₂) anticommutes, where ♯ stands either for a ∗ or for no symbol (see the detailed definitions in [27]).

The operator 𝑎_𝜖(𝜉₃) (resp., 𝑎^∗𝜖(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 𝑏^♯ℓ,𝜖(𝜉₁) and 𝑐^♯ℓ_′,𝜖′(𝜉₂).

The following canonical anticommutation and commutation relations hold: 𝑏𝑐_{ℓ,𝜖ℓ,𝜖}𝜉₁,𝑏^∗ℓ′_,𝜖′𝜉^1=𝛿_ℓℓ^′𝛿_𝜖𝜖′𝛿𝜉₁−𝜉^1,𝑎_𝜖𝜉₂,𝑐^∗ℓ_′,𝜖′𝜉^2=𝛿_ℓℓ^′𝛿_𝜖𝜖′𝛿𝜉₂−𝜉^2,𝑏_ℓ,𝜖𝜉₃,𝑎^∗𝜖_′𝜉^3=𝛿_𝜖𝜖^′𝛿𝜉₃−𝜉^3,𝜉₁,𝑏_ℓ^′_,𝜖^′𝑎𝜉^1=𝑐_ℓ,𝜖𝜉₂,𝑐_ℓ^′_,𝜖^′𝜉^2=0,_𝜖𝑏𝜉₃,𝑎_𝜖^′𝜉^3=0,_ℓ,𝜖𝑏𝜉₁,𝑐_ℓ^′_,𝜖^′𝜉₂=𝑏_ℓ,𝜖𝜉₁,𝑐^∗ℓ_′,𝜖′𝜉₂=0,_ℓ,𝜖𝜉₁,𝑎_𝜖^′𝜉₃=𝑏_ℓ,𝜖𝜉₁,𝑎^∗𝜖_′𝜉₃=𝑐_ℓ,𝜖𝜉₂,𝑎_𝜖^′𝜉₃=𝑐_ℓ,𝜖𝜉₂,𝑎^∗𝜖_′𝜉₃=0,(2.17) where we used the notation 𝛿(𝜉_𝑗−𝜉^𝑗)=𝛿_𝜆𝜆^′𝛿(𝑘−𝑘^).

We recall that the following operators, with 𝜑∈𝐿²(Σ₁), 𝑏𝑏_ℓ,𝜖(𝜑)=_Σ1𝑏_ℓ,𝜖(𝜉)𝜑(𝜉)d𝜉,𝑐_ℓ,𝜖(𝜑)=_Σ1𝑐_ℓ,𝜖(𝜉)𝜑(𝜉)d𝜉,_∗ℓ,𝜖(𝜑)=_Σ1𝑏^∗ℓ,𝜖(𝜉)𝜑(𝜉)d𝜉,𝑐^∗ℓ,𝜖(𝜑)=_Σ1𝑐^∗ℓ,𝜖(𝜉)𝜑(𝜉)d𝜉(2.18) are bounded operators in 𝔉 such that ‖‖𝑏^♯ℓ,𝜖(𝜑)‖‖=‖‖𝑐^♯ℓ,𝜖(𝜑)‖‖=‖𝜑‖_𝐿²,(2.19) where 𝑏^♯ (resp., 𝑐^♯) is 𝑏 (resp., 𝑐) or 𝑏^∗ (resp., 𝑐^∗).

The operators 𝑏^♯ℓ,𝜖(𝜑) and 𝑐^♯ℓ,𝜖(𝜑) satisfy similar anticommutaion relations (see, e.g., [28]).

The free Hamiltonian 𝐻₀ is given by 𝐻₀=𝐻⁰⁽¹⁾+𝐻⁰⁽²⁾+𝐻⁰⁽³⁾=³_ℓ=10𝑥0200𝑑_𝜖=±𝑤^ℓ(1)𝜉₁𝑏^∗ℓ,𝜖𝜉₁𝑏_ℓ,𝜖𝜉₁d𝜉₁+³_ℓ=10𝑥0200𝑑_𝜖=±𝑤^ℓ(2)𝜉₂𝑐^∗ℓ,𝜖𝜉₂𝑐_ℓ,𝜖𝜉₂d𝜉₂+_𝜖=±𝑤⁽³⁾𝜉₃𝑎^∗𝜖𝜉₃𝑎_𝜖𝜉₃d𝜉₃,(2.20) where 𝑤𝑤^ℓ(1)_ℓ(2)𝜉₁=||𝑝₁||²+𝑚^2ℓ^1/2,with0<𝑚₁<𝑚₂<𝑚₃,𝑤₍₃₎𝜉₂=||𝑝₂||,𝜉₃=||𝑘||²+𝑚^2𝑊^1/2,(2.21) where 𝑚_𝑊 is the mass of the bosons 𝑊⁺ and 𝑊⁻ such that 𝑚_𝑊>𝑚₃.

The spectrum of 𝐻₀ is [0,∞) and 0 is a simple eigenvalue with Ω as eigenvector. The set of thresholds of 𝐻₀, denoted by 𝑇, is given by 𝑇=𝑝𝑚₁+𝑞𝑚₂+𝑟𝑚₃+𝑠𝑚_𝑊;(𝑝,𝑞,𝑟,𝑠)∈ℕ⁴,𝑝+𝑞+𝑟+𝑠≥1,(2.22) and each set [𝑡,∞), 𝑡∈𝑇, is a branch of absolutely continuous spectrum for 𝐻₀.

The interaction, denoted by 𝐻_𝐼, is given by 𝐻_𝐼=²_𝛼=1𝐻^𝐼(𝛼),(2.23) where 𝐻^𝐼(1)=³_ℓ=10𝑥0200𝑑_{𝜖≠𝜖′}𝐺^{(1)ℓ,𝜖,𝜖′}𝜉₁,𝜉₂,𝜉₃𝑏^∗ℓ,𝜖𝜉₁𝑐^∗ℓ,𝜖_′𝜉₂𝑎_𝜖𝜉₃d𝜉₁d𝜉₂d𝜉₃+³_ℓ=10𝑥0200𝑑_{𝜖≠𝜖′}𝐻_𝐼(2)𝐺^{(1)ℓ,𝜖,𝜖}_′(𝜉₁,𝜉₂,𝜉₃)𝑎^∗𝜖𝜉₃𝑐_ℓ,𝜖^′𝜉₂𝑏_ℓ,𝜖𝜉₁d𝜉₁d𝜉₂d𝜉₃,=³_ℓ=1_{𝜖≠𝜖′}𝐺^{(2)ℓ,𝜖,𝜖′}𝜉₁,𝜉₂,𝜉₃𝑏^∗ℓ,𝜖𝜉₁𝑐^∗ℓ,𝜖_′𝜉₂𝑎^∗𝜖𝜉₃d𝜉₁d𝜉₂d𝜉₃+³_ℓ=1_{𝜖≠𝜖′}𝐺^{(2)ℓ,𝜖,𝜖}_′𝜉₁,𝜉₂,𝜉₃𝑎_𝜖𝜉₃𝑐_ℓ,𝜖^′𝜉₂𝑏_ℓ,𝜖𝜉₁d𝜉₁d𝜉₂d𝜉₃.(2.24) The kernels 𝐺^{(2)ℓ,𝜖,𝜖}_′(⋅,⋅,⋅), 𝛼=1,2, are supposed to be functions.

The total Hamiltonian is then 𝐻=𝐻₀+𝑔𝐻_𝐼,𝑔>0,(2.25) where 𝑔 is a coupling constant.

The operator 𝐻^𝐼(1) describes the decay of the bosons 𝑊⁺ and 𝑊⁻ into leptons. Because of 𝐻^𝐼(2) the bare vacuum will not be an eigenvector of the total Hamiltonian for every 𝑔>0 as we expect from the physics.

Every kernel 𝐺_{ℓ,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 kernels 𝐺^{(𝛼)ℓ,𝜖,𝜖}_′ are supposed to satisfy the following hypothesis.

Hypothesis 2.1. For 𝛼=1,2, ℓ=1,2,3, 𝜖,𝜖^=±, we assume 𝐺^{(𝛼)ℓ,𝜖,𝜖}_′𝜉₁,𝜉₂,𝜉₃∈𝐿²Σ₁×Σ₁×Σ₂.(2.26)

Remark 2.2. A similar model can be written down for the weak decay of pions 𝜋⁻ 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 by 𝔉^, is isomorphic to 𝔉^𝐿⊗𝔉_𝑊, with 𝔉^𝐿=³_ℓ=1𝔉_ℓ𝔉^ℓ,=²_𝑎𝐿²Σ₁⊗²_𝑎𝐿²ℝ³.(2.27) The free Hamiltonian, now denoted by 𝐻^0, is then given by 𝐻^0=³_ℓ=1_𝜖=±𝑤^ℓ(1)𝜉₁𝑏^∗ℓ,𝜖𝜉₁𝑏_ℓ,𝜖𝜉₁d𝜉₁+³_ℓ=1_𝜖=±_ℝ3||𝑝₂||𝑐^∗ℓ,𝜖𝑝₂𝑐_ℓ,𝜖𝑝₂d𝑝₂+_𝜖=±𝑤⁽³⁾𝜉₃𝑎^∗𝜖𝜉₃𝑎_𝜖𝜉₃d𝜉₃,(2.28) and the interaction, now denoted by 𝐻^𝐼, is the one obtained from 𝐻_𝐼 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^=𝐻^0+𝑔𝐻^𝐼. The results obtained in this paper for 𝐻 hold true for 𝐻^ with obvious modifications.

Under Hypothesis 2.1 a well-defined operator on 𝔇 corresponds to the formal interaction 𝐻_𝐼 as it follows.

The formal operator 𝐺^{(1)ℓ,𝜖,𝜖′}𝜉₁,𝜉₂,𝜉₃𝑏^∗ℓ,𝜖𝜉₁𝑐^∗ℓ,𝜖_′𝜉₂𝑎_𝜖𝜉₃d𝜉₁d𝜉₂d𝜉₃(2.29) is defined as a quadratic form on (𝔇_ℓ⊗𝔇_𝑊)×(𝔇_ℓ⊗𝔇_𝑊) as 𝑐^ℓ,𝜖′𝜉₂𝑏_ℓ,𝜖𝜉₁𝜓,𝐺^{(1)ℓ,𝜖,𝜖}_′𝑎_𝜖𝜉₃𝜙d𝜉₁d𝜉₂d𝜉₃,(2.30) where 𝜓, 𝜙∈𝔇_ℓ⊗𝔇_𝑊.

By mimicking the proof of [29, Theorem X.44], we get a closed operator, denoted by 𝐻^{(1)𝐼,ℓ,𝜖,𝜖}_′, associated with the quadratic form such that it is the unique operator in 𝔉_ℓ⊗𝔉_𝑊 such that 𝔇_ℓ⊗𝔇_𝑊⊂𝒟(𝐻^{(1)𝐼,ℓ,𝜖,𝜖}_′) is a core for 𝐻^{(1)𝐼,ℓ,𝜖,𝜖}_′ and 𝐻^{(1)𝐼,ℓ,𝜖,𝜖}_′=𝐺^{(1)ℓ,𝜖,𝜖}_′𝜉₁,𝜉₂,𝜉₃𝑏^∗ℓ,𝜖𝜉₁𝑐^∗ℓ,𝜖_′𝜉₂𝑎_𝜖𝜉₃d𝜉₁d𝜉₂d𝜉₃(2.31) as quadratic forms on (𝔇_ℓ⊗𝔇_𝑊)×(𝔇_ℓ⊗𝔇_𝑊).

Similarly for the operator (𝐻^{(1)𝐼,ℓ,𝜖,𝜖}_′)^∗, we have the equality as quadratic forms 𝐻^{(1)𝐼,ℓ,𝜖,𝜖′}^∗=𝐺^{(1)ℓ,𝜖,𝜖}_′(𝜉₁,𝜉₂,𝜉₃)𝑎^∗𝜖𝜉₃𝑐_ℓ,𝜖^′𝜉₂𝑏_ℓ,𝜖𝜉₁d𝜉₁d𝜉₂d𝜉₃.(2.32)

Again, there exists two closed operators 𝐻^{(2)𝐼,ℓ,𝜖,𝜖}_′ and (𝐻^{(2)𝐼,ℓ,𝜖,𝜖}_′)^∗ such that 𝔇_ℓ⊗𝔇_𝑊⊂𝒟(𝐻^{(2)𝐼,ℓ,𝜖,𝜖}_′), 𝔇_ℓ⊗𝔇_𝑊⊂𝒟((𝐻^{(2)𝐼,ℓ,𝜖,𝜖}_′)^∗), and 𝔇_ℓ⊗𝔇_𝑊 is a core for 𝐻^{(2)𝐼,ℓ,𝜖,𝜖}_′ and (𝐻^{(2)𝐼,ℓ,𝜖,𝜖}_′)^∗ and such that 𝐻𝐻^{(2)𝐼,ℓ,𝜖,𝜖}_{′(2)𝐼,ℓ,𝜖,𝜖′}=𝐺^{(2)ℓ,𝜖,𝜖}_′𝜉₁,𝜉₂,𝜉₃𝑏^∗ℓ,𝜖𝜉₁𝑐^∗ℓ,𝜖_′𝜉₂𝑎^∗𝜖𝜉₃d𝜉₁d𝜉₂d𝜉₃,^∗=𝐺^{(2)ℓ,𝜖,𝜖}_′𝜉₁,𝜉₂,𝜉₃𝑎_𝜖𝜉₃𝑐_ℓ,𝜖^′𝜉₂𝑏_ℓ,𝜖𝜉₁d𝜉₁d𝜉₂d𝜉₃(2.33) as quadratic forms on (𝔇_ℓ⊗𝔇_𝑊)×(𝔇_ℓ⊗𝔇_𝑊).

We will still denote by 𝐻^{(𝛼)𝐼,ℓ,𝜖,𝜖}_′ and (𝐻^{(𝛼)𝐼,ℓ,𝜖,𝜖}_′)^∗ (𝛼=1,2) their extensions to 𝔉. The set 𝔇 is then a core for 𝐻^{(𝛼)𝐼,ℓ,𝜖,𝜖}_′ and (𝐻^{(𝛼)𝐼,ℓ,𝜖,𝜖}_′)^∗.

Thus 𝐻=𝐻₀+𝑔_3𝛼=1,2_ℓ=1_{𝜖≠𝜖′}𝐻^{(𝛼)𝐼,ℓ,𝜖,𝜖′}+𝐻^{(2)𝐼,ℓ,𝜖,𝜖}_′^∗(2.34) is a symmetric operator defined on 𝔇.

We now want to prove that 𝐻 is essentially self-adjoint on 𝔇 by showing that 𝐻^{(𝛼)𝐼,ℓ,𝜖,𝜖}_′ and (𝐻^{(𝛼)𝐼,ℓ,𝜖,𝜖}_′)^∗ are relatively 𝐻₀-bounded.

Once again, as above, for almost every 𝜉₃∈Σ₂, there exists closed operators in 𝔉_𝐿, denoted by 𝐵^{(𝛼)ℓ,𝜖,𝜖}_′(𝜉₃) and (𝐵^{(𝛼)ℓ,𝜖,𝜖}_′(𝜉₃))^∗ such that 𝐵𝐵^{(1)ℓ,𝜖,𝜖}_{′(1)ℓ,𝜖,𝜖′}𝜉₃=−𝐺^{(1)ℓ,𝜖,𝜖}_′(𝜉₁,𝜉₂,𝜉₃)𝑏_ℓ,𝜖𝜉₁𝑐_ℓ,𝜖^′𝜉₂d𝜉₁d𝜉₂,𝐵𝜉_{3(2)ℓ,𝜖,𝜖′}^∗=𝐺^{(1)ℓ,𝜖,𝜖}_′𝜉₁,𝜉₂,𝜉₃𝑏^∗ℓ,𝜖𝜉₁𝑐^∗ℓ,𝜖_′𝜉₂d𝜉₁d𝜉₂,𝐵_{(2)ℓ,𝜖,𝜖′}𝜉₃=𝐺^{(2)ℓ,𝜖,𝜖}_′𝜉₁,𝜉₂,𝜉₃𝑏^∗ℓ,𝜖𝜉₁𝑐^∗ℓ,𝜖_′𝜉₂d𝜉₁d𝜉₂,𝜉₃^∗=−𝐺^{(2)ℓ,𝜖,𝜖}_′(𝜉₁,𝜉₂,𝜉₃)𝑏_ℓ,𝜖𝜉₁𝑐_ℓ,𝜖^′𝜉₂d𝜉₁d𝜉₂(2.35) as quadratic forms on 𝔇_ℓ×𝔇_ℓ.

We have that 𝔇_ℓ⊂𝒟(𝐵^{(𝛼)ℓ,𝜖,𝜖}_′(𝜉₃)) (resp., 𝔇_ℓ⊂𝒟((𝐵^{(𝛼)ℓ,𝜖,𝜖}_′(𝜉₃))^∗) is a core for 𝐵^{(𝛼)ℓ,𝜖,𝜖}_′(𝜉₃) (resp., for (𝐵^{(𝛼)ℓ,𝜖,𝜖}_′(𝜉₃))^∗). We still denote by 𝐵^{(𝛼)ℓ,𝜖,𝜖}_′(𝜉₃)) and (𝐵^{(𝛼)ℓ,𝜖,𝜖}_′(𝜉₃))^∗) their extensions to 𝔉_𝐿.

It then follows that the operator 𝐻_𝐼 with domain 𝔇 is symmetric and can be written in the following form: 𝐻_𝐼=_3𝛼=1,2_ℓ=1_{𝜖≠𝜖′}𝐻^{(𝛼)𝐼,ℓ,𝜖,𝜖′}+𝐻^{(𝛼)𝐼,ℓ,𝜖,𝜖}_′^∗=_3𝛼=1,2_ℓ=1_{𝜖≠𝜖′}𝐵^{(𝛼)ℓ,𝜖,𝜖′}𝜉₃⊗𝑎^∗𝜖𝜉₃d𝜉₃+_3𝛼=1,2_ℓ=1_{𝜖≠𝜖′}𝐵^{(𝛼)ℓ,𝜖,𝜖′}𝜉₃^∗⊗𝑎_𝜖𝜉₃d𝜉₃.(2.36) Let 𝑁_ℓ denote the operator number of massive leptons ℓ in 𝔉_ℓ, that is, 𝑁_ℓ=_𝜖𝑏^∗ℓ,𝜖𝜉₁𝑏_ℓ,𝜖𝜉₁d𝜉₁.(2.37) The operator 𝑁_ℓ is a positive self-adjoint operator in 𝔉_ℓ. We still denote by 𝑁_ℓ its extension to 𝔉_𝐿. The set 𝔇_𝐿 is a core for 𝑁_ℓ.

We then have the following.

Proposition 2.4. For almost every 𝜉₃∈Σ₂, 𝒟(𝐵^{(𝛼)ℓ,𝜖,𝜖}_′(𝜉₃)), 𝒟((𝐵^{(𝛼)ℓ,𝜖,𝜖}_′(𝜉₃))^∗)⊃𝒟(𝑁^ℓ1/2), and for Φ∈𝒟(𝑁^ℓ1/2)⊂𝔉_𝐿 one has ‖‖‖𝐵‖‖𝐵^{(𝛼)ℓ,𝜖,𝜖}_{′(𝛼)ℓ,𝜖,𝜖′}(𝜉₃)Φ‖‖_𝔉𝐿≤‖‖𝐺^{(𝛼)ℓ,𝜖,𝜖}_′(⋅,⋅,𝜉₃)‖‖_{𝐿2(Σ1×Σ1)}‖‖𝑁^ℓ1/2Φ‖‖_𝔉𝐿,𝜉₃^∗Φ‖‖‖_𝔉𝐿≤‖‖𝐺^{(𝛼)ℓ,𝜖,𝜖}_′(⋅,⋅,𝜉₃)‖‖_{𝐿2(Σ1×Σ1)}‖‖𝑁^ℓ1/2Φ‖‖_𝔉𝐿.(2.38)

Proof. The estimates (2.38) are examples of 𝑁_𝜏 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 𝐻^(3)0,𝜖=𝑤⁽³⁾𝜉₃𝑎^∗𝜖𝜉₃𝑎_𝜖𝜉₃d𝜉₃.(2.39) Then 𝐻^(3)0,𝜖 is a self-adjoint operator in 𝔉_𝑊, and 𝔇_𝑊 is a core for 𝐻^(3)0,𝜖.

We get the following.

Proposition 2.5. One has ‖‖‖𝐵_{(𝛼)ℓ,𝜖,𝜖′}𝜉₃^∗⊗𝑎_𝜖𝜉₃d𝜉₃Ψ‖‖‖²≤⎛⎜⎜⎜⎝_{Σ1×Σ1×Σ2}|||𝐺_{(𝛼)ℓ,𝜖,𝜖′}𝜉₁,𝜉₂,𝜉₃|||²𝑤⁽³⁾𝜉₃d𝜉₁d𝜉₂d𝜉₃⎞⎟⎟⎟⎠‖‖‖𝑁_ℓ+1^1/2⊗𝐻^(3)0,𝜖^1/2Ψ‖‖‖²,(2.40)‖‖‖𝐵_{(𝛼)ℓ,𝜖,𝜖′}(𝜉₃)⊗𝑎^∗𝜖(𝜉₃)d𝜉₃Ψ‖‖‖²≤⎛⎜⎜⎜⎝_{Σ1×Σ1×Σ2}|||𝐺_{(𝛼)ℓ,𝜖,𝜖′}𝜉₁,𝜉₂,𝜉₃|||²𝑤⁽³⁾𝜉₃d𝜉₁d𝜉₂d𝜉₃⎞⎟⎟⎟⎠‖‖‖𝑁_ℓ+1^1/2⊗𝐻^(3)0,𝜖^1/2Ψ‖‖‖²+_{Σ1×Σ1×Σ2}|||𝐺_{(𝛼)ℓ,𝜖,𝜖′}𝜉₁,𝜉₂,𝜉₃|||²d𝜉₁d𝜉₂d𝜉₃𝜂‖‖𝑁^ℓ+1^1/2⊗1Ψ‖‖²+41𝜂‖Ψ‖²(2.41) for every Ψ∈𝒟(𝐻₀) and every 𝜂>0.

Proof. Suppose that Ψ∈𝒟(𝑁^ℓ1/2)⊗𝒟((𝐻^(3)0,𝜖)^1/2). Let Ψ_𝜖𝜉₃=𝑤⁽³⁾𝜉₃^1/2𝑁^ℓ+1^1/2⊗𝑎_𝜖𝜉₃Φ.(2.42) We have 𝐵^{(𝛼)ℓ,𝜖,𝜖′}𝜉₃^∗⊗𝑎_𝜖𝜉₃d𝜉₃Ψ=_Σ21𝑤⁽³⁾𝜉₃^1/2𝐵^{(𝛼)ℓ,𝜖,𝜖′}𝜉₃^∗𝑁_ℓ+1^−1/2⊗1Ψ_𝜖𝜉₃d𝜉₃.(2.43) Therefore, for Ψ∈𝒟(𝑁^ℓ1/2)⊗𝒟((𝐻^(3)0,𝜖)^1/2), (2.40) follows from Proposition 2.4.
We now have ‖‖‖𝐵_{(𝛼)ℓ,𝜖,𝜖′}(𝜉₃)⊗𝑎^∗𝜖(𝜉₃)Ψd𝜉₃‖‖‖^2𝔉=𝐵^{(𝛼)ℓ,𝜖,𝜖}_′𝜉₃⊗𝑎_𝜖𝜉^3Ψ,𝐵^{(𝛼)ℓ,𝜖,𝜖}_′𝜉^3⊗𝑎_𝜖𝜉₃Ψd𝜉₃d𝜉^3+‖‖𝐵^{(𝛼)ℓ,𝜖,𝜖}_′(𝜉₃)⊗1Ψ‖‖²d𝜉₃,(2.44)_Σ2×Σ2𝐵^{(𝛼)ℓ,𝜖,𝜖′}𝜉₃⊗𝑎_𝜖𝜉^3Ψ,𝐵^{(𝛼)ℓ,𝜖,𝜖}_′𝜉^3⊗𝑎_𝜖𝜉₃Ψd𝜉₃d𝜉^3=_Σ2×Σ21𝑤⁽³⁾𝜉₃^1/2𝑤⁽³⁾𝜉^3^1/2×𝐵^{(𝛼)ℓ,𝜖,𝜖}_′𝜉₃𝑁_ℓ+1^−1/2⊗1Ψ_𝜖𝜉^3,𝐵^{(𝛼)ℓ,𝜖,𝜖}_′𝜉^3𝑁_ℓ+1^−1/2⊗1Ψ_𝜖𝜉₃d𝜉₃d𝜉^3≤_Σ21𝑤⁽³⁾𝜉₃^1/2‖‖𝐵^{(𝛼)ℓ,𝜖,𝜖}_′𝜉₃𝑁_ℓ+1^−1/2‖‖_𝔉𝐿‖‖Ψ_𝜖𝜉₃‖‖d𝜉₃²≤_{Σ1×Σ1×Σ2}||𝐺^(𝛼)𝜉₁,𝜉₂,𝜉₃||²𝑤⁽³⁾𝜉₃d𝜉₁d𝜉₂d𝜉₃‖‖‖𝑁^ℓ+1^1/2⊗𝐻^(3)0,𝜖^1/2Ψ‖‖‖².(2.45) Furthermore _Σ2‖‖𝐵^{(𝛼)ℓ,𝜖,𝜖}_′𝜉₃⊗1Ψ‖‖²d𝜉₃=_Σ2‖‖𝐵^{(𝛼)ℓ,𝜖,𝜖}_′𝜉₃𝑁_ℓ+1^−1/2⊗1𝑁_ℓ+1^1/2⊗1Ψ‖‖²d𝜉₃≤_{Σ1×Σ1×Σ2}|||𝐺_{(𝛼)ℓ,𝜖,𝜖′}𝜉₁,𝜉₂,𝜉₃|||²d𝜉₁d𝜉₂d𝜉₃𝜂‖‖𝑁^ℓ+1Ψ‖‖²+41𝜂‖Ψ‖²(2.46) for every 𝜂>0.
By (2.40), (2.45), and (2.46), we finally get (2.41) for every Ψ∈𝒟(𝑁^ℓ1/2)⊗𝒟(𝐻^(3)0,𝜖). It then follows that (2.40) and (2.41) are verified for every Ψ∈𝒟(𝐻₀).

We now prove that 𝐻 is a self-adjoint operator in 𝔉 for 𝑔 sufficiently small.

Theorem 2.6. Let 𝑔₁>0 be such that 3𝑔²¹𝑚_𝑊𝑚1²¹+1_3𝛼=1,2_ℓ=1_{𝜖≠𝜖′}‖‖𝐺^{(𝛼)ℓ,𝜖,𝜖}_′‖‖^2𝐿_{2(Σ1×Σ1×Σ2)}<1.(2.47) 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. Let Ψ be in 𝔇. We have ‖‖𝐻_𝐼Ψ‖‖²≤12₃_𝛼=1,2_ℓ=1_{𝜖≠𝜖′}‖‖‖𝐵^{(𝛼)ℓ,𝜖,𝜖′}𝜉₃^∗⊗𝑎_𝜖𝜉₃Ψd𝜉₃‖‖‖²+‖‖‖𝐵^{(𝛼)ℓ,𝜖,𝜖}_′𝜉₃⊗𝑎^∗𝜖𝜉₃Ψd𝜉₃‖‖‖².(2.48) Note that ‖‖𝑁_ℓ‖‖𝐻^(3)0,𝜖Ψ‖‖≤‖‖𝐻⁰⁽³⁾Ψ‖‖≤‖‖𝐻₀Ψ‖‖,Ψ‖‖≤1𝑚_ℓ‖‖𝐻_0,ℓΨ‖‖≤1𝑚₁‖‖𝐻_0,ℓΨ‖‖≤1𝑚₁‖‖𝐻₀Ψ‖‖,(2.49) where 𝐻_0,ℓ=_𝜖𝑤^ℓ(1)𝜉₁𝑏^∗ℓ,𝜖𝜉₁𝑏_ℓ,𝜖𝜉₁d𝜉₁+_𝜖𝑤^ℓ(2)𝜉₂𝑐^∗ℓ,𝜖𝜉₂𝑐_ℓ,𝜖𝜉₂d𝜉₂.(2.50) We further note that ‖‖‖𝑁_ℓ+1^1/2⊗𝐻^(3)0,𝜖^1/2Ψ‖‖‖²≤12𝑚1²¹+1‖‖𝐻₀Ψ‖‖²+𝛽2𝑚²¹‖‖𝐻₀Ψ‖‖²+12+81𝛽‖Ψ‖²(2.51) for 𝛽>0, and 𝜂‖‖𝑁_ℓ+1⊗1Ψ‖‖²+14𝜂‖Ψ‖²≤𝜂𝑚²¹‖‖𝐻₀Ψ‖‖²+𝜂𝛽𝑚²¹‖‖𝐻₀Ψ‖‖²+𝜂1+14𝛽‖Ψ‖²+14𝜂‖Ψ‖².(2.52) Combining (2.48) with (2.40), (2.41), (2.51), and (2.52) we get for 𝜂>0, 𝛽>0‖‖𝐻_𝐼Ψ‖‖²≤6⎛⎜⎜⎝_3𝛼=1,2_ℓ=1_{𝜖≠𝜖′}‖‖𝐺^{(𝛼)ℓ,𝜖,𝜖}_′‖‖²×1⎞⎟⎟⎠2𝑚_𝑊𝑚1²¹+1‖‖𝐻₀Ψ‖‖²+𝛽2𝑚_𝑊𝑚²¹‖‖𝐻₀Ψ‖‖²+1+12⎛⎜⎜⎝_3𝛼=1,22𝑚_𝑊1+41𝛽‖Ψ‖²_ℓ=1_{𝜖≠𝜖′}‖‖𝐺^{(𝛼)ℓ,𝜖,𝜖}_′‖‖²⎞⎟⎟⎠𝜂𝑚²¹(1+𝛽)‖‖𝐻₀Ψ‖‖²+𝜂1+14𝛽+41𝜂‖Ψ‖²,(2.53) by noting _{Σ1×Σ1×Σ2}||𝐺_{ℓ,𝜖,𝜖′}𝜉₁,𝜉₂,𝜉₃||²𝑤⁽³⁾𝜉₃d𝜉₁d𝜉₂d𝜉₃≤1𝑚_𝑊‖‖𝐺^{(𝛼)ℓ,𝜖,𝜖}_′‖‖².(2.54) 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 kernels 𝐺^{(𝛼)ℓ,𝜖,𝜖}_′.

Hypothesis 3.1. (i) For 𝛼=1,2, ℓ=1,2,3, 𝜖,𝜖^=±, _{Σ1×Σ1×Σ2}|||𝐺_{(𝛼)ℓ,𝜖,𝜖′}𝜉₁,𝜉₂,𝜉₃|||²||𝑝₂||²d𝜉₁d𝜉₂d𝜉₃<∞.(3.1)
(ii) There exists 𝐶>0 such that for 𝛼=1,2, ℓ=1,2,3, 𝜖,𝜖^=±, _{Σ1×{|𝑝2|≤𝜎}×Σ2}|||𝐺_{(𝛼)ℓ,𝜖,𝜖′}𝜉₁,𝜉₂,𝜉₃|||²d𝜉₁d𝜉₂d𝜉₃^1/2≤𝐶𝜎².(3.2)
(iii) For 𝛼=1,2, ℓ=1,2,3, 𝜖,𝜖^=±, and 𝑖,𝑗=1,2,3(iii.a)_{Σ1×Σ1×Σ2}|||𝑝₂(⋅∇_𝑝2𝐺^{(𝛼)ℓ,𝜖,𝜖}_′(𝜉¹,𝜉₂,𝜉₃)|||²d𝜉₁d𝜉₂d𝜉₃<∞,iii.b)_{Σ1×Σ1×Σ2}𝑝^22,𝑖𝑝^22,𝑗|||||𝜕²𝐺^{(𝛼)ℓ,𝜖,𝜖}_′𝜕𝑝_2,𝑖𝜕𝑝_2,𝑗(𝜉₁,𝜉₂,𝜉₃)|||||²d𝜉₁d𝜉₂d𝜉₃<∞.(3.3)
(iv) There exists Λ>𝑚₁ such that for 𝛼=1,2, ℓ=1,2,3, 𝜖,𝜖^=±, 𝐺^{(𝛼)ℓ,𝜖,𝜖}_′𝜉₁,𝜉₂,𝜉₃=0if||𝑝₂||≥Λ.(3.4)

Remark 3.2. Hypothesis 3.1(ii) is nothing but an infrared regularization of the kernels 𝐺^{(𝛼)ℓ,𝜖,𝜖}_′. In order to satisfy this hypothesis it is, for example, sufficient to suppose that 𝐺^{(𝛼)ℓ,𝜖,𝜖}_′𝜉₁,𝜉₂,𝜉₃=||𝑝₂||^1/2𝐺^{(𝛼)ℓ,𝜖,𝜖}_′𝜉₁,𝜉₂,𝜉₃,(3.5) where 𝐺^{(𝛼)ℓ,𝜖,𝜖}_′ is a smooth function of (𝑝₁,𝑝₂,𝑝₃) in the Schwartz space.
Hypothesis 3.1(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 for 𝐻 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 kernels 𝐺^{(𝛼)ℓ,𝜖,𝜖}_′ 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 𝜎(𝐻)=𝜎_ac(𝐻)=[inf𝜎(𝐻),∞)(3.6) with inf𝜎(𝐻)≤0.

According to Theorem 3.3 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.

Let 𝑎 denote the following operator in 𝐿²(Σ₁): 𝑎=12𝑝²⋅𝑖∇_𝑝2+𝑖∇_𝑝2⋅𝑝₂.(3.7) The operator 𝑎 is essentially self-adjoint on 𝐶^∞0(ℝ³,ℂ²). Its second quantized version dΓ(𝑎) is a self-adjoint operator in 𝔉_𝑎(𝐿²(Σ₁)). From the definition (2.8) of the space 𝔉_ℓ, the following operator in 𝔉_ℓ𝐴_ℓ=1⊗1⊗dΓ(𝑎)⊗1+1⊗1⊗1⊗dΓ(𝑎)(3.8) is essentially self-adjoint on 𝔇_𝐿.

Let now 𝐴 be the following operator in 𝔉_𝐿: 𝐴=𝐴₁⊗1₂⊗1₃+1₁⊗𝐴₂⊗1₃+1₁⊗1₂⊗𝐴₃.(3.9) Then 𝐴 is essentially self-adjoint on 𝔇_𝐿.

We will denote again by 𝐴 its extension to 𝔉. Thus 𝐴 is essentially self-adjoint on 𝔇 and we still denote by 𝐴 its closure.

We also set ⟨𝐴⟩=1+𝐴²^1/2.(3.10)

We then have the following.

Theorem 3.4. Suppose that the kernels 𝐺^{(𝛼)ℓ,𝜖,𝜖}_′ satisfy Hypotheses 2.1 and 3.1. For any 𝛿>0 satisfying 0<𝛿<𝑚₁ there exists 0<𝑔_𝛿≤𝑔₂ such that, for 0<𝑔≤𝑔_𝛿, the following points are satisfied. (i)The spectrum of 𝐻 in (inf𝜎(𝐻),𝑚₁−𝛿] is purely absolutely continuous.(ii)Limiting absorption principle.
For every 𝑠>1/2 and 𝜑, 𝜓 in 𝔉, the limits lim_𝜀→0𝜑,⟨𝐴⟩^−𝑠(𝐻−𝜆±𝑖𝜀)⟨𝐴⟩^−𝑠𝜓(3.11) exist uniformly for 𝜆 in any compact subset of (inf𝜎(𝐻),𝑚₁−𝛿]. (iii) Pointwise decay in time.
Suppose 𝑠∈(1/2,1) and 𝑓∈𝐶^∞0(ℝ) with supp𝑓⊂(inf𝜎(𝐻),𝑚₁−𝛿). Then ‖‖⟨𝐴⟩^−𝑠e^{−𝑖𝑡𝐻}𝑓(𝐻)⟨𝐴⟩^−𝑠‖‖=𝒪𝑡^1/2−𝑠(3.12) as 𝑡→∞.

The proof of Theorem 3.4 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.

Let 𝜒₀(⋅), 𝜒_∞(⋅)∈𝐶^∞(ℝ,[0,1]) with 𝜒₀=1 on (−∞,1], 𝜒_∞=1 on [2,∞) and 𝜒₀₂+𝜒_∞2=1.

For 𝜎>0 we set 𝜒_𝜎(𝑝)=𝜒₀𝜒_𝜎||𝜎𝑝||,(𝑝)=𝜒_∞𝜒_𝜎||𝜎𝑝||,(𝑝)=1−𝜒_𝜎(𝑝),(3.13) where 𝑝∈ℝ³.

The operator 𝐻_𝐼,𝜎 is the interaction given by (2.23) and (2.24) and associated with the kernels 𝜒^𝜎(𝑝₂)𝐺^{(𝛼)ℓ,𝜖,𝜖}_′(𝜉₁,𝜉₂,𝜉₃). We then set 𝐻_𝜎∶=𝐻₀+𝑔𝐻_𝐼,𝜎.(3.14)

Let ΣΣ_1,𝜎=Σ₁∩𝑝₂,𝑠₂;||𝑝₂||<𝜎_𝜎1,𝔉=Σ₁∩𝑝₂,𝑠₂;||𝑝₂||≥𝜎_ℓ,2,𝜎𝔉=𝔉_𝑎𝐿²Σ_1,𝜎⊗𝔉_𝑎𝐿²Σ_1,𝜎_𝜎ℓ,2,𝔉=𝔉_𝑎𝐿²Σ^𝜎1⊗𝔉_𝑎𝐿²Σ^𝜎1_ℓ,2,𝔉_ℓ,1=𝔉_ℓ,2,𝜎⊗𝔉^𝜎ℓ,2,=²𝔉^𝑎𝐿²Σ₁.(3.15) The space 𝔉_ℓ,1 is the Fock space for the massive leptons ℓ, and 𝔉_ℓ,2 is the Fock space for the neutrinos and antineutrinos ℓ.

Set 𝔉^𝜎ℓ=𝔉_ℓ,1⊗𝔉^𝜎ℓ,2𝔉_ℓ,𝜎=𝔉_ℓ,2,𝜎.(3.16) We have 𝔉_ℓ≃𝔉^𝜎ℓ⊗𝔉_ℓ,𝜎.(3.17) Set 𝔉^𝜎𝐿=³_ℓ=1𝔉^𝜎ℓ𝔉_𝐿,𝜎=³_ℓ=1𝔉_ℓ,𝜎.(3.18) We have 𝔉_𝐿≃𝔉^𝜎𝐿⊗𝔉_𝐿,𝜎.(3.19) Set 𝔉^𝜎=𝔉^𝜎𝐿⊗𝔉_𝑊.(3.20) We have 𝔉≃𝔉_𝐿,𝜎⊗𝔉^𝜎.(3.21) Set 𝐻⁰⁽¹⁾=³_ℓ=1_𝜖=±𝑤^ℓ(1)𝜉₁𝑏^∗ℓ,𝜖𝜉₁𝑏_ℓ,𝜖𝜉₁𝐻₀₍₂₎d𝜉₁,=³_ℓ=1_𝜖=±𝑤^ℓ(2)𝜉₂𝑐^∗ℓ,𝜖𝜉₂𝑐_ℓ,𝜖𝜉₂𝐻₀₍₃₎d𝜉₂,=_𝜖=±𝑤⁽³⁾𝜉₃𝑎^∗𝜖𝜉₃𝑎_𝜖𝜉₃𝐻_0(2)𝜎d𝜉₃,=³_ℓ=1_𝜖=±_|𝑝2|>𝜎𝐻_(2)0,𝜎𝑤^ℓ(2)𝜉₂𝑐^∗ℓ,𝜖𝜉₂𝑐_ℓ,𝜖𝜉₂d𝜉₂,=³_ℓ=1_𝜖=±_{|𝑝2|≤𝜎}𝑤^ℓ(2)𝜉₂𝑐^∗ℓ,𝜖𝜉₂𝑐_ℓ,𝜖𝜉₂d𝜉₂.(3.22) We have on 𝔉^𝜎⊗𝔉_𝜎𝐻⁰⁽²⁾=𝐻^0(2)𝜎⊗1_𝜎+1^𝜎⊗𝐻^(2)0,𝜎.(3.23) Here, 1^𝜎 (resp., 1_𝜎) is the identity operator on 𝔉^𝜎 (resp., 𝔉_𝜎).

Define 𝐻^𝜎=𝐻_𝜎||_𝔉𝜎,𝐻^𝜎0=𝐻₀||_𝔉𝜎.(3.24)

We get 𝐻𝐻^𝜎_𝜎=𝐻⁰⁽¹⁾+𝐻^0(2)𝜎+𝐻⁰⁽³⁾+𝑔𝐻_𝐼,𝜎on𝔉^𝜎,=𝐻^𝜎⊗1_𝜎+1^𝜎⊗𝐻^(2)0,𝜎on𝔉^𝜎⊗𝔉_𝜎.(3.25) In order to implement the conjugate operator theory, we have to show that 𝐻^𝜎 has a gap in its spectrum above its ground state.

We now set, for 𝛽>0 and 𝜂>0, 𝐶_𝛽𝜂=3𝑚_𝑊1+𝑚1₁₂+𝑚_𝑊3𝛽𝑚₁₂+12𝜂𝑚₁₂𝐵_𝛽𝜂(1+𝛽)^1/2,=3𝑚_𝑊1+41𝛽+12𝜂1+41𝛽+41𝜂^1/2.(3.26) Let 𝐺=𝐺^{(𝛼)ℓ,𝜖,𝜖}_′(⋅,⋅,⋅)_{𝛼=1,2;ℓ=1,2,3;𝜖,𝜖′=±,𝜖≠𝜖′},(3.27) and set 𝐾(𝐺)=⎛⎜⎜⎝_3𝛼=1,2_ℓ=1_{𝜖≠𝜖′}‖‖𝐺^{(𝛼)ℓ,𝜖,𝜖}_′‖‖^2𝐿_{2(Σ1×Σ1×Σ2)}⎞⎟⎟⎠^1/2.(3.28) Let 𝐶_𝛽𝜂=𝐶_𝛽𝜂1+𝑔¹𝐾(𝐺)𝐶_𝛽𝜂𝐵_𝛽𝜂1−𝑔₁𝐾(𝐺)𝐶_𝛽𝜂,=1+𝑔¹𝐾(𝐺)𝐶_𝛽𝜂1−𝑔₁𝐾(𝐺)𝐶_𝛽𝜂2+𝑔¹𝐾(𝐺)𝐵_𝛽𝜂𝐶_𝛽𝜂1−𝑔₁𝐾(𝐺)𝐶_𝛽𝜂𝐵^𝛽𝜂.(3.29)

Let 𝐾(𝐺)=⎛⎜⎜⎜⎝_3𝛼=1,2_ℓ=1_{𝜖≠𝜖′}_{Σ1×Σ1×Σ2}|||𝐺_{(𝛼)ℓ,𝜖,𝜖′}𝜉₁,𝜉₂,𝜉₃|||²|𝑝₂|²d𝜉₁d𝜉₂d𝜉₃⎞⎟⎟⎟⎠^1/2.(3.30) Let 𝛿∈ℝ be such that 0<𝛿<𝑚₁.(3.31)

We set 𝐷=sup4Λ𝛾2𝑚₁−𝛿,1𝐾(𝐺)2𝑚¹𝐶_𝛽𝜂+𝐵_𝛽𝜂,(3.32) where Λ>𝑚₁ has been introduced in Hypothesis 3.1(iv).

Let us define the sequence (𝜎_𝑛)_𝑛≥0 by 𝜎𝜎₀=Λ,₁=𝑚₁−𝛿𝜎₂2,𝜎=_𝑛+1𝑚₁−𝛿=𝛾𝜎₁,=𝛾𝜎_𝑛,𝑛≥1,(3.33) where 𝛾=1−𝛿/(2𝑚₁−𝛿).

Let 𝑔^𝛿(1) be such that 0<𝑔^𝛿(1)<inf1,𝑔₁,𝛾−𝛾²3𝐷.(3.34) For 0<𝑔≤𝑔^𝛿(1) we have 0<𝛾<1−3𝑔𝐷𝛾,(3.35)0<𝜎_𝑛+1<1−3𝑔𝐷𝛾𝜎^𝑛,𝑛≥1.(3.36) Set 𝐻^𝑛𝐸=𝐻^𝜎𝑛;𝐻^𝑛0=𝐻^𝜎𝑛₀,𝑛≥0,_𝑛=inf𝜎(𝐻^𝑛),𝑛≥0.(3.37) We then get the following.

Proposition 3.5. Suppose that the kernels 𝐺^{(𝛼)ℓ,𝜖,𝜖}_′ satisfy Hypotheses 2.1, 3.1(i), and 3.1(iv). Then there exists 0<𝑔̃_𝛿≤𝑔^𝛿(1) such that, for 𝑔≤𝑔̃_𝛿 and 𝑛≥1, 𝐸^𝑛 is a simple eigenvalue of 𝐻^𝑛 and 𝐻^𝑛 does not have spectrum in (𝐸^𝑛,𝐸^𝑛+(1−3𝑔𝐷/𝛾)𝜎_𝑛).

The proof of Proposition 3.5 is given in the appendix.

We now introduce the positive commutator estimates and the regularity property of 𝐻 with respect to 𝐴 in order to prove Theorem 3.4.

The operator 𝐴 has to be split into two pieces depending on 𝜎.

Let 𝜂𝜂_𝜎𝜎𝑝₂=𝜒_2𝜎𝑝₂,𝑎_𝜎𝑝₂=𝜒^2𝜎𝑝₂,𝑎_𝜎=𝜂_𝜎𝑝₂𝑎𝜂_𝜎𝑝₂,=𝜂^𝜎𝑝₂𝑎𝜂^𝜎𝑝₂.(3.38) Since 𝜂^2𝜎+(𝜂^𝜎)²=1, and [𝜂_𝜎,[𝜂_𝜎,𝑎]]=0=[𝜂^𝜎,[𝜂^𝜎,𝑎]], we obtain (see [4]) 𝑎=𝑎^𝜎+𝑎_𝜎.(3.39)

Note that we also have 𝑎_𝜎=12𝜂^𝜎𝑝₂²𝑎_𝜎𝑝₂⋅𝑖∇𝑝₂+𝑖∇𝑝₂⋅𝜂_𝜎𝑝₂²𝑝₂,=12𝜂^𝜎𝑝₂²𝑝₂⋅𝑖∇𝑝₂+𝑖∇𝑝₂⋅𝜂^𝜎𝑝₂²𝑝₂.(3.40)

The operators 𝑎, 𝑎_𝜎, and 𝑎^𝜎 are essentially self-adjoint on 𝐶^∞0(ℝ³,ℂ²) (see [5, Proposition  4.2.3]). We still denote by 𝑎, 𝑎_𝜎, and 𝑎^𝜎 their closures. If 𝑎̃ denotes any of the operator 𝑎, 𝑎_𝜎, and 𝑎^𝜎, we have 𝒟(𝑎)̃=𝑢∈𝐿²Σ₁;𝑎𝑢̃∈𝐿²Σ₁.(3.41)

The operators dΓ(𝑎), dΓ(𝑎^𝜎), and dΓ(𝑎_𝜎) are self-adjoint operators in 𝔉_𝑎(𝐿²(Σ₁)), and we have dΓ(𝑎)=dΓ(𝑎^𝜎)+dΓ𝑎_𝜎.(3.42)

By (2.8), the following operators in 𝔉_ℓ, denoted by 𝐴^𝜎ℓ and A_𝜎ℓ, respectively, 𝐴𝐴^𝜎ℓ=1⊗1⊗dΓ(𝑎^𝜎)⊗1+1⊗1⊗1⊗dΓ(𝑎^𝜎),_𝜎ℓ=1⊗1⊗dΓ𝑎_𝜎⊗1+1⊗1⊗1⊗dΓ𝑎_𝜎(3.43) are essentially self-adjoint on 𝔇_ℓ.

Let 𝐴^𝜎 and 𝐴_𝜎 be the following two operators in 𝔉_𝐿: 𝐴𝐴^𝜎_𝜎=𝐴^𝜎1⊗1₂⊗1₃+1₁⊗𝐴^𝜎2⊗1₃+1₁⊗1₂⊗𝐴^𝜎3,=𝐴_𝜎1⊗1₂⊗1₃+1₁⊗𝐴_𝜎2⊗1₃+1₁⊗1₂⊗𝐴_𝜎3.(3.44) The operators 𝐴^𝜎 and 𝐴_𝜎 are essentially self-adjoint on 𝔇_𝐿. Still denoting by 𝐴^𝜎 and 𝐴_𝜎 their extensions to 𝔉, 𝐴^𝜎 and 𝐴_𝜎 are essentially self-adjoint on 𝔇 and we still denote by 𝐴^𝜎 and 𝐴_𝜎 their closures.

We have 𝐴=𝐴^𝜎+𝐴_𝜎.(3.45)

The operators 𝑎, 𝑎^𝜎, and 𝑎_𝜎 are associated to the following 𝐶^∞-vector fields in ℝ³, respectively: 𝑣_𝜎𝑣𝑝₂=𝑝₂,𝑣_𝜎𝑝₂=𝜂^𝜎𝑝₂²𝑝₂,𝑝₂=𝜂_𝜎𝑝₂²𝑝₂.(3.46) Let 𝒱(𝑝) be any of these vector fields. We have ||𝒱(𝑝)||≤Γ||𝑝||(3.47) for some Γ>0, and we also have 𝒱(𝑝)=𝑣̃||𝑝||𝑝,(3.48) where the 𝑣̃'s are defined by (3.46) and (3.48) and fulfil |𝑝|^𝛼(d^𝛼/d|𝑝|^𝛼)𝑣(̃|𝑝|) bounded for 𝛼=0,1,2.

Let 𝜓_𝑡(⋅)∶ℝ³→ℝ³ be the corresponding flow generated by 𝒱: dd𝑡𝜓^𝑡(𝜓𝑝)=𝒱𝜓_𝑡(𝑝)₀,(𝑝)=𝑝.(3.49)𝜓_𝑡(𝑝) is a 𝐶^∞-flow and we have e^−Γ|𝑡|||𝑝||≤||𝜓_𝑡(𝑝)||≤e^Γ|𝑡|||𝑝||.(3.50)𝜓_𝑡(𝑝) induces a one-parameter group of unitary operators 𝑈(𝑡) in 𝐿²(Σ₁)≃𝐿²(ℝ³,ℂ²) defined by (𝑈(𝑡)𝑓)(𝑝)=𝑓𝜓_𝑡(𝑝)det∇𝜓_𝑡(𝑝)^1/2.(3.51) Let 𝜙_𝑡(⋅), 𝜙^𝜎𝑡(⋅), and 𝜙_𝜎𝑡(⋅) be the flows associated with the vector fields 𝑣(⋅), 𝑣^𝜎(⋅), and 𝑣_𝜎(⋅), respectively.

Let 𝑈(𝑡), 𝑈^𝜎(𝑡), and 𝑈_𝜎(𝑡) be the corresponding one-parameter groups of unitary operators in 𝐿²(Σ₁). The operators 𝑎, 𝑎^𝜎, and 𝑎_𝜎 are the generators of 𝑈(𝑡), 𝑈^𝜎(𝑡), and 𝑈_𝜎(𝑡), respectively, that is, 𝑈𝑈_𝜎(𝑡)=e^{−𝑖𝑎𝑡},𝑈_𝜎(𝑡)=e^{−𝑖𝑎𝜎𝑡},(𝑡)=e^{−𝑖𝑎𝜎𝑡}.(3.52)

Let 𝑤⁽²⁾𝜉₂=𝑤^ℓ(2)𝜉₂_ℓ=1,2,3,dΓ𝑤⁽²⁾=³_ℓ=1_𝜖𝑤^ℓ(2)𝜉₂𝑐^∗ℓ,𝜖𝜉₂𝑐_ℓ𝜖𝜉₂d𝜉₂.(3.53) Let 𝑉(𝑡) be any of the one-parameter groups 𝑈(𝑡), 𝑈^𝜎(𝑡), and 𝑈_𝜎(𝑡). We set 𝑉(𝑡)𝑤⁽²⁾𝑉(𝑡)^∗=𝑉(𝑡)𝑤^ℓ(2)𝑉(𝑡)^∗_ℓ=1,2,3,(3.54) and we have 𝑉(𝑡)𝑤⁽²⁾𝑉(𝑡)^∗=𝑤⁽²⁾𝜓_𝑡.(3.55) Here 𝜓_𝑡 is the flow associated to 𝑉(𝑡).

This yields, for any 𝜑∈𝔇, (see [11, Lemma  2.8]) e^{−𝑖𝐴𝑡}𝐻₀e^𝑖𝐴𝑡𝜑−𝐻₀𝜑==dΓe^{−𝑖𝑎𝑡}𝑤⁽²⁾e^𝑖𝑎𝑡−dΓ𝑤⁽²⁾𝜑dΓw⁽²⁾∘𝜙_𝑡−𝑤⁽²⁾𝜑,e(3.56)_{−𝑖𝐴𝜎𝑡}𝐻₀e^{𝑖𝐴𝜎𝑡}𝜑−𝐻₀𝜑==dΓe^{−𝑖𝑎𝜎𝑡}𝑤⁽²⁾e^{𝑖𝑎𝜎𝑡}−dΓ𝑤⁽²⁾𝜑dΓ𝑤⁽²⁾∘𝜙^𝜎𝑡−𝑤⁽²⁾𝜑,e(3.57)_{−𝑖𝐴𝜎𝑡}𝐻₀e^{𝑖𝐴𝜎𝑡}𝜑−𝐻₀𝜑==dΓe^{−𝑖𝑎𝜎𝑡}𝑤⁽²⁾e^{𝑖𝑎𝜎𝑡}−dΓ𝑤⁽²⁾𝜑dΓ𝑤⁽²⁾∘𝜙_𝜎𝑡−𝑤⁽²⁾𝜑.(3.58)

Proposition 3.6. Suppose that the kernels 𝐺^{(𝛼)ℓ,𝜖,𝜖}_′ satisfy Hypothesis 2.1.
For every 𝑡∈ℝ one has, for 𝑔≤𝑔₁, ((i)e^𝑖𝑡𝐴𝒟𝐻₀=e^𝑖𝑡𝐴𝒟(𝐻)⊂𝒟𝐻₀=𝒟(𝐻),(ii)e^{𝑖𝑡𝐴𝜎}𝒟𝐻₀=e^{𝑖𝑡𝐴𝜎}𝒟(𝐻)⊂𝒟𝐻₀=𝒟(𝐻),iii)e^{𝑖𝑡𝐴𝜎}𝒟𝐻₀=e^{𝑖𝑡𝐴𝜎}𝒟(𝐻)⊂𝒟𝐻₀=𝒟(𝐻).(3.59)

Proof. We only prove (i), since (ii) and (iii) can be proved similarly. By (3.56) we have, for 𝜑∈𝔇, e^{−𝑖𝑡𝐴}𝐻₀e^𝑖𝑡𝐴𝜑=𝐻⁰⁽¹⁾+𝐻⁰⁽³⁾+dΓ𝑤⁽²⁾∘𝜙_𝑡𝜑.(3.60) It follows from (3.50) and (3.60) that ‖‖𝐻₀e^𝑖𝑡𝐴𝜑‖‖≤e^Γ|𝑡|‖‖𝐻₀𝜑‖‖.(3.61) This yields (i) because 𝔇 is a core for 𝐻₀. Moreover we get ‖‖𝐻₀e^𝑖𝑡𝐴𝐻₀+1⁻¹‖‖≤e^Γ|𝑡|.(3.62) 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 ‖‖𝐻e^𝑖𝑡𝐴(𝐻+𝑖)⁻¹‖‖≤𝐶e^Γ|𝑡|.(3.63) Similarly, we also get ‖‖𝐻₀e^{𝑖𝑡𝐴𝜎}𝐻₀+1⁻¹‖‖𝐻₀‖‖≤e^Γ|𝑡|,‖‖𝐻e^{𝑖𝑡𝐴𝜎}𝐻₀+1⁻¹‖‖≤e^Γ|𝑡|,‖‖𝐻e^{𝑖𝑡𝐴𝜎}(𝐻+𝑖)⁻¹‖‖≤𝐶e^Γ|𝑡|,e^{𝑖𝑡𝐴𝜎}(𝐻+𝑖)⁻¹‖‖≤𝐶e^Γ|𝑡|.(3.64)

Let 𝐻_𝐼(𝐺) be the interaction associated with the kernels 𝐺=(𝐺^{(𝛼)ℓ,𝜖,𝜖}_′)_{𝛼=1,2;ℓ=1,2,3;𝜖≠𝜖′=±}, where the kernels (𝐺^{(𝛼)ℓ,𝜖,𝜖}_′) satisfy Hypothesis 2.1.

We set 𝑉(𝑡)𝐺=𝑉(𝑡)𝐺^{(𝛼)ℓ,𝜖,𝜖}_′_{𝛼=1,2;ℓ=1,2,3;𝜖≠𝜖′=±}.(3.65) We have for 𝜑∈𝔇 (see [11, Lemma  2.7]), e^{−𝑖𝐴𝑡}𝐻_𝐼(𝐺)e^𝑖𝐴𝑡𝜑=𝐻_𝐼e^{−𝑖𝑎𝑡}𝐺𝜑,e^{−𝑖𝐴𝜎𝑡}𝐻_𝐼(𝐺)e^{𝑖𝐴𝜎𝑡}𝜑=𝐻_𝐼e^{−𝑖𝑎𝜎𝑡}𝐺𝜑,e^{−𝑖𝐴𝜎𝑡}𝐻_𝐼(𝐺)e^{𝑖𝐴𝜎𝑡}𝜑=𝐻_𝐼e^{−𝑖𝑎𝜎𝑡}𝐺𝜑.(3.66)

According to [5, 6], in order to prove Theorem 3.4 we must prove that 𝐻 is locally of class 𝐶²(𝐴^𝜎), 𝐶²(𝐴_𝜎), and 𝐶²(𝐴) in (−∞,𝑚₁−𝛿/2) and that 𝐴 and 𝐴_𝜎 are locally strictly conjugate to 𝐻 in (𝐸,𝑚₁−𝛿/2).

Recall that 𝐻 is locally of class 𝐶²(𝐴) in (−∞,𝑚₁−𝛿/2) if, for any 𝜑∈𝐶^∞0((−∞,𝑚₁−𝛿/2)), 𝜑(𝐻) is of class 𝐶²(𝐴); that is, 𝑡→e^{−𝑖𝐴𝑡}𝜑(𝐻)e^𝑖𝑡𝐴𝜓 is twice continuously differentiable for all 𝜑∈𝐶^∞0((−∞,𝑚₁−𝛿/2)) and all 𝜓∈𝔉.

Thus, one of our main results is the following one.

Theorem 3.7. Suppose that the kernels 𝐺^{(𝛼)ℓ,𝜖,𝜖}_′ satisfy Hypotheses 2.1, 3.1(i) and 3.1(iii). (a)𝐻 is locally of class 𝐶²(𝐴), 𝐶²(𝐴^𝜎), and 𝐶²(𝐴_𝜎) in (−∞,𝑚₁−𝛿/2). (b)𝐻^𝜎 is locally of class 𝐶²(𝐴^𝜎) in (−∞,𝑚₁−𝛿/2).

It follows from Theorem 3.7 that [𝐻,𝑖𝐴], [𝐻,𝑖𝐴_𝜎], [𝐻,𝑖𝐴^𝜎], and [𝐻^𝜎,𝑖𝐴^𝜎] are defined as sesquilinear forms on ⋃_𝐾𝐸_𝐾(𝐻)𝔉, where the union is taken over all the compact subsets 𝐾 of (−∞,𝑚₁−𝛿/2).

Furthermore, by Proposition 3.6, Theorem 3.7 and [4, Lemma  29], we get for all 𝜑∈𝐶^∞0((𝐸,𝑚₁−𝛿/2)) and all 𝜓∈𝔉, 𝜑(𝐻)[𝐻,𝑖𝐴]𝜑(𝐻)𝜓=lim_𝑡→0𝜑(𝐻)𝐻,e^𝑖𝑡𝐴−1𝜑(𝐻)𝐻,𝑖𝐴_𝜎𝑡𝜑(𝐻)𝜓,𝜑(𝐻)𝜓=lim_𝑡→0𝜑(𝐻)𝐻,e^{𝑖𝑡𝐴𝜎}−1𝜑(𝐻)[𝐻,𝑖𝐴_𝜎𝑡𝜑(𝐻)𝜓,]𝜑(𝐻)𝜓=lim_𝑡→0𝜑(𝐻)𝐻,e^{𝑖𝑡𝐴𝜎}−1𝜑(𝐻_𝜎𝑡𝜑(𝐻)𝜓,)[𝐻^𝜎,𝑖𝐴^𝜎]𝜑(𝐻^𝜎)𝜓=lim_𝑡→0𝜑(𝐻^𝜎)𝐻^𝜎,e^{𝑖𝑡𝐴𝜎}−1𝑡𝜑(𝐻^𝜎)𝜓.(3.67)

The following proposition allows us to compute [𝐻,𝑖𝐴], [𝐻,𝑖𝐴^𝜎], [𝐻,𝑖𝐴_𝜎], and [𝐻^𝜎,𝑖𝐴^𝜎] as sesquilinear forms. By Hypotheses 2.1 and 3.1(iii.a), the kernels 𝐺^{(𝛼)ℓ,𝜖,𝜖}_′(𝜉₁,⋅,𝜉₃) belong to the domains of 𝑎, 𝑎^𝜎, and 𝑎_𝜎.

Proposition 3.8. Suppose that the kernels 𝐺^{(𝛼)ℓ,𝜖,𝜖}_′ satisfy Hypotheses 2.1 and 3.1(iii.a). Then (a)for all 𝜓∈𝒟(𝐻) one has (i)lim_𝑡→0[𝐻,(e^𝑖𝑡𝐴−1)/𝑡]𝜓=(dΓ(𝑤⁽²⁾)+𝑔𝐻_𝐼(−𝑖𝑎𝐺))𝜓, (ii)lim_𝑡→0[𝐻,(e^{𝑖𝑡𝐴𝜎}−1)/𝑡]𝜓=(dΓ((𝜂^𝜎)²𝑤⁽²⁾)+𝑔𝐻_𝐼(−𝑖𝑎^𝜎𝐺))𝜓, (iii)lim_𝑡→0[𝐻,(e^{𝑖𝑡𝐴𝜎}−1)/𝑡]𝜓=(dΓ((𝜂_𝜎)²𝑤⁽²⁾)+𝑔𝐻_𝐼(−𝑖𝑎_𝜎𝐺))𝜓, (iv)lim_𝑡→0[𝐻^𝜎,(e^{𝑖𝑡𝐴𝜎}−1)/𝑡]𝜓=(dΓ((𝜂^𝜎)²𝑤⁽²⁾)+𝑔𝐻_𝐼(−𝑖𝑎^𝜎(𝜒^𝜎(𝑝₂)𝐺)))𝜓; (b)and(i)sup_0<|𝑡|≤1‖[𝐻,(e^𝑖𝑡𝐴−1)/𝑡](𝐻+𝑖)⁻¹‖<∞, (ii)sup_0<|𝑡|≤1‖[𝐻,(e^{𝑖𝑡𝐴𝜎}−1)/𝑡](𝐻+𝑖)⁻¹‖<∞, (iii)sup_0<|𝑡|≤1‖[𝐻,(e^{𝑖𝑡𝐴𝜎}−1)/𝑡](𝐻+𝑖)⁻¹‖<∞, (iv)sup_0<|𝑡|≤1‖[𝐻^𝜎,(e^{𝑖𝑡𝐴𝜎}−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 1|𝑡|||𝑤^ℓ(2)𝜙_𝑡𝑝₂−𝑤^ℓ(2)𝑝₂||≤|1𝑡|e^Γ|𝑡|−1𝑤^ℓ(2)𝑝₂(3.68) for ℓ=1,2,3.
By (3.56)–(3.58) and Lebesgue's Theorem we then get for all 𝜓∈𝒟(𝐻₀)lim_𝑡→0𝐻⁰,e^𝑖𝑡𝐴−1𝑡𝜓=lim^𝑡→01𝑡lim_𝑡→0e^{−𝑖𝑡𝐴}𝐻₀e^𝑖𝑡𝐴−𝐻₀𝜓=dΓ𝑤⁽²⁾𝜓,𝐻⁰,e^{𝑖𝑡𝐴𝜎}−1𝑡𝜓=lim^𝑡→01𝑡lim_𝑡→0e^{−𝑖𝑡𝐴𝜎}𝐻₀e^{𝑖𝑡𝐴𝜎}−𝐻₀𝜓=dΓ(𝜂^𝜎)²𝑤⁽²⁾𝜓,𝐻⁰,e^{𝑖𝑡𝐴𝜎}−1𝑡𝜓=lim^𝑡→01𝑡e^{−𝑖𝑡𝐴𝜎}𝐻₀e^{𝑖𝑡𝐴𝜎}−𝐻₀𝜓=dΓ𝜂_𝜎²𝑤⁽²⁾𝜓.(3.69) By (3.66), we obtain, for all 𝜓∈𝒟(𝐻), lim_𝑡→0𝐻^𝐼(𝐺),e^𝑖𝑡𝐴−1𝑡𝜓=lim^𝑡→01𝑡lim_𝑡→0e^{−𝑖𝑡𝐴}𝐻_𝐼(𝐺)e^𝑖𝑡𝐴−𝐻_𝐼(𝐺)𝜓=𝐻_𝐼(−𝑖(𝑎𝐺))𝜓,𝐻^𝐼(𝐺),e^{𝑖𝑡𝐴𝜎}−1𝑡𝜓=lim^𝑡→01𝑡lim_𝑡→0e^{−𝑖𝑡𝐴𝜎}𝐻_𝐼(𝐺)e^{𝑖𝑡𝐴𝜎}−𝐻_𝐼(𝐺)𝜓=𝐻_𝐼(−𝑖(𝑎^𝜎𝐺))𝜓,𝐻^𝐼(𝐺),e^{𝑖𝑡𝐴𝜎}−1𝑡𝜓=lim^𝑡→01𝑡lim_𝑡→0e^{−𝑖𝑡𝐴𝜎}𝐻_𝐼(𝐺)e^{𝑖𝑡𝐴𝜎}−𝐻_𝐼(𝐺)𝜓=𝐻_𝐼−𝑖𝑎_𝜎𝐺𝜓,𝐻^𝐼𝜒^𝜎𝑝₂𝐺,e^{𝑖𝑡𝐴𝜎}−1𝑡𝜓=lim^𝑡→01𝑡=𝐻_𝐼e^{−𝑖𝑡𝐴𝜎}𝐻_𝐼𝜒^𝜎𝑝₂𝐺e^{𝑖𝑡𝐴𝜎}−𝐻_𝐼𝜒^𝜎𝑝₂𝐺𝜓−𝑖𝑎^𝜎𝜒^𝜎𝑝₂𝐺𝜓.(3.70) This concludes the proof of Proposition 3.8.

Combining (3.67) with Proposition 3.8, we finally get for every 𝜑∈𝐶^∞0((−∞,𝑚₁−𝛿/2)) and every 𝜓∈𝔉𝜑(𝐻)[𝐻,𝑖𝐴]𝜑(𝐻)𝜓=𝜑(𝐻)dΓ𝑤⁽²⁾+𝑔𝐻_𝐼(−𝑖(𝑎𝐺))𝜑(𝐻)𝜓,(3.71)𝜑(𝐻)[𝐻,𝑖𝐴^𝜎]𝜑(𝐻)𝜓=𝜑(𝐻)dΓ(𝜂^𝜎)²𝑤⁽²⁾+𝑔𝐻_𝐼(−𝑖(𝑎^𝜎𝐺))𝜑(𝐻)𝜓,(3.72)𝜑(𝐻)𝐻,𝑖𝐴_𝜎𝜑(𝐻)𝜓=𝜑(𝐻)dΓ𝜂_𝜎²𝑤⁽²⁾+𝑔𝐻^𝐼−𝑖𝑎_𝜎𝐺𝜑(𝐻)𝜓,(3.73)𝜑(𝐻^𝜎)[𝐻^𝜎,𝑖𝐴^𝜎]𝜑(𝐻^𝜎)𝜓=𝜑(𝐻^𝜎)dΓ(𝜂^𝜎)²𝑤⁽²⁾+𝑔𝐻_𝐼−𝑖𝑎^𝜎𝜒^𝜎𝐺𝜑(𝐻^𝜎)𝜓.(3.74)

We now introduce the Mourre inequality.

Let 𝑁 be the smallest integer such that 𝑁𝛾≥1.(3.75)

We have, for 𝑔≤𝑔^𝛿(1), 𝛾<𝛾+1𝛾𝑁1−3𝑔𝛾𝐷−𝛾<1−3𝑔𝛾𝐷,𝑁≤𝛾−𝑁11−3𝑔𝛾𝐷−𝛾<𝛾.(3.76) Let 𝜖_𝛾=12𝑁1−3𝑔^𝛿(1)𝐷𝛾−𝛾.(3.77) We choose 𝑓∈𝐶^∞0(ℝ) such that 1≥𝑓≥0 and 𝑓(𝜆)=⎧⎪⎪⎪⎨⎪⎪⎪⎩10if𝜆∈𝛾−𝜖_𝛾²,𝛾+𝜖_𝛾,if𝜆>𝛾+1𝑁1−3𝑔^𝛿(1)𝐷𝛾−𝛾=𝛾+2𝜖^𝛾0,if𝜆<𝛾−1𝑁1−3𝑔^𝛿(1)𝐷𝛾−𝛾²=𝛾−2𝜖_𝛾².(3.78) Note that 𝛾+2𝜖_𝛾<1−3𝑔𝐷/𝛾 for 𝑔≤𝑔^𝛿(1) and 𝛾−𝜖_𝛾>𝛾/𝑁.

We set, for 𝑛≥1, f_𝑛(𝜆)=𝑓𝜆𝜎_𝑛.(3.79)

Let 𝐸_𝑛𝐻_𝑛=𝐻_𝜎𝑛,𝐻=_(2)0𝑛inf𝜎𝐻_𝑛,=𝐻^(2)0𝜎_𝑛.(3.80) Let 𝑃^𝑛 denote the ground state projection of 𝐻^𝑛. It follows from Proposition 3.5 that, for 𝑛≥1 and 𝑔≤𝑔̃_𝛿≤𝑔^𝛿(1), 𝑓_𝑛𝐻_𝑛−𝐸_𝑛=𝑃^𝑛⊗𝑓_𝑛𝐻^(2)0,𝑛.(3.81) Note that 𝐸_𝑛=𝐸^𝑛=inf𝜎(𝐻^𝑛).(3.82) Set 𝑎𝑎^𝑛_𝑛=𝑎^𝜎𝑛,𝐴_𝑛=𝑎_𝜎𝑛,𝐴_𝑛=𝐴^𝜎𝑛,𝔉_𝑛=𝐴_𝜎𝑛,𝔉_𝑛=𝔉^𝜎𝑛,=𝔉_𝜎𝑛.(3.83) We have 𝔉𝐴≃=𝔉𝐴^𝑛_𝑛⊗𝔉_𝑛,+𝐴_𝑛.(3.84) We further note that 𝑎^𝑛𝜒^𝜎𝑛𝑝₂=𝑎^𝑛.(3.85) By (3.72), (3.74), and (3.85), we obtain 𝐻,𝑖𝐴^𝑛=𝐻^𝑛,𝑖𝐴^𝑛⊗1(3.86) as sesquilinear forms with respect to 𝔉=𝔉^𝑛⊗𝔉_𝑛.

Furthermore, it follows from the virial theorem (see [6, Proposition  3.2] and Proposition 6.1) that 𝑃^𝑛𝐻^𝑛,i𝐴^𝑛𝑃^𝑛=0.(3.87) By (3.81) and (3.87) we then get, for 𝑔≤𝑔̃_𝛿≤𝑔^𝛿(1), 𝑓_𝑛𝐻_𝑛−𝐸_𝑛𝐻,𝑖𝐴^𝑛𝑓_𝑛𝐻_𝑛−𝐸_𝑛=0.(3.88)

We then have the following.

Proposition 3.9. Suppose that the kernels 𝐺^{(𝛼)ℓ,𝜖,𝜖}_′ satisfy Hypotheses 2.1 and 3.1. Then there exists 𝐶_𝛿>0 and 𝑔̃^𝛿(1)>0 such that 𝑔̃^𝛿(1)≤𝑔̃_𝛿 and 𝑓_𝑛𝐻_𝑛−𝐸_𝑛𝐻,𝑖𝐴_𝑛𝑓_𝑛𝐻_𝑛−𝐸_𝑛≥𝐶_𝛿𝛾²𝑁²𝜎_𝑛𝑓_𝑛𝐻_𝑛−𝐸_𝑛²(3.89) for 𝑛≥1 and 𝑔≤𝑔̃^𝛿(1).

Let 𝐸_Δ(𝐻−𝐸) be the spectral projection for the operator 𝐻−𝐸 associated with the interval Δ, and let Δ_𝑛=𝛾−𝜖_𝛾²𝜎_𝑛,𝛾+𝜖_𝛾𝜎_𝑛,𝑛≥1.(3.90) Note that 𝜎_𝑛+2,𝜎_𝑛+1⊂𝛾−𝜖_𝛾²𝜎_𝑛,𝛾+𝜖_𝛾𝜎_𝑛,𝑛≥1.(3.91)

Theorem 3.10. Suppose that the kernels 𝐺^{(𝛼)ℓ,𝜖,𝜖}_′ satisfy Hypotheses 2.1 and 3.1. Then there exists 𝐶_𝛿>0 and 𝑔̃^𝛿(2)>0 such that 𝑔̃^𝛿(2)≤𝑔̃^𝛿(1) and 𝐸_Δ𝑛(𝐻−𝐸)[𝐻,𝑖𝐴]𝐸_Δ𝑛(𝐻−𝐸)≥𝐶_𝛿𝛾²𝑁²𝜎_𝑛𝐸_Δ𝑛(𝐻−𝐸)(3.92) for 𝑛≥1 and 𝑔≤𝑔̃^𝛿(2).

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

We now prove Theorem 3.3. The scheme of the proof is quite well known (see [9, 31]). It follows from Proposition 3.5 that 𝐻^𝑛 has a unique ground state, denoted by 𝜙^𝑛, in 𝔉^𝑛: 𝐻^𝑛𝜙^𝑛=𝐸^𝑛𝜙^𝑛,𝜙^𝑛∈𝒟(𝐻^𝑛),‖𝜙^𝑛‖=1,𝑛≥1.(4.1) Therefore 𝐻_𝑛 has an unique normalized ground state in 𝔉, given by 𝜙_𝑛=𝜙^𝑛⊗Ω_𝑛, where Ω_𝑛 is the vacuum state in 𝔉_𝑛: 𝐻_𝑛𝜙_𝑛=𝐸^𝑛𝜙_𝑛,𝜙_𝑛∈𝒟𝐻_𝑛,‖‖𝜙_𝑛‖‖=1,𝑛≥1.(4.2) Since ‖𝜙_𝑛‖=1, there exists a subsequence (𝑛_𝑘)_𝑘≥1, converging to ∞ such that (𝜙_𝑛𝑘)_𝑘≥1 converges weakly to a state 𝜙∈𝔉. We have to prove that 𝜙≠0. By adapting the proof of Theorem 4.1 in [22] (see also [20]), the key point is to estimate ‖𝑐_ℓ,𝜖(𝜉₂)Φ_𝑛‖_𝔉 in order to show that ³_ℓ=1_𝜖‖‖𝑐^ℓ,𝜖(𝜉₂)𝜙_𝑛‖‖²d𝜉₂=𝒪𝑔²,(4.3) uniformly with respect to 𝑛.

The estimate (4.3) is a consequence of the so-called “pull-through’’ formula as it follows.

Let 𝐻_𝐼,𝑛 denote the interaction 𝐻_𝐼 associated with the kernels 1_{{|𝑝2|≥𝜎𝑛}}(𝑝₂)𝐺^{(𝛼)ℓ,𝜖,𝜖}_′. We thus have 𝐻𝑔𝐻₀𝑐_{ℓ,𝜖𝐼,𝑛}𝜉₂𝜙_𝑛=𝑐_ℓ,𝜖𝜉₂𝐻₀𝜙_𝑛−𝑤^ℓ(2)𝜉₂𝑐_ℓ,𝜖𝜉₂𝜙_𝑛,𝑐_ℓ,𝜖𝜉₂𝜙_𝑛=𝑐_ℓ,𝜖𝜉₂𝑔𝐻_𝐼,𝑛𝜙_𝑛+𝑔𝑉_{ℓ,𝜖,𝜖′}𝜉₂𝜙_𝑛(4.4) with 𝑉_{ℓ,𝜖,𝜖′}𝜉₂=𝑔𝐺^(1)ℓ,𝜖_′𝜖𝜉₁,𝜉₂,𝜉₃𝑏^{∗ℓ,𝜖′}𝜉₁𝑎_𝜖𝜉₃d𝜉₁d𝜉₃+𝑔𝐺^(2)ℓ,𝜖_′𝜖𝜉₁,𝜉₂,𝜉₃𝑏^{∗ℓ,𝜖′}𝜉₁𝑎^∗𝜖𝜉₃d𝜉₁d𝜉₃.(4.5) This yields 𝐻^𝑛−𝐸_𝑛+𝑤^ℓ(2)𝜉₂𝑐_ℓ,𝜖𝜉₂𝜙_𝑛=𝑉_{ℓ,𝜖,𝜖′}𝜉₂𝜙_𝑛.(4.6) By adapting the proof of Propositions 2.4 and 2.5 we easily get ‖‖𝑉_{ℓ,𝜖,𝜖′}𝜓‖‖_𝔉≤𝑔𝑚_𝑊1/2_𝛼=1,2‖‖𝐺^{(𝛼)ℓ,𝜖,𝜖}_′(⋅,𝜉₂,⋅)‖‖_{𝐿2(Σ1×Σ2)}‖‖𝐻^01/2+𝑔‖‖𝐺_{(2)ℓ,𝜖,𝜖′}𝜓‖‖(⋅,𝜉₂,⋅)‖‖_{𝐿2(Σ1×Σ2)}‖𝜓‖,(4.7) where 𝜓∈𝒟(𝐻₀).

Let us estimate ‖𝐻₀𝜙_𝑛‖. By (2.53), (2.54), (3.26), and (3.28) we have 𝑔‖‖𝐻_𝐼,𝑛‖‖𝜙𝐻^𝑛‖‖₀≤𝑔𝐾(𝐺)𝐶_𝛽𝜂‖‖𝐻₀𝜙_𝑛‖‖+𝐵_𝛽𝜂,𝜙_𝑛‖‖≤||𝐸_𝑛||+𝑔‖‖𝐻_𝐼,𝑛𝜙_𝑛‖‖.(4.8) Therefore ‖‖𝐻₀𝜙_𝑛‖‖≤||𝐸^𝑛||1−𝑔₁𝐾(𝐺)𝐶_𝛽𝜂+𝑔𝐾(𝐺)𝐵^𝛽𝜂1−𝑔₁𝐾(𝐺)𝐶_𝛽𝜂.(4.9) By (3.82), (A.11), and (4.9), there exists 𝐶>0 such that ‖‖𝐻₀𝜙_𝑛‖‖≤𝐶,(4.10) uniformly in 𝑛 and 𝑔≤𝑔₁.

By (4.6), (4.7), and (4.10) we get ‖‖𝑐_ℓ,𝜖𝜙_𝑛‖‖≤𝑔||𝑝₂||𝐶^1/2²_𝛼=1‖‖𝐺^{(𝛼)ℓ,𝜖,𝜖}_′(⋅,𝜉₂,⋅)‖‖_{𝐿2(Σ1×Σ2)}+‖‖𝐺^{(2)ℓ,𝜖,𝜖′}(⋅,𝜉₂,⋅)‖‖_{𝐿2(Σ1×Σ2)}.(4.11) By Hypothesis 3.1(i), there exists a constant 𝐶(𝐺)>0 depending on the kernels 𝐺=(𝐺^{(𝛼)ℓ,𝜖,𝜖}_′)_{ℓ=1,2,3;𝛼=1,2;𝜖≠𝜖′=±} and such that ³_ℓ=1_𝜖‖‖𝑐^ℓ,𝜖(𝜉₂)𝜙_𝑛‖‖²d𝜉₂≤𝐶(𝐺)²𝑔².(4.12) The existence of a ground state 𝜙 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 about 𝜎(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 𝑐^♯ℓ,𝜖(𝜉₂)'s. For 𝑓∈𝐿²(ℝ³,ℂ²), we define on 𝒟(𝐻₀) the operators 𝑐^{♯𝑡ℓ,𝜖}(𝑓)=e^𝑖𝑡𝐻e^{−𝑖𝑡𝐻0}𝑐^♯ℓ,𝜖(𝑓)e^i𝑡𝐻0e^{−𝑖𝑡𝐻}.(4.13) By mimicking the proof given in [21, 31] one proves, under the hypothesis of Theorem 3.3 and for 𝑓∈𝐶^∞0(ℝ³,ℂ²), that the strong limits of 𝑐^{♯𝑡ℓ,𝜖}(𝑓) when 𝑡→±∞ exist for 𝜓∈𝒟(𝐻₀): lim_𝑡→±∞𝑐^{♯𝑡ℓ,𝜖}(𝑓)𝜓∶=𝑐^{♯±ℓ,𝜖}(𝑓)𝜓.(4.14) The operators 𝑐^{♯±ℓ,𝜖}(𝑓) satisfy the CAR and we have 𝑐^±ℓ,𝜖(𝑓)𝜙=0,𝑓∈𝐶^∞0ℝ³,ℂ²,(4.15) where 𝜙 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 Proposition 3.9. In view of Proposition 3.8(a)(iii) and (3.73), we have, as sesquilinear forms, 𝐻,𝑖𝐴_𝜎=(1−𝑔)dΓ𝜂_𝜎²𝑤⁽²⁾+𝑔dΓ𝜂_𝜎²𝑤⁽²⁾+𝑔𝐻^𝐼−𝑖𝑎_𝜎𝐺.(5.1) Let 𝔉^ℓ(1) (resp., 𝔉^ℓ(2)) be the Fock space for the massive leptons ℓ (resp., the neutrinos and antineutrinos ℓ).

We have 𝔉_ℓ≃𝔉^ℓ(1)⊗𝔉^ℓ(2).(5.2)

Let 𝔉⁽¹⁾=𝔉_𝑊⊗³_ℓ=1𝔉^ℓ(1),𝔉⁽²⁾=³_ℓ=1𝔉^ℓ(2).(5.3)

We have 𝔉≃𝔉⁽¹⁾⊗𝔉⁽²⁾,(5.4)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), dΓ=𝜂_𝜎²𝑤⁽²⁾+𝐻^𝐼−𝑖𝑎_𝜎𝐺₃_ℓ=1_𝜖𝜂^𝜎𝑝₂²||𝑝₂||𝑐^∗ℓ,𝜖𝜉₂𝑐_ℓ,𝜖𝜉₂d𝜉₂+³_ℓ=1_{𝜖≠𝜖′}||𝑝²||⎛⎜⎜⎝1₁⊗𝜂_𝜎𝑝₂𝑐^∗ℓ,𝜖𝜉₂+_𝛼=1,2ℳ^{(𝛼)∗ℓ,𝜖,𝜖}_′,𝜎𝜉₂||𝑝₂||⊗1²×⎛⎜⎜⎝1₁⎞⎟⎟⎠⊗𝜂_𝜎𝑝₂𝑐_ℓ,𝜖𝜉₂+_𝛼=1,2ℳ^{(𝛼)ℓ,𝜖,𝜖}_′,𝜎𝜉₂||𝑝₂||⊗1²⎞⎟⎟⎠d𝜉₂−³_ℓ=1_{𝜖≠𝜖′}⎛⎜⎜⎝_𝛼=1,2ℳ^{(𝛼)∗ℓ,𝜖,𝜖}_′,𝜎𝜉₂||𝑝₂||^1/2⊗1₂⎞⎟⎟⎠⎛⎜⎜⎝_𝛼=1,2ℳ^{(𝛼)ℓ,𝜖,𝜖}_′,𝜎𝜉₂||𝑝₂||^1/2⊗1₂⎞⎟⎟⎠d𝜉₂,(5.5) where ℳ^{(𝛼)ℓ,𝜖,𝜖}_′,𝜎𝜉₂=𝑖_𝛼=1,2𝑎𝜂^𝜎𝑝₂𝐺^{(𝛼)ℓ,𝜖,𝜖}_′𝜉₁,𝜉₂,𝜉₃𝑏^{∗ℓ,𝜖′}𝜉₁𝑎_𝜖^′𝜉₃d𝜉₁d𝜉₃,(5.6) and where 1_𝑗 is the identity operator in 𝔉^(𝑗).

By mimicking the proofs of Propositions 2.4 and 2.5, we get, for every 𝜓∈𝔇, ³_ℓ=1_{𝜖≠𝜖′}⎛⎜⎜⎝𝜓,⎛⎜⎜⎝_𝛼=1,2ℳ^{(𝛼)∗ℓ,𝜖,𝜖}_′,𝜎𝜉₂||𝑝₂||^1/2⊗1₂⎞⎟⎟⎠⎛⎜⎜⎝_𝛼=1,2ℳ^{(𝛼)ℓ,𝜖,𝜖}_′,𝜎𝜉₂||𝑝₂||^1/2⊗1₂⎞⎟⎟⎠𝜓d𝜉₂=₃⎞⎟⎟⎠_ℓ=1_{𝜖≠𝜖′}‖‖‖‖_𝛼=1,2ℳ^{𝛼ℓ,𝜖,𝜖}_′,𝜎𝜉₂||𝑝₂||^1/2⊗1₂𝜓d𝜉₂‖‖‖‖²≤⎛⎜⎜⎜⎝|||∑^𝛼=1,2𝑎𝜂^𝜎𝑝₂𝐺^{(𝛼)ℓ,𝜖,𝜖}_′𝜉¹,𝜉₂,𝜉₃|||²𝑤⁽³⁾𝜉₃||𝑝₂||d𝜉₁d𝜉₂d𝜉₃⎞⎟⎟⎟⎠‖‖‖𝐻₀₍₃₎^1/2𝜓‖‖‖.(5.7) Noting that |(𝑎𝜂_𝜎)(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 |||∑^𝛼=1,2𝑎𝜂^𝜎𝑝₂𝐺^{(𝛼)ℓ,𝜖,𝜖}_′𝜉¹,𝜉₂,𝜉₃|||²𝑤⁽³⁾𝜉₃||𝑝₂||d𝜉₁d𝜉₂d𝜉₃≤𝐶(𝐺)𝜎.(5.8) This yields −⎛⎜⎜⎝_𝛼=1,2ℳ^{(𝛼)∗ℓ,𝜖,𝜖}_′,𝜎𝜉₂||𝑝₂||^1/2⊗1₂⎞⎟⎟⎠⎛⎜⎜⎝_𝛼=1,2ℳ^{(𝛼)ℓ,𝜖,𝜖}_′,𝜎𝜉₂||𝑝₂||^1/2⊗1₂⎞⎟⎟⎠d𝜉₂≥−𝐶(𝐺)𝜎.(5.9)

Combining (5.1), (5.5) with (5.9), we obtain 𝐻,𝑖𝐴_𝑛≥(1−𝑔)dΓ𝜂_𝜎𝑛²𝑤⁽²⁾−𝑔𝐶(𝐺)𝜎^𝑛.(5.10)

We have dΓ𝜂_𝜎𝑛²𝑤⁽²⁾≥𝐻^(2)0𝑛.(5.11)

By (3.76), (3.81), and (5.11) we get 𝑓_𝑛𝐻≥_𝑛𝑃−_𝑛𝐸_𝑛dΓ𝜂_𝜎𝑛²𝑤⁽²⁾𝑓_𝑛𝐻_𝑛−𝐸_𝑛≥𝛾₂⊗𝑓_𝑛𝐻^(2)0𝑛𝐻^(2)0𝑛𝑓_𝑛𝐻^(2)0𝑛𝑁²𝜎_𝑛𝑓_𝑛𝐻_𝑛−𝐸_𝑛²(5.12) for 𝑔≤𝑔^𝛿(1).

This, together with (5.10), yields for 𝑔≤𝑔^𝛿(1)𝑓_𝑛𝐻≥_𝑛−1𝐸−_𝑛𝑔𝐻_𝛿(1),𝑖𝐴_𝑛𝑓_𝑛𝐻_𝑛−𝐸_𝑛𝑁𝛾²₂𝜎_𝑛𝑓_𝑛𝐻_𝑛−𝐸_𝑛²−𝑔𝐶(𝐺)𝜎_𝑛𝑓_𝑛𝐻_𝑛−𝐸_𝑛².(5.13) Setting 𝑔^𝛿(2)=inf𝑔^𝛿(1),1−𝑔^𝛿(1)2𝐶(𝐺)𝑁𝛾²²,(5.14) we get 𝑓_𝑛𝐻_𝑛−𝐸_𝑛H,𝑖𝐴_𝑛𝑓_𝑛𝐻_𝑛−𝐸_𝑛≥1−𝑔^𝛿(1)2𝑁𝛾²²𝜎_𝑛𝑓_𝑛𝐻_𝑛−𝐸_𝑛²(5.15) for 𝑔≤𝑔^𝛿(2).

Proposition 3.9 is proved by setting 𝑔̃^𝛿(1)=𝑔^𝛿(2) and 𝐶_𝛿=(1−𝑔^𝛿(1))/2.

The proof of Theorem 3.10 is the consequence of the following two lemmas.

Lemma 5.1. Assume that the kernels 𝐺^{(𝛼)ℓ,𝜖,𝜖}_′ satisfy Hypotheses 2.1 and 3.1(ii). Then there exists a constant 𝐷>0 such that ||𝐸−𝐸_𝑛||≤𝑔𝐷𝜎_𝑛2(5.16) for 𝑛≥1 and 𝑔≤𝑔⁽²⁾.

Proof. Let 𝜙 (resp., 𝜙_𝑛) be the unique normalized ground state of 𝐻 (resp., 𝐻_𝑛). We have 𝐸𝐸−_𝑛𝐸_𝑛≤𝜙_𝑛,𝐻−𝐻_𝑛𝜙_𝑛,−𝐸≤𝜙,𝐻_𝑛−𝐻𝜙(5.17) with 𝐻−𝐻_𝑛=𝑔𝐻_𝐼𝜒_𝜎𝑛𝑝₂𝐺.(5.18) Combining (2.53) and (2.54) with (3.26)–(3.28) and (5.18), we get ‖‖𝐻−𝐻_𝑛𝜙_𝑛‖‖≤𝑔𝐾𝜒_𝜎𝑛𝑝₂𝐺𝐶_𝛽𝜂‖‖𝐻₀𝜙_𝑛‖‖+𝐵_𝛽𝜂‖‖𝐻−𝐻_𝑛,𝜙‖‖≤𝑔𝐾𝜒_𝜎𝑛𝑝₂𝐺𝐶_𝛽𝜂‖‖𝐻₀𝜙‖‖+𝐵_𝛽𝜂.(5.19) It follows from Hypothesis 3.1(ii), (4.10), and (5.19) that there exists a constant 𝐷>0 such that max‖‖𝐻−𝐻_𝑛𝜙_𝑛‖‖,‖‖𝐻−𝐻_𝑛𝜙‖‖≤𝑔𝐷𝜎_𝑛2(5.20) for 𝑛≥1 and 𝑔≤𝑔⁽²⁾.
By (5.17), this proves Lemma 5.1.

Lemma 5.2. Suppose that the kernels 𝐺^{(𝛼)ℓ,𝜖,𝜖}_′ satisfy Hypotheses 2.1 and 3.1(ii). Then there exists a constant 𝐶>0 such that ‖‖𝑓_𝑛(𝐻−𝐸)−𝑓_𝑛𝐻_𝑛−𝐸_𝑛‖‖≤𝑔𝐶𝜎_𝑛(5.21) for 𝑛≥1 and 𝑔≤𝑔⁽²⁾.

Proof. Let 𝑓(⋅) be an almost analytic extension of 𝑓(⋅) given by (3.78) satisfying ||𝜕_𝑧𝑓(𝑥+𝑖𝑦)||≤𝐶𝑦².(5.22) Note that 𝑓(𝑥+𝑖𝑦)∈𝐶^∞0(ℝ²). We thus have 𝑓(𝑠)=d𝑓(𝑧)𝑧−𝑠,d𝑓(𝑧)=−1𝜋𝜕𝜕𝑓𝑧d𝑥d𝑦.(5.23) Using the functional calculus based on this representation of 𝑓(𝑠), we get 𝑓_𝑛(𝐻−𝐸)−𝑓_𝑛𝐻_𝑛−𝐸_𝑛=𝜎_𝑛𝐻−𝐸1−𝑧𝜎_𝑛𝐻−𝐻_𝑛+𝐸_𝑛−𝐸1𝐻_𝑛−𝐸_𝑛−𝑧𝜎_𝑛d𝑓(𝑧).(5.24) 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⁾, 𝑔‖‖𝐻_𝐼𝜒_𝜎𝑛𝐺𝜓‖‖≤2𝑔𝐶𝜎_𝑛2𝐾(𝐺)𝐶_𝛽𝜂‖‖𝐻₀+1𝜓‖‖+𝐶_𝛽𝜂+𝐵_𝛽𝜂‖𝜓‖.(5.25) This yields 𝑔‖‖𝐻_𝐼𝜒_𝜎𝑛𝑝₂𝐺𝐻₀+1⁻¹‖‖≤𝑔𝐶₁𝜎_𝑛2(5.26) 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 ‖‖𝐻₀+1𝐻_𝑛−𝐸_𝑛−𝑧𝜎_𝑛⁻¹‖‖≤𝐶₂1+||Im1𝑧||𝜎_𝑛(5.27) or 𝑔≤𝑔⁽¹⁾.
Combining Lemma 5.1 and (5.24) with (5.25)–(5.27) we obtain ‖‖𝑓_𝑛(𝐻−𝐸)−𝑓_𝑛𝐻_𝑛−𝐸_𝑛‖‖≤𝑔𝐶𝜎_𝑛|||𝜕𝑓/𝜕𝑧(𝑥+𝑖𝑦)|||𝑦²d𝑥d𝑦(5.28) for some constant 𝐶>0 and for 𝑔≤𝑔⁽²⁾.
Using (5.22) and 𝑓(𝑥+𝑖𝑦)∈𝐶^∞0(ℝ²) one concludes the proof of Lemma 5.2.

We now prove Theorem 3.10.

Proof. It follows from Proposition 3.9 that 𝑓_𝑛𝐻=_𝑛𝑓−_𝑛𝐸_𝑛[𝐻,𝑖𝐴]𝑓_𝑛𝐻_𝑛−𝐸_𝑛𝐻_𝑛−𝐸_𝑛𝐻,𝑖𝐴_𝑛𝑓_𝑛𝐻_𝑛−𝐸_𝑛≥𝐶_𝛿𝛾²𝑁²𝜎_𝑛𝑓_𝑛𝐻_𝑛−𝐸_𝑛²(5.29) for 𝑛≥1 and 𝑔≤𝑔̃^𝛿(1).
This yields 𝑓_𝑛(𝐻≥−𝐶𝐸)𝐻,𝑖𝐴_𝑛𝑓_𝑛(𝐻−𝐸)_𝛿𝛾²𝑁²𝜎_𝑛𝑓_𝑛(𝐻−𝐸)²−−𝑓𝑓_𝑛(𝐻−𝐸)[𝐻,𝑖𝐴]𝑓_𝑛𝐻_𝑛−𝐸_𝑛−𝑓_𝑛(𝐻−𝐸)_𝑛+𝐶_𝛿𝐻_𝑛−𝐸_𝑛−𝑓_𝑛(𝐻−𝐸)[𝐻,𝑖𝐴]𝑓_𝑛𝐻_𝑛−𝐸_𝑛𝛾²𝑁²𝜎_𝑛𝑓_𝑛𝐻_𝑛−𝐸_𝑛−𝑓_𝑛(𝐻−𝐸)²+𝐶_𝛿𝛾²𝑁²+𝐶𝜎_𝑛𝑓_𝑛(𝐻−𝐸)𝑓_𝑛𝐻_𝑛−𝐸_𝑛−𝑓_𝑛(𝐻−𝐸)_𝛿𝛾²𝑁²𝜎_𝑛𝑓_𝑛𝐻_𝑛−𝐸_𝑛−𝑓_𝑛(𝐻−𝐸)𝑓_𝑛(𝐻−𝐸).(5.30) Combining Proposition 3.8(i) and (5.23) with (5.26) and (5.27) we show that [𝐻,𝑖𝐴]𝑓_𝑛(𝐻_𝑛−𝐸_𝑛) and 𝑓_𝑛(𝐻−𝐸)[𝐻,𝑖𝐴] are bounded operators uniformly with respect to 𝑛. This, together with Lemma 5.2, yields 𝑓_𝑛(𝐻−𝐸)[𝐻,𝑖𝐴]𝑓_𝑛(𝐻−𝐸)≥𝐶_𝛿𝛾²𝑁²𝜎_𝑛𝑓_𝑛(𝐻−𝐸)²−𝐶𝑔𝜎_𝑛(5.31) for some constant 𝐶>0 and for 𝑔≤inf(𝑔⁽²⁾,𝑔̃^𝛿(1)).
Multiplying both sides of (5.31) with 𝐸_Δ𝑛(𝐻−𝐸) we then get 𝐸_Δ𝑛(𝐻−𝐸)[𝐻,𝑖𝐴]𝐸_Δ𝑛(𝐻−𝐸)≥𝐶_𝛿𝛾²𝑁²𝜎_𝑛𝐸_Δ𝑛(𝐻−𝐸)−𝐶𝑔𝜎_𝑛𝐸_Δ𝑛(𝐻−𝐸).(5.32) Setting 𝑔̃^𝛿(2)<inf𝐶^𝛿𝐶𝑁𝛾²²,𝑔⁽²⁾,𝑔̃^𝛿(1),(5.33) Theorem 3.10 is proved with 𝐶_𝛿=𝐶_𝛿−𝐶(𝑁²/𝛾²)𝑔̃^𝛿(2)>0.

6. Proof of Theorem 3.7

We set 𝐴_𝑡=e^{𝑖𝑡𝐴−1}𝑡,𝐴ad_𝐴𝑡⋅=𝐴_𝑡,⋅_𝜎𝑡,=e^{𝑖𝑡𝐴𝜎}−1𝐴_𝜎𝑡𝑡,=e^{𝑖𝑡𝐴𝜎}−1𝑡.(6.1) The fact that 𝐻 is of class 𝐶¹(𝐴), 𝐶¹(𝐴^𝜎), and 𝐶¹(𝐴_𝜎) in (−∞,𝑚₁−𝛿/2) is the consequence of the following proposition.

Proposition 6.1. Suppose that the kernels 𝐺^{(𝛼)ℓ,𝜖,𝜖}_′ satisfy Hypotheses 2.1 and 3.1(iii.a). For every 𝜑∈𝐶^∞0((−∞,𝑚₁−𝛿/2)) and 𝑔≤𝑔₁, one then has ssuupp_{0<|𝑡|≤10<|𝑡|≤1}‖‖𝜑(𝐻),𝐴_𝑡‖‖<∞,sup_0<|𝑡|≤1‖‖𝜑(𝐻),𝐴^𝜎𝑡‖‖<∞,sup_0<|𝑡|≤1‖‖𝜑(𝐻),𝐴_𝜎𝑡‖‖<∞,‖‖𝜑(𝐻^𝜎),𝐴^𝜎𝑡‖‖<∞.(6.2)

Proof. We use the representation 𝜑(𝐻)=d𝜙(𝑧)(𝑧−𝐻)⁻¹,(6.3) where 𝜙(𝑧) is an almost analytic extension of 𝜑 with ||𝜕_𝑧𝜙(𝑥+𝑖𝑦)||≤𝐶||𝑦||²,d𝜙(𝑧)=−1𝜋𝜕𝜕𝑧𝜙(𝑧)d𝑥d𝑦.(6.4) Note that 𝜙(𝑥+𝑖𝑦)∈𝐶^∞0(ℝ²).
We get ad_𝐴𝑡𝜑(𝐻)=d𝜙(𝑧)(𝑧−𝐻)⁻¹𝐴_𝑡,𝐻(𝑧−𝐻)⁻¹.(6.5) This yields ‖‖ad_𝐴𝑡𝜑(𝐻)‖‖≤sup_0<|𝑡|≤1‖‖𝐴_𝑡,𝐻(𝑖−𝐻)⁻¹‖‖||d𝜙(𝑧)||‖‖(𝑧−𝐻)⁻¹‖‖‖‖(𝑖−𝐻)(𝑧−𝐻)⁻¹‖‖.(6.6) It is easy to prove that ||d𝜙(𝑧)||‖‖(𝑧−𝐻)⁻¹‖‖‖‖(𝑖−𝐻)(𝑧−𝐻)⁻¹‖‖≤𝐶||d𝜙(𝑧)||||Im𝑧||²<∞.(6.7)
By Proposition 3.8(b)(i) and (6.7) we finally get, for 𝑔≤𝑔₁, sup_0<|𝑡|≤1‖‖ad_𝐴𝑡𝜑(𝐻)‖‖<∞.(6.8) In a similar way we obtain, for 𝑔≤𝑔₁, ssuupp_{0<|𝑡|≤10<|𝑡|≤1}‖‖𝐴^𝜎𝑡,𝜑(𝐻)‖‖<∞,sup_0<|𝑡|≤1‖‖𝐴_𝜎𝑡,𝜑(𝐻)‖‖<∞,‖‖𝐴^𝜎𝑡,𝜑(𝐻^𝜎)‖‖<∞.(6.9)

The proof of Theorem 3.7 is the consequence of the following proposition.

Proposition 6.2. Suppose that the kernels 𝐺^{(𝛼)ℓ,𝜖,𝜖}_′ satisfy Hypotheses 2.1, 3.1(i) and 3.1(iii). One then has, for 𝑔≤𝑔₁, ssuupp_{0<|𝑡|≤10<|𝑡|≤1}‖‖𝐴_𝑡,𝐴_𝑡,𝐻(𝐻+𝑖)⁻¹‖‖<∞,sup_0<|𝑡|≤1‖‖𝐴^𝜎𝑡,𝐴^𝜎𝑡,𝐻(𝐻+𝑖)⁻¹‖‖<∞,sup_0<|𝑡|≤1‖‖𝐴_𝜎𝑡,𝐴_𝜎𝑡,𝐻(𝐻+𝑖)⁻¹‖‖<∞,‖‖𝐴^𝜎𝑡,𝐴^𝜎𝑡,𝐻^𝜎(𝐻^𝜎+𝑖)⁻¹‖‖<∞.(6.10)

Proof. We have, for every 𝜓∈𝒟(𝐻), 𝐴_𝑡,𝐴_𝑡,𝐻𝜓=1𝑡²e^{2𝑖𝑡𝐴}e^{−2𝑖𝑡𝐴}𝐻e^{2𝑖𝑡𝐴}−2e^{−𝑖𝑡𝐴}𝐻e^𝑖𝑡𝐴+𝐻𝜓.(6.11) By (3.56) we get 𝐴_𝑡,𝐴_𝑡,𝐻₀𝜓=1𝑡²e^{2𝑖𝑡𝐴}dΓ𝑤⁽²⁾∘𝜙_2𝑡−2𝑤⁽²⁾∘𝜙_𝑡+𝑤⁽²⁾𝜓,(6.12) where, for ℓ=1,2,3, 𝑤^ℓ(2)∘𝜙_2𝑡𝑝²−2𝑤^ℓ(2)∘𝜙_𝑡𝑝²+𝑤^ℓ(2)𝑝₂=||𝜙_2𝑡𝑝₂||−2||𝜙_𝑡𝑝₂||+||𝑝₂||.(6.13) We further note that 1𝑡²||||𝜙_2𝑡𝑝₂||−2||𝜙_𝑡𝑝₂||+||𝑝₂||||≤sup_{|𝑠|≤2|𝑡|}||||𝜕²𝜕𝑠²𝜕₂||𝜙_𝑠𝑝₂||||||,𝜕𝑠²||𝜙_𝑠𝑝₂||=||𝜙_𝑠𝑝₂||≤e^Γ|𝑠|||𝑝₂||.(6.14) Combining (6.12) with (6.13) and (6.14) we get ‖‖𝐴_𝑡,𝐴_𝑡,𝐻₀𝐻₀+1⁻¹sup_0<|𝑡|≤1‖‖≤e^2Γ|𝑡|,‖‖𝐴_𝑡,𝐴_𝑡,𝐻₀𝐻₀+1⁻¹‖‖≤e^2Γ.(6.15) In a similar way we obtain ssuupp_{0<|𝑡|≤10<|𝑡|≤1}‖‖𝐴^𝜎𝑡,𝐴^𝜎𝑡,𝐻₀𝐻₀+1⁻¹‖‖≤𝐶e^2Γ,‖‖𝐴_𝜎𝑡,𝐴_𝜎𝑡,𝐻₀𝐻₀+1⁻¹‖‖≤𝐶e^2Γ.(6.16) Here 𝐶 is a positive constant.
Let us now prove that sup_0<|𝑡|≤1‖‖𝐴_𝑡,𝐴_𝑡,𝐻_𝐼(𝐺)(𝐻+𝑖)⁻¹‖‖<∞.(6.17) By (3.66) and (6.11) we get, for every 𝜓∈𝒟(𝐻), 𝐴_𝑡,=𝐴_𝑡,𝐻_𝐼(𝐺)_𝛼=1,2𝜓_ℓ=_1,2,3_{𝜖≠𝜖′}e^{2𝑖𝑡𝐴}𝑡²=e^{−2𝑖𝑡𝐴}𝐻_𝐼𝐺^{(𝛼)ℓ,𝜖,𝜖′}e^{2𝑖𝑡𝐴}−2e^{−𝑖𝑡𝐴}𝐻_𝐼𝐺^{(𝛼)ℓ,𝜖,𝜖′}e^𝑖𝑡𝐴+𝐻_𝐼𝐺^{(𝛼)ℓ,𝜖,𝜖′}_𝛼=1,2𝜓_ℓ=_1,2,3_{𝜖≠𝜖′}e^{2𝑖𝑡𝐴}𝑡²𝐻^𝐼𝐺^{(𝛼)ℓ,𝜖,𝜖′}_;2𝑡−2𝐻^𝐼𝐺^{(𝛼)ℓ,𝜖,𝜖′}_;𝑡+𝐻^𝐼𝐺^{(𝛼)ℓ,𝜖,𝜖′}_;0𝜓,(6.18) where 𝐺^{(𝛼)ℓ,𝜖,𝜖}_′;𝑡𝜉₁,𝜉₂,𝜉₃==𝐷𝑒𝜙_{𝑡−𝑖𝑡𝑎}𝑝₂^1/2𝐺^{(𝛼)ℓ,𝜖,𝜖}_′𝜉₁;𝜙_𝑡𝑝₂,𝑠₂;𝜉₃𝐺^{(𝛼)ℓ,𝜖,𝜖}_′𝜉¹,𝜉₂,𝜉₃.(6.19) Combining (2.53) and (2.54) with (3.26)–(3.28) and (6.18) we get ‖‖𝐴_𝑡,𝐴_𝑡,𝐻_𝐼(𝐺)𝜓‖‖≤𝑔𝐾𝐺_𝑡𝐶_𝛽𝜂‖‖𝐻₀+𝐼𝜓‖‖+𝐶_𝛽𝜂+𝐵_𝛽𝜂‖𝜓‖.(6.20) Here 𝐾(𝐺_𝑡)>0 and 𝐾𝐺_𝑡²=_𝛼=1,2ℓ=_1,2,3_{𝜖≠𝜖′}1𝑡²‖‖𝐺^{(𝛼)ℓ,𝜖,𝜖}_′;2𝑡−2𝐺^{(𝛼)ℓ,𝜖,𝜖}_′;𝑡+𝐺^{(𝛼)ℓ,𝜖,𝜖}_′‖‖^2𝐿_{2(Σ1×Σ1×Σ2)}.(6.21) We further note that, for 0<|𝑡|≤1, 𝐾𝐺_𝑡≤sup_0<|𝑠|≤2⎛⎜⎜⎝_𝛼=1,2ℓ=_1,2,3_{𝜖≠𝜖′}‖‖‖𝜕²𝜕𝑠²𝐺^{(𝛼)ℓ,𝜖,𝜖}_′;𝑠‖‖‖^2𝐿_{2Σ1×Σ1×Σ2}⎞⎟⎟⎠^1/2.(6.22) We get 𝜕𝜕𝜕𝑡𝐺^{(𝛼)ℓ,𝜖,𝜖′;𝑡}=32e^{−𝑖𝑡𝑎}𝐺^{(𝛼)ℓ,𝜖,𝜖}_′+e^{−𝑖𝑡𝑎}𝑝²⋅∇_𝑝2𝐺^{(𝛼)ℓ,𝜖,𝜖}_′₂,𝜕𝑡²𝐺^{(𝛼)ℓ,𝜖,𝜖}_′;𝑡=94e^{−𝑖𝑡𝑎}𝐺^{(𝛼)ℓ,𝜖,𝜖}_′+72+e^{−𝑖𝑡𝑎}𝑝²⋅∇_𝑝2𝐺^{(𝛼)ℓ,𝜖,𝜖}_′_{𝑖,𝑗=1,2,3}e^{−𝑖𝑡𝑎}𝑝^2,𝑖𝑝_2,𝑗𝜕^2𝑝_{2,𝑖𝑝2,𝑗}𝐺^{(𝛼)ℓ,𝜖,𝜖}_′.(6.23) Recall that e^{−𝑖𝑡𝑎} is an one parameter group of unitary operators in 𝐿²(Σ₁×Σ₁×Σ₂).
Combining Hypothesis 3.1(iii.a) and (iii.b) with (6.20)–(6.23) we finally get sup_0<|𝑡|≤1‖‖𝐴_𝑡,𝐴_𝑡,𝐻_𝐼(𝐺)𝐻₀+1⁻¹‖‖<∞.(6.24) In view of 𝒟(𝐻)=𝒟(𝐻₀) the operators 𝐻₀(𝐻+𝑖)⁻¹ and 𝐻(𝐻₀−1)⁻¹ are bounded and we obtain suspu_0<|𝑡|≤1p_0<|𝑡|≤1‖‖𝐴_𝑡,𝐴_𝑡,𝐻₀(𝐻+𝑖)⁻¹‖‖<∞,‖‖𝐴_𝑡,𝐴_𝑡,𝐻_𝐼(𝐺)(𝐻+𝑖)⁻¹‖‖<∞.(6.25) This yields sup_0<|𝑡|≤1‖‖𝐴_𝑡,𝐴_𝑡,𝐻(𝐻+𝑖)⁻¹‖‖<∞(6.26) for 𝑔≤𝑔₁.
Let 𝑉(𝑝₂) denote any of the two 𝐶^∞-vector fields 𝑣^𝜎(𝑝₂) and 𝑣_𝜎(𝑝₂) and let 𝑎̃ denote the corresponding 𝑎^𝜎 and 𝑎_𝜎 operators. We get 𝜕𝜕𝑡²²=1e^−𝑖𝑎_𝑡̃𝐺^{(𝛼)ℓ,𝜖,𝜖}_′𝜉¹,𝜉₂,𝜉₃4+1e^−𝑖𝑎_𝑡̃div𝑉𝑝²²𝐺^{(𝛼)ℓ,𝜖,𝜖}_′𝜉¹,𝜉₂,𝜉₃2+1e^−𝑖𝑎_𝑡̃div𝑉𝑝²𝑉𝑝₂⋅∇_𝑝2𝐺^{(𝛼)ℓ,𝜖,𝜖}_′𝜉¹,𝜉₂,𝜉₃2e^−𝑖𝑎_𝑡̃³_𝑖,𝑗=1𝑉^𝑖𝑝₂𝜕^2𝑝_{2,𝑖𝑝2,𝑗}𝑉_𝑗𝑝₂𝐺^{(𝛼)ℓ,𝜖,𝜖}_′𝜉¹+1,𝜉₂,𝜉₃2e^−𝑖𝑎_𝑡̃³_𝑖,𝑗=1𝑉_𝑖𝑝₂𝜕𝑉^𝑗𝜕𝑝_2,𝑖𝑝₂𝜕𝑝𝜕_2,𝑗𝐺^{(𝛼)ℓ,𝜖,𝜖}_′𝜉¹+1,𝜉₂,𝜉₃2e^−𝑖𝑎_𝑡̃³_𝑖,𝑗=1𝑉_𝑖𝑝₂𝑉_𝑗𝑝₂𝜕²𝜕𝑝_2,𝑖𝜕𝑝_2,𝑗𝐺^{(𝛼)ℓ,𝜖,𝜖}_′𝜉¹,𝜉₂,𝜉₃.(6.27) Combining the properties of the 𝐶^∞ fields 𝑣^𝜎(𝑝₂) and 𝑣_𝜎(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), ssuupp_{0<|𝑡|≤10<|𝑡|≤1}‖‖𝐴^𝜎𝑡,𝐴^𝜎𝑡,𝐻(𝐻+𝑖)⁻¹‖‖<∞,‖‖𝐴_𝜎𝑡,𝐴_𝜎𝑡,𝐻(𝐻+𝑖)⁻¹‖‖<∞(6.28) for 𝑔≤𝑔₁.
Similarly, by mimicking the proof of (6.28), we easily get, for 𝑔≤𝑔₁, sup_0<|𝑡|≤1‖‖𝐴^𝜎𝑡,𝐴^𝜎𝑡,𝐻^𝜎(𝐻^𝜎+𝑖)⁻¹‖‖<∞.(6.29) This concludes the proof of Proposition 6.2.

We now prove Theorem 3.7.

Proof of Theorem 3.7. 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: sup_0<|𝑡|≤1‖‖𝐴_𝑡,𝐴_𝑡,𝜑(𝐻)‖‖<∞,(6.30)ssuupp_{0<|𝑡|≤10<|𝑡|≤1}‖‖𝐴^𝜎𝑡,𝐴^𝜎𝑡,𝜑(𝐻)‖‖<∞,sup_0<|𝑡|≤1‖‖𝐴_𝜎𝑡,𝐴_𝜎𝑡,𝜑(𝐻)‖‖<∞,‖‖𝐴^𝜎𝑡,𝐴^𝜎𝑡,𝜑(𝐻^𝜎)‖‖<∞(6.31) for every 𝜑∈𝐶^∞0((−∞,𝑚₁−𝛿/2)) and for 𝑔≤𝑔₁.
Let us prove (6.30). The estimates (6.31) can be proved similarly.
To this end, let 𝜙 be an almost analytic extension of 𝜑 satisfying 𝜑(𝐻)=(𝑧−𝐻||)𝜕_𝑧−1𝜙(𝑥+𝑖𝑦)||≤𝐶||𝑦||³,d𝜙(𝑧),d𝜙(𝑧)=−1𝜋𝜕𝜕𝑧𝜙(𝑧)d𝑥d𝑦.(6.32) It follows that =𝐴_𝑡𝐴(_𝑡𝑧,𝜑−(𝐻𝐻))₋₁𝐴_𝑡𝐴_𝑡,𝐻(𝑧−𝐻)⁻¹+2(𝑧−𝐻)⁻¹𝐴_𝑡,𝐻(𝑧−𝐻)⁻¹𝐴_𝑡,𝐻(𝑧−𝐻)⁻¹d𝜙(𝑧).(6.33)
We note that ‖‖(𝐻+𝑖)(𝐻−𝑧)⁻¹‖‖≤𝐶||Im𝑧||,for𝑧∈supp𝜙.(6.34)
We also have sup_0<|𝑡|≤1‖‖‖(𝑧−𝐻)₋₁≤sup_0<|𝑡|≤1𝐴_𝑡𝐴_𝑡,𝐻(𝑧−𝐻)⁻¹d𝜙(𝑧)‖‖‖‖‖𝐴^𝑡𝐴_𝑡,𝐻(𝐻+𝑖)⁻¹‖‖‖‖(𝐻+𝑖)(𝑧−𝐻)⁻¹‖‖||d𝜙(𝑧)||≤𝐶sup_0<|𝑡|≤1||Im𝑧||‖‖𝐴_𝑡,𝐴_𝑡,𝐻(𝐻+𝑖)⁻¹‖‖||d𝜙(𝑧)||||Im𝑧||².(6.35)
Therefore, combining Proposition 3.8(b)(i) and (6.34) we obtain sup_0<|𝑡|≤1‖‖‖d𝜙(𝑧)(𝐻−𝑧)⁻¹𝐴_𝑡,𝐻(𝐻−𝑧)⁻¹𝐴_𝑡,𝐻(𝐻−𝑧)⁻¹=sup_0<|𝑡|≤1‖‖‖‖‖‖(𝐻−𝑧)₋₁𝐴_𝑡,𝐻(𝐻+𝑖)⁻¹(𝐻+𝑖)(𝐻−𝑧)⁻¹𝐴_𝑡,𝐻(𝐻+𝑖)⁻¹(𝐻+𝑖)(𝐻−𝑧)⁻¹‖‖‖≤𝐶||d𝜙(𝑧)d𝜙(𝑧)||||𝑦||³sup_0<|𝑡|≤1‖‖[𝐴_𝑡,𝐻](𝐻+𝑖)⁻¹‖‖²<∞.(6.36) 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 Propositions 3.8(b), 6.1, and 6.2 that 𝐻 is locally of class 𝐶²(𝐴^𝜎) and 𝐶²(𝐴_𝜎) in (−∞,𝑚₁−𝛿/2) and that 𝐻^𝜎 is locally of class 𝐶²(𝐴^𝜎) in (−∞,𝑚₁−𝛿/2), for 𝑔≤𝑔₁. This ends the proof of Theorem 3.7.

7. Proof of Theorem 3.4

By (3.91), ⋃_𝑛≥1((𝛾−𝜖_𝛾)²𝜎_𝑛,(𝛾+𝜖_𝛾)𝜎_𝑛)) 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 𝑔_𝛿=𝑔̃^𝛿(2), where 𝑔̃^𝛿(2) is given in Theorem 3.10. Theorem 3.4(iii) follows from [30, Theorem  25].

Appendix

In this appendix, we will prove Proposition 3.5. We apply the method developed in [3] because every infrared cutoff Hamiltonian that one considers has a ground state energy which is a simple eigenvalue.

Let, for 𝑛≥0, Σ_𝑛+11𝑛𝔉^𝜎𝑛=𝔉^𝑛,𝔉_{𝑛+1ℓ,2,𝑛}=Σ₁∩𝑝₂;𝜎_𝑛+1≤||𝑝₂||<𝜎_𝑛,=𝔉_𝑎𝔉𝐿²Σ^𝑛+11𝑛⊗𝔉_𝑎𝐿²Σ^𝑛+11𝑛_𝑛𝑛+1,=³_ℓ=1𝔉^{𝑛+1ℓ,2,𝑛}.(A.1)

We have 𝔉^𝑛+1≃𝔉^𝑛⊗𝔉^𝑛𝑛+1.(A.2) Let Ω^𝑛 (resp., Ω^𝑛𝑛+1) be the vacuum state in 𝔉^𝑛 (resp., in 𝔉^𝑛𝑛+1). We now set 𝐻^𝑛+10𝑛=𝐻⁰⁽¹⁾+𝐻⁰⁽³⁾+³_ℓ=1_𝜖=±_{𝜎𝑛+1≤|𝑝2|<𝜎𝑛}𝑤^ℓ(2)𝜉₂𝑐^∗ℓ,𝜖𝜉₂𝑐_ℓ,𝜖𝜉₂d𝜉₂.(A.3) The operator 𝐻^𝑛+10𝑛 is a self-adjoint operator in 𝔉^𝑛𝑛+1.

Let us denote by 𝐻^𝑛𝐼 and 𝐻^{𝑛+1𝐼𝑛} the interaction 𝐻_𝐼 given by (2.23) and (2.24) but associated with the following kernels: 𝜒^𝜎𝑛𝑝₂𝐺^{(𝛼)ℓ,𝜖,𝜖}_′𝜉₁,𝜉₂,𝜉₃,(A.4)𝜒^𝜎𝑛+1𝑝₂−𝜒^𝜎𝑛𝑝₂𝐺^{(𝛼)ℓ,𝜖,𝜖}_′𝜉₁,𝜉₂,𝜉₃,(A.5) respectively, where 𝜒^𝜎𝑛+1 is defined by (3.13).

Let for 𝑛≥0, 𝐻𝐻^𝑛+_𝑛+=𝐻^𝑛−𝐸^𝑛,=𝐻^𝑛+⊗1^𝑛𝑛+1+1_𝑛⊗𝐻^𝑛+10𝑛.(A.6) The operators 𝐻^𝑛+ and 𝐻^𝑛+ are self-adjoint operators in 𝔉^𝑛 and 𝔉^𝑛+1, respectively. Here 1^𝑛 and 1^𝑛𝑛+1 are the identity operators in 𝔉^𝑛 and 𝔉^𝑛𝑛+1, respectively.

Combining (2.53) and (2.54) with (3.26)–(3.28) we obtain for 𝑛≥0, 𝑔‖‖𝐻^𝑛𝐼𝜓‖‖≤𝑔𝐾(𝐺)𝐶_𝛽𝜂‖‖𝐻₀𝜓‖‖+𝐵_𝛽𝜂‖𝜓‖(A.7) for every 𝜓∈𝒟(𝐻^𝑛0)⊂𝔉^𝑛.

It follows from [33, Section  V, Theorem  4.11] that 𝐻^𝑛≥−𝑔𝐾(𝐺)𝐵^𝛽𝜂1−𝑔₁𝐾(𝐺)𝐶_𝛽𝜂≥−𝑔¹𝐾(𝐺)𝐵_𝛽𝜂1−𝑔₁𝐾(𝐺)𝐶_𝛽𝜂𝐸_𝑛,≥−𝑔𝐾(𝐺)𝐵^𝛽𝜂1−𝑔₁𝐾(𝐺)𝐶_𝛽𝜂.(A.8)

We have (Ω^𝑛,𝐻^𝑛Ω^𝑛)=0.(A.9) Therefore 𝐸^𝑛≤0,(A.10)||𝐸^𝑛||≤𝑔𝐾(𝐺)𝐵^𝛽𝜂1−𝑔₁𝐾(𝐺)𝐶_𝛽𝜂.(A.11)

Let 𝐾^𝑛𝑛+1(𝐺)=𝐾1_{𝜎𝑛+1≤||𝑝2||≤2𝜎𝑛}𝐺.(A.12)

Combining (2.53) and (2.54) with (3.26) and (A.12) we obtain for 𝑛≥0𝑔‖‖𝐻^{𝑛+1𝐼𝑛}𝜓‖‖≤𝑔𝐾^𝑛𝑛+1(𝐺)𝐶_𝛽𝜂‖‖𝐻^0𝑛+1𝜓‖‖+𝐵_𝛽𝜂‖𝜓‖(A.13) for 𝜓∈𝒟(𝐻^0𝑛+1)⊂𝔉^𝑛+1, where we remind that 𝐻^0𝑛+1=𝐻₀|_𝔉^𝜎𝑛+1 as defined in (3.24).

We have, for every 𝜓∈𝒟(𝐻^0𝑛+1), 𝐻^0𝑛+1𝜓=𝐻^𝑛+𝜓+𝐸^𝑛𝜓−𝑔𝐻^𝑛𝐼⊗1^𝑛𝑛+1𝜓,(A.14) and by (A.7) 𝑔‖‖𝐻^𝑛𝐼⊗1^𝑛𝑛+1𝜓‖‖≤𝑔𝐾(𝐺)𝐶_𝛽𝜂‖‖𝐻^0𝑛+1𝜓‖‖+𝐵_𝛽𝜂‖𝜓‖.(A.15)

In view of (A.11) and (A.14) it follows from (A.15) that 𝑔‖‖𝐻≤^𝑛𝐼⊗𝑔𝐾1(𝐺^𝑛𝑛+1)𝜓𝐶‖‖_𝛽𝜂1−𝑔₁𝐾(𝐺)𝐶_𝛽𝜂‖‖𝐻^𝑛+𝜓‖‖+𝑔𝐾(𝐺)𝐵^𝛽𝜂1−𝑔₁𝐾(𝐺)𝐶_𝛽𝜂1+1−𝑔𝐾𝑔₁(𝐾𝐺()𝐺𝐵)𝐶^𝛽𝜂_𝛽𝜂‖𝜓‖.(A.16) By (3.29), (A.13), (A.14), and (A.16) we finally get 𝑔‖‖𝐻^{𝑛+1𝐼𝑛}𝜓‖‖≤𝑔𝐾^𝑛𝑛+1(𝐺)𝐶_𝛽𝜂‖‖𝐻^𝑛+𝜓‖‖+𝐵_𝛽𝜂‖𝜓‖.(A.17)

For 𝑛≥0, a straightforward computation yields 𝐾^𝑛𝑛+1(𝐺)≤𝜎_𝑛𝐾(𝐺)≤sup4Λ𝛾2𝑚₁−𝛿,1𝐾(𝐺)𝜎𝛾^𝑛+1.(A.18)

Recall that, for 𝑛≥0, 𝜎_𝑛+1<𝑚₁.(A.19)

By (A.17), (A.18), and (A.19), we get, for 𝜓∈𝒟(𝐻₀), 𝑔‖‖𝐻^{𝑛+1𝐼𝑛}𝜓‖‖≤𝑔𝐾^𝑛𝑛+1(𝐺)𝐶_𝛽𝜂‖‖𝐻^𝑛++𝜎_𝑛+1𝜓‖‖+𝐶^𝛽𝜂𝑚₁+𝐵_𝛽𝜂‖𝜓‖,(A.20) and for 𝜙∈𝔉, 𝑔‖‖‖𝐻^{𝑛+1𝐼𝑛}𝐻^𝑛++𝜎_𝑛+1⁻¹𝜙‖‖‖≤𝑔𝐾^𝑛𝑛+1(𝐺)𝐶_𝛽𝜂+𝑚¹𝐶_𝛽𝜂+𝐵_𝛽𝜂≤𝑔𝜎_𝑛+1‖𝜙‖𝛾sup2𝑚4₁Λ−𝛾𝛿,1𝐾(𝐺)2𝑚¹𝐶_𝛽𝜂+𝐵_𝛽𝜂‖𝜙‖.(A.21)

Thus, by (A.21), the operator 𝐻^{𝑛+1𝐼𝑛}(𝐻^𝑛++𝜎_𝑛+1)⁻¹ is bounded and 𝑔‖‖‖𝐻^{𝑛+1𝐼𝑛}𝐻^𝑛++𝜎_𝑛+1⁻¹‖‖‖≤𝑔𝐷𝛾,(A.22) where 𝐷 is given by (see (3.32)) 𝐷=sup4Λ𝛾2𝑚₁−𝛿,1𝐾(𝐺)2𝑚¹𝐶_𝛽𝜂+𝐵_𝛽𝜂.(A.23)

This yields, for 𝜓∈𝒟(𝐻^𝑛+), 𝑔‖‖𝐻^{𝑛+1𝐼𝑛}𝜓‖‖≤𝑔𝐷𝛾‖‖𝐻^𝑛++𝜎_𝑛+1𝜓‖‖.(A.24) Hence it follows from [33, Section  V, Theorems  4.11 and  4.12] that 𝑔||𝐻^{𝑛+1𝐼𝑛}𝜓,𝜓||≤𝑔𝐷𝛾𝐻^𝑛++𝜎_𝑛+1𝜓,𝜓.(A.25) Let 𝑔^𝛿(2)>0 be such that 𝑔^𝛿(2)𝐷𝛾<1,𝑔^𝛿(2)≤𝑔^𝛿(1).(A.26) By (A.25) we get, for 𝑔≤𝑔^𝛿(2), 𝐻^𝑛+1=𝐻^𝑛++𝐸^𝑛+𝑔𝐻^{𝑛+1𝐼𝑛}≥𝐸^𝑛−𝑔𝐷𝛾𝜎^𝑛+1+1−𝑔𝐷𝛾𝐻^𝑛+.(A.27) Because (1−𝑔𝐷/𝛾)𝐻^𝑛+≥0 we get from (A.27) 𝐸^𝑛+1≥𝐸^𝑛−𝑔𝐷𝛾𝜎^𝑛+1,𝑛≥0.(A.28) Suppose that 𝜓^𝑛∈𝔉^𝑛 satisfies ‖𝜓^𝑛‖=1 and, for 𝜖>0, (𝜓^𝑛,𝐻^𝑛𝜓^𝑛)≤𝐸^𝑛+𝜖.(A.29) Let 𝜓^𝑛+1=𝜓^𝑛⊗Ω^𝑛𝑛+1∈𝔉^𝑛+1.(A.30) We obtain 𝐸^𝑛+1≤𝜓^𝑛+1,𝐻^𝑛+1𝜓^𝑛+1≤𝐸^𝑛+𝜖+𝑔𝜓^𝑛+1,𝐻^{𝑛+1𝐼𝑛}𝜓^𝑛+1.(A.31) By (A.25), (A.29), (A.30), and (A.31) we get, for every 𝜖>0, 𝐸^𝑛+1≤𝐸^𝑛+𝜖1+𝑔𝐷𝛾+𝑔𝛾𝐷𝜎^𝑛+1,(A.32) where 𝑔≤𝑔^𝛿(2).

This yields 𝐸^𝑛+1≤𝐸^𝑛+𝑔𝐷𝛾𝜎^𝑛+1,(A.33) and by (A.28), we obtain ||𝐸^𝑛−𝐸^𝑛+1||≤𝑔𝐷𝛾𝜎^𝑛+1.(A.34)

For 𝑛=0, since 𝜎₀=Λ, remind that 𝐻⁰⁰=𝐻^0𝑛=0=𝐻^𝜎0₀=𝐻₀|_𝔉^Λ. Thus, the ground state energy of 𝐻⁰⁰ is 0 and it is a simple isolated eigenvalue of 𝐻⁰⁰ with Ω⁰, the vacuum in 𝔉⁰, as eigenvector. Moreover, since Λ>𝑚₁, inf𝜎𝐻⁰⁰⧵{0}=𝑚₁,(A.35) thus (0,𝑚₁) belongs to the resolvent set of 𝐻⁰⁰.

By Hypothesis 3.1(iv) we have 𝐻⁰=𝐻⁰⁰. Hence 𝐸⁰={0} is a simple isolated eigenvalue of 𝐻⁰ and 𝐻⁰=𝐻⁰⁺. We finally get inf𝜎𝐻⁰⁺−{0}=𝑚₁>𝑚₁−𝛿2=𝜎¹.(A.36)

We now prove Proposition 3.5 by induction in 𝑛∈ℕ^∗. Suppose that 𝐸^𝑛 is a simple isolated eigenvalue of 𝐻^𝑛 such that inf𝜎𝐻^𝑛+⧵{0}≥1−3𝑔𝐷𝛾𝜎^𝑛,𝑛≥1.(A.37) Since (3.36) gives 𝜎_𝑛+1<(1−3𝑔𝐷/𝛾)𝜎_𝑛 for 𝑔≤𝑔^𝛿(2), 0 is also a simple isolated eigenvalue of 𝐻^𝑛+ such that inf𝜎𝐻^𝑛+⧵{0}≥𝜎^𝑛+1.(A.38) We must now prove that 𝐸^𝑛+1 is a simple isolated eigenvalue of 𝐻^𝑛+1 such that inf𝜎𝐻^+𝑛+1⧵{0}≥1−3𝑔𝐷𝛾𝜎^𝑛+1.(A.39) Let 𝜆^(𝑛+1)=sup_{𝜓∈𝔉𝑛+1;𝜓≠0}inf_{(𝜙,𝜓)=0;𝜙∈𝒟(𝐻𝑛+1);‖𝜙‖=1}𝜙,𝐻^+𝑛+1𝜙.(A.40) By (A.27) and (A.33), we obtain, in 𝔉^𝑛+1, 𝐻^+𝑛+1≥𝐸^𝑛−𝐸^𝑛+1−𝑔𝐷𝛾𝜎^𝑛+1+1−𝑔𝐷𝛾𝐻^𝑛+≥1−𝑔𝐷𝛾𝐻^𝑛+−2𝑔𝐷𝛾𝜎^𝑛+1.(A.41) By (A.30), 𝜓^𝑛+1 is the unique ground state of 𝐻^𝑛+, and by (A.38) and (A.41), we have, for 𝑔≤𝑔^𝛿(2), 𝜆^(𝑛+1)≥inf_(𝜙,𝜓^𝑛+1_{)=0;𝜙∈𝒟(𝐻}^𝑛+1_{);‖𝜙‖=1}≥1−𝑔𝐷𝜙,𝐻^+𝑛+1𝜙𝛾𝜎^𝑛+1−2𝑔𝐷𝛾𝜎^𝑛+1=1−3𝑔𝐷𝛾𝜎^𝑛+1>0.(A.42) This concludes the proof of Proposition 3.5 by choosing 𝑔_𝛿=𝑔^𝛿(2), if one proves that 𝐻¹ satisfies Proposition 3.5. By noting that 0 is a simple isolated eigenvalue of 𝐻⁰⁺ such that inf(𝜎(𝐻⁰⁺)⧵{0})=𝜎₁, we prove that 𝐸¹ is indeed an isolated simple eigenvalue of 𝐻¹ such that inf(𝜎(𝐻¹⁺)⧵{0})≥(1−3𝑔𝐷/𝛾)𝜎₁ by mimicking the proof given above for 𝐻^+𝑛+1.

Acknowledgments

The second author wishes to thank Laurent Amour and Benoît Grébert for helpful discussions. The authors also thank Walter Aschbacher for valuable remarks. The work was done partially while J.-M. B. was visiting the Institute for Mathematical Sciences, National University of Singapore in 2008. The visit was supported by the institute.