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Advances in Mathematical Physics
VolumeΒ 2009Β (2009), Article IDΒ 978903, 52 pages
Research Article

Spectral Theory for a Mathematical Model of the Weak Interactionβ€”Part I: The Decay of the Intermediate Vector Bosons π‘ŠΒ±

1Centre de Physique ThΓ©orique, Centre National de la Recherche Scientique (CNRS), Luminy Case 907, 13288 Marseille Cedex 9, France
2DΓ©partement de MathΓ©matiques, UniversitΓ© du Sud Toulon-Var, 83957 La Garde Cedex, France
3Centre de MathΓ©matiques AppliquΓ©es, Γ‰cole Polytechnique, UMR-CNRS 7641, 91128 Palaiseau Cedex, France

Received 28 April 2009; Accepted 13 August 2009

Academic Editor: ValentinΒ Zagrebnov

Copyright Β© 2009 J.-M. Barbaroux and J.-C. Guillot. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 π‘ŠΒ± 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 π‘Š+ and π‘Šβˆ’ 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 π‘’βˆ’ and the positron 𝑒+, together with the associated neutrino πœˆπ‘’ and antineutrino πœˆπ‘’, the muons πœ‡βˆ’ and πœ‡+ together with the associated neutrino πœˆπœ‡ and antineutrino πœˆπœ‡, and the tau leptons πœβˆ’ and 𝜏+ together with the associated neutrino 𝜈𝜏 and antineutrino 𝜈𝜏.

It follows from the Standard Model that neutrinos and antineutrinos are massless particles. Neutrinos are left handed, that is, neutrinos have helicity βˆ’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 π‘Š+ and π‘Šβˆ’ 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 Β±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 π‘ŠΒ± 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 SchrΓΆdinger representation the interaction is given by (see [1, page 159, equation (4.139)] and [2, page 308, equation (21.3.20)]) ξ€œπΌ=d3π‘₯Ψ𝑒(π‘₯)𝛾𝛼1βˆ’π›Ύ5ξ€ΈΞ¨πœˆπ‘’(π‘₯)π‘Šπ›Ό(ξ€œπ‘₯)+d3π‘₯Ξ¨πœˆπ‘’(π‘₯)𝛾𝛼1βˆ’π›Ύ5Ψ𝑒(π‘₯)π‘Šπ›Ό(π‘₯)βˆ—,(2.1) where 𝛾𝛼, 𝛼=0,1,2,3 and 𝛾5 are the Dirac matrices and Ξ¨β‹…(π‘₯) and Ξ¨β‹…(π‘₯) are the Dirac fields for π‘’βˆ’, 𝑒+, πœˆπ‘’, and πœˆπ‘’.

We have Ψ𝑒1(π‘₯)=2πœ‹3/2𝑠=Β±1/2ξ€œd3𝑝𝑏𝑒,+(𝑝,𝑠)𝑒(𝑝,𝑠)βˆšπ‘0e𝑖𝑝⋅π‘₯+π‘βˆ—π‘’,βˆ’(𝑝,𝑠)𝑣(𝑝,𝑠)βˆšπ‘0eβˆ’π‘–π‘β‹…π‘₯ξƒͺ,Ψ𝑒(π‘₯)=Ψ𝑒(π‘₯)†𝛾0.(2.2) Here 𝑝0=(|𝑝|2+π‘š2𝑒)1/2 where π‘šπ‘’>0 is the mass of the electron, and 𝑒(𝑝,𝑠) and 𝑣(𝑝,𝑠) are the normalized solutions to the Dirac equation (see [1, Appendix]).

The operators 𝑏𝑒,+(𝑝,𝑠) and π‘βˆ—π‘’,+(𝑝,𝑠) (resp., 𝑏𝑒,βˆ’(𝑝,𝑠) and π‘βˆ—π‘’,βˆ’(𝑝,𝑠)) are the annihilation and creation operators for the electrons (resp., the positrons) satisfying the anticommutation relations (see below).

Similarly we define Ξ¨πœˆπ‘’(π‘₯) and Ξ¨πœˆπ‘’(π‘₯) by substituting the operators π‘πœˆπ‘’,Β±(𝑝,𝑠) and π‘βˆ—πœˆπ‘’,Β±(𝑝,𝑠) for 𝑏𝑒,Β±(𝑝,𝑠) and π‘βˆ—π‘’,Β±(𝑝,𝑠) with 𝑝0=|𝑝|. The operators π‘πœˆπ‘’,+(𝑝,𝑠) and π‘βˆ—πœˆπ‘’,+(𝑝,𝑠) (resp., π‘πœˆπ‘’,βˆ’(𝑝,𝑠) and π‘βˆ—πœˆπ‘’,βˆ’(𝑝,𝑠)) are the annihilation and creation operators for the neutrinos associated with the electrons (resp., the antineutrinos).

For the π‘Šπ›Ό fields we have (see [24, Section  5.3]) π‘Šπ›Όξ‚€1(π‘₯)=2πœ‹3/2ξ“πœ†=βˆ’1,0,1ξ€œd3π‘˜βˆš2π‘˜0ξ€·πœ–π›Ό(π‘˜,πœ†)π‘Ž+(π‘˜,πœ†)eπ‘–π‘˜β‹…π‘₯+πœ–βˆ—π›Ό(π‘˜,πœ†)π‘Žβˆ—βˆ’(π‘˜,πœ†)eβˆ’π‘–π‘˜β‹…π‘₯ξ€Έ.(2.3) Here π‘˜0=(|π‘˜|2+π‘š2π‘Š)1/2 where π‘šπ‘Š>0 is the mass of the bosons π‘ŠΒ±. π‘Š+ is the antiparticule of π‘Šβˆ’. The operators π‘Ž+(π‘˜,πœ†) and π‘Žβˆ—+(π‘˜,πœ†) (resp., π‘Žβˆ’(π‘˜,πœ†) and π‘Žβˆ—βˆ’(π‘˜,πœ†)) are the annihilation and creation operators for the bosons π‘Šβˆ’ (resp., π‘Š+) satisfying the canonical commutation relations. The vectors πœ–π›Ό(π‘˜,πœ†) are the polarizations of the massive spin 1 bosons π‘ŠΒ± (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(π‘₯) such that π‘”βˆˆπΏ1(ℝ3), together with momentum cutoffs πœ’(𝑝) and 𝜌(π‘˜) for the Dirac fields and the π‘Šπœ‡ 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 πœ’ξ€·π‘1ξ€Έπœ’ξ€·π‘2ξ€ΈπœŒξ€·π‘(π‘˜)̂𝑔1+𝑝2ξ€Έ,βˆ’π‘˜(2.4) where ̂𝑔 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: β€œThe 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, β„“βˆˆ{1,2,3} denotes each species of leptons. β„“=1 denotes the electron π‘’βˆ’ the positron 𝑒+ and the neutrinos πœˆπ‘’, πœˆπ‘’. β„“=2 denotes the muons πœ‡βˆ’, πœ‡+ and the neutrinos πœˆπœ‡ and πœˆπœ‡, and β„“=3 denotes the tau-leptons and the neutrinos 𝜈𝜏 and 𝜈𝜏.

Let πœ‰1=(𝑝1,𝑠1) be the quantum variables of a massive lepton, where 𝑝1βˆˆβ„3 and 𝑠1∈{βˆ’1/2,1/2} is the spin polarization of particles and antiparticles. Let πœ‰2=(𝑝2,𝑠2) be the quantum variables of a massless lepton where 𝑝2βˆˆβ„3 and 𝑠2∈{βˆ’1/2,1/2} is the helicity of particles and antiparticles, and, finally, let πœ‰3=(π‘˜,πœ†) be the quantum variables of the spin 1 bosons π‘Š+ and π‘Šβˆ’ where π‘˜βˆˆβ„3 and πœ†βˆˆ{βˆ’1,0,1} is the polarization of the vector bosons (see [24, Section  5]). We set Ξ£1=ℝ3Γ—{βˆ’1/2,1/2} for the leptons and Ξ£2=ℝ3Γ—{βˆ’1,0,1} for the bosons. Thus 𝐿2(Ξ£1) is the Hilbert space of each lepton and 𝐿2(Ξ£2) is the Hilbert space of each boson. The scalar product in 𝐿2(Σ𝑗), 𝑗=1,2 is defined by (ξ€œπ‘“,𝑔)=Σ𝑗𝑓(πœ‰)𝑔(πœ‰)dπœ‰,𝑗=1,2.(2.5) Here ξ€œΞ£1dξ“πœ‰=𝑠=+1/2,βˆ’1/2ξ€œdξ€œπ‘,Ξ£2dξ“πœ‰=πœ†=0,1,βˆ’1ξ€œdξ€·π‘˜,𝑝,π‘˜βˆˆβ„3ξ€Έ.(2.6)

The Hilbert space for the weak decay of the vector bosons π‘Š+ and π‘Šβˆ’ is the Fock space for leptons and bosons that we now describe.

Let 𝔖 be any separable Hilbert space. Let β¨‚π‘›π‘Žπ”– (resp., ⨂𝑛𝑠𝔖) denote the antisymmetric (resp., symmetric) 𝑛th tensor power of 𝔖. The fermionic (resp., bosonic) Fock space over 𝔖, denoted by π”‰π‘Ž(𝔖) (resp., 𝔉𝑠(𝔖)), is the direct sum π”‰π‘Ž(𝔖)=βˆžξΆπ‘›π‘›=0ξ·π‘Žπ”–ξƒ©resp.,𝔉𝑠(𝔖)=βˆžξΆπ‘›π‘›=0𝑠𝔖ξƒͺ,(2.7) where ⨂0π‘Žβ¨‚π”–=0𝑠𝔖≑ℂ. The state Ξ©=(1,0,0,…,0,…) denotes the vacuum state in π”‰π‘Ž(𝔖) and in 𝔉𝑠(𝔖).

For every β„“, 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, 𝔉ℓ=4ξ·π”‰π‘Žξ€·πΏ2ξ€·Ξ£1ξ€Έξ€Έβ„“=1,2,3.(2.8) We have 𝔉ℓ=ξΆπ‘žβ„“β‰₯0,π‘žβ„“β‰₯0,π‘Ÿβ„“β‰₯0,π‘Ÿβ„“β‰₯0𝔉(π‘žβ„“,π‘žβ„“,π‘Ÿβ„“,π‘Ÿβ„“)β„“(2.9) with 𝔉(π‘žβ„“,π‘žβ„“,π‘Ÿβ„“,π‘Ÿβ„“)β„“=ξƒ©π‘žβ„“ξ·π‘ŽπΏ2ξ€·Ξ£1ξ€ΈξƒͺβŠ—βŽ›βŽœβŽœβŽπ‘žβ„“ξ·π‘ŽπΏ2ξ€·Ξ£1ξ€ΈβŽžβŽŸβŽŸβŽ βŠ—ξƒ©π‘Ÿβ„“ξ·π‘ŽπΏ2ξ€·Ξ£1ξ€ΈξƒͺβŠ—βŽ›βŽœβŽœβŽπ‘Ÿβ„“ξ·π‘ŽπΏ2ξ€·Ξ£1ξ€ΈβŽžβŽŸβŽŸβŽ .(2.10) Here π‘žβ„“ (resp., π‘žβ„“) is the number of massive fermionic particle (resp., antiparticles) and π‘Ÿβ„“ (resp., π‘Ÿβ„“) is the number of neutrinos (resp., antineutrinos). The vector Ξ©β„“ is the associated vacuum state. The fermionic Fock space denoted by 𝔉𝐿 for the leptons is then 𝔉𝐿=3ℓ=1𝔉ℓ,(2.11) and Ω𝐿=⨂3β„“=1Ξ©β„“ is the vacuum state.

The bosonic Fock space for the vector bosons π‘Š+ and π‘Šβˆ’, denoted by π”‰π‘Š, is then π”‰π‘Š=𝔉𝑠𝐿2ξ€·Ξ£2ξ€Έξ€ΈβŠ—π”‰π‘ ξ€·πΏ2ξ€·Ξ£2≃𝔉𝑠𝐿2ξ€·Ξ£2ξ€ΈβŠ•πΏ2ξ€·Ξ£2.ξ€Έξ€Έ(2.12) We have π”‰π‘Š=𝑑β‰₯0,𝑑β‰₯0𝔉(𝑑,π‘Šπ‘‘),(2.13) where 𝔉(𝑑,π‘Šπ‘‘)⨂=(𝑑𝑠𝐿2(Ξ£2⨂))βŠ—(𝑑𝑠𝐿2(Ξ£2)). Here 𝑑 (resp., 𝑑) is the number of bosons π‘Šβˆ’ (resp., π‘Š+). The vector Ξ©π‘Š is the corresponding vacuum.

The Fock space for the weak decay of the vector bosons π‘Š+ and π‘Šβˆ’, denoted by 𝔉, is thus 𝔉=π”‰πΏβŠ—π”‰π‘Š,(2.14) and Ξ©=Ξ©πΏβŠ—Ξ©π‘Š is the vacuum state.

For every β„“βˆˆ{1,2,3} let 𝔇ℓ denote the set of smooth vectors πœ“β„“βˆˆπ”‰β„“ for which πœ“(π‘žβ„“,π‘žβ„“,π‘Ÿβ„“,π‘Ÿβ„“)β„“ has a compact support and πœ“(π‘žβ„“,π‘žβ„“,π‘Ÿβ„“,π‘Ÿβ„“)β„“=0 for all but finitely many (π‘žβ„“,π‘žβ„“,π‘Ÿβ„“,π‘Ÿβ„“). Let 𝔇𝐿=ξ‚Šβ¨‚3β„“=1𝔇ℓ.(2.15) Here ξ‚Šβ¨‚ is the algebraic tensor product.

Let π”‡π‘Š denote the set of smooth vectors πœ™βˆˆπ”‰π‘Š for which πœ™(𝑑,𝑑) has a compact support and πœ™(𝑑,𝑑)=0 for all but finitely many (𝑑,𝑑).

Let 𝔇=π”‡πΏξβŠ—π”‡π‘Š.(2.16) The set 𝔇 is dense in 𝔉.

Let 𝐴ℓ 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 𝐴1βŠ—12βŠ—13 with domain 𝔇𝐿 when β„“=1, of the operator 11βŠ—π΄2βŠ—13 with domain 𝔇𝐿 when β„“=2, and of the operator 11βŠ—12βŠ—π΄3 with domain 𝔇𝐿 when β„“=3. Here 1β„“ is the operator identity on 𝔉ℓ.

The extension of 𝐴ℓ 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 ξ‚π΄β„“βŠ—1π‘Š with domain 𝔇, where 𝐴ℓ is the extension of 𝐴ℓ to 𝔉𝐿. The extension of 𝐴ℓ to 𝔉 is a self-adjoint operator for which 𝔇 is a core.

Let 𝐡 be a self-adjoint operator in π”‰π‘Š for which π”‡π‘Š is a core. The extension of the self-adjoint operator π΄β„“βŠ—π΅ is, by definition, the closure in 𝔉 of the operator 𝐴1βŠ—12βŠ—13βŠ—π΅ with domain 𝔇 when β„“=1, of the operator 11βŠ—π΄2βŠ—13βŠ—π΅ with domain 𝔇 when β„“=2, and of the operator 11βŠ—12βŠ—π΄3βŠ—π΅ with domain 𝔇 when β„“=3. The extension of π΄β„“βŠ—π΅ to 𝔉 is a self-adjoint operator for which 𝔇 is a core.

We now define the creation and annihilation operators.

For each β„“=1,2,3, 𝑏ℓ,πœ–(πœ‰1) (resp., π‘βˆ—β„“,πœ–(πœ‰1)) is the fermionic annihilation (resp., fermionic creation) operator for the corresponding species of massive particle when πœ–=+ and for the corresponding species of massive antiparticle when πœ–=βˆ’. The operators 𝑏ℓ,πœ–(πœ‰1) and π‘βˆ—β„“,πœ–(πœ‰1) are defined as usually (see, e.g., [20, 26]; see also the detailed definitions in [27]).

Similarly, for each β„“=1,2,3, 𝑐ℓ,πœ–(πœ‰2) (resp., π‘βˆ—β„“,πœ–(πœ‰2)) is the fermionic annihilation (resp., fermionic creation) operator for the corresponding species of neutrino when πœ–=+ and for the corresponding species of antineutrino when πœ–=βˆ’. The operators 𝑐ℓ,πœ–(πœ‰2) and π‘βˆ—β„“,πœ–(πœ‰2) are defined in a standard way, but with the additional requirements that for any β„“, β„“ξ…ž, πœ– and πœ–ξ…ž, the operators 𝑏♯ℓ,πœ–(πœ‰1) and 𝑐♯ℓ′,πœ–β€²(πœ‰2) anticommutes, where β™― stands either for a βˆ— or for no symbol (see the detailed definitions in [27]).

The operator π‘Žπœ–(πœ‰3) (resp., π‘Žβˆ—πœ–(πœ‰3)) is the bosonic annihilation (resp., bosonic creation) operator for the boson π‘Šβˆ’ when πœ–=+ and for the boson π‘Š+ when πœ–=βˆ’ (see, e.g., [20, 26], or [27]). Note that π‘Žβ™―(πœ‰3) commutes with 𝑏♯ℓ,πœ–(πœ‰1) and 𝑐♯ℓ′,πœ–β€²(πœ‰2).

The following canonical anticommutation and commutation relations hold: 𝑏ℓ,πœ–ξ€·πœ‰1ξ€Έ,π‘βˆ—β„“β€²,πœ–β€²ξ€·πœ‰ξ…ž1=π›Ώβ„“β„“β€²π›Ώπœ–πœ–β€²π›Ώξ€·πœ‰1βˆ’πœ‰ξ…ž1ξ€Έ,𝑐ℓ,πœ–ξ€·πœ‰2ξ€Έ,π‘βˆ—β„“β€²,πœ–β€²ξ€·πœ‰ξ…ž2=π›Ώβ„“β„“β€²π›Ώπœ–πœ–β€²π›Ώξ€·πœ‰2βˆ’πœ‰ξ…ž2ξ€Έ,ξ€Ίπ‘Žπœ–ξ€·πœ‰3ξ€Έ,π‘Žβˆ—πœ–β€²ξ€·πœ‰ξ…ž3ξ€Έξ€»=π›Ώπœ–πœ–β€²π›Ώξ€·πœ‰3βˆ’πœ‰ξ…ž3ξ€Έ,𝑏ℓ,πœ–ξ€·πœ‰1ξ€Έ,𝑏ℓ′,πœ–β€²ξ€·πœ‰ξ…ž1=𝑐ℓ,πœ–ξ€·πœ‰2ξ€Έ,𝑐ℓ′,πœ–β€²ξ€·πœ‰ξ…ž2ξ€Ίπ‘Žξ€Έξ€Ύ=0,πœ–ξ€·πœ‰3ξ€Έ,π‘Žπœ–β€²ξ€·πœ‰ξ…ž3𝑏=0,β„“,πœ–ξ€·πœ‰1ξ€Έ,𝑐ℓ′,πœ–β€²ξ€·πœ‰2=𝑏ℓ,πœ–ξ€·πœ‰1ξ€Έ,π‘βˆ—β„“β€²,πœ–β€²ξ€·πœ‰2𝑏=0,β„“,πœ–ξ€·πœ‰1ξ€Έ,π‘Žπœ–β€²ξ€·πœ‰3=𝑏ℓ,πœ–ξ€·πœ‰1ξ€Έ,π‘Žβˆ—πœ–β€²ξ€·πœ‰3=𝑐ℓ,πœ–ξ€·πœ‰2ξ€Έ,π‘Žπœ–β€²ξ€·πœ‰3=𝑐ℓ,πœ–ξ€·πœ‰2ξ€Έ,π‘Žβˆ—πœ–β€²ξ€·πœ‰3ξ€Έξ€»=0,(2.17) where we used the notation 𝛿(πœ‰π‘—βˆ’πœ‰ξ…žπ‘—)=π›Ώπœ†πœ†β€²π›Ώ(π‘˜βˆ’π‘˜ξ…ž).

We recall that the following operators, with πœ‘βˆˆπΏ2(Ξ£1), 𝑏ℓ,πœ–(ξ€œπœ‘)=Ξ£1𝑏ℓ,πœ–(πœ‰)πœ‘(πœ‰)dπœ‰,𝑐ℓ,πœ–(ξ€œπœ‘)=Ξ£1𝑐ℓ,πœ–(πœ‰)πœ‘(πœ‰)dπ‘πœ‰,βˆ—β„“,πœ–ξ€œ(πœ‘)=Ξ£1π‘βˆ—β„“,πœ–(πœ‰)πœ‘(πœ‰)dπœ‰,π‘βˆ—β„“,πœ–ξ€œ(πœ‘)=Ξ£1π‘βˆ—β„“,πœ–(πœ‰)πœ‘(πœ‰)dπœ‰(2.18) are bounded operators in 𝔉 such that ‖‖𝑏♯ℓ,πœ–β€–β€–=‖‖𝑐(πœ‘)β™―β„“,πœ–β€–β€–(πœ‘)=β€–πœ‘β€–πΏ2,(2.19) where 𝑏♯ (resp., 𝑐♯) is 𝑏 (resp., 𝑐) or π‘βˆ— (resp., π‘βˆ—).

The operators 𝑏♯ℓ,πœ–(πœ‘) and 𝑐♯ℓ,πœ–(πœ‘) satisfy similar anticommutaion relations (see, e.g., [28]).

The free Hamiltonian 𝐻0 is given by 𝐻0=𝐻0(1)+𝐻0(2)+𝐻0(3)=3ℓ=10π‘₯0200π‘‘πœ–=Β±ξ€œπ‘€β„“(1)ξ€·πœ‰1ξ€Έπ‘βˆ—β„“,πœ–ξ€·πœ‰1𝑏ℓ,πœ–ξ€·πœ‰1ξ€Έdπœ‰1+3ℓ=10π‘₯0200π‘‘πœ–=Β±ξ€œπ‘€β„“(2)ξ€·πœ‰2ξ€Έπ‘βˆ—β„“,πœ–ξ€·πœ‰2𝑐ℓ,πœ–ξ€·πœ‰2ξ€Έdπœ‰2+ξ“πœ–=Β±ξ€œπ‘€(3)ξ€·πœ‰3ξ€Έπ‘Žβˆ—πœ–ξ€·πœ‰3ξ€Έπ‘Žπœ–ξ€·πœ‰3ξ€Έdπœ‰3,(2.20) where 𝑀ℓ(1)ξ€·πœ‰1ξ€Έ=ξ‚€||𝑝1||2+π‘š2ℓ1/2,with0<π‘š1<π‘š2<π‘š3,𝑀ℓ(2)ξ€·πœ‰2ξ€Έ=||𝑝2||,𝑀(3)ξ€·πœ‰3ξ€Έ=ξ‚€||π‘˜||2+π‘š2π‘Šξ‚1/2,(2.21) where π‘šπ‘Š is the mass of the bosons π‘Š+ and π‘Šβˆ’ such that π‘šπ‘Š>π‘š3.

The spectrum of 𝐻0 is [0,∞) and 0 is a simple eigenvalue with Ξ© as eigenvector. The set of thresholds of 𝐻0, denoted by 𝑇, is given by 𝑇=π‘π‘š1+π‘žπ‘š2+π‘Ÿπ‘š3+π‘ π‘šπ‘Š;(𝑝,π‘ž,π‘Ÿ,𝑠)βˆˆβ„•4ξ€Ύ,,𝑝+π‘ž+π‘Ÿ+𝑠β‰₯1(2.22) and each set [𝑑,∞), π‘‘βˆˆπ‘‡, is a branch of absolutely continuous spectrum for 𝐻0.

The interaction, denoted by 𝐻𝐼, is given by 𝐻𝐼=2𝛼=1𝐻𝐼(𝛼),(2.23) where 𝐻𝐼(1)=3ℓ=10π‘₯0200π‘‘πœ–β‰ πœ–β€²ξ€œπΊ(1)β„“,πœ–,πœ–β€²ξ€·πœ‰1,πœ‰2,πœ‰3ξ€Έπ‘βˆ—β„“,πœ–ξ€·πœ‰1ξ€Έπ‘βˆ—β„“,πœ–β€²ξ€·πœ‰2ξ€Έπ‘Žπœ–ξ€·πœ‰3ξ€Έdπœ‰1dπœ‰2dπœ‰3+3ℓ=10π‘₯0200π‘‘πœ–β‰ πœ–β€²ξ€œπΊ(1)β„“,πœ–,πœ–β€²(πœ‰1,πœ‰2,πœ‰3)π‘Žβˆ—πœ–ξ€·πœ‰3𝑐ℓ,πœ–β€²ξ€·πœ‰2𝑏ℓ,πœ–ξ€·πœ‰1ξ€Έdπœ‰1dπœ‰2dπœ‰3,𝐻𝐼(2)=3ℓ=1ξ“πœ–β‰ πœ–β€²ξ€œπΊ(2)β„“,πœ–,πœ–β€²ξ€·πœ‰1,πœ‰2,πœ‰3ξ€Έπ‘βˆ—β„“,πœ–ξ€·πœ‰1ξ€Έπ‘βˆ—β„“,πœ–β€²ξ€·πœ‰2ξ€Έπ‘Žβˆ—πœ–ξ€·πœ‰3ξ€Έdπœ‰1dπœ‰2dπœ‰3+3ℓ=1ξ“πœ–β‰ πœ–β€²ξ€œπΊ(2)β„“,πœ–,πœ–β€²ξ€·πœ‰1,πœ‰2,πœ‰3ξ€Έπ‘Žπœ–ξ€·πœ‰3𝑐ℓ,πœ–β€²ξ€·πœ‰2𝑏ℓ,πœ–ξ€·πœ‰1ξ€Έdπœ‰1dπœ‰2dπœ‰3.(2.24) The kernels 𝐺(2)β„“,πœ–,πœ–β€²(β‹…,β‹…,β‹…), 𝛼=1,2, are supposed to be functions.

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

The operator 𝐻𝐼(1) describes the decay of the bosons π‘Š+ and π‘Šβˆ’ 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 kernel 𝐺ℓ,πœ–,πœ–β€²(πœ‰1,πœ‰2,πœ‰3), 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)β„“,πœ–,πœ–β€² are supposed to satisfy the following hypothesis.

Hypothesis 2.1. For 𝛼=1,2, β„“=1,2,3, πœ–,πœ–ξ…ž=Β±, we assume 𝐺(𝛼)β„“,πœ–,πœ–β€²ξ€·πœ‰1,πœ‰2,πœ‰3ξ€ΈβˆˆπΏ2ξ€·Ξ£1Γ—Ξ£1Γ—Ξ£2ξ€Έ.(2.26)

Remark 2.2. A similar model can be written down for the weak decay of pions πœ‹βˆ’ and πœ‹+ (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 βˆ’1/2 and antineutrinos have helicity 1/2.
In the physical case, the Fock space, denoted by π”‰ξ…ž, is isomorphic to π”‰ξ…žπΏβŠ—π”‰π‘Š, with π”‰ξ…žπΏ=3ℓ=1π”‰ξ…žβ„“,π”‰ξ…žβ„“=2ξ·π‘ŽπΏ2ξ€·Ξ£1ξ€ΈξƒͺβŠ—ξƒ©2ξ·π‘ŽπΏ2ℝ3ξ€Έξƒͺ.(2.27) The free Hamiltonian, now denoted by π»ξ…ž0, is then given by π»ξ…ž0=3ℓ=1ξ“πœ–=Β±ξ€œπ‘€β„“(1)ξ€·πœ‰1ξ€Έπ‘βˆ—β„“,πœ–ξ€·πœ‰1𝑏ℓ,πœ–ξ€·πœ‰1ξ€Έdπœ‰1+3ℓ=1ξ“πœ–=Β±ξ€œβ„3||𝑝2||π‘βˆ—β„“,πœ–ξ€·π‘2𝑐ℓ,πœ–ξ€·π‘2ξ€Έd𝑝2+ξ“πœ–=Β±ξ€œπ‘€(3)ξ€·πœ‰3ξ€Έπ‘Žβˆ—πœ–ξ€·πœ‰3ξ€Έπ‘Žπœ–ξ€·πœ‰3ξ€Έdπœ‰3,(2.28) and the interaction, now denoted by π»ξ…žπΌ, is the one obtained from 𝐻𝐼 by supposing that 𝐺(𝛼)(πœ‰1,(𝑝2,𝑠2),πœ‰3)=0 if 𝑠2=πœ–(1/2). The total Hamiltonian, denoted by π»ξ…ž, is then given by π»ξ…ž=π»ξ…ž0+π‘”π»ξ…žπΌ. The results obtained in this paper for 𝐻 hold true for π»ξ…ž with obvious modifications.

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

The formal operator ξ€œπΊ(1)β„“,πœ–,πœ–β€²ξ€·πœ‰1,πœ‰2,πœ‰3ξ€Έπ‘βˆ—β„“,πœ–ξ€·πœ‰1ξ€Έπ‘βˆ—β„“,πœ–β€²ξ€·πœ‰2ξ€Έπ‘Žπœ–ξ€·πœ‰3ξ€Έdπœ‰1dπœ‰2dπœ‰3(2.29) is defined as a quadratic form on (π”‡β„“βŠ—π”‡π‘Š)Γ—(π”‡β„“βŠ—π”‡π‘Š) as ξ€œξ‚€π‘β„“,πœ–β€²ξ€·πœ‰2𝑏ℓ,πœ–ξ€·πœ‰1ξ€Έπœ“,𝐺(1)β„“,πœ–,πœ–β€²π‘Žπœ–ξ€·πœ‰3ξ€Έπœ™ξ‚dπœ‰1dπœ‰2dπœ‰3,(2.30) where πœ“, πœ™βˆˆπ”‡β„“βŠ—π”‡π‘Š.

By mimicking the proof of [29, Theorem X.44], we get a closed operator, denoted by 𝐻(1)𝐼,β„“,πœ–,πœ–β€², associated with the quadratic form such that it is the unique operator in π”‰β„“βŠ—π”‰π‘Š such that π”‡β„“βŠ—π”‡π‘ŠβŠ‚π’Ÿ(𝐻(1)𝐼,β„“,πœ–,πœ–β€²) is a core for 𝐻(1)𝐼,β„“,πœ–,πœ–β€² and 𝐻(1)𝐼,β„“,πœ–,πœ–β€²=ξ€œπΊ(1)β„“,πœ–,πœ–β€²ξ€·πœ‰1,πœ‰2,πœ‰3ξ€Έπ‘βˆ—β„“,πœ–ξ€·πœ‰1ξ€Έπ‘βˆ—β„“,πœ–β€²ξ€·πœ‰2ξ€Έπ‘Žπœ–ξ€·πœ‰3ξ€Έdπœ‰1dπœ‰2dπœ‰3(2.31) as quadratic forms on (π”‡β„“βŠ—π”‡π‘Š)Γ—(π”‡β„“βŠ—π”‡π‘Š).

Similarly for the operator (𝐻(1)𝐼,β„“,πœ–,πœ–β€²)βˆ—, we have the equality as quadratic forms 𝐻(1)𝐼,β„“,πœ–,πœ–β€²ξ‚βˆ—=ξ€œπΊ(1)β„“,πœ–,πœ–β€²(πœ‰1,πœ‰2,πœ‰3)π‘Žβˆ—πœ–ξ€·πœ‰3𝑐ℓ,πœ–β€²ξ€·πœ‰2𝑏ℓ,πœ–ξ€·πœ‰1ξ€Έdπœ‰1dπœ‰2dπœ‰3.(2.32)

Again, there exists two closed operators 𝐻(2)𝐼,β„“,πœ–,πœ–β€² and (𝐻(2)𝐼,β„“,πœ–,πœ–β€²)βˆ— such that π”‡β„“βŠ—π”‡π‘ŠβŠ‚π’Ÿ(𝐻(2)𝐼,β„“,πœ–,πœ–β€²), π”‡β„“βŠ—π”‡π‘ŠβŠ‚π’Ÿ((𝐻(2)𝐼,β„“,πœ–,πœ–β€²)βˆ—), and π”‡β„“βŠ—π”‡π‘Š is a core for 𝐻(2)𝐼,β„“,πœ–,πœ–β€² and (𝐻(2)𝐼,β„“,πœ–,πœ–β€²)βˆ— and such that 𝐻(2)𝐼,β„“,πœ–,πœ–β€²=ξ€œπΊ(2)β„“,πœ–,πœ–β€²ξ€·πœ‰1,πœ‰2,πœ‰3ξ€Έπ‘βˆ—β„“,πœ–ξ€·πœ‰1ξ€Έπ‘βˆ—β„“,πœ–β€²ξ€·πœ‰2ξ€Έπ‘Žβˆ—πœ–ξ€·πœ‰3ξ€Έdπœ‰1dπœ‰2dπœ‰3,𝐻(2)𝐼,β„“,πœ–,πœ–β€²ξ‚βˆ—=ξ€œπΊ(2)β„“,πœ–,πœ–β€²ξ€·πœ‰1,πœ‰2,πœ‰3ξ€Έπ‘Žπœ–ξ€·πœ‰3𝑐ℓ,πœ–β€²ξ€·πœ‰2𝑏ℓ,πœ–ξ€·πœ‰1ξ€Έdπœ‰1dπœ‰2dπœ‰3(2.33) as quadratic forms on (π”‡β„“βŠ—π”‡π‘Š)Γ—(π”‡β„“βŠ—π”‡π‘Š).

We will still denote by 𝐻(𝛼)𝐼,β„“,πœ–,πœ–β€² and (𝐻(𝛼)𝐼,β„“,πœ–,πœ–β€²)βˆ— (𝛼=1,2) their extensions to 𝔉. The set 𝔇 is then a core for 𝐻(𝛼)𝐼,β„“,πœ–,πœ–β€² and (𝐻(𝛼)𝐼,β„“,πœ–,πœ–β€²)βˆ—.

Thus 𝐻=𝐻0+𝑔3𝛼=1,2ℓ=1ξ“πœ–β‰ πœ–β€²ξ‚€π»(𝛼)𝐼,β„“,πœ–,πœ–β€²+𝐻(2)𝐼,β„“,πœ–,πœ–β€²ξ‚βˆ—ξ‚(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,β„“,πœ–,πœ–β€² and (𝐻(𝛼)𝐼,β„“,πœ–,πœ–β€²)βˆ— are relatively 𝐻0-bounded.

Once again, as above, for almost every πœ‰3∈Σ2, there exists closed operators in 𝔉𝐿, denoted by 𝐡(𝛼)β„“,πœ–,πœ–β€²(πœ‰3) and (𝐡(𝛼)β„“,πœ–,πœ–β€²(πœ‰3))βˆ— such that 𝐡(1)β„“,πœ–,πœ–β€²ξ€·πœ‰3ξ€Έξ€œ=βˆ’πΊ(1)β„“,πœ–,πœ–β€²(πœ‰1,πœ‰2,πœ‰3)𝑏ℓ,πœ–ξ€·πœ‰1𝑐ℓ,πœ–β€²ξ€·πœ‰2ξ€Έdπœ‰1dπœ‰2,𝐡(1)β„“,πœ–,πœ–β€²ξ€·πœ‰3ξ€Έξ‚βˆ—=ξ€œπΊ(1)β„“,πœ–,πœ–β€²ξ€·πœ‰1,πœ‰2,πœ‰3ξ€Έπ‘βˆ—β„“,πœ–ξ€·πœ‰1ξ€Έπ‘βˆ—β„“,πœ–β€²ξ€·πœ‰2ξ€Έdπœ‰1dπœ‰2,𝐡(2)β„“,πœ–,πœ–β€²ξ€·πœ‰3ξ€Έ=ξ€œπΊ(2)β„“,πœ–,πœ–β€²ξ€·πœ‰1,πœ‰2,πœ‰3ξ€Έπ‘βˆ—β„“,πœ–ξ€·πœ‰1ξ€Έπ‘βˆ—β„“,πœ–β€²ξ€·πœ‰2ξ€Έdπœ‰1dπœ‰2,𝐡(2)β„“,πœ–,πœ–β€²ξ€·πœ‰3ξ€Έξ‚βˆ—ξ€œ=βˆ’πΊ(2)β„“,πœ–,πœ–β€²(πœ‰1,πœ‰2,πœ‰3)𝑏ℓ,πœ–ξ€·πœ‰1𝑐ℓ,πœ–β€²ξ€·πœ‰2ξ€Έdπœ‰1dπœ‰2(2.35) as quadratic forms on 𝔇ℓ×𝔇ℓ.

We have that π”‡β„“βŠ‚π’Ÿ(𝐡(𝛼)β„“,πœ–,πœ–β€²(πœ‰3)) (resp., π”‡β„“βŠ‚π’Ÿ((𝐡(𝛼)β„“,πœ–,πœ–β€²(πœ‰3))βˆ—) is a core for 𝐡(𝛼)β„“,πœ–,πœ–β€²(πœ‰3) (resp., for (𝐡(𝛼)β„“,πœ–,πœ–β€²(πœ‰3))βˆ—). We still denote by 𝐡(𝛼)β„“,πœ–,πœ–β€²(πœ‰3)) and (𝐡(𝛼)β„“,πœ–,πœ–β€²(πœ‰3))βˆ—) their extensions to 𝔉𝐿.

It then follows that the operator 𝐻𝐼 with domain 𝔇 is symmetric and can be written in the following form: 𝐻𝐼=3𝛼=1,2ℓ=1ξ“πœ–β‰ πœ–β€²ξ‚€π»(𝛼)𝐼,β„“,πœ–,πœ–β€²+𝐻(𝛼)𝐼,β„“,πœ–,πœ–β€²ξ‚βˆ—ξ‚=3𝛼=1,2ℓ=1ξ“πœ–β‰ πœ–β€²ξ€œπ΅(𝛼)β„“,πœ–,πœ–β€²ξ€·πœ‰3ξ€ΈβŠ—π‘Žβˆ—πœ–ξ€·πœ‰3ξ€Έdπœ‰3+3𝛼=1,2ℓ=1ξ“πœ–β‰ πœ–β€²ξ€œξ‚€π΅(𝛼)β„“,πœ–,πœ–β€²ξ€·πœ‰3ξ€Έξ‚βˆ—βŠ—π‘Žπœ–ξ€·πœ‰3ξ€Έdπœ‰3.(2.36) Let 𝑁ℓ denote the operator number of massive leptons β„“ in 𝔉ℓ, that is, 𝑁ℓ=ξ“πœ–ξ€œπ‘βˆ—β„“,πœ–ξ€·πœ‰1𝑏ℓ,πœ–ξ€·πœ‰1ξ€Έdπœ‰1.(2.37) 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 every πœ‰3∈Σ2, π’Ÿ(𝐡(𝛼)β„“,πœ–,πœ–β€²(πœ‰3)), π’Ÿ((𝐡(𝛼)β„“,πœ–,πœ–β€²(πœ‰3))βˆ—)βŠƒπ’Ÿ(𝑁ℓ1/2), and for Ξ¦βˆˆπ’Ÿ(𝑁ℓ1/2)βŠ‚π”‰πΏ one has ‖‖𝐡(𝛼)β„“,πœ–,πœ–β€²(πœ‰3β€–β€–)Φ𝔉𝐿≀‖‖𝐺(𝛼)β„“,πœ–,πœ–β€²(β‹…,β‹…,πœ‰3)‖‖𝐿2(Ξ£1Γ—Ξ£1)‖‖𝑁ℓ1/2Φ‖‖𝔉𝐿,‖‖‖𝐡(𝛼)β„“,πœ–,πœ–β€²ξ€·πœ‰3ξ€Έξ‚βˆ—Ξ¦β€–β€–β€–π”‰πΏβ‰€β€–β€–πΊ(𝛼)β„“,πœ–,πœ–β€²(β‹…,β‹…,πœ‰3)‖‖𝐿2(Ξ£1Γ—Ξ£1)‖‖𝑁ℓ1/2Φ‖‖𝔉𝐿.(2.38)

Proof. The estimates (2.38) are examples of π‘πœ 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,πœ–=ξ€œπ‘€(3)ξ€·πœ‰3ξ€Έπ‘Žβˆ—πœ–ξ€·πœ‰3ξ€Έπ‘Žπœ–ξ€·πœ‰3ξ€Έdπœ‰3.(2.39) Then 𝐻(3)0,πœ– is a self-adjoint operator in π”‰π‘Š, and π”‡π‘Š is a core for 𝐻(3)0,πœ–.

We get the following.

Proposition 2.5. One has β€–β€–β€–ξ€œξ‚€π΅(𝛼)β„“,πœ–,πœ–β€²ξ€·πœ‰3ξ€Έξ‚βˆ—βŠ—π‘Žπœ–ξ€·πœ‰3ξ€Έdπœ‰3Ξ¨β€–β€–β€–2β‰€βŽ›βŽœβŽœβŽœβŽξ€œΞ£1Γ—Ξ£1Γ—Ξ£2|||𝐺(𝛼)β„“,πœ–,πœ–β€²ξ€·πœ‰1,πœ‰2,πœ‰3ξ€Έ|||2𝑀(3)ξ€·πœ‰3ξ€Έdπœ‰1dπœ‰2dπœ‰3βŽžβŽŸβŽŸβŽŸβŽ β€–β€–β€–ξ€·π‘β„“ξ€Έ+11/2βŠ—ξ‚€π»(3)0,πœ–ξ‚1/2Ξ¨β€–β€–β€–2,(2.40)β€–β€–β€–ξ€œπ΅(𝛼)β„“,πœ–,πœ–β€²(πœ‰3)βŠ—π‘Žβˆ—πœ–(πœ‰3)dπœ‰3Ξ¨β€–β€–β€–2β‰€βŽ›βŽœβŽœβŽœβŽξ€œΞ£1Γ—Ξ£1Γ—Ξ£2|||𝐺(𝛼)β„“,πœ–,πœ–β€²ξ€·πœ‰1,πœ‰2,πœ‰3ξ€Έ|||2𝑀(3)ξ€·πœ‰3ξ€Έdπœ‰1dπœ‰2dπœ‰3βŽžβŽŸβŽŸβŽŸβŽ β€–β€–β€–ξ€·π‘β„“ξ€Έ+11/2βŠ—ξ‚€π»(3)0,πœ–ξ‚1/2Ξ¨β€–β€–β€–2+ξ‚΅ξ€œΞ£1Γ—Ξ£1Γ—Ξ£2|||𝐺(𝛼)β„“,πœ–,πœ–β€²ξ€·πœ‰1,πœ‰2,πœ‰3ξ€Έ|||2dπœ‰1dπœ‰2dπœ‰3πœ‚β€–β€–ξ€·π‘ξ‚Άξ‚΅β„“ξ€Έ+11/2β€–β€–βŠ—1Ξ¨2+14πœ‚β€–Ξ¨β€–2ξ‚Ά(2.41) for every Ξ¨βˆˆπ’Ÿ(𝐻0) and every πœ‚>0.

Proof. Suppose that Ξ¨βˆˆπ’Ÿ(𝑁ℓ1/2)ξβŠ—π’Ÿ((𝐻(3)0,πœ–)1/2). Let Ξ¨πœ–ξ€·πœ‰3ξ€Έ=𝑀(3)ξ€·πœ‰3ξ€Έ1/2𝑁ℓ+11/2βŠ—π‘Žπœ–ξ€·πœ‰3Φ.(2.42) We have ξ€œξ‚€π΅(𝛼)β„“,πœ–,πœ–β€²ξ€·πœ‰3ξ€Έξ‚βˆ—βŠ—π‘Žπœ–ξ€·πœ‰3ξ€Έdπœ‰3ξ€œΞ¨=Ξ£21𝑀(3)ξ€·πœ‰3ξ€Έξ€Έ1/2𝐡(𝛼)β„“,πœ–,πœ–β€²ξ€·πœ‰3ξ€Έξ‚βˆ—ξ€·π‘β„“ξ€Έ+1βˆ’1/2ξ‚Ξ¨βŠ—1πœ–ξ€·πœ‰3ξ€Έdπœ‰3.(2.43) Therefore, for Ξ¨βˆˆπ’Ÿ(𝑁ℓ1/2)ξβŠ—π’Ÿ((𝐻(3)0,πœ–)1/2), (2.40) follows from Proposition 2.4.
We now have β€–β€–β€–ξ€œπ΅(𝛼)β„“,πœ–,πœ–β€²(πœ‰3)βŠ—π‘Žβˆ—πœ–(πœ‰3)Ξ¨dπœ‰3β€–β€–β€–2𝔉=ξ€œξ‚€π΅(𝛼)β„“,πœ–,πœ–β€²ξ€·πœ‰3ξ€ΈβŠ—π‘Žπœ–ξ€·πœ‰ξ…ž3ξ€ΈΞ¨,𝐡(𝛼)β„“,πœ–,πœ–β€²ξ€·πœ‰ξ…ž3ξ€ΈβŠ—π‘Žπœ–ξ€·πœ‰3Ψdπœ‰3dπœ‰ξ…ž3+ξ€œβ€–β€–ξ‚€π΅(𝛼)β„“,πœ–,πœ–β€²(πœ‰3Ψ‖‖)βŠ—12dπœ‰3,(2.44)ξ€œΞ£2Γ—Ξ£2𝐡(𝛼)β„“,πœ–,πœ–β€²ξ€·πœ‰3ξ€ΈβŠ—π‘Žπœ–ξ€·πœ‰ξ…ž3ξ€ΈΞ¨,𝐡(𝛼)β„“,πœ–,πœ–β€²ξ€·πœ‰ξ…ž3ξ€ΈβŠ—π‘Žπœ–ξ€·πœ‰3Ψdπœ‰3dπœ‰ξ…ž3=ξ€œΞ£2Γ—Ξ£21𝑀(3)ξ€·πœ‰3ξ€Έ1/2𝑀(3)ξ€·πœ‰ξ…ž3ξ€Έ1/2×𝐡(𝛼)β„“,πœ–,πœ–β€²ξ€·πœ‰3𝑁ℓ+1βˆ’1/2ξ‚Ξ¨βŠ—1πœ–ξ€·πœ‰ξ…ž3ξ€Έ,𝐡(𝛼)β„“,πœ–,πœ–β€²ξ€·πœ‰ξ…ž3𝑁ℓ+1βˆ’1/2ξ‚Ξ¨βŠ—1πœ–ξ€·πœ‰3dπœ‰3dπœ‰ξ…ž3β‰€ξƒ©ξ€œΞ£21𝑀(3)ξ€·πœ‰3ξ€Έ1/2‖‖𝐡(𝛼)β„“,πœ–,πœ–β€²ξ€·πœ‰3𝑁ℓ+1βˆ’1/2β€–β€–π”‰πΏβ€–β€–Ξ¨πœ–ξ€·πœ‰3ξ€Έβ€–β€–dπœ‰3ξƒͺ2β‰€ξƒ©ξ€œΞ£1Γ—Ξ£1Γ—Ξ£2||𝐺(𝛼)ξ€·πœ‰1,πœ‰2,πœ‰3ξ€Έ||2𝑀(3)ξ€·πœ‰3ξ€Έdπœ‰1dπœ‰2dπœ‰3ξƒͺ‖‖‖𝑁ℓ+11/2βŠ—ξ‚€π»(3)0,πœ–ξ‚1/2Ξ¨β€–β€–β€–2.(2.45) Furthermore ξ€œΞ£2‖‖𝐡(𝛼)β„“,πœ–,πœ–β€²ξ€·πœ‰3ξ€Έξ‚Ξ¨β€–β€–βŠ—12dπœ‰3=ξ€œΞ£2‖‖𝐡(𝛼)β„“,πœ–,πœ–β€²ξ€·πœ‰3𝑁ℓ+1βˆ’1/2ξ€·π‘βŠ—1ℓ+11/2ξ‚Ξ¨β€–β€–βŠ—12dπœ‰3β‰€ξ‚΅ξ€œΞ£1Γ—Ξ£1Γ—Ξ£2|||𝐺(𝛼)β„“,πœ–,πœ–β€²ξ€·πœ‰1,πœ‰2,πœ‰3ξ€Έ|||2dπœ‰1dπœ‰2dπœ‰3πœ‚β€–β€–ξ€·π‘ξ‚Άξ‚΅β„“ξ€ΈΞ¨β€–β€–+12+14πœ‚β€–Ξ¨β€–2ξ‚Ά(2.46) for every πœ‚>0.
By (2.40), (2.45), and (2.46), we finally get (2.41) for every Ξ¨βˆˆπ’Ÿ(𝑁ℓ1/2)ξβŠ—π’Ÿ(𝐻(3)0,πœ–). It then follows that (2.40) and (2.41) are verified for every Ξ¨βˆˆπ’Ÿ(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 3𝑔21π‘šπ‘Šξƒ©1π‘š21ξƒͺ+13𝛼=1,2ℓ=1ξ“πœ–β‰ πœ–β€²β€–β€–πΊ(𝛼)β„“,πœ–,πœ–β€²β€–β€–2𝐿2(Ξ£1Γ—Ξ£1Γ—Ξ£2)<1.(2.47) Then for every 𝑔 satisfying 𝑔≀𝑔1, 𝐻 is a self-adjoint operator in 𝔉 with domain π’Ÿ(𝐻)=π’Ÿ(𝐻0), and 𝔇 is a core for 𝐻.

Proof. Let Ξ¨ be in 𝔇. We have ‖‖𝐻𝐼Ψ‖‖2≀123𝛼=1,2ℓ=1ξ“πœ–β‰ πœ–β€²ξ‚»β€–β€–β€–ξ€œξ‚€π΅(𝛼)β„“,πœ–,πœ–β€²ξ€·πœ‰3ξ€Έξ‚βˆ—βŠ—π‘Žπœ–ξ€·πœ‰3ξ€ΈΞ¨dπœ‰3β€–β€–β€–2+β€–β€–β€–ξ€œξ‚€π΅(𝛼)β„“,πœ–,πœ–β€²ξ€·πœ‰3ξ€Έξ‚βŠ—π‘Žβˆ—πœ–ξ€·πœ‰3ξ€ΈΞ¨dπœ‰3β€–β€–β€–2ξ‚Ό.(2.48) Note that ‖‖𝐻(3)0,πœ–Ξ¨β€–β€–β‰€β€–β€–π»0(3)Ψ‖‖≀‖‖𝐻0Ξ¨β€–β€–,‖‖𝑁ℓΨ‖‖≀1π‘šβ„“β€–β€–π»0,ℓΨ‖‖≀1π‘š1‖‖𝐻0,ℓΨ‖‖≀1π‘š1‖‖𝐻0Ξ¨β€–β€–,(2.49) where 𝐻0,β„“=ξ“πœ–ξ€œπ‘€β„“(1)ξ€·πœ‰1ξ€Έπ‘βˆ—β„“,πœ–ξ€·πœ‰1𝑏ℓ,πœ–ξ€·πœ‰1ξ€Έdπœ‰1+ξ“πœ–ξ€œπ‘€β„“(2)ξ€·πœ‰2ξ€Έπ‘βˆ—β„“,πœ–ξ€·πœ‰2𝑐ℓ,πœ–ξ€·πœ‰2ξ€Έdπœ‰2.(2.50) We further note that ‖‖‖𝑁ℓ+11/2βŠ—ξ‚€π»(3)0,πœ–ξ‚1/2Ξ¨β€–β€–β€–2≀121π‘š21ξƒͺ‖‖𝐻+10Ξ¨β€–β€–2+𝛽2π‘š21‖‖𝐻0Ξ¨β€–β€–2+ξ‚΅12+1ξ‚Άβ€–8𝛽Ψ‖2(2.51) for 𝛽>0, and πœ‚β€–β€–π‘ξ€·ξ€·β„“ξ€Έξ€ΈΞ¨β€–β€–+1βŠ—12+14πœ‚β€–Ξ¨β€–2β‰€πœ‚π‘š21‖‖𝐻0Ξ¨β€–β€–2+πœ‚π›½π‘š21‖‖𝐻0Ξ¨β€–β€–2ξ‚΅1+πœ‚1+ξ‚Ά4𝛽‖Ψ‖2+14πœ‚β€–Ξ¨β€–2.(2.52) Combining (2.48) with (2.40), (2.41), (2.51), and (2.52) we get for πœ‚>0, 𝛽>0‖‖𝐻𝐼Ψ‖‖2βŽ›βŽœβŽœβŽξ“β‰€63𝛼=1,2ℓ=1ξ“πœ–β‰ πœ–β€²β€–β€–πΊ(𝛼)β„“,πœ–,πœ–β€²β€–β€–2βŽžβŽŸβŽŸβŽ Γ—ξƒ©12π‘šπ‘Šξƒ©1π‘š21ξƒͺ‖‖𝐻+10Ξ¨β€–β€–2+𝛽2π‘šπ‘Šπ‘š21‖‖𝐻0Ξ¨β€–β€–2+12π‘šπ‘Šξ‚΅11+ξ‚Ά4𝛽‖Ψ‖2ξƒͺβŽ›βŽœβŽœβŽξ“+123𝛼=1,2ℓ=1ξ“πœ–β‰ πœ–β€²β€–β€–πΊ(𝛼)β„“,πœ–,πœ–β€²β€–β€–2βŽžβŽŸβŽŸβŽ ξƒ©πœ‚π‘š21‖‖𝐻(1+𝛽)0Ξ¨β€–β€–2+ξ‚΅πœ‚ξ‚΅11+ξ‚Ά+14𝛽4πœ‚β€–Ξ¨β€–2ξƒͺ,(2.53) by noting ξ€œΞ£1Γ—Ξ£1Γ—Ξ£2||𝐺ℓ,πœ–,πœ–β€²ξ€·πœ‰1,πœ‰2,πœ‰3ξ€Έ||2𝑀(3)ξ€·πœ‰3ξ€Έdπœ‰1dπœ‰2dπœ‰3≀1π‘šπ‘Šβ€–β€–πΊ(𝛼)β„“,πœ–,πœ–β€²β€–β€–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)β„“,πœ–,πœ–β€².

Hypothesis 3.1. (i) For 𝛼=1,2, β„“=1,2,3, πœ–,πœ–ξ…ž=Β±, ξ€œΞ£1Γ—Ξ£1Γ—Ξ£2|||𝐺(𝛼)β„“,πœ–,πœ–β€²ξ€·πœ‰1,πœ‰2,πœ‰3ξ€Έ|||2||𝑝2||2dπœ‰1dπœ‰2dπœ‰3<∞.(3.1)
(ii) There exists 𝐢>0 such that for 𝛼=1,2, β„“=1,2,3, πœ–,πœ–ξ…ž=Β±, ξ‚΅ξ€œΞ£1Γ—{|𝑝2|β‰€πœŽ}Γ—Ξ£2|||𝐺(𝛼)β„“,πœ–,πœ–β€²ξ€·πœ‰1,πœ‰2,πœ‰3ξ€Έ|||2dπœ‰1dπœ‰2dπœ‰3ξ‚Ά1/2β‰€πΆπœŽ2.(3.2)
(iii) For 𝛼=1,2, β„“=1,2,3, πœ–,πœ–ξ…ž=Β±, and 𝑖,𝑗=1,2,3(iii.a)ξ€œΞ£1Γ—Ξ£1Γ—Ξ£2|||𝑝2β‹…βˆ‡π‘2𝐺(𝛼)β„“,πœ–,πœ–β€²ξ‚„(πœ‰1,πœ‰2,πœ‰3)|||2dπœ‰1dπœ‰2dπœ‰3(<∞,iii.b)ξ€œΞ£1Γ—Ξ£1Γ—Ξ£2𝑝22,𝑖𝑝22,𝑗|||||πœ•2𝐺(𝛼)β„“,πœ–,πœ–β€²πœ•π‘2,π‘–πœ•π‘2,𝑗(πœ‰1,πœ‰2,πœ‰3)|||||2dπœ‰1dπœ‰2dπœ‰3<∞.(3.3)
(iv) There exists Ξ›>π‘š1 such that for 𝛼=1,2, β„“=1,2,3, πœ–,πœ–ξ…ž=Β±, 𝐺(𝛼)β„“,πœ–,πœ–β€²ξ€·πœ‰1,πœ‰2,πœ‰3ξ€Έ=0if||𝑝2||β‰₯Ξ›.(3.4)

Remark 3.2. Hypothesis 3.1(ii) is nothing but an infrared regularization of the kernels 𝐺(𝛼)β„“,πœ–,πœ–β€². In order to satisfy this hypothesis it is, for example, sufficient to suppose that 𝐺(𝛼)β„“,πœ–,πœ–β€²ξ€·πœ‰1,πœ‰2,πœ‰3ξ€Έ=||𝑝2||1/2𝐺(𝛼)β„“,πœ–,πœ–β€²ξ€·πœ‰1,πœ‰2,πœ‰3ξ€Έ,(3.5) where 𝐺(𝛼)β„“,πœ–,πœ–β€² 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)β„“,πœ–,πœ–β€² 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)=𝜎ac[(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 π‘Ž denote the following operator in 𝐿2(Ξ£1): 1π‘Ž=2𝑝2β‹…π‘–βˆ‡π‘2+π‘–βˆ‡π‘2⋅𝑝2ξ€Έ.(3.7) The operator π‘Ž is essentially self-adjoint on 𝐢∞0(ℝ3,β„‚2). Its second quantized version dΞ“(π‘Ž) is a self-adjoint operator in π”‰π‘Ž(𝐿2(Ξ£1)). From the definition (2.8) of the space 𝔉ℓ, the following operator in 𝔉ℓ𝐴ℓ=1βŠ—1βŠ—dΞ“(π‘Ž)βŠ—1+1βŠ—1βŠ—1βŠ—dΞ“(π‘Ž)(3.8) is essentially self-adjoint on 𝔇𝐿.

Let now 𝐴 be the following operator in 𝔉𝐿: 𝐴=𝐴1βŠ—12βŠ—13+11βŠ—π΄2βŠ—13+11βŠ—12βŠ—π΄3.(3.9) Then 𝐴 is essentially self-adjoint on 𝔇𝐿.

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 ξ€·βŸ¨π΄βŸ©=1+𝐴2ξ€Έ1/2.(3.10)

We then have the following.

Theorem 3.4. Suppose that the kernels 𝐺(𝛼)β„“,πœ–,πœ–β€² satisfy Hypotheses 2.1 and 3.1. For any 𝛿>0 satisfying 0<𝛿<π‘š1 there exists 0<𝑔𝛿≀𝑔2 such that, for 0<𝑔≀𝑔𝛿, the following points are satisfied. (i)The spectrum of 𝐻 in (inf𝜎(𝐻),π‘š1βˆ’π›Ώ] is purely absolutely continuous.(ii)Limiting absorption principle.
For every 𝑠>1/2 and πœ‘, πœ“ in 𝔉, the limits limπœ€β†’0ξ€·πœ‘,βŸ¨π΄βŸ©βˆ’π‘ (π»βˆ’πœ†Β±π‘–πœ€)βŸ¨π΄βŸ©βˆ’π‘ πœ“ξ€Έ(3.11) exist uniformly for πœ† in any compact subset of (inf𝜎(𝐻),π‘š1βˆ’π›Ώ]. (iii) Pointwise decay in time.
Suppose π‘ βˆˆ(1/2,1) and π‘“βˆˆπΆβˆž0(ℝ) with suppπ‘“βŠ‚(inf𝜎(𝐻),π‘š1βˆ’π›Ώ). Then β€–β€–βŸ¨π΄βŸ©βˆ’π‘ eβˆ’π‘–π‘‘π»π‘“(𝐻)βŸ¨π΄βŸ©βˆ’π‘ β€–β€–ξ€·π‘‘=π’ͺ1/2βˆ’π‘ ξ€Έ(3.12) as π‘‘β†’βˆž.

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 πœ’0(β‹…), πœ’βˆž(β‹…)∈𝐢∞(ℝ,[0,1]) with πœ’0=1 on (βˆ’βˆž,1], πœ’βˆž=1 on [2,∞) and πœ’02+πœ’βˆž2=1.

For 𝜎>0 we set πœ’πœŽ(𝑝)=πœ’0ξ‚΅||𝑝||πœŽξ‚Ά,πœ’πœŽ(𝑝)=πœ’βˆžξ‚΅||𝑝||πœŽξ‚Ά,ξ‚πœ’πœŽ(𝑝)=1βˆ’πœ’πœŽ(𝑝),(3.13) where π‘βˆˆβ„3.

The operator 𝐻𝐼,𝜎 is the interaction given by (2.23) and (2.24) and associated with the kernels ξ‚πœ’πœŽ(𝑝2)𝐺(𝛼)β„“,πœ–,πœ–β€²(πœ‰1,πœ‰2,πœ‰3). We then set 𝐻𝜎∢=𝐻0+𝑔𝐻𝐼,𝜎.(3.14)

Let Ξ£1,𝜎=Ξ£1βˆ©π‘ξ€½ξ€·2,𝑠2ξ€Έ;||𝑝2||ξ€Ύ,Ξ£<𝜎𝜎1=Ξ£1βˆ©π‘ξ€½ξ€·2,𝑠2ξ€Έ;||𝑝2||𝔉β‰₯πœŽβ„“,2,𝜎=π”‰π‘Žξ€·πΏ2ξ€·Ξ£1,πœŽξ€Έξ€ΈβŠ—π”‰π‘Žξ€·πΏ2ξ€·Ξ£1,𝜎,π”‰ξ€Έξ€ΈπœŽβ„“,2=π”‰π‘Žξ€·πΏ2ξ€·Ξ£πœŽ1ξ€Έξ€ΈβŠ—π”‰π‘Žξ€·πΏ2ξ€·Ξ£πœŽ1,𝔉ℓ,2=𝔉ℓ,2,πœŽβŠ—π”‰πœŽβ„“,2,𝔉ℓ,1=2ξ·π”‰π‘Žξ€·πΏ2ξ€·Ξ£1.ξ€Έξ€Έ(3.15) The space 𝔉ℓ,1 is the Fock space for the massive leptons β„“, and 𝔉ℓ,2 is the Fock space for the neutrinos and antineutrinos β„“.

Set π”‰πœŽβ„“=𝔉ℓ,1βŠ—π”‰πœŽβ„“,2𝔉ℓ,𝜎=𝔉ℓ,2,𝜎.(3.16) We have π”‰β„“β‰ƒπ”‰πœŽβ„“βŠ—π”‰β„“,𝜎.(3.17) Set π”‰πœŽπΏ=3ℓ=1π”‰πœŽβ„“π”‰πΏ,𝜎=3ℓ=1𝔉ℓ,𝜎.(3.18) We have π”‰πΏβ‰ƒπ”‰πœŽπΏβŠ—π”‰πΏ,𝜎.(3.19) Set π”‰πœŽ=π”‰πœŽπΏβŠ—π”‰π‘Š.(3.20) We have 𝔉≃𝔉𝐿,πœŽβŠ—π”‰πœŽ.(3.21) Set 𝐻0(1)=3ℓ=1ξ“πœ–=Β±ξ€œπ‘€β„“(1)ξ€·πœ‰1ξ€Έπ‘βˆ—β„“,πœ–ξ€·πœ‰1𝑏ℓ,πœ–ξ€·πœ‰1ξ€Έdπœ‰1,𝐻0(2)=3ℓ=1ξ“πœ–=Β±ξ€œπ‘€β„“(2)ξ€·πœ‰2ξ€Έπ‘βˆ—β„“,πœ–ξ€·πœ‰2𝑐ℓ,πœ–ξ€·πœ‰2ξ€Έdπœ‰2,𝐻0(3)=ξ“πœ–=Β±ξ€œπ‘€(3)ξ€·πœ‰3ξ€Έπ‘Žβˆ—πœ–ξ€·πœ‰3ξ€Έπ‘Žπœ–ξ€·πœ‰3ξ€Έdπœ‰3,𝐻0(2)𝜎=3ℓ=1ξ“πœ–=Β±ξ€œ|𝑝2|>πœŽπ‘€β„“(2)ξ€·πœ‰2ξ€Έπ‘βˆ—β„“,πœ–ξ€·πœ‰2𝑐ℓ,πœ–ξ€·πœ‰2ξ€Έdπœ‰2,𝐻(2)0,𝜎=3ℓ=1ξ“πœ–=Β±ξ€œ|𝑝2|β‰€πœŽπ‘€β„“(2)ξ€·πœ‰2ξ€Έπ‘βˆ—β„“,πœ–ξ€·πœ‰2𝑐ℓ,πœ–ξ€·πœ‰2ξ€Έdπœ‰2.(3.22) We have on π”‰πœŽβŠ—π”‰πœŽπ»0(2)=𝐻0(2)πœŽβŠ—1𝜎+1πœŽβŠ—π»(2)0,𝜎.(3.23) Here, 1𝜎 (resp., 1𝜎) is the identity operator on π”‰πœŽ (resp., π”‰πœŽ).

Define 𝐻𝜎=𝐻𝜎||π”‰πœŽ,𝐻𝜎0=𝐻0||π”‰πœŽ.(3.24)

We get 𝐻𝜎=𝐻0(1)+𝐻0(2)𝜎+𝐻0(3)+𝑔𝐻𝐼,𝜎onπ”‰πœŽ,𝐻𝜎=π»πœŽβŠ—1𝜎+1πœŽβŠ—π»(2)0,𝜎onπ”‰πœŽβŠ—π”‰πœŽ.(3.25) 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, for 𝛽>0 and πœ‚>0, πΆπ›½πœ‚=ξ‚΅3π‘šπ‘Šξ‚΅11+π‘š12ξ‚Ά+3π›½π‘šπ‘Šπ‘š12+12πœ‚π‘š12ξ‚Ά(1+𝛽)1/2,π΅π›½πœ‚=ξ‚΅3π‘šπ‘Šξ‚΅11+ξ‚Άξ‚΅πœ‚ξ‚΅14𝛽+121+ξ‚Ά+14𝛽4πœ‚ξ‚Άξ‚Ά1/2.(3.26) Let 𝐺𝐺=(𝛼)β„“,πœ–,πœ–β€²ξ‚(β‹…,β‹…,β‹…)𝛼=1,2;β„“=1,2,3;πœ–,πœ–β€²=Β±,πœ–β‰ πœ–β€²,(3.27) and set βŽ›βŽœβŽœβŽξ“πΎ(𝐺)=3𝛼=1,2ℓ=1ξ“πœ–β‰ πœ–β€²β€–β€–πΊ(𝛼)β„“,πœ–,πœ–β€²β€–β€–2𝐿2(Ξ£1Γ—Ξ£1Γ—Ξ£2)⎞⎟⎟⎠1/2.(3.28) Let ξ‚πΆπ›½πœ‚=πΆπ›½πœ‚ξ‚΅π‘”1+1𝐾(𝐺)πΆπ›½πœ‚1βˆ’π‘”1𝐾(𝐺)πΆπ›½πœ‚ξ‚Ά,ξ‚π΅π›½πœ‚=𝑔1+1𝐾(𝐺)πΆπ›½πœ‚1βˆ’π‘”1𝐾(𝐺)πΆπ›½πœ‚ξ‚΅π‘”2+1𝐾(𝐺)π΅π›½πœ‚πΆπ›½πœ‚1βˆ’π‘”1𝐾(𝐺)πΆπ›½πœ‚π΅ξ‚Άξ‚Άπ›½πœ‚.(3.29)

Let ξ‚πΎβŽ›βŽœβŽœβŽœβŽξ“(𝐺)=3𝛼=1,2ℓ=1ξ“πœ–β‰ πœ–β€²ξ€œΞ£1Γ—Ξ£1Γ—Ξ£2|||𝐺(𝛼)β„“,πœ–,πœ–β€²ξ€·πœ‰1,πœ‰2,πœ‰3ξ€Έ|||2|𝑝2|2dπœ‰1dπœ‰2dπœ‰3⎞⎟⎟⎟⎠1/2.(3.30) Let π›Ώβˆˆβ„ be such that 0<𝛿<π‘š1.(3.31)

We set 𝐷=sup4Λ𝛾2π‘š1ξ‚Άξ‚ξ‚€βˆ’π›Ώ,1𝐾(𝐺)2π‘š1ξ‚πΆπ›½πœ‚+ξ‚π΅π›½πœ‚ξ‚,(3.32) where Ξ›>π‘š1 has been introduced in Hypothesis 3.1(iv).

Let us define the sequence (πœŽπ‘›)𝑛β‰₯0 by 𝜎0𝜎=Ξ›,1=π‘š1βˆ’π›Ώ2,𝜎2=π‘š1βˆ’π›Ώ=π›ΎπœŽ1,πœŽπ‘›+1=π›ΎπœŽπ‘›,𝑛β‰₯1,(3.33) where 𝛾=1βˆ’π›Ώ/(2π‘š1βˆ’π›Ώ).

Let 𝑔𝛿(1) be such that 0<𝑔𝛿(1)ξ‚΅<inf1,𝑔1,π›Ύβˆ’π›Ύ23𝐷.(3