Spectral Theory for a Mathematical Model of the Weak Interaction—Part I: The Decay of the

Advances in Mathematical Physics Volume 2009, Article ID 978903,52pages doi:10.1155/2009/978903

Research Article

Spectral Theory for a Mathematical Model of the Weak Interaction—Part I: The Decay of the

Intermediate Vector Bosons W

J.-M. Barbaroux

and J.-C. Guillot

1Centre de Physique Th´eorique, Centre National de la Recherche Scientique (CNRS), Luminy Case 907, 13288 Marseille Cedex 9, France

2D´epartement de Math´ematiques, Universit´e du Sud Toulon-Var, 83957 La Garde Cedex, France

3Centre de Math´ematiques Appliqu´ees, ´Ecole Polytechnique, UMR-CNRS 7641, 91128 Palaiseau Cedex, France

Correspondence should be addressed to J.-M. Barbaroux,barbarou@univ-tln.fr Received 28 April 2009; Accepted 13 August 2009

Recommended by Valentin Zagrebnov

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.

Copyrightq2009 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.

1. Introduction

In this article, we consider a mathematical model of the weak interaction as patterned accord- ing to the Standard Model in Quantum Field Theorysee1,2. We choose the example of the weak decay of the intermediate vector bosonsW^±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 from3,4and are based on5,6. Some of the results of this article have been announced in7.

For other applications of the conjugate operator method see8–19.

For related results about models in Quantum Field Theory see20,21in the case of the Quantum Electrodynamics and22in the case of the weak interaction.

The paper is organized as follows. InSection 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 bosonsWandW⁻involves the full family of leptons together with the bosons themselves, according to the Standard Model see 1, formula 4.139and2.

The full family of leptons involves the electrone⁻and the positrone, together with the associated neutrinoν_eand antineutrinoν_e, the muonsμ⁻andμtogether with the associated neutrinoν_μand antineutrinoν_μ,and the tau leptonsτ⁻andτ together with the associated neutrinoντ and antineutrinoντ.

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 helicity1/2.

In what follows, the mathematical model for the weak decay of the vector bosons W andW⁻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 mixing23.

Let us sketch how we define a mathematical model for the weak decay of the vector bosonsW^±into the full family of leptons.

The energy of the free leptons and bosons is a self-adjoint operator in the correspond- ing Fock spacesee 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 Schr ¨odinger representation the interaction is given bysee1, page 159, equation4.139and2, page 308, equation21.3.20

I

d³xΨexγ^α 1−γ₅

ΨνexWαx

d³xΨνexγ^α 1−γ₅

ΨexWαx^∗, 2.1

whereγ^α,α 0,1,2,3 andγ5are the Dirac matrices andΨ·xandΨ·xare the Dirac fields fore₋,e,ν_e,andν_e.

We have Ψex

1 2π

_3/2

s ±1/2

d³p

b_e,

p, su p, s

√p0 e^ip·xb^∗_e,−

p, sv p, s

√p0 e^−ip·x , Ψex Ψex^†γ⁰.

2.2

Herep₀ |p|²m²_e^1/2wherem_e >0 is the mass of the electron, andup, sandvp, sare the normalized solutions to the Dirac equationsee1, Appendix.

The operatorsb_e,p, sandb^∗_e,p, s resp.,b_e,−p, sandb^∗_e,−p, sare the annihilation and creation operators for the electronsresp., the positronssatisfying the anticommutation relationssee below.

Similarly we define Ψνex and Ψνex by substituting the operators cνe,±p, s and c^∗_νe,±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, sandc^∗_νe,−p, sare the annihilation and creation operators for the neutrinos associated with the electronsresp., the antineutrinos.

For theWαfields we havesee24, Section 5.3

W_αx 1

2π

_3/2

λ −1,0,1

d³k 2k0

_αk, λak, λe^ik·x∗_αk, λa^∗₋k, λe^−ik·x

. 2.3

Herek0 |k|²m²_W^1/2wheremW >0 is the mass of the bosonsW^±.Wis the antiparticule ofW⁻. The operatorsak, λanda^∗k, λ resp.,a₋k, λanda^∗₋k, λare the annihilation and creation operators for the bosonsW⁻resp.,Wsatisfying the canonical commutation relations. The vectorsαk, λare the polarizations of the massive spin 1 bosonsW^±see24, Section 5.2.

The interaction2.1is 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 Hamiltoniansee25. For that sake, we have to introduce a spatial cutoffgxsuch thatg ∈L¹R³, together with momentum cutoffsχpandρkfor the Dirac fields and the Wμfields, 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

ρkg

p₁p₂−k

, 2.4

wheregis the Fourier transform ofg. 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 from24, Section 4.2, fermionic creation and annihilation operators of different species of leptons will always anticommute.

Concerning our notations, from now on, ∈ {1,2,3}denotes each species of leptons.

1 denotes the electrone⁻the positroneand the neutrinosνe,νe. 2 denotes the muons μ⁻,μ and the neutrinosν_μandν_μ, and 3 denotes the tau-leptons and the neutrinosν_τ andν_τ.

Letξ1 p1, s1be the quantum variables of a massive lepton, where p1 ∈ R³ and s₁ ∈ {−1/2,1/2}is the spin polarization of particles and antiparticles. Letξ₂ p2, s₂be the quantum variables of a massless lepton wherep₂∈R³ands₂∈ {−1/2,1/2}is the helicity of particles and antiparticles, and, finally, letξ3 k, λbe the quantum variables of the spin 1 bosonsWandW⁻wherek ∈R³andλ ∈ {−1,0,1}is the polarization of the vector bosons see24, Section 5. We setΣ1 R³× {−1/2,1/2}for the leptons andΣ2 R³× {−1,0,1}for the bosons. ThusL²Σ1is the Hilbert space of each lepton andL²Σ2is the Hilbert space of each boson. The scalar product inL²Σj,j 1,2 is defined by

f, g

Σj

fξgξdξ, j 1,2. 2.5

Here

Σ1

dξ

s 1/2,−1/2

dp,

Σ2

dξ

λ 0,1,−1

dk,

p, k∈R³

. 2.6

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

LetSbe any separable Hilbert space. Let_n

aSresp.,_n

sSdenote the antisymmetric resp., symmetricnth tensor power ofS. The fermionicresp., bosonicFock space overS, denoted byFaS resp.,FsS, is the direct sum

FaS ^∞

n 0

n a S

resp., FsS ^∞

n 0

n

s S , 2.7

where₀

aS ₀

sS ≡C. The stateΩ 1,0,0, . . . ,0, . . .denotes the vacuum state inFaS and inFsS.

For every,F 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

4

Fa

L²Σ1

1,2,3. 2.8

We have

F

q≥0,q≥0,r≥0,r≥0

F^q,q,r,r 2.9

with

F^q,q,r,r _q

a

L²Σ1 ⊗

⎛

⎝^q

a

L²Σ1

⎞

⎠⊗ _r

a

L²Σ1 ⊗ _r

a

L²Σ1 . 2.10

Hereq resp.,q is the number of massive fermionic particleresp., antiparticlesandr

resp.,ris the number of neutrinosresp., antineutrinos. The vectorΩ is the associated vacuum state. The fermionic Fock space denoted byFLfor the leptons is then

FL

3 1

F, 2.11

andΩL ₃

1Ωis the vacuum state.

The bosonic Fock space for the vector bosonsWandW⁻, denoted byFW, is then FW Fs

L²Σ2

⊗Fs

L²Σ2 Fs

L²Σ2⊕L²Σ2

. 2.12

We have

FW

t≥0,t≥0

F^t,t_W , 2.13

whereF^t,t_{W t}

sL²Σ2⊗_t

sL²Σ2. Heretresp.,tis the number of bosonsW⁻resp., W. The vectorΩWis the corresponding vacuum.

The Fock space for the weak decay of the vector bosonsWandW⁻, denoted byF, is thus

F FL⊗FW, 2.14

andΩ ΩL⊗ΩW is the vacuum state.

For every ∈ {1,2,3} let D denote the set of smooth vectorsψ ∈ F for which ψ^q,q,r,rhas a compact support andψ^q,q,r,r 0 for all but finitely manyq, q, r, r. Let

DL 3

1D. 2.15

Here

is the algebraic tensor product.

LetDWdenote the set of smooth vectorsφ∈FWfor whichφ^t,thas a compact support andφ^t,t 0 for all but finitely manyt, t.

Let

D DL⊗D W. 2.16

The setDis dense inF.

LetAbe a self-adjoint operator inFsuch thatD is a core forA. Its extension toFL

is, by definition, the closure inFLof the operatorA₁⊗1₂⊗1₃with domainDLwhen 1, of the operator 11⊗A2⊗13 with domainDLwhen 2, and of the operator 11⊗12⊗A3with domainDLwhen 3. Here 1is the operator identity onF.

The extension ofA toFLis a self-adjoint operator for whichDLis a core and it can be extended toF. The extension ofA toFis, by definition, the closure inFof the operator A⊗1W with domainD, whereAis the extension ofA toFL. The extension ofAtoFis a self-adjoint operator for whichDis a core.

LetB be a self-adjoint operator inFW for which DW is a core. The extension of the self-adjoint operatorA⊗Bis, by definition, the closure inFof the operatorA₁⊗1₂⊗1₃⊗B with domainDwhen 1, of the operator 11⊗A2⊗13⊗Bwith domainDwhen 2, and of the operator 11⊗12⊗A₃⊗Bwith domainDwhen 3. The extension ofA⊗BtoFis a self-adjoint operator for whichDis a core.

We now define the creation and annihilation operators.

For each 1,2,3, b_,ξ1 resp., b^∗_,ξ1 is the fermionic annihilation resp., fermionic creation operator for the corresponding species of massive particle when and for the corresponding species of massive antiparticle when −. The operatorsb,ξ1 andb_,^∗ ξ1are defined as usuallysee, e.g.,20,26; see also the detailed definitions in27.

Similarly, for each 1,2,3,c_,ξ2 resp.,c^∗_,ξ2is the fermionic annihilationresp., fermionic creationoperator for the corresponding species of neutrino when and for the corresponding species of antineutrino when −. The operatorsc_,ξ2andc^∗_,ξ2are defined in a standard way, but with the additional requirements that for any,,and, the operatorsb_,ξ1andc,ξ2anticommutes, wherestands either for a∗or for no symbol see the detailed definitions in27.

The operatoraξ3 resp.,a^∗ξ3is the bosonic annihilationresp., bosonic creation operator for the bosonW⁻when and for the bosonWwhen −see, e.g.,20,26, or27. Note thataξ3commutes withb_, ξ1andc,ξ2.

The following canonical anticommutation and commutation relations hold:

b_,ξ1, b^∗,

ξ₁

δδδ ξ₁−ξ₁

, c_,ξ2, c^∗,

ξ₂

δδδ ξ₂−ξ₂

, aξ3, a^∗

ξ₃

δδ ξ₃−ξ₃

, b_,ξ₁, b,

ξ₁

c_,ξ₂, c, ξ₂

0, aξ3, a

ξ₃ 0, {b,ξ1, c,ξ2}

b_,ξ1, c^∗, ξ2 0, b,ξ1, aξ3

b,ξ1, a^∗ξ3

c,ξ2, aξ3

c,ξ2, a^∗ ξ3 0,

2.17

where we used the notationδξj−ξ_j δ_λλδk−k.

We recall that the following operators, withϕ∈L²Σ1, b_,

ϕ

Σ1

b_,ξϕξdξ, c_,

ϕ

Σ1

c_,ξϕξdξ,

b_,^∗

ϕ

Σ1

b^∗_,ξϕξdξ, c^∗_,

ϕ

Σ1

c^∗_,ξϕξdξ

2.18

are bounded operators inFsuch that b_,

ϕ c_,

ϕ ϕ_L2, 2.19

wherebresp.,cisbresp.,corb^∗resp.,c^∗.

The operators b_, ϕand c_,ϕsatisfy similar anticommutaion relations see, e.g., 28.

The free HamiltonianH0is given by H₀ H₀¹H₀²H₀³

3 1

±

w¹ ξ1b^∗_,ξ1b,ξ1dξ1³

1

±

w² ξ2c_,^∗ ξ2c,ξ2dξ2

±

w³ξ3a^∗ξ3aξ3dξ3,

2.20

where

w¹ ξ1 p₁²m²_1/2

, with 0< m₁ < m₂< m₃, w² ξ2 p₂,

w³ξ3

|k|²m²_W1/2 ,

2.21

wheremW is the mass of the bosonsWandW⁻such thatmW > m3.

The spectrum ofH₀is0,∞and 0 is a simple eigenvalue withΩas eigenvector. The set of thresholds ofH₀, denoted byT, is given by

T

pm₁qm₂rm₃sm_W;

p, q, r, s

∈N⁴, pqrs≥1

, 2.22

and each sett,∞,t∈T, is a branch of absolutely continuous spectrum forH0. The interaction, denoted byH_I, is given by

HI

2 α 1

H_I^α, 2.23

where

H_I^{1 3}

1

/

G¹_,,ξ1, ξ2, ξ3b_,^∗ ξ1c_,^∗ ξ2aξ3dξ1dξ2dξ3

³

1

/

G¹_,,ξ1, ξ₂, ξ₃a^∗ξ3c,ξ2b,ξ1dξ1dξ₂dξ₃,

H_I^{2 3}

1

/

G²_,,ξ1, ξ2, ξ3b^∗_,ξ1c^∗_,ξ2a^∗ξ3dξ1dξ2dξ3

³

1

/

G²_,,ξ1, ξ₂, ξ₃aξ3c,ξ2b,ξ1dξ1dξ₂dξ₃.

2.24

The kernelsG²_,,·,·,·,α 1,2, are supposed to be functions.

The total Hamiltonian is then

H H0gHI, g >0, 2.25

wheregis a coupling constant.

The operatorH_I¹describes the decay of the bosonsWandW⁻into leptons. Because ofH_I² the bare vacuum will not be an eigenvector of the total Hamiltonian for everyg >0 as we expect from the physics.

Every kernelG_,,ξ1, ξ₂, ξ₃, computed in theoretical physics, contains aδ-distribution because of the conservation of the momentumsee1and24, Section 4.4. In what follows, we approximate the singular kernels by square integrable functions.

Thus, from now on, the kernelsG^α_,,are supposed to satisfy the following hypothesis.

Hypothesis 2.1. Forα 1,2, 1,2,3,, ±, we assume

G^α_,,ξ1, ξ2, ξ3∈L²Σ1×Σ1×Σ2. 2.26 Remark 2.2. A similar model can be written down for the weak decay of pionsπ⁻andπsee 1, Section 6.2.

Remark 2.3. The total Hamiltonian is more general than the one involved in the theory of weak interactions because, in the Standard Model, neutrinos have helicity−1/2 and antineutrinos have helicity 1/2.

In the physical case, the Fock space, denoted byF, is isomorphic toF_L⊗FW, with

F_L ³

1

F,

F

₂

a

L²Σ1 ⊗ ₂

a

L² R³ .

2.27

The free Hamiltonian, now denoted byH₀, is then given by

H₀ ³

1

±

w¹ξ₁b^∗_,ξ₁b_,ξ₁dξ1³

1

±

R³

p₂c^∗_, p₂

c_, p₂

dp₂

±

w³ξ3a^∗ξ3aξ3dξ3,

2.28

and the interaction, now denoted by H_I, is the one obtained from HI by supposing that G^αξ1,p2, s2, ξ3 0 ifs2 1/2. The total Hamiltonian, denoted by H, is then given byH H₀ gH_I. The results obtained in this paper forH hold true forHwith obvious modifications.

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

The formal operator

G¹_,,ξ1, ξ₂, ξ₃b^∗_,ξ1c^∗_,ξ2aξ3dξ1dξ₂dξ₃ 2.29

is defined as a quadratic form onD⊗DW×D⊗DWas c,ξ2b,ξ1ψ, G¹_,,aξ3φ

dξ₁dξ₂dξ₃, 2.30

whereψ,φ∈D⊗DW.

By mimicking the proof of29, Theorem X.44, we get a closed operator, denoted by H_I,,,¹ , associated with the quadratic form such that it is the unique operator inF⊗FW such thatD⊗DW ⊂ DH_I,,,¹ is a core forH_I,,,¹ and

H_I,,,¹

G¹_,,ξ1, ξ₂, ξ₃b^∗_,ξ1c_,^∗ ξ2aξ3dξ1dξ₂dξ₃ 2.31

as quadratic forms onD⊗DW×D⊗DW.

Similarly for the operatorH_I,,,^{1 ∗}, we have the equality as quadratic forms H_I,,,¹

_∗

G¹_,,ξ1, ξ2, ξ3a^∗ξ3c,ξ2b,ξ1dξ1dξ2dξ3. 2.32

Again, there exists two closed operatorsH_I,,,² andH_I,,,^{2 ∗} such that D ⊗DW ⊂ DH_I,,,² ,D ⊗DW ⊂ DH_I,,,^{2 ∗}, andD ⊗DW is a core forH_I,,,² and H_I,,,^{2 ∗}and

such that

H_I,,,²

G²_,,ξ1, ξ₂, ξ₃b_,^∗ ξ1c_,^∗ ξ2a^∗ξ3dξ1dξ₂dξ₃, H_I,,,²

_∗

G²_,,ξ1, ξ2, ξ3aξ3c,ξ2b,ξ1dξ1dξ2dξ3

2.33

as quadratic forms onD⊗DW×D⊗DW.

We will still denote byH_I,,,^α andH_I,,,^{α ∗}α 1,2their extensions toF. The setD is then a core forH_I,,,^α andH_I,,,^{α ∗}.

Thus

H H0g

α 1,2

3 1

/

H_I,,,^α H_I,,,²

_∗

2.34

is a symmetric operator defined onD.

We now want to prove thatHis essentially self-adjoint onDby showing thatH_I,,,^α

andH_I,,,^{α ∗}are relativelyH0-bounded.

Once again, as above, for almost everyξ₃ ∈ Σ2, there exists closed operators inFL, denoted byB^α_,,ξ3andB^α_,,ξ3^∗such that

B_,,¹ ξ3 −

G¹_,,ξ1, ξ2, ξ3b,ξ1c,ξ2dξ1dξ2, B¹_,,ξ3_∗

G¹_,,ξ1, ξ₂, ξ₃b^∗_,ξ1c^∗_,ξ2dξ1dξ₂, B_,,² ξ3

G²_,,ξ1, ξ2, ξ3b^∗_,ξ1c^∗_,ξ2dξ1dξ2, B²_,,ξ3_∗

−

G²_,,ξ1, ξ₂, ξ₃b,ξ1c,ξ2dξ1dξ₂

2.35

as quadratic forms onD×D.

We have that D ⊂ DB^α_,,ξ3 resp.,D ⊂ DB_,,^α ξ3^∗ is a core for B_,,^α ξ3 resp., forB^α_,,ξ3^∗. We still denote byB^α_,,ξ3andB_,,^α ξ3^∗their extensions toFL.

It then follows that the operatorH_I with domainDis symmetric and can be written in the following form:

HI

α 1,2

3 1

/

H_I,,,^α H_I,,,^α

_∗

α 1,2

3 1

/

B^α_,,ξ3⊗a^∗ξ3dξ3

α 1,2

3 1

/

B^α_,,ξ3_∗

⊗aξ3dξ3.

2.36

LetNdenote the operator number of massive leptonsinF, that is,

N

b^∗_,ξ1b,ξ1dξ1. 2.37

The operatorN is a positive self-adjoint operator inF. We still denote byN its extension toFL. The setDLis a core forN.

We then have the following.

Proposition 2.4. For almost every ξ₃ ∈ Σ2,DB^α_,,ξ3,DB^α_,,ξ3^∗ ⊃ DN^1/2 , and for Φ∈ DN^1/2 ⊂FLone has

B_,,^α ξ3Φ

FL≤G^α_,,·,·, ξ3

L²Σ1×Σ1

N^1/2 Φ

FL,

B^α_,,ξ3_∗ Φ

FL≤G^α_,,·,·, ξ3_L2

Σ1×Σ1

N^1/2 Φ

FL.

2.38

Proof. The estimates2.38are examples ofN_τestimatessee25. The proof is quite similar to the proof of20, Proposition 3.7. Details can be found in27but are omitted here.

Let

H_0,³

w³ξ₃a^∗ξ₃aξ₃dξ3. 2.39

ThenH_0,³is a self-adjoint operator inFW, andDW is a core forH_0,³. We get the following.

Proposition 2.5. One has B^α_,,ξ3_∗

⊗aξ3dξ3Ψ ²

≤

⎛

⎜⎝

Σ1×Σ1×Σ2

G^α_,,ξ1, ξ2, ξ3²

w³ξ3 dξ₁dξ₂dξ₃

⎞

⎟⎠

N1^1/2⊗

H_0,³1/2

Ψ ²,

2.40

B^α_,,ξ3⊗a^∗ξ3dξ3Ψ ²

≤

⎛

⎜⎝

Σ1×Σ1×Σ2

G^α_,,ξ₁, ξ₂, ξ₃²

w³ξ3 dξ₁dξ₂dξ₃

⎞

⎟⎠

N1^1/2⊗

H_0,³_1/2 Ψ

²

Σ1×Σ1×Σ2

G^α_,,ξ1, ξ₂, ξ₃²dξ₁dξ₂dξ₃ ηN1^1/2⊗1Ψ² 1 4ηΨ²

2.41

for everyΨ∈ DH0and everyη >0.

Proof. Suppose thatΨ∈ DN^1/2⊗DH_0,^31/2. Let Ψξ3 w³ξ3^1/2

N1^1/2⊗aξ3

Φ. 2.42

We have

B^α_,,ξ3_∗

⊗aξ3dξ3Ψ

Σ2

1

w³ξ3_1/2

B_,,^α ξ3_∗

N1^−1/2⊗1

Ψξ3dξ3. 2.43 Therefore, forΨ∈ DN^1/2 ⊗DH_0,^31/2,2.40follows fromProposition 2.4.

We now have

B_,,^α ξ3⊗a^∗ξ3Ψdξ3

² F

B_,,^α ξ3⊗a ξ₃

Ψ, B^α_,,

ξ₃

⊗aξ3Ψ

dξ₃dξ₃

B_,,^α ξ3⊗1

Ψ²dξ₃, 2.44

Σ2×Σ2

B^α_,,ξ3⊗a ξ₃

Ψ, B_,,^α

ξ₃

⊗aξ3Ψ dξ₃dξ₃

Σ2×Σ2

1 w³ξ3^1/2w³

ξ_31/2

×

B^α_,,ξ3N1^−1/2⊗1 Ψ

ξ₃ ,

B_,,^α

ξ₃

N1^−1/2⊗1

Ψξ3 dξ₃dξ₃

≤

Σ2

1 w³ξ3^1/2

B^α_,,ξ₃N1^−1/2

FLΨξ₃dξ3 2

≤

⎛

⎝

Σ1×Σ1×Σ2

G^αξ1, ξ2, ξ3²

w³ξ3 dξ₁dξ₂dξ₃

⎞

⎠

N1^1/2⊗

H_0,³_1/2 Ψ

².

2.45 Furthermore

Σ2

B_,,^α ξ3⊗1 Ψ²dξ₃

Σ2

B^α_,,ξ3N1^−1/2⊗1

N1^1/2⊗1 Ψ²dξ₃

≤

Σ1×Σ1×Σ2

G^α_,,ξ1, ξ2, ξ3²dξ1dξ2dξ3 ηN1Ψ² 1 4ηΨ²

2.46

for everyη >0.

By2.40,2.45, and2.46, we finally get2.41for everyΨ ∈ DN^1/2⊗DH_0,³. It then follows that2.40and2.41are verified for everyΨ∈ DH0.

We now prove thatHis a self-adjoint operator inFforgsufficiently small.

Theorem 2.6. Letg₁>0 be such that 3g₁²

mW

1

m²₁ 1

α 1,2

3 1

/

G^α_,,²_L2

Σ1×Σ1×Σ2<1. 2.47

Then for everyg satisfyingg ≤g1,His a self-adjoint operator inFwith domainDH DH0, andDis a core forH.

Proof. LetΨbe inD. We have

HIΨ²≤12

α 1,2

3 1

/

"

B^α_,,ξ3_∗

⊗aξ3Ψdξ3

²

B_,,^α ξ3

⊗a^∗ξ3Ψdξ3

²

# . 2.48

Note that

H_0,³Ψ≤H₀³Ψ≤ H0Ψ, NΨ ≤ 1

mH0,Ψ ≤ 1

m₁H0,Ψ ≤ 1

m₁H0Ψ,

2.49

where H0,

w¹ ξ1b_,^∗ ξ1b,ξ1dξ1

w² ξ2c^∗_,ξ2c,ξ2dξ2. 2.50

We further note that N1^1/2⊗

H_0,³1/2

Ψ ²≤ 1

2 1

m²₁ 1 H0Ψ² β

2m²₁H0Ψ² 1

2 1 8β

Ψ² 2.51

forβ >0, and

ηN1⊗1Ψ² 1

4ηΨ²≤ η

m²₁H0Ψ² ηβ

m²₁H0Ψ²η

1 1 4β

Ψ² 1 4ηΨ².

2.52

Combining2.48with2.40,2.41,2.51, and2.52we get forη >0,β >0

HIΨ² ≤6

α 1,2

3 1

/

G^α_,,²

× 1

2mW

1

m²₁ 1 H0Ψ² β

2m_Wm²₁H0Ψ² 1 2mW

1 1

4β

Ψ² 12

α 1,2

3 1

/

G^α_,,² η m²₁

1β

H0Ψ²

η

1 1 4β

1

4η

Ψ² , 2.53

by noting

Σ1×Σ1×Σ2

|G,,ξ1, ξ₂, ξ₃|²

w³ξ3 dξ₁dξ₂dξ₃≤ 1

m_WG^α_,,². 2.54 By2.53the theorem follows from the Kato-Rellich theorem.

3. Main Results

In the sequel, we will make the following additional assumptions on the kernelsG^α_,,. Hypothesis 3.1. iForα 1,2, 1,2,3,, ±,

Σ1×Σ1×Σ2

G^α_,,ξ1, ξ₂, ξ₃²

p₂² dξ1dξ2dξ3<∞. 3.1

iiThere existsC >0 such that forα 1,2, 1,2,3,, ±,

Σ1×{|p2|≤σ}×Σ2

G^α_,,ξ1, ξ₂, ξ₃²dξ₁dξ₂dξ₃

1/2

≤Cσ². 3.2

iiiForα 1,2, 1,2,3,, ±, andi, j 1,2,3

iii.a

Σ1×Σ1×Σ2

$p₂· ∇p2

G^α_,,%

ξ1, ξ₂, ξ₃²dξ₁dξ₂dξ₃ <∞, iii.b

Σ1×Σ1×Σ2

p_2,i² p_2,j²

∂²G^α_,,

∂p2,i∂p2,jξ1, ξ2, ξ3

2

dξ1dξ2dξ3<∞.

3.3