Hindawi Publishing Corporation
Advances in Mathematical Physics
Volume 2009, Article ID 978903, 52 pages
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. Barbaroux1, 2 and J.-C. Guillot3
1
Centre de Physique Théorique, Centre National de la Recherche Scientique (CNRS), Luminy Case 907,
13288 Marseille Cedex 9, France
2
Département de Mathématiques, Université du Sud Toulon-Var, 83957 La Garde Cedex, France
3
Centre de Mathématiques Appliquées, École 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.
Copyright q 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.
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
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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 νe and antineutrino ν e , the muons μ− and μ together with the associated
neutrino νμ and antineutrino ν μ , and the tau leptons τ − 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 −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
3
α
2.1
I d xΨe xγ 1 − γ5 Ψνe xWα x d3 xΨνe xγ α 1 − γ5 Ψe xWα x∗ ,
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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 , νe , and νe .
We have
Ψe x 1
2π
3/2 s±1/2
3
d p be,
u p, s ip·x
v p, s −ip·x
∗
,
e be,− p, s √
e
p, s √
p0
p0
2.2
Ψe x Ψe x† γ 0 .
Here p0 |p|2 m2e 1/2 where me > 0 is the mass of the electron, and up, s and vp, s are
the normalized solutions to the Dirac equation see 1, Appendix.
∗
∗
p, s resp., be,− p, s and be,−
p, s are the annihilation
The operators be, p, s and be,
and creation operators for the electrons resp., the positrons satisfying the anticommutation
relations see below.
Similarly we define Ψνe x and Ψνe x by substituting the operators cνe ,± p, s and
∗
p, s with p0 |p|. The operators cνe , p, s and cν∗e , p, s
cν∗e ,± p, s for be,± p, s and be,±
∗
resp., cνe ,− p, s and cνe ,− p, s are the annihilation and creation operators for the neutrinos
associated with the electrons resp., the antineutrinos.
For the Wα fields we have see 24, Section 5.3
Wα x 1
2π
3/2 3 dk
α k, λa k, λeik·x α∗ k, λa∗− k, λe−ik·x .
2k0
λ−1,0,1
2.3
Here k0 |k|2 m2W 1/2 where mW > 0 is the mass of the bosons W ± . W is the antiparticule
of W − . The operators a k, λ and a∗ k, λ resp., a− k, λ and a∗− k, λ are the annihilation
and creation operators for the bosons W − resp., W satisfying the canonical commutation
relations. The vectors 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 gx such
that g ∈ L1 R3 , together with momentum cutoffs χp and ρk for the Dirac fields and the
Wμ 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 ρk
g p1 p2 − k ,
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: “The statevector 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.”
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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, ∈ {1, 2, 3} denotes each species of leptons.
1 denotes the electron e− the positron e and the neutrinos νe , νe . 2 denotes the muons
μ− , μ and the neutrinos νμ and νμ , and 3 denotes the tau-leptons and the neutrinos ντ
and ντ .
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 Σj , j 1, 2 is defined by
f, g Σj
fξgξdξ,
j 1, 2.
2.5
Here
Σ1
dξ dp,
s1/2,−1/2
Σ2
dξ p, k ∈ R3 .
dk,
λ0,1,−1
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 na S resp., ns S denote the antisymmetric
resp., symmetric nth tensor power of S. The fermionic resp., bosonic Fock space over S,
denoted by Fa S resp., Fs S, is the direct sum
Fa S ∞ n
n0
S
a
resp., Fs S ∞ n
n0
S ,
2.7
s
where 0a S 0s S ≡ C. The state Ω 1, 0, 0, . . . , 0, . . . denotes the vacuum state in Fa S
and in Fs S.
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 L2 Σ1 1, 2, 3.
2.8
q ,q ,r ,r 2.9
We have
F F q ≥0,q ≥0,r ≥0,r ≥0
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with
q ,q ,r ,r F q
a
L Σ1 2
⎛
⊗⎝
q
a
⎞
L Σ1 ⎠ ⊗
2
r
a
L Σ1 2
⊗
r
a
L Σ1 .
2
2.10
Here q resp., q is the number of massive fermionic particle resp., antiparticles and r
resp., r is the number of neutrinos resp., antineutrinos. The vector Ω is the associated
vacuum state. The fermionic Fock space denoted by FL for the leptons is then
FL 3
F ,
2.11
1
and ΩL 31 Ω is the vacuum state.
The bosonic Fock space for the vector bosons W and W − , denoted by FW , is then
FW Fs L2 Σ2 ⊗ Fs L2 Σ2 Fs L2 Σ2 ⊕ L2 Σ2 .
2.12
We have
FW t,t
FW ,
2.13
t≥0,t≥0
t,t
where FW ts L2 Σ2 ⊗ ts L2 Σ2 . Here t resp., t is the number of bosons W − resp.,
W . The vector ΩW is the corresponding vacuum.
The Fock space for the weak decay of the vector bosons W and W − , denoted by F, is
thus
F FL ⊗ FW ,
2.14
and Ω ΩL ⊗ ΩW is the vacuum state.
For every ∈ {1, 2, 3} let D denote the set of smooth vectors ψ ∈ F for which
q ,q ,r ,r ψ Let
q ,q ,r ,r has a compact support and ψ
0 for all but finitely many q , q , r , r .
3
DL 1 D .
2.15
Here is the algebraic tensor product.
Let DW denote the set of smooth vectors φ ∈ FW for which φt,t has a compact support
t,t
and φ
0 for all but finitely many t, t.
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Let
DW .
D DL ⊗
2.16
The set D is dense in F.
Let A be a self-adjoint operator in F such that D is a core for A . Its extension to FL
is, by definition, the closure in FL of the operator A1 ⊗ 12 ⊗ 13 with domain DL when 1, of
the operator 11 ⊗ A2 ⊗ 13 with domain DL when 2, and of the operator 11 ⊗ 12 ⊗ A3 with
domain DL when 3. Here 1 is the operator identity on F .
The extension of A to FL is a self-adjoint operator for which DL is a core and it can
be extended to F. The extension of A to F is, by definition, the closure in F of the operator
is the extension of A to FL . The extension of A to F is a
⊗ 1W with domain D, where A
A
self-adjoint operator for which D is a core.
Let B be a self-adjoint operator in FW for which DW is a core. The extension of the
self-adjoint operator A ⊗ B is, by definition, the closure in F of the operator A1 ⊗ 12 ⊗ 13 ⊗ B
with domain D when 1, of the operator 11 ⊗ A2 ⊗ 13 ⊗ B with domain D when 2, and
of the operator 11 ⊗ 12 ⊗ A3 ⊗ B with domain D when 3. The extension of A ⊗ B to F is a
self-adjoint operator for which D is a core.
We now define the creation and annihilation operators.
∗
ξ1 is the fermionic annihilation resp.,
For each 1, 2, 3, b, ξ1 resp., b,
fermionic creation operator for the corresponding species of massive particle when and for the corresponding species of massive antiparticle when −. The operators b, ξ1 ∗
ξ1 are defined as usually see, e.g., 20, 26; see also the detailed definitions in 27.
and b,
∗
ξ2 is the fermionic annihilation resp.,
Similarly, for each 1, 2, 3, c, ξ2 resp., c,
fermionic creation operator for the corresponding species of neutrino when and for
∗
ξ2 are
the corresponding species of antineutrino when −. The operators c, ξ2 and c,
defined in a standard way, but with the additional requirements that for any , , and , the
operators b, ξ1 and c , ξ2 anticommutes, where stands either for a ∗ or for no symbol
see the detailed definitions in 27.
The operator a ξ3 resp., a∗ ξ3 is the bosonic annihilation resp., bosonic creation
operator for the boson W − when and for the boson W when − see, e.g., 20, 26,
or 27. Note that a ξ3 commutes with b, ξ1 and c , ξ2 .
The following canonical anticommutation and commutation relations hold:
δ δ δ ξ1 − ξ1 ,
b, ξ1 , b∗ , ξ1
δ δ δ ξ2 − ξ2 ,
c, ξ2 , c∗ , ξ2
a ξ3 , a∗ ξ3 δ δ ξ3 − ξ3 ,
2.17
b, ξ1 , b , ξ1 c, ξ2 , c , ξ2 0,
a ξ3 , a ξ3 0,
{b, ξ1 , c , ξ2 } b, ξ1 , c∗ , ξ2 0,
b, ξ1 , a ξ3 b, ξ1 , a∗ ξ3 c, ξ2 , a ξ3 c, ξ2 , a∗ ξ3 0,
where we used the notation δξj − ξj δλλ δk − k .
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We recall that the following operators, with ϕ ∈ L2 Σ1 ,
b, ϕ ∗
b,
ϕ Σ1
b, ξϕξdξ,
c, ϕ ∗
b,
ξϕξdξ,
∗
c,
Σ1
ϕ Σ1
c, ξϕξdξ,
Σ1
2.18
∗
c,
ξϕξdξ
are bounded operators in F such that
b, ϕ c, ϕ ϕL2 ,
2.19
where b resp., c is b resp., c or b∗ resp., c∗ .
The operators b, ϕ and c, ϕ satisfy similar anticommutaion relations see, e.g.,
28.
The free Hamiltonian H0 is given by
1
2
3
H0 H0 H0 H0
3 1 ±
±
1
∗
w ξ1 b,
ξ1 b, ξ1 dξ1
3 1 ±
2
∗
w ξ2 c,
ξ2 c, ξ2 dξ2
2.20
w3 ξ3 a∗ ξ3 a ξ3 dξ3 ,
where
1/2
2
1
w ξ1 p1 m2
,
with 0 < m1 < m2 < m3 ,
2
w ξ2 p2 ,
2.21
1/2
w3 ξ3 |k|2 m2W
,
where mW is the mass of the bosons W and W − such that mW > 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 smW ; 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 HI , is given by
HI 2
α
HI ,
α1
2.23
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where
1
HI
3 1
/
1
∗
∗
G,, ξ1 , ξ2 , ξ3 b,
ξ1 c,
ξ2 a ξ3 dξ1 dξ2 dξ3
3 1 / 2
HI
3 1 /
1
G,, ξ1 , ξ2 , ξ3 a∗ ξ3 c, ξ2 b, ξ1 dξ1 dξ2 dξ3 ,
2.24
2
∗
∗
∗
G,, ξ1 , ξ2 , ξ3 b,
ξ1 c,
ξ2 a ξ3 dξ1 dξ2 dξ3
3 1 /
2
G,, ξ1 , ξ2 , ξ3 a ξ3 c, ξ2 b, ξ1 dξ1 dξ2 dξ3 .
2
The kernels G,, ·, ·, ·, 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.
1
The operator HI describes the decay of the bosons W and W − into leptons. Because
2
of HI 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,, 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.
α
Thus, from now on, the kernels G,, are supposed to satisfy the following hypothesis.
Hypothesis 2.1. For α 1, 2, 1, 2, 3, , ±, we assume
α
2.26
G,, ξ1 , ξ2 , ξ3 ∈ L2 Σ1 × Σ1 × Σ2 .
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 FL ⊗ FW , with
FL F 2
a
3
1
L2 Σ1 ⊗
F ,
2
a
L2 R3
2.27
.
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The free Hamiltonian, now denoted by H0 , is then given by
H0 3 1 ±
±
1
∗
w ξ1 b,
ξ1 b, ξ1 dξ1 3 p2 c∗ p2 c, p2 dp2
,
1 ±
R3
2.28
w3 ξ3 a∗ ξ3 a ξ3 dξ3 ,
and the interaction, now denoted by HI , is the one obtained from HI by supposing that
Gα ξ1 , p2 , s2 , ξ3 0 if s2 1/2. The total Hamiltonian, denoted by H , is then given
by H H0 gHI . The results obtained in this paper for H hold true for H with obvious
modifications.
Under Hypothesis 2.1 a well-defined operator on D corresponds to the formal
interaction HI as it follows.
The formal operator
1
∗
∗
G,, ξ1 , ξ2 , ξ3 b,
ξ1 c,
ξ2 a ξ3 dξ1 dξ2 dξ3
2.29
is defined as a quadratic form on D ⊗ DW × D ⊗ DW as
1
c, ξ2 b, ξ1 ψ, G,, a ξ3 φ dξ1 dξ2 dξ3 ,
2.30
where ψ, φ ∈ D ⊗ DW .
By mimicking the proof of 29, Theorem X.44, we get a closed operator, denoted by
1
HI,,, , associated with the quadratic form such that it is the unique operator in F ⊗ FW such
1
1
that D ⊗ DW ⊂ DHI,,, is a core for HI,,, and
1
HI,,, 1
∗
∗
G,, ξ1 , ξ2 , ξ3 b,
ξ1 c,
ξ2 a ξ3 dξ1 dξ2 dξ3
2.31
as quadratic forms on D ⊗ DW × D ⊗ DW .
1
Similarly for the operator HI,,, ∗ , we have the equality as quadratic forms
1
HI,,,
∗
1
G,, ξ1 , ξ2 , ξ3 a∗ ξ3 c, ξ2 b, ξ1 dξ1 dξ2 dξ3 .
2
2.32
2
Again, there exists two closed operators HI,,, and HI,,, ∗ such that D ⊗ DW ⊂
2
2
2
2
DHI,,, , D ⊗ DW ⊂ DHI,,, ∗ , and D ⊗ DW is a core for HI,,, and HI,,, ∗ and
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such that
2
HI,,,
2
∗
∗
∗
G,, ξ1 , ξ2 , ξ3 b,
ξ1 c,
ξ2 a ξ3 dξ1 dξ2 dξ3 ,
∗ 2
2
HI,,, G,, ξ1 , ξ2 , ξ3 a ξ3 c, ξ2 b, ξ1 dξ1 dξ2 dξ3
2.33
as quadratic forms on D ⊗ DW × D ⊗ DW .
α
α
We will still denote by HI,,, and HI,,, ∗ α 1, 2 their extensions to F. The set D
α
α
is then a core for HI,,, and HI,,, ∗ .
Thus
H H0 g
3 α1,2 1 /
∗ α
2
HI,,, HI,,,
2.34
is a symmetric operator defined on D.
α
We now want to prove that H is essentially self-adjoint on D by showing that HI,,,
α
and HI,,, ∗ are relatively H0 -bounded.
Once again, as above, for almost every ξ3 ∈ Σ2 , there exists closed operators in FL ,
α
α
denoted by B,, ξ3 and B,, ξ3 ∗ such that
1
B,, ξ3 −
∗
1
B,, ξ3 2
B,, ξ3 ∗
2
B,, ξ3 1
G,, ξ1 , ξ2 , ξ3 b, ξ1 c, ξ2 dξ1 dξ2 ,
1
∗
∗
G,, ξ1 , ξ2 , ξ3 b,
ξ1 c,
ξ2 dξ1 dξ2 ,
2.35
2
∗
∗
G,, ξ1 , ξ2 , ξ3 b,
ξ1 c,
ξ2 dξ1 dξ2 ,
−
2
G,, ξ1 , ξ2 , ξ3 b, ξ1 c, ξ2 dξ1 dξ2
as quadratic forms on D × D .
α
α
α
We have that D ⊂ DB,, ξ3 resp., D ⊂ DB,, ξ3 ∗ is a core for B,, ξ3 α
α
α
resp., for B,, ξ3 ∗ . We still denote by B,, ξ3 and B,, ξ3 ∗ 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 3 α1,2 1 /
α
HI,,,
3 α1,2 1 /
∗ α
HI,,,
∗
α
B,, ξ3 3 α1,2 1 /
⊗ a ξ3 dξ3 .
α
B,, ξ3 ⊗ a∗ ξ3 dξ3
2.36
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Let N denote the operator number of massive leptons in F , that is,
N ∗
b,
ξ1 b, ξ1 dξ1 .
2.37
The operator N is a positive self-adjoint operator in F . We still denote by N its extension
to FL . The set DL is a core for N .
We then have the following.
α
α
Proposition 2.4. For almost every ξ3 ∈ Σ2 , DB,, ξ3 , DB,, ξ3 ∗ ⊃ DN1/2 , and for
Φ ∈ DN1/2 ⊂ FL one has
α
B,, ξ3 Φ
α
≤ G,, ·, ·, ξ3 1/2 N Φ ,
∗ α
B,, ξ3 Φ
α
≤ G,, ·, ·, ξ3 1/2 N Φ .
FL
L2 Σ1 ×Σ1 L2 Σ1 ×Σ1 FL
FL
2.38
FL
Proof. The estimates 2.38 are examples of Nτ estimates see 25. The proof is quite similar
to the proof of 20, Proposition 3.7. Details can be found in 27 but are omitted here.
Let
3
H0,
w3 ξ3 a∗ ξ3 a ξ3 dξ3 .
3
2.39
3
Then H0, is a self-adjoint operator in FW , and DW is a core for H0, .
We get the following.
Proposition 2.5. One has
2
∗
Bα ξ3 ⊗ a ξ3 dξ3 Ψ
,,
⎛
⎜
≤⎝
2
α
G,, ξ1 , ξ2 , ξ3 Σ1 ×Σ1 ×Σ2
w3 ξ3 ⎞
2
⎟
3 1/2 1/2
,
1
⊗
H
Ψ
dξ1 dξ2 dξ3 ⎠
N
0,
2.40
2
Bα ξ3 ⊗ a∗ ξ3 dξ3 Ψ
,,
⎛
⎜
≤⎝
2
α
G,, ξ1 , ξ2 , ξ3 Σ1 ×Σ1 ×Σ2
w3 ξ3 ⎞
2
⎟
3 1/2 1/2
1
⊗
H
Ψ
dξ1 dξ2 dξ3 ⎠
N
0,
2
2
1
α
ηN 11/2 ⊗ 1Ψ Ψ2
G,, ξ1 , ξ2 , ξ3 dξ1 dξ2 dξ3
4η
Σ1 ×Σ1 ×Σ2
for every Ψ ∈ DH0 and every η > 0.
2.41
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DH0, 1/2 . Let
Proof. Suppose that Ψ ∈ DN1/2 ⊗
Ψ ξ3 w3 ξ3 1/2 N 11/2 ⊗ a ξ3 Φ.
We have
∗
α
B,, ξ3 ⊗ a ξ3 dξ3 Ψ Σ2
1
w3 ξ3 1/2
2.42
∗
α
B,, ξ3 N 1−1/2 ⊗ 1 Ψ ξ3 dξ3 .
2.43
3
DH0, 1/2 , 2.40 follows from Proposition 2.4.
Therefore, for Ψ ∈ DN1/2 ⊗
We now have
2
Bα ξ3 ⊗ a∗ ξ3 Ψdξ3 ,,
F
α
B,, ξ3 α ⊗ a ξ3 Ψ, B,, ξ3 ⊗ a ξ3 Ψ dξ3 dξ3 2
α
B,, ξ3 ⊗ 1 Ψ dξ3 ,
2.44
Σ2 ×Σ2
α
α B,, ξ3 ⊗ a ξ3 Ψ, B,, ξ3 ⊗ a ξ3 Ψ dξ3 dξ3
1
Σ2 ×Σ2
×
1
Σ2
⎛
≤⎝
1/2
w3 ξ3
α
α B,, ξ3 N 1−1/2 ⊗ 1 Ψ ξ3 , B,, ξ3 N 1−1/2 ⊗ 1 Ψ ξ3 dξ3 dξ3
≤
w3 ξ3 1/2
w3 ξ3 α
−1/2 1
ξ
N
B
Ψ ξ3 dξ3
3
,,
1/2
FL
α
G ξ1 , ξ2 , ξ3 2
Σ1 ×Σ1 ×Σ2
2
w3 ξ3 ⎞
2
3 1/2 1/2
⎠
dξ1 dξ2 dξ3 N 1 ⊗ H0,
Ψ
.
2.45
Furthermore
2
α
B,, ξ3 ⊗ 1 Ψ dξ3
Σ2
2
α
B,, ξ3 N 1−1/2 ⊗ 1 N 11/2 ⊗ 1 Ψ dξ3
Σ2
≤
for every η > 0.
2
1
α
ηN 1Ψ2 Ψ2
G,, ξ1 , ξ2 , ξ3 dξ1 dξ2 dξ3
4η
Σ1 ×Σ1 ×Σ2
2.46
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13
3
DH0, . It
By 2.40, 2.45, and 2.46, we finally get 2.41 for every Ψ ∈ DN1/2 ⊗
then follows that 2.40 and 2.41 are verified for every Ψ ∈ DH0 .
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
3g12
mW
3 1
α 2
1
< 1.
G
,, L2 Σ ×Σ ×Σ 1
1
2
m21
α1,2 1 /
2.47
Then for every g satisfying g ≤ g1 , H is a self-adjoint operator in F with domain DH DH0 ,
and D is a core for H.
Proof. Let Ψ be in D. We have
" 2 2 #
3 ∗
α
α
∗
HI Ψ ≤ 12
B,, ξ3 ⊗ a ξ3 Ψdξ3 B,, ξ3 ⊗ a ξ3 Ψdξ3 .
2
α1,2 1 /
2.48
Note that
3 3 H0, Ψ ≤ H0 Ψ ≤ H0 Ψ,
N Ψ ≤
1
1
1
H0, Ψ ≤
H0, Ψ ≤
H0 Ψ,
m
m1
m1
2.49
where
H0, 1
∗
w ξ1 b,
ξ1 b, ξ1 dξ1 2
∗
w ξ2 c,
ξ2 c, ξ2 dξ2 .
2.50
We further note that
1/2 2 1
β
1
1
1
2
2
N 11/2 ⊗ H 3
Ψ
≤
1
Ψ
Ψ
Ψ2 2.51
H
H
0
0
0,
2 m21
2 8β
2m21
for β > 0, and
η
ηβ
1
1
1
2
2
2
ηN 1 ⊗ 1Ψ Ψ2 Ψ ≤ 2 H0 Ψ 2 H0 Ψ η 1 Ψ2 .
4η
4β
4η
m1
m1
2.52
2
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Advances in Mathematical Physics
Combining 2.48 with 2.40, 2.41, 2.51, and 2.52 we get for η > 0, β > 0
2
HI Ψ ≤ 6
3 α 2
G,, α1,2 1 /
1
2mW
×
12
β
1
1
1
2
2
2
1 H0 Ψ 1
Ψ
H0 Ψ 2mW
4β
m21
2mW m21
3 α 2
G,, α1,2 1 /
1
1
2
2
1 β H0 Ψ η 1 Ψ ,
4β
4η
η m21
2.53
by noting
|G,, ξ1 , ξ2 , ξ3 |2
1 α 2
dξ
dξ
≤
.
dξ
G
1
2
3
,, mW
w3 ξ3 Σ1 ×Σ1 ×Σ2
2.54
By 2.53 the theorem follows from the Kato-Rellich theorem.
3. Main Results
α
In the sequel, we will make the following additional assumptions on the kernels G,, .
Hypothesis 3.1. i For α 1, 2, 1, 2, 3, , ±,
2
α
G,, ξ1 , ξ2 , ξ3 dξ1 dξ2 dξ3 < ∞.
2
p2 Σ1 ×Σ1 ×Σ2
3.1
ii There exists C > 0 such that for α 1, 2, 1, 2, 3, , ±,
Σ1 ×{|p2 |≤σ}×Σ2
2
α
G,, ξ1 , ξ2 , ξ3 dξ1 dξ2 dξ3
1/2
≤ Cσ 2 .
3.2
iii For α 1, 2, 1, 2, 3, , ±, and i, j 1, 2, 3
iii.a
Σ1 ×Σ1 ×Σ2
iii.b
$
2
α %
p2 · ∇p2 G,, ξ1 , ξ2 , ξ3 dξ1 dξ2 dξ3 < ∞,
2
∂2 Gα
,,
2 2 p2,i
p2,j ξ1 , ξ2 , ξ3 dξ1 dξ2 dξ3 < ∞.
∂p2,i ∂p2,j
Σ1 ×Σ1 ×Σ2
3.3
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15
iv There exists Λ > m1 such that for α 1, 2, 1, 2, 3, , ±,
α
G,, ξ1 , ξ2 , ξ3 0
if p2 ≥ Λ.
3.4
α
Remark 3.2. Hypothesis 3.1ii is nothing but an infrared regularization of the kernels G,, .
In order to satisfy this hypothesis it is, for example, sufficient to suppose that
1/2 α
α
ξ1 , ξ2 , ξ3 ,
G,, ξ1 , ξ2 , ξ3 p2 G
,,
3.5
α
is a smooth function of p1 , p2 , p3 in the Schwartz space.
where G
,,
Hypothesis 3.1iv, 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.
α
Theorem 3.3. Suppose that the kernels G,, 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
σH σac H inf σH, ∞
3.6
with inf σH ≤ 0.
According to Theorem 3.3 the ground state energy E inf σH 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
1
p2 · i∇p2 i∇p2 · p2 .
2
3.7
The operator a is essentially self-adjoint on C0∞ R3 , C2 . Its second quantized version dΓa is
a self-adjoint operator in Fa L2 Σ1 . From the definition 2.8 of the space F , the following
operator in F
A 1 ⊗ 1 ⊗ dΓa ⊗ 1 1 ⊗ 1 ⊗ 1 ⊗ dΓa
is essentially self-adjoint on DL .
3.8
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Let now A be the following operator in FL :
A A1 ⊗ 12 ⊗ 13 11 ⊗ A2 ⊗ 13 11 ⊗ 12 ⊗ A3 .
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
1/2
.
A 1 A2
3.10
We then have the following.
α
Theorem 3.4. Suppose that the kernels G,, satisfy Hypotheses 2.1 and 3.1. For any δ > 0 satisfying
0 < δ < m1 there exists 0 < gδ ≤ g2 such that, for 0 < g ≤ gδ , the following points are satisfied.
i The spectrum of H in inf σH, m1 − δ is purely absolutely continuous.
ii Limiting absorption principle.
For every s > 1/2 and ϕ, ψ in F, the limits
lim ϕ, A−s H − λ ± iεA−s ψ
ε→0
3.11
exist uniformly for λ in any compact subset of inf σH, m1 − δ.
iii Pointwise decay in time.
Suppose s ∈ 1/2, 1 and f ∈ C0∞ R with supp f ⊂ inf σH, m1 − δ. Then
A−s e−itH fHA−s O t1/2−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 χ0 2 χ∞ 2 1.
For σ > 0 we set
p
,
χσ p χ0
σ
p
3.13
σ
,
χ p χ∞
σ
χσ p 1 − χσ p ,
where p ∈ R3 .
Advances in Mathematical Physics
17
The operator HI,σ is the interaction given by 2.23 and 2.24 and associated with the
α
kernels χσ p2 G,, ξ1 , ξ2 , ξ3 . We then set
Hσ : H0 gHI,σ .
3.14
Let
p2 , s2 ; p2 < σ ,
Σσ1 Σ1 ∩ p2 , s2 ; p2 ≥ σ
F,2,σ Fa L2 Σ1,σ ⊗ Fa L2 Σ1,σ ,
Σ1,σ Σ1 ∩
Fσ,2 Fa L2 Σσ1 ⊗ Fa L2 Σσ1 ,
3.15
F,2 F,2,σ ⊗ Fσ,2 ,
F,1 2
Fa L2 Σ1 .
The space F,1 is the Fock space for the massive leptons , and F,2 is the Fock space for the
neutrinos and antineutrinos .
Set
Fσ F,1 ⊗ Fσ,2
F,σ F,2,σ .
3.16
We have
F Fσ ⊗ F,σ .
3.17
Set
FσL 3
1
FL,σ Fσ
3
3.18
F,σ .
1
We have
FL FσL ⊗ FL,σ .
3.19
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Advances in Mathematical Physics
Set
Fσ FσL ⊗ FW .
3.20
F FL,σ ⊗ Fσ .
3.21
We have
Set
1
H0 2
H0
3
2
1 ±
3 1 ±
H0 2σ
H0
3 H0,σ ±
1
∗
w ξ1 b,
ξ1 b, ξ1 dξ1 ,
2
∗
w ξ2 c,
ξ2 c, ξ2 dξ2 ,
w3 ξ3 a∗ ξ3 a ξ3 dξ3 ,
3 1 ±
2
|p2 |>σ
3 1 ±
3.22
∗
w ξ2 c,
ξ2 c, ξ2 dξ2 ,
2
|p2 |≤σ
∗
w ξ2 c,
ξ2 c, ξ2 dξ2 .
We have on Fσ ⊗ Fσ
2
2σ
H0 H0
2
⊗ 1σ 1σ ⊗ H0,σ .
3.23
Here, 1σ resp., 1σ is the identity operator on Fσ resp., Fσ .
Define
H σ Hσ |Fσ ,
H0σ H0 |Fσ .
3.24
We get
1
2σ
H σ H0 H0
3
H0 gHI,σ
2
Hσ H σ ⊗ 1σ 1σ ⊗ H0,σ
on Fσ ,
on Fσ ⊗ Fσ .
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.
Advances in Mathematical Physics
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We now set, for β > 0 and η > 0,
3
1
mW
3
1
mW
Cβη Bβη
1/2
12η 3β
1
,
1
β
m1 2
mW m1 2 m1 2
1/2
1
1
1
.
12 η 1 4β
4β
4η
3.26
Let
α
G G,, ·, ·, ·
α1,2;1,2,3;, ±, /
,
3.27
and set
KG 3 α 2
G,, 2
1/2
L Σ1 ×Σ1 ×Σ2 α1,2 1 /
3.28
.
Let
βη
C
Bβη
g1 KGCβη
Cβη 1 ,
1 − g1 KGCβη
g1 KGBβη Cβη
g1 KGCβη
2
Bβη .
1
1 − g1 KGCβη
1 − g1 KGCβη
3.29
Let
⎛
3
⎜ KG
⎝
α1,2 1 /
2
α
G,, ξ1 , ξ2 , ξ3 Σ1 ×Σ1 ×Σ2
|p2 |2
⎞1/2
⎟
dξ1 dξ2 dξ3 ⎠
.
3.30
Let δ ∈ R be such that
0 < δ < m1 .
3.31
We set
sup
D
4Λγ
βη Bβη ,
2m1 C
, 1 KG
2m1 − δ
where Λ > m1 has been introduced in Hypothesis 3.1iv.
3.32
20
Advances in Mathematical Physics
Let us define the sequence σn n≥0 by
σ0 Λ,
σ1 m1 −
δ
,
2
3.33
σ2 m1 − δ γσ1 ,
σn1 γσn ,
n ≥ 1,
where γ 1 − δ/2m1 − δ.
1
Let gδ be such that
0<
1
gδ
γ − γ2
< inf 1, g1 ,
3D
.
3.34
1
For 0 < g ≤ gδ we have
3g D
0<γ < 1−
,
γ
3g D
σn , n ≥ 1.
0 < σn1 < 1 −
γ
3.35
3.36
Set
H n H σn ;
H0n H0σn ,
En inf σH n ,
n ≥ 0,
n ≥ 0.
3.37
We then get the following.
α
Proposition 3.5. Suppose that the kernels G,, satisfy Hypotheses 2.1, 3.1(i), and 3.1(iv). Then
1
there exists 0 < gδ ≤ gδ such that, for g ≤ gδ and n ≥ 1, En is a simple eigenvalue of H n and H n
does not have spectrum in En , En 1 − 3g D/γσ
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.
Advances in Mathematical Physics
21
Let
ησ p2 χ2σ p2 ,
ησ p2 χ2σ p2 ,
aσ ησ p2 aησ p2 ,
aσ ησ p2 aησ p2 .
3.38
Since ησ2 ησ 2 1, and ησ , ησ , a 0 ησ , ησ , a, we obtain see 4
a aσ aσ .
3.39
2 1 2
ησ p2 p2 · i∇p2 i∇p2 · ησ p2 p2 ,
2
2 1 σ 2
η p2 p2 · i∇p2 i∇p2 · ησ p2 p2 .
aσ 2
3.40
Note that we also have
aσ The operators a, aσ , and aσ are essentially self-adjoint on C0∞ R3 , C2 see 5,
Proposition 4.2.3. We still denote by a, aσ , and aσ their closures. If a denotes any of the
operator a, aσ , and aσ , we have
D
a u ∈ L2 Σ1 ; au ∈ L2 Σ1 .
3.41
The operators dΓa, dΓaσ , and dΓaσ are self-adjoint operators in Fa L2 Σ1 , and
we have
dΓa dΓaσ dΓaσ .
3.42
By 2.8, the following operators in F , denoted by Aσ and Aσ , respectively,
Aσ 1 ⊗ 1 ⊗ dΓaσ ⊗ 1 1 ⊗ 1 ⊗ 1 ⊗ dΓaσ ,
Aσ 1 ⊗ 1 ⊗ dΓaσ ⊗ 1 1 ⊗ 1 ⊗ 1 ⊗ dΓaσ 3.43
are essentially self-adjoint on D .
Let Aσ and Aσ be the following two operators in FL :
Aσ Aσ1 ⊗ 12 ⊗ 13 11 ⊗ Aσ2 ⊗ 13 11 ⊗ 12 ⊗ Aσ3 ,
Aσ Aσ1 ⊗ 12 ⊗ 13 11 ⊗ Aσ2 ⊗ 13 11 ⊗ 12 ⊗ Aσ3 .
3.44
22
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The operators Aσ and Aσ are essentially self-adjoint on DL . Still denoting by Aσ and Aσ their
extensions to F, Aσ and Aσ are essentially self-adjoint on D and we still denote by Aσ and Aσ
their closures.
We have
A A σ Aσ .
3.45
The operators a, aσ , and aσ are associated to the following C∞ -vector fields in R3 ,
respectively:
v p 2 p2 ,
2
v σ p2 η σ p2 p2 ,
2
vσ p2 ησ p2 p2 .
3.46
Let Vp be any of these vector fields. We have
V p ≤ Γp
3.47
V p v p p,
3.48
for some Γ > 0, and we also have
where the v ’s are defined by 3.46 and 3.48 and fulfil |p|α dα /d|p|α v|p| bounded for
α 0, 1, 2.
Let ψt · : R3 → R3 be the corresponding flow generated by V:
d ψt p V ψt p ,
dt
ψ0 p p.
3.49
e−Γ|t| p ≤ ψt p ≤ eΓ|t| p.
3.50
ψt p is a C∞ -flow and we have
ψt p induces a one-parameter group of unitary operators Ut in L2 Σ1 L2 R3 , C2 defined
by
Utf
1/2
.
p f ψt p det ∇ψt p
3.51
Let φt ·, φtσ ·, and φσt · be the flows associated with the vector fields v·, vσ ·, and vσ ·,
respectively.
Advances in Mathematical Physics
23
Let Ut, Uσ t, and Uσ t be the corresponding one-parameter groups of unitary
operators in L2 Σ1 . The operators a, aσ , and aσ are the generators of Ut, Uσ t, and Uσ t,
respectively, that is,
Ut e−iat ,
σ
Uσ t e−ia t ,
3.52
Uσ t e−iaσ t .
Let
2
w2 ξ2 w ξ2 1,2,3
,
3 2
∗
dΓ w2 w ξ2 c,
ξ2 c ξ2 dξ2 .
3.53
1 Let V t be any of the one-parameter groups Ut, Uσ t, and Uσ t. We set
2
V tw2 V t∗ V tw V t∗
1,2,3
,
3.54
and we have
V tw2 V t∗ w2 ψt .
3.55
Here ψt is the flow associated to V t.
This yields, for any ϕ ∈ D, see 11, Lemma 2.8
e−iAt H0 eiAt ϕ − H0 ϕ dΓ e−iat w2 eiat − dΓ w2 ϕ
dΓ w2 ◦ φt − w2 ϕ,
3.56
σ
σ
σ
σ
e−iA t H0 eiA t ϕ − H0 ϕ dΓ e−ia t w2 eia t − dΓ w2 ϕ
dΓ w2 ◦ φtσ − w2 ϕ,
3.57
e−iAσ t H0 eiAσ t ϕ − H0 ϕ dΓ e−iaσ t w2 eiaσ t − dΓ w2 ϕ
dΓ w2 ◦ φσt − w2 ϕ.
α
Proposition 3.6. Suppose that the kernels G,, satisfy Hypothesis 2.1.
3.58
24
Advances in Mathematical Physics
For every t ∈ R one has, for g ≤ g1 ,
i
eitA DH0 eitA DH ⊂ DH0 DH,
σ
σ
ii eitA DH0 eitA DH ⊂ DH0 DH,
3.59
iii eitAσ DH0 eitAσ DH ⊂ DH0 DH.
Proof. We only prove i, since ii and iii can be proved similarly. By 3.56 we have, for
ϕ ∈ D,
1
3
e−itA H0 eitA ϕ H0 H0 dΓ w2 ◦ φt ϕ.
3.60
It follows from 3.50 and 3.60 that
H0 eitA ϕ ≤ eΓ|t| H0 ϕ.
3.61
This yields i because D is a core for H0 . Moreover we get
H0 eitA H0 1−1 ≤ eΓ|t| .
3.62
In view of DH0 DH, the operators H0 H i−1 and HH0 i−1 are bounded, and there
exists a constant C > 0 such that
itA
He H i−1 ≤ CeΓ|t| .
3.63
Similarly, we also get
σ
H0 eitA H0 1−1 ≤ eΓ|t| ,
H0 eitAσ H0 1−1 ≤ eΓ|t| ,
3.64
itAσ
He H i−1 ≤ CeΓ|t| ,
itAσ
He H i−1 ≤ CeΓ|t| .
α
Let HI G be the interaction associated with the kernels G G,, α1,2;1,2,3; / ± ,
α
where the kernels G,, satisfy Hypothesis 2.1.
We set
α
V tG V tG,,
α1,2;1,2,3; /
±
.
3.65
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25
We have for ϕ ∈ D see 11, Lemma 2.7,
e−iAt HI GeiAt ϕ HI e−iat G ϕ,
σ
σ
σ
e−iA t HI GeiA t ϕ HI e−ia t G ϕ,
e−iAσ t HI GeiAσ t ϕ HI e−iaσ t 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 Aσ , C2 Aσ , and C2 A in −∞, m1 − δ/2 and that A and Aσ are locally strictly
conjugate to H in E, m1 − δ/2.
Recall that H is locally of class C2 A in −∞, m1 − δ/2 if, for any ϕ ∈ C0∞ −∞, m1 −
δ/2, ϕH is of class C2 A; that is, t → e−iAt ϕHeitA ψ is twice continuously differentiable
for all ϕ ∈ C0∞ −∞, m1 − δ/2 and all ψ ∈ F.
Thus, one of our main results is the following one.
α
Theorem 3.7. Suppose that the kernels G,, satisfy Hypotheses 2.1, 3.1(i) and 3.1(iii).
a H is locally of class C2 A, C2 Aσ , and C2 Aσ in −∞, m1 − δ/2.
b H σ is locally of class C2 Aσ in −∞, m1 − δ/2.
It follows from Theorem 3.7 that H, iA, H, iAσ , H, iAσ , and H σ , iAσ are defined
&
as sesquilinear forms on K EK HF, 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 ∈
C0∞ E, m1 − δ/2 and all ψ ∈ F,
'
(
eitA − 1
ϕHH, iAϕHψ lim ϕH H,
ϕHψ,
t→0
t
'
(
eitAσ − 1
ϕHH, iAσ ϕHψ lim ϕH H,
ϕHψ,
t→0
t
'
(
σ
eitA − 1
σ
ϕHψ,
ϕHH, iA ϕHψ lim ϕH H,
t→0
t
3.67
'
(
σ
eitA − 1
ϕH H , iA ϕH ψ lim ϕH H ,
ϕH σ ψ.
t→0
t
σ
σ
σ
σ
σ
σ
The following proposition allows us to compute H, iA, H, iAσ , H, iAσ , and
α
H σ , iAσ as sesquilinear forms. By Hypotheses 2.1 and 3.1iii.a, the kernels G,, ξ1 , ·, ξ3 belong to the domains of a, aσ , and aσ .
26
Advances in Mathematical Physics
α
Proposition 3.8. Suppose that the kernels G,, satisfy Hypotheses 2.1 and 3.1(iii.a). Then
a for all ψ ∈ DH one has
i limt → 0 H, eitA − 1/tψ dΓw2 gHI −iaGψ,
σ
ii limt → 0 H, eitA − 1/tψ dΓησ 2 w2 gHI −iaσ Gψ,
iii limt → 0 H, eitAσ − 1/tψ dΓησ 2 w2 gHI −iaσ Gψ,
σ
iv limt → 0 H σ , eitA − 1/tψ dΓησ 2 w2 gHI −iaσ χσ p2 Gψ;
b and
i sup0<|t|≤1 H, eitA − 1/tH i−1 < ∞,
ii sup0<|t|≤1 H, eitA − 1/tH i−1 < ∞,
σ
iii sup0<|t|≤1 H, eitAσ − 1/tH i−1 < ∞,
iv sup0<|t|≤1 H σ , eitA − 1/tH i−1 < ∞.
σ
Proof. Part b follows from part a by the uniform boundedness principle. For part a, we
only prove ai, since other statements can be proved similarly.
By 3.50, we obtain
1
1 2 2 2 eΓ|t| − 1 w p2
w φt p2 − w p2 ≤
|t|
|t|
3.68
for 1, 2, 3.
By 3.56–3.58 and Lebesgue’s Theorem we then get for all ψ ∈ DH0 '
(
%
1$
eitA − 1
ψ lim e−itA H0 eitA − H0 ψ dΓ w2 ψ,
lim H0 ,
t→0
t→0 t
t
'
(
σ
%
1$
eitA − 1
σ
σ
2
ψ lim e−itA H0 eitA − H0 ψ dΓ ησ w2 ψ,
lim H0 ,
t→0
t→0 t
t
'
(
%
1$
eitAσ − 1
2
ψ lim e−itAσ H0 eitAσ − H0 ψ dΓ ησ w2 ψ.
lim H0 ,
t→0
t→0 t
t
By 3.66, we obtain, for all ψ ∈ DH,
'
(
%
1$
eitA − 1
lim HI G,
ψ lim e−itA HI GeitA − HI G ψ HI −iaGψ,
t→0
t→0 t
t
3.69
Advances in Mathematical Physics
27
'
(
σ
%
1$
eitA − 1
σ
σ
ψ lim e−itA HI GeitA − HI G ψ HI −iaσ Gψ,
lim HI G,
t→0
t→0 t
t
'
(
%
1$
eitAσ − 1
lim HI G,
ψ lim e−itAσ HI GeitAσ − HI G ψ HI −iaσ Gψ,
t→0
t→0 t
t
'
(
σ
eitA − 1
σ
%
1$
σ
lim HI χ p2 G ,
ψ lim e−itA HI χσ p2 G eitA − HI χσ p2 G ψ
t→0
t→0 t
t
σ
HI −i aσ χσ 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 ∈ C0∞ −∞, m1 −δ/2
and every ψ ∈ F
$ %
ϕHH, iAϕHψ ϕH dΓ w2 gHI −iaG ϕHψ,
3.71
%
$ 2
ϕHH, iAσ ϕHψ ϕH dΓ ησ w2 gHI −iaσ G ϕHψ,
3.72
%
$ 2
ϕHH, iAσ ϕHψ ϕH dΓ ησ w2 gHI −iaσ G ϕHψ,
3.73
$ %
2
ϕH σ ψ.
ϕH σ H σ , iAσ ϕH σ ψ ϕH σ dΓ ησ w2 gHI −i aσ χσ G
3.74
We now introduce the Mourre inequality.
Let N be the smallest integer such that
Nγ ≥ 1.
3.75
1
We have, for g ≤ gδ ,
1
γ <γ
N
3g D
1−
−γ
γ
γ
1
≤γ−
N
N
<1−
3g D
1−
−γ
γ
3g D
,
γ
3.76
< γ.
Let
⎛
γ 1 ⎝
1−
2N
1 3gδ D
γ
⎞
− γ ⎠.
3.77
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We choose f ∈ C0∞ R such that 1 ≥ f ≥ 0 and
⎧
⎪
1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
0
fλ ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩0
if λ ∈
$
γ − γ
2
%
, γ γ ,
⎛
⎞
1 3gδ D
1⎝
if λ > γ 1−
− γ ⎠ γ 2γ ,
N
γ
⎛
⎞⎞2
⎛
1 3gδ D
2
1
− γ ⎠⎠ γ − 2γ .
if λ < ⎝γ − ⎝1 −
N
γ
3.78
1
for g ≤ gδ and γ − γ > γ/N.
Note that γ 2γ < 1 − 3g D/γ
We set, for n ≥ 1,
fn λ f
λ
.
σn
3.79
Let
Hn Hσn ,
En inf σHn ,
2
H0n
3.80
2
H0σn .
Let P n denote the ground state projection of H n . It follows from Proposition 3.5 that, for n ≥ 1
1
and g ≤ gδ ≤ gδ ,
2
fn Hn − En P n ⊗ fn H0,n .
3.81
En En inf σH n .
3.82
Note that
Set
an aσn ,
an aσn ,
An A σ n ,
A n Aσ n ,
F n F σn ,
F n F σn .
3.83
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29
We have
F Fn ⊗ Fn ,
3.84
A An An .
We further note that
an χσn p2 an .
3.85
H, iAn H n , iAn ⊗ 1
3.86
By 3.72, 3.74, and 3.85, we obtain
as sesquilinear forms with respect to F Fn ⊗ Fn .
Furthermore, it follows from the virial theorem see 6, Proposition 3.2 and
Proposition 6.1 that
P n H n , iAn P n 0.
3.87
1
By 3.81 and 3.87 we then get, for g ≤ gδ ≤ gδ ,
fn Hn − En H, iAn fn Hn − En 0.
3.88
We then have the following.
α
Proposition 3.9. Suppose that the kernels G,, satisfy Hypotheses 2.1 and 3.1. Then there exists
δ > 0 and g 1 > 0 such that g 1 ≤ gδ and
C
δ
δ
δ
fn Hn − En H, iAn fn Hn − En ≥ C
γ2
σn fn Hn − En 2
N2
3.89
1
for n ≥ 1 and g ≤ gδ .
Let EΔ H − E be the spectral projection for the operator H − E associated with the
interval Δ, and let
Δn $
γ − γ
2
%
σn , γ γ σn ,
n ≥ 1.
3.90
Note that
σn2 , σn1 ⊂
γ − γ
2
σn , γ γ σn ,
n ≥ 1.
3.91
30
Advances in Mathematical Physics
α
Theorem 3.10. Suppose that the kernels G,, satisfy Hypotheses 2.1 and 3.1. Then there exists
2
2
1
Cδ > 0 and gδ > 0 such that gδ ≤ gδ and
EΔn H − EH, iAEΔn H − E ≥ Cδ
γ2
σn EΔn H − E
N2
3.92
2
for n ≥ 1 and g ≤ gδ .
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 φn , in Fn :
H n φn En φn ,
φn ∈ DH n ,
n
φ 1,
n ≥ 1.
4.1
Therefore Hn has an unique normalized ground state in F, given by φn φn ⊗ Ωn , where Ωn
is the vacuum state in Fn :
Hn φn En φn ,
φn ∈ DHn ,
φn 1,
n ≥ 1.
4.2
Since φn 1, there exists a subsequence nk k≥1 , converging to ∞ such that φnk k≥1
converges weakly to a state φ ∈ F. We have to prove that φ / 0. By adapting the proof of
n F in order to show
Theorem 4.1 in 22 see also 20, the key point is to estimate c, ξ2 Φ
that
3 2
c, ξ2 φn dξ2 O g 2 ,
1 4.3
uniformly with respect to n.
The estimate 4.3 is a consequence of the so-called “pull-through” formula as it
follows.
α
Let HI, n denote the interaction HI associated with the kernels 1{|p2 |≥σn } p2 G,, . We
thus have
H0 c, ξ2 φn c, ξ2 H0 φn − w ξ2 c, ξ2 φn ,
2
gHI,n c, ξ2 φn c, ξ2 gHI,n φn gV,, ξ2 φn
4.4
Advances in Mathematical Physics
31
with
V,, ξ2 g
1
∗
G, ξ1 , ξ2 , ξ3 b,
ξ1 a ξ3 dξ1 dξ3
g
4.5
2
∗
∗
G, ξ1 , ξ2 , ξ3 b,
ξ1 a ξ3 dξ1 dξ3 .
This yields
2
Hn − En w ξ2 c, ξ2 φn V,, ξ2 φn .
4.6
By adapting the proof of Propositions 2.4 and 2.5 we easily get
V,, ψ ≤
F
g
α
G,, ·, ξ2 , ·
mW 1/2 α1,2
2
g G,, ·, ξ2 , ·
L2 Σ1 ×Σ2 L2 Σ1 ×Σ2 1/2 H0 ψ 4.7
ψ ,
where ψ ∈ DH0 .
Let us estimate H0 φn . By 2.53, 2.54, 3.26, and 3.28 we have
g HI,n φn ≤ gKG Cβη H0 φn Bβη ,
4.8
H0 φn ≤ |En | g HI,n φn .
Therefore
H0 φn ≤
gKGBβη
|En |
.
1 − g1 KGCβη 1 − g1 KGCβη
4.9
By 3.82, A.11, and 4.9, there exists C > 0 such that
H0 φn ≤ C,
4.10
uniformly in n and g ≤ g1 .
By 4.6, 4.7, and 4.10 we get
g
c, φn ≤ p2
C
1/2
2 α
G,, ·, ξ2 , ·
α1
L2 Σ1 ×Σ2 2
G,, ·, ξ2 , ·
L2 Σ1 ×Σ2 .
4.11
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By Hypothesis 3.1i, there exists a constant CG > 0 depending on the kernels G α
G,, 1,2,3;α1,2; / ± and such that
3 2
c, ξ2 φn dξ2 ≤ CG2 g 2 .
1 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 σH 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, ξ2 ’s. For f ∈ L2 R3 , C2 , we define on DH0 the operators
t c, f eitH e−itH0 c, f eitH0 e−itH .
4.13
By mimicking the proof given in 21, 31 one proves, under the hypothesis of Theorem 3.3
t
and for f ∈ C0∞ R3 , C2 , that the strong limits of c, f when t → ±∞ exist for ψ ∈ DH0 :
t lim c f ψ
t → ±∞ ,
± : c, f ψ.
4.14
±
The operators c, f satisfy the CAR and we have
±
c,
f φ 0,
f ∈ C0∞ 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 inf σH, ∞. We omit the details see 21, 31.
5. Proof of the Mourre Inequality
We first prove Proposition 3.9. In view of Proposition 3.8aiii and 3.73, we have, as
sesquilinear forms,
2
2
H, iAσ 1 − g dΓ ησ w2 g dΓ ησ w2 gHI −iaσ G.
1
5.1
2
Let F resp., F be the Fock space for the massive leptons resp., the neutrinos and
antineutrinos .
We have
1
2
F F ⊗ F .
5.2
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33
Let
F
1
FW ⊗
3
1
1
F
,
F2 3
1
2
F .
5.3
We have
F F1 ⊗ F2 ,
5.4
F1 is the Fock space for the massive leptons and the bosons W ± , and F2 is the Fock space
for the neutrinos and antineutrinos.
We have, as sesquilinear forms and with respect to 5.4,
2
dΓ ησ w2 HI −iaσ G
3 2 ∗
ησ p2 p2 c,
ξ2 c, ξ2 dξ2
1 ⎛
⎞
α∗
3 M,, ,σ ξ2 ∗
p2 ⎝11 ⊗ ησ p2 c ξ2 ⊗ 12 ⎠
,
p2 1 /
α1,2
5.5
⎛
⎞
α
M,, ,σ ξ2 × ⎝11 ⊗ ησ p2 c, ξ2 ⊗ 12 ⎠dξ2
p2 α1,2
−
3 1 /
⎛
⎞⎛
⎞
α∗
α
M,, ,σ ξ2 M,, ,σ ξ2 ⎝
1/2 ⊗ 12 ⎠⎝
1/2 ⊗ 12 ⎠dξ2 ,
p2 p
α1,2
α1,2
2
where
α
M,, ,σ ξ2 i
α1,2
α
∗
aησ p2 G,, ξ1 , ξ2 , ξ3 b,
ξ1 a ξ3 dξ1 dξ3 ,
and where 1j is the identity operator in Fj .
5.6
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Advances in Mathematical Physics
By mimicking the proofs of Propositions 2.4 and 2.5, we get, for every ψ ∈ D,
⎛
⎛
⎞⎛
⎞
⎞
α∗
α
M,, ,σ ξ2 M,, ,σ ξ2 ⎝ψ, ⎝
1/2 ⊗ 12 ⎠⎝
1/2 ⊗ 12 ⎠ψdξ2 ⎠
p2 p2 α1,2
α1,2
1 /
3 2
3 Mα,, ,σ ξ2 ψdξ
⊗
1
2
2
1/2
p
α1,2
1 /
2
5.7
⎞
⎛ 2
α aη
,
ξ
,
ξ
p
G
ξ
σ
2
1 2 3
α1,2
,,
⎟
⎜
3 1/2 dξ1 dξ2 dξ3 ⎠ H0
≤⎝
ψ
.
w3 ξ3 p2 Noting that |aησ p2 | ≤ C uniformly with respect to σ, it follows from Hypotheses 2.1 and
3.1 that there exists a constant CG > 0 such that
2
α α1,2 aησ p2 G,, ξ1 , ξ2 , ξ3 dξ1 dξ2 dξ3 ≤ CGσ.
w3 ξ3 p2 5.8
⎞⎛
⎞
α∗
α
M,, ,σ ξ2 M,, ,σ ξ2 − ⎝
1/2 ⊗ 12 ⎠⎝
1/2 ⊗ 12 ⎠dξ2 ≥ −CGσ.
p2 p2 α1,2
α1,2
5.9
This yields
⎛
Combining 5.1, 5.5 with 5.9, we obtain
2
H, iAn ≥ 1 − g dΓ ησn w2 − gCGσn .
5.10
2
2
dΓ ησn w2 ≥ H0n .
5.11
We have
By 3.76, 3.81, and 5.11 we get
fn Hn − En dΓ ησn 2 w2 fn Hn − En 2
2
2
≥ Pn ⊗ fn H0n H0n fn H0n
≥
1
for g ≤ gδ .
γ2
σn fn Hn − En 2
N2
5.12
Advances in Mathematical Physics
35
1
This, together with 5.10, yields for g ≤ gδ
fn Hn − En H, iAn fn Hn − En 2
1 γ
≥ 1 − gδ
σn fn Hn − En 2 − gCGσn fn Hn − En 2 .
N2
5.13
Setting
⎛
2
gδ
1
inf⎝gδ ,
⎞
γ2 ⎠
,
2CG N 2
1
1 − gδ
5.14
we get
1
fn Hn − En H, iAn fn Hn − En ≥
1 − gδ
2
γ2
σn fn Hn − En 2
N2
5.15
2
for g ≤ gδ .
1
2
δ 1 − g 1 /2.
Proposition 3.9 is proved by setting gδ gδ and C
δ
The proof of Theorem 3.10 is the consequence of the following two lemmas.
α
Lemma 5.1. Assume that the kernels G,, satisfy Hypotheses 2.1 and 3.1(ii). Then there exists a
constant D > 0 such that
|E − En | ≤ gDσn 2
5.16
for n ≥ 1 and g ≤ g 2 .
Proof. Let φ resp., φn be the unique normalized ground state of H resp., Hn . We have
E − En ≤ φn , H − Hn φn ,
En − E ≤ φ, Hn − Hφ
5.17
H − Hn gHI χσn p2 G .
5.18
with
Combining 2.53 and 2.54 with 3.26–3.28 and 5.18, we get
H − Hn φn ≤ gK χσn p2 G Cβη H0 φn Bβη ,
H − Hn φ ≤ gK χσ p2 G Cβη H0 φ Bβη .
n
5.19
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Advances in Mathematical Physics
It follows from Hypothesis 3.1ii, 4.10, and 5.19 that there exists a constant D > 0 such
that
max H − Hn φn , H − Hn φ ≤ gDσn 2
5.20
for n ≥ 1 and g ≤ g 2 .
By 5.17, this proves Lemma 5.1.
α
Lemma 5.2. Suppose that the kernels G,, satisfy Hypotheses 2.1 and 3.1(ii). Then there exists a
constant C > 0 such that
fn H − E − fn Hn − En ≤ gCσn
5.21
for n ≥ 1 and g ≤ g 2 .
be an almost analytic extension of f· given by 3.78 satisfying
Proof. Let f·
∂z f x iy ≤ Cy2 .
5.22
iy ∈ C∞ R2 . We thus have
Note that fx
0
fs dfz
,
z−s
dfz
−
1 ∂f
dx dy.
π ∂z
5.23
Using the functional calculus based on this representation of fs, we get
fn H − E − fn Hn − En σn
1
1
dfz.
H − Hn En − E
H − E − zσn
Hn − En − zσn
5.24
Combining 2.53 and 2.54 with 3.26–3.28 and Hypothesis 3.1ii, we get, for every ψ ∈
DH0 and for g ≤ g 2 ,
g HI χσn G ψ ≤ 2gCσn 2 KG Cβη H0 1ψ Cβη Bβη ψ .
5.25
g HI χσn p2 G H0 1−1 ≤ gC1 σn 2
5.26
This yields
for some constant C1 > 0 and for g ≤ g 2 .
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37
By mimicking the proof of A.21 we show that there exists a constant C2 > 0 such that
−1 H0 1Hn − En − zσn ≤ C2 1 1
|Im z|σn
5.27
or g ≤ g 1 .
Combining Lemma 5.1 and 5.24 with 5.25–5.27 we obtain
fn H − E − fn Hn − En ≤ gCσn
∂f/∂z
x iy y2
dx dy
5.28
for some constant C > 0 and for g ≤ g 2 .
iy ∈ C∞ R2 one concludes the proof of Lemma 5.2.
Using 5.22 and fx
0
We now prove Theorem 3.10.
Proof. It follows from Proposition 3.9 that
fn Hn − En H, iAfn Hn − En δ
fn Hn − En H, iAn fn Hn − En ≥ C
γ2
σn fn Hn − En 2
N2
5.29
1
for n ≥ 1 and g ≤ gδ .
This yields
fn H − EH, iAn fn H − E
2
δ γ σn fn H − E2
≥C
N2
− fn H − EH, iA fn Hn − En − fn H − E
− fn Hn − En − fn H − E H, iAfn Hn − En 2
δ γ σn fn Hn − En − fn H − E 2
C
2
N
δ
C
γ2
σn fn H − E fn Hn − En − fn H − E
2
N
δ
C
γ2 σn fn Hn − En − fn H − E fn H − E.
2
N
5.30
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Advances in Mathematical Physics
Combining Proposition 3.8i and 5.23 with 5.26 and 5.27 we show that H, iAfn Hn −
En and fn H − EH, iA are bounded operators uniformly with respect to n. This, together
with Lemma 5.2, yields
2
n
δ γ σn fn H − E2 − Cgσ
fn H − EH, iAfn H − E ≥ C
N2
5.31
1
> 0 and for g ≤ infg 2 , g .
for some constant C
δ
Multiplying both sides of 5.31 with EΔn H − E we then get
2
δ γ σn EΔn H − E − Cgσ
n EΔn H − E.
EΔn H − EH, iAEΔn H − E ≥ C
N2
5.32
Setting
2
gδ
< inf
δ γ 2 2 1
C
, g , gδ
,
N2
C
5.33
2
δ − CN
2 /γ 2 gδ > 0.
Theorem 3.10 is proved with Cδ C
6. Proof of Theorem 3.7
We set
eitA−1
,
t
adAt · At , ·,
At σ
eitA − 1
,
t
itAσ
−1
e
.
Aσt t
Aσt
6.1
The fact that H is of class C1 A, C1 Aσ , and C1 Aσ in −∞, m1 − δ/2 is the consequence
of the following proposition.
α
Proposition 6.1. Suppose that the kernels G,, satisfy Hypotheses 2.1 and 3.1(iii.a). For every ϕ ∈
C0∞ −∞, m1 − δ/2 and g ≤ g1 , one then has
sup ϕH, At < ∞,
0<|t|≤1
sup ϕH, Aσt < ∞,
0<|t|≤1
sup ϕH, Aσt < ∞,
0<|t|≤1
sup ϕH σ , Aσt < ∞.
0<|t|≤1
6.2
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39
Proof. We use the representation
ϕH dφzz − H−1 ,
6.3
where φz is an almost analytic extension of ϕ with
∂z φ x iy ≤ Cy2 ,
dφz −
1 ∂
φzdx dy.
π ∂z
6.4
Note that φx iy ∈ C0∞ R2 .
We get
adAt ϕH dφzz − H−1 At , Hz − H−1 .
6.5
This yields
−1 adA ϕH ≤ sup ,
Hi
−
H
A
dφzz − H−1 i − Hz − H−1 .
t
t
6.6
0<|t|≤1
It is easy to prove that
dφz
−1 −1 dφz
< ∞.
z − H i − Hz − H ≤ C
|Im z|2
6.7
By Proposition 3.8bi and 6.7 we finally get, for g ≤ g1 ,
sup adAt ϕH < ∞.
0<|t|≤1
6.8
In a similar way we obtain, for g ≤ g1 ,
sup Aσt , ϕH < ∞,
0<|t|≤1
sup Aσt , ϕH < ∞,
0<|t|≤1
sup Aσ , ϕH σ < ∞.
0<|t|≤1
t
The proof of Theorem 3.7 is the consequence of the following proposition.
6.9
40
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α
Proposition 6.2. Suppose that the kernels G,, satisfy Hypotheses 2.1, 3.1(i) and 3.1(iii). One then
has, for g ≤ g1 ,
sup At , At , HH i−1 < ∞,
0<|t|≤1
sup Aσt , Aσt , H H i−1 < ∞,
0<|t|≤1
sup Aσt , Aσt , HH i−1 < ∞,
6.10
0<|t|≤1
sup Aσt , Aσt , H σ H σ i−1 < ∞.
0<|t|≤1
Proof. We have, for every ψ ∈ DH,
1 2itA −2itA
e
e
He2itA − 2e−itA HeitA H ψ.
2
t
6.11
1 2itA 2
dΓ w ◦ φ2t − 2w2 ◦ φt w2 ψ,
e
2
t
6.12
At , At , Hψ By 3.56 we get
At , At , H0 ψ where, for 1, 2, 3,
2
2
2 w ◦ φ2t p2 − 2 w ◦ φt p2 w p2 φ2t p2 − 2φt p2 p2 .
6.13
We further note that
∂2 1 φ2t p2 − 2 φt p2 p2 ≤ sup 2 φs p2 ,
t2
|s|≤2|t| ∂s
∂2 φs p2 φs p2 ≤ eΓ|s| p2 .
2
∂s
6.14
Combining 6.12 with 6.13 and 6.14 we get
At , At , H0 H0 1−1 ≤ e2Γ|t| ,
sup At , At , H0 H0 1−1 ≤ e2Γ .
0<|t|≤1
6.15
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41
In a similar way we obtain
sup Aσt , Aσt , H0 H0 1−1 ≤ Ce2Γ ,
0<|t|≤1
sup Aσt , Aσt , H0 H0 1−1 ≤ Ce2Γ .
6.16
0<|t|≤1
Here C is a positive constant.
Let us now prove that
sup At , At , HI GH i−1 < ∞.
0<|t|≤1
6.17
By 3.66 and 6.11 we get, for every ψ ∈ DH,
At , At , HI Gψ
e2itA α1,2 1,2,3 /
t2
α
α
α
e−2itA HI G,, e2itA − 2e−itA HI G,, eitA HI G,, ψ
e2itA α α
α
HI G,, ;2t − 2HI G,, ;t HI G,, ;0 ψ,
2
t
α1,2 1,2,3 /
6.18
where
1/2 α α
G,, ;t ξ1 , ξ2 , ξ3 Dφt p2
G,, ξ1 ; φt p2 , s2 ; ξ3
α
e−ita G,, ξ1 , ξ2 , ξ3 .
6.19
Combining 2.53 and 2.54 with 3.26–3.28 and 6.18 we get
At , At , HI Gψ ≤ gKGt Cβη H0 Iψ Cβη Bβη ψ .
6.20
Here KGt > 0 and
KGt 2 1
α
α
α 2
−
2G
G
.
G
,, ;2t
,, ;t
,, L2 Σ ×Σ ×Σ t2
1
1
2
α1,2 1,2,3 /
6.21
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We further note that, for 0 < |t| ≤ 1,
⎛
2
2
∂
α
KGt ≤ sup ⎝
2 G,, ;s 2
∂s
0<|s|≤2
α1,2 1,2,3 /
⎞1/2
⎠
.
6.22
L Σ1 ×Σ1 ×Σ2 We get
∂ α
G,, ;t ∂t
∂2 α
G ∂t2 ,, ;t
3 −ita α −ita α
e G,, e
p2 · ∇p2 G,, ,
2
9 −ita α 7 −ita α
p2 · ∇p2 G,,
e G,, e
4
2
α
e−ita p2,i p2,j ∂2p2,i p2,j G,, .
6.23
i,j1,2,3
Recall that e−ita is an one parameter group of unitary operators in L2 Σ1 × Σ1 × Σ2 .
Combining Hypothesis 3.1iii.a and iii.b with 6.20–6.23 we finally get
sup At , At , HI GH0 1−1 < ∞.
0<|t|≤1
6.24
In view of DH DH0 the operators H0 H i−1 and HH0 − 1−1 are bounded and we
obtain
sup At , At , H0 H i−1 < ∞,
0<|t|≤1
sup At , At , HI GH i−1 < ∞.
6.25
0<|t|≤1
This yields
sup At , At , HH i−1 < ∞
0<|t|≤1
for g ≤ g1 .
6.26
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43
Let V p2 denote any of the two C∞ -vector fields vσ p2 and vσ p2 and let a denote
the corresponding aσ and aσ operators. We get
∂2 −iat α e G,,
ξ1 , ξ2 , ξ3 ∂t2
2 α 1 −iat e
div V p2 G,, ξ1 , ξ2 , ξ3 4
1 −iat α
e
div V p2 V p2 · ∇p2 G,, ξ1 , ξ2 , ξ3 2
⎛
⎞⎞
⎛
3 2
α
1 ⎝ −iat ⎝ G,, ⎠⎠ξ1 , ξ2 , ξ3 Vi p2 ∂p2,i p2,j Vj p2
e
2
i,j1
6.27
⎛
⎞⎞
⎛
3
∂Vj ∂
1 ⎝ −iat ⎝ α
e
Vi p2
G,, ⎠⎠ξ1 , ξ2 , ξ3 p2
2
∂p
∂p
2,i
2,j
i,j1
⎛
⎞⎞
⎛
3
∂2
1 ⎝ −iat ⎝ α ⎠⎠
e
Vi p2 Vj p2
G ξ1 , ξ2 , ξ3 .
2
∂p2,i ∂p2,j ,,
i,j1
Combining the properties of the C∞ fields vσ p2 and vσ p2 together with Hypotheses 2.1,
3.1i and 3.1iii we get, from 6.25 and by mimicking the proof of 6.26,
sup Aσt , Aσt , H H i−1 < ∞,
0<|t|≤1
sup Aσt , Aσt , HH i−1 < ∞
6.28
0<|t|≤1
for g ≤ g1 .
Similarly, by mimicking the proof of 6.28, we easily get, for g ≤ g1 ,
sup Aσt , Aσt , H σ H σ i−1 < ∞.
0<|t|≤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 At , At , ϕH < ∞,
0<|t|≤1
6.30
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sup Aσt , Aσt , ϕH < ∞,
0<|t|≤1
sup Aσt , Aσt , ϕH < ∞,
0<|t|≤1
6.31
sup Aσt , Aσt , ϕH σ < ∞
0<|t|≤1
for every ϕ ∈ C0∞ −∞, m1 − δ/2 and for g ≤ g1 .
Let us prove 6.30. The estimates 6.31 can be proved similarly.
To this end, let φ be an almost analytic extension of ϕ satisfying
∂z φ x iy ≤ Cy3 ,
ϕH −1
z − H dφz,
1 ∂
φzdx dy.
dφz −
π ∂z
6.32
It follows that
At At , ϕH
z − H−1 At At , Hz − H−1 2z − H−1 At , Hz − H−1 At , Hz − H−1 dφz.
6.33
We note that
H iH − z−1 ≤
C
,
|Im z|
for z ∈ supp φ.
6.34
We also have
−1
−1
−
H
sup ,
Hz
−
H
dφz
z
A
A
t
t
0<|t|≤1
−1 −1 dφz
≤ sup At At , HH i H iz − H |Im z|
0<|t|≤1
dφz
−1 ≤ C sup At , At , HH i .
|Im z|2
0<|t|≤1
6.35
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45
Therefore, combining Proposition 3.8bi and 6.34 we obtain
−1
−1
−1 sup dφzH − z At , HH − z At , HH − z 0<|t|≤1
−1
−1
−1
−1
−1 sup −
z
,
HH
i
,
HH
i
iH
−
z
iH
−
z
H
A
H
A
H
t
t
dφz
0<|t|≤1
2
dφz
≤C
sup At , HH i−1 < ∞.
3
y
0<|t|≤1
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 3.8b, 6.1, and 6.2 that H is locally of class
C2 Aσ and C2 Aσ in −∞, m1 −δ/2 and that H σ is locally of class C2 Aσ 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 − γ 2 σn , γ γ σn is a covering by open sets of any compact subset
of E, m1 − δ and of the interval E, m1 − δ itself. Theorem 3.4i and ii follow from 6,
2
2
Theorems 0.1 and 0.2 and Theorems 3.7 and 3.10 above with gδ gδ , where gδ is given in
Theorem 3.10. Theorem 3.4iii 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,
F σn F n ,
Σn1
1n Σ1 ∩ p2 ; σn1 ≤ p2 < σn ,
2
n1
2
n1
L
Σ
⊗
F
L
Σ
,
F
Fn1
a
a
1n
1n
,2,n
Fn1
n
3
1
A.1
Fn1
,2,n .
We have
Fn1 Fn ⊗ Fn1
n .
A.2
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Advances in Mathematical Physics
n
n1
Let Ωn resp., Ωn1
n be the vacuum state in F resp., in Fn . We now set
n1
H0n
1
H0
3
H0
3 1 ±
2
σn1 ≤|p2 |<σn
∗
w ξ2 c,
ξ2 c, ξ2 dξ2 .
A.3
n1
The operator H0n
is a self-adjoint operator in Fn1
n .
n1
the interaction HI given by 2.23 and 2.24 but
Let us denote by HIn and HIn
associated with the following kernels:
α
χσn p2 G,, ξ1 , ξ2 , ξ3 ,
A.4
α
χσn1 p2 − χσn p2 G,, ξ1 , ξ2 , ξ3 ,
A.5
respectively, where χσn1 is defined by 3.13.
Let for n ≥ 0,
Hn H n − En ,
n1
.n Hn ⊗ 1n1
H
n 1n ⊗ H0n .
A.6
.n are self-adjoint operators in Fn and Fn1 , respectively. Here 1n and
The operators Hn and H
n1
1n are the identity operators in Fn and Fn1
n , respectively.
Combining 2.53 and 2.54 with 3.26–3.28 we obtain for n ≥ 0,
g HIn ψ ≤ gKG Cβη H0 ψ Bβη ψ A.7
for every ψ ∈ DH0n ⊂ Fn .
It follows from 33, Section V, Theorem 4.11 that
Hn ≥ −
gKGBβη
g1 KGBβη
≥−
,
1 − g1 KGCβη
1 − g1 KGCβη
gKGBβη
E ≥−
.
1 − g1 KGCβη
A.8
n
We have
Ωn , H n Ωn 0.
A.9
En ≤ 0,
A.10
Therefore
|En | ≤
gKGBβη
.
1 − g1 KGCβη
A.11
Advances in Mathematical Physics
47
Let
Knn1 G K 1σn1 ≤|p2 |≤2σn G .
A.12
Combining 2.53 and 2.54 with 3.26 and A.12 we obtain for n ≥ 0
n1 ψ ≤ gKnn1 G Cβη H0n1 ψ Bβη ψ g HIn
A.13
for ψ ∈ DH0n1 ⊂ Fn1 , where we remind that H0n1 H0 |Fσn1 as defined in 3.24.
We have, for every ψ ∈ DH0n1 ,
.n ψ En ψ − g H n ⊗ 1n1
ψ,
H0n1 ψ H
n
I
A.14
ψ ≤ gKG Cβη H0n1 ψ Bβη ψ .
g HIn ⊗ 1n1
n
A.15
and by A.7
In view of A.11 and A.14 it follows from A.15 that
g HIn ⊗ 1n1
ψ
n
gKGCβη gKGBβη
.n ≤
H ψ 1 − g1 KGCβη
1 − g1 KGCβη
gKGBβη
1
1 − g1 KGCβη
ψ .
A.16
By 3.29, A.13, A.14, and A.16 we finally get
n1 .n ψ βη g HIn
ψ ≤ gKnn1 G C
H
Bβη ψ .
A.17
For n ≥ 0, a straightforward computation yields
Knn1 G ≤ σn KG
≤ sup
4Λγ
σn1
, 1 KG
.
2m1 − δ
γ
A.18
Recall that, for n ≥ 0,
σn1 < m1 .
A.19
By A.17, A.18, and A.19, we get, for ψ ∈ DH0 ,
n1 .n σn1 ψ βη βη m1 Bβη ψ ,
ψ ≤ gKnn1 G C
g HIn
H
C
A.20
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Advances in Mathematical Physics
and for φ ∈ F,
−1 βη Bβη n1 n
m1 C
n1
φ .
g HIn H σn1 φ ≤ gKn G Cβη σn1
g
4Λγ
βη Bβη φ.
≤ sup
2m1 C
, 1 KG
γ
2m1 − δ
A.21
n1 .n
Thus, by A.21, the operator HIn
H σn1 −1 is bounded and
−1 n1 n
D
.
g HIn H σn1 ≤gγ ,
A.22
is given by see 3.32
where D
sup
D
4Λγ
βη Bβη .
2m1 C
, 1 KG
2m1 − δ
A.23
.n ,
This yields, for ψ ∈ DH
D
.n
n1 g HIn
ψ ≤ g H
σn1 ψ .
γ
A.24
Hence it follows from 33, Section V, Theorems 4.11 and 4.12 that
n
D
n1
. σn1 ψ, ψ .
g HIn
ψ, ψ ≤ g
H
γ
A.25
2
Let gδ > 0 be such that
2 D
gδ
γ
2
< 1,
1
A.26
gδ ≤ gδ .
2
By A.25 we get, for g ≤ gδ ,
H
n1
.n En gH n1 ≥ En − g D σn1 H
In
γ
gD
1−
γ
.n .
H
A.27
.n ≥ 0 we get from A.27
Because 1 − g D/γ
H
En1 ≥ En −
gD
σn1 ,
γ
n ≥ 0.
A.28
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49
Suppose that ψ n ∈ Fn satisfies ψ n 1 and, for > 0,
ψ n , H n ψ n ≤ En .
A.29
ψ n1 ψ n ⊗ Ωn1
∈ Fn1 .
n
A.30
n1 n1
En1 ≤ ψ n1 , H n1 ψ n1 ≤ En g ψ n1 , HIn
.
ψ
A.31
Let
We obtain
By A.25, A.29, A.30, and A.31 we get, for every > 0,
E
n1
gD
≤E 1
γ
n
gD
σn1 ,
γ
A.32
2
where g ≤ gδ .
This yields
En1 ≤ En gD
σn1 ,
γ
A.33
and by A.28, we obtain
gD
n
σn1 .
E − En1 ≤
γ
A.34
For n 0, since σ0 Λ, remind that H00 H0n0 H0σ0 H0 |FΛ . 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 3.1iv we have H 0 H00 . Hence E0 {0} is a simple isolated
eigenvalue of H 0 and H 0 H0 . We finally get
δ
inf σ H0 − {0} m1 > m1 − σ1 .
2
A.36
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Advances in Mathematical Physics
We now prove Proposition 3.5 by induction in n ∈ N∗ . Suppose that En is a simple
isolated eigenvalue of H n such that
infσHn 3g D
1−
γ
\ {0} ≥
σn ,
n ≥ 1.
A.37
2
Since 3.36 gives σn1 < 1 − 3g D/γσ
n for g ≤ gδ , 0 is also a simple isolated eigenvalue of
n
.
H such that
.n \ {0} ≥ σn1 .
inf σ H
A.38
We must now prove that En1 is a simple isolated eigenvalue of H n1 such that
inf σ Hn1 \ {0} ≥
3g D
1−
γ
A.39
σn1 .
Let
λn1 sup
inf
ψ∈Fn1 ;ψ /
0 φ,ψ0;φ∈DH
n1 ;φ1
φ, Hn1 φ .
A.40
By A.27 and A.33, we obtain, in Fn1 ,
Hn1
gD
gD
.n
σn1 1 −
≥E −E −
H
γ
γ
gD
.n − 2g D σn1 .
≥ 1−
H
γ
γ
n
n1
A.41
.n , and by A.38 and A.41, we have, for
By A.30, ψ n1 is the unique ground state of H
2
g ≤ gδ ,
λn1 ≥
inf
φ,ψ n1 0;φ∈DH n1 ;φ1
≥
gD
1−
γ
φ, Hn1 φ
2g D
σn1 −
σn1 γ
3g D
1−
γ
A.42
σn1 > 0.
2
This concludes the proof of Proposition 3.5 by choosing gδ gδ , if one proves that H 1
.0 such that
satisfies Proposition 3.5. By noting that 0 is a simple isolated eigenvalue of H
.0 \ {0} σ1 , we prove that E1 is indeed an isolated simple eigenvalue of H 1 such
infσH
n1
that infσH1 \ {0} ≥ 1 − 3g D/γσ
1 by mimicking the proof given above for H .
Advances in Mathematical Physics
51
Acknowledgments
The second author wishes to thank Laurent Amour and Benoı̂t 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.
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