The confined hydrogenoı̈d ion in non-relativistic quantum
electrodynamics
Laurent AMOUR, Jérémy FAUPIN
Laboratoire de Mathématiques, UMR-CNRS 6056, Université de Reims,
Moulin de la Housse - BP 1039, 51687 REIMS Cedex 2, France.
laurent.amour@univ-reims.fr, jeremy.faupin@univ-reims.fr
Abstract
We consider a system of a nucleus with an electron together with the quantized electromagnetic field. Instead of fixing the nucleus, the system is confined by its center of mass.
This model is used in theoretical physics to explain the Lamb-Dicke and the Mössbauer
effects (see [CTDRG]). When an ultraviolet cut-off is imposed we initiate the spectral analysis of the Hamiltonian describing the system and we derive the existence of a ground state.
This is achieved without conditions on the fine structure constant.
1
Contents
1 Introduction and statements of results
2 Definition of the Hamiltonian
2.1 Fock space, operators of creation and annihilation
2.2 Definition of the Hamiltonian . . . . . . . . . . . .
2.3 Self-adjointness and domains of quadratic forms . .
2.4 Massive and massless ground state energy . . . . .
3
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4
4
7
9
12
3 Binding conditions
3.1 Proof of condition (i) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Proof of condition (ii) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
13
15
4 Existence of a ground state for HUV
4.1 Existence of a ground state for HUV (m) . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Exponential decay of the ground state Φm . . . . . . . . . . . . . . . . . . . . . .
4.3 Convergence of the ground state Φm when m → 0 . . . . . . . . . . . . . . . . .
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1
Introduction and statements of results
In this paper we address the problem of the existence of a ground state for the hydrogen atom and
more generally for the hydrogenoı̈d ion confined by its center of mass. The fact that the nucleus
is confined but not fixed is important since intense rays appearing in the scattering spectrum for
dynamical nucleus, disappears when the nucleus is fixed (see [CTDRG]). This model is used to
explain the Lamb-Dicke effect and it is also related to the Mössbauer effect (see [CTDRG]).
We consider a system of one nucleus and one electron, together with the electromagnetic
field. Here the nucleus is dynamical and our Hamiltonian acts on an Hilbert spaces describing
the nucleus, the electrons and the photons. The center of mass of the system is confined by an
external potential. Let us denote by U (resp. V ) the external confining potential (resp. the
attractive Coulomb potential). The Hamiltonian of the system is HUV and HUV (m) is the corresponding operator if we decide that the photons have a positive mass m > 0.
Our results are stated in Theorems 1.1-1.3 below. We first define HUV precisely using quadratic
forms:
Theorem 1.1 HUV defines a self-adjoint operator.
Remark 1.1
(i) The hypotheses on the confining potential U are stated in section 2.
(ii) Throughout the proof of theorem 1.1 we note that HUV (m) also defined a self-adjoint operator.
The point is that Q(HUV (m)) = Q(HUV ) ∩ Q(N ) where N is the number operator for the photons
(see section 2). The last equality is fully used in the sequel.
For a semi-bounded self-adjoint operator A, set E(A) be the infimum of the spectrum of A.
Let us denote by HU0 (resp. H0V ) the operator HUV when V (resp. U ) is put to zero.
Theorem 1.2 Assume that E(HUV ) < min(E(HU0 ), E(H0V )) then HUV has a ground state.
The assumption in theorem 1.2 are the so-called binding conditions.
Here we obtain the existence of the ground state without assuming any conditions on the
smallness for the different charges. We follow the fundamental strategy of [G], [GLL] and [LL]
where the authors consider a similar system (actually more general) but with fixed nuclei and
succeed to deal with the quantized electromagnetic field in a non perturbative way. The heart
of the proof is to specify correctly the binding conditions. These conditions need to be properly
chosen so that, on one side we are able to prove them and on the other side they imply the
existence of a ground state.
The main result of the paper is the following theorem.
3
Theorem 1.3
(i) E(HUV ) < E(H0V ).
(ii) E(HUV ) < E(HU0 ).
The proof of the existence of the ground state once the binding conditions being assumed
(theorem 1.2) and the proof of the first binding condition (theorem 1.3(i)) are derived in the
same way as in [GLL]. The validity of the remaining binding condition (theorem 1.3(ii)) is more
difficult and its proof borrows ideas of [LL] and still use tools given in [LL].
For the sake of simplicity the spin of the electron is not taken into account in this work. This
and the case of several electrons should be treated in a similar manner.
As it is already mentioned intense rays should appear in the spectrum of HUV . This may
provide resonances with a very small imaginary part among other resonances. The study of the
resonances for HUV is a work in progress.
Let us also mention another case of a similar system with a dynamical nucleus: the case of
free atoms and ions with quantized electromagnetic field. It is analyzed in [AGG].
The paper is organized as follows. In section 2 we verify theorem 1.1. In section 3, theorem
1.3 is derived. Finally, we prove theorem 1.2 in section 4.
2
2.1
Definition of the Hamiltonian
Fock space, operators of creation and annihilation
The Hilbert space in which operate the Hamiltonian considered in this paper is
H := L2 (R6 ) ⊗ Fs ' L2 (R6 ; Fs ),
where L2 (R6 ) ' L2 (R3 ) ⊗ L2 (R3 ) is the space of states describing the nucleus together with the
electron, and where Fs is the bosonic Fock space over L2 (R3 ; C2 ). This Fock space describes the
states of the polarized radiation field and is defined by
Fs = Fs (L2 (R3 )) ⊗ Fs (L2 (R3 )),
where Fs (L2 (R3 )) = C ⊕
L
Sn L2 (R3n ; C), and where Sn L2 (R3n ; C) is the set of all elements
n≥1
(k1 , . . . , kn ) 7→ Φ(k1 , . . . , kn ) in L2 (R3n ; C) which are invariant under any permutations of
{k1 , . . . , kn }. Note that
M
Fs ' C ⊕
Sn ⊗nk=1 L2 (R3 ; C2 ).
n≥1
4
Moreover Fs0 (L2 (R3 )) (respectively Fs0 ) is defined as the set of all Φ = (Φ(0) , Φ(1) , Φ(2) , . . . ) in
Fs (L2 (R3 )) (respectively in Fs ) such that Φ(n) = 0 except for a finite number of terms.
For any f ∈ L2 (R3 ), the creation operator a∗ (f ) and the annihilation operator a(f ) are
defined for all Φ ∈ Fs0 (L2 (R3 )) by
n
1 Xb
(a (f )Φ) (k1 , . . . , kn ) := √
f (kj )Φ(n−1) (k1 , . . . , kˆj , . . . , kn ),
n j=1
Z
√
fb(k)Φ(n+1) (k, k1 , . . . , kn )dk,
(a(f )Φ)(n) (k1 , . . . , kn ) := n + 1
∗
(n)
R3
where kˆj means that the variable kj is missing in Φ(n−1) , and where fb is the Fourier transform
of f .
These operators are closable on Fs0 (L2 (R3 )) (their closed extensions are denoted by the same
symbols) and they verify on Fs0 (L2 (R3 ))
[a(f ), a∗ (g)] = (f, g),
[a(f ), a(g)] = [a∗ (f ), a∗ (g)] = 0,
(a(f )Φ, Ψ) = (Φ, a∗ (f )Ψ).
Let
n
o
DS := Φ ∈ Fs0 (L2 (R3 )), Φ(n) ∈ S(R3n ) for all n ,
where S(R3n ) denotes the Schwartz space over R3n , and let
(b
a(k)Φ)(n) (k1 , . . . , kn ) :=
√
n + 1Φ(n+1) (k, k1 , . . . , kn ),
n
1 X
δ(k − kl )Φ(n−1) (k1 , . . . , k̂l , . . . , kn )
(b
a∗ (k)Φ)(n) (k1 , . . . , kn ) := √
n
l=1
as quadratic forms on DS × DS .
Then in the sense of quadratic forms on DS × DS we have :
Z
Z
a∗ (f ) =
b
a∗ (k)fb(k)dk, a(f ) =
b
a(k)fb(k)dk.
R3
R3
Now for λ = 1, 2 and f ∈ L2 (R3 ), a#
λ (f ) are defined to be the closures in Fs of
#
a#
1 (f ) = a (f ) ⊗ I,
#
a#
2 (f ) = I ⊗ a (f ),
where a# stands for a or a∗ .
2
b
a#
λ (k) is defined as a quadratic form on (DS ⊗ DS ) similarly.
It follows that on Fs0
[aλ (f ), a∗λ0 (g)] = δλλ0 (f, g),
[aλ (f ), aλ0 (g)] = [a∗λ (f ), a∗λ0 (g)] = 0.
5
If f ∈ L2 (R3 ; C2 ), we can write f = (f1 , f2 ) with f1 and f2 in L2 (R3 ), and a# (f ) is defined by
X #
a# (f ) =
aλ (fλ ).
λ=1,2
Finally, for λ = 1, 2, define the creation and annihilation operators acting in the configuration
space by
Z
Z
1
1
∗
−ik.y
b
a
b
aλ (k)eik.y dk,
(k)e
dk,
a
(y)
:=
a∗λ (y) :=
λ
(2π)3/2 R3 λ
(2π)3/2 R3
as quadratic forms on (DS ⊗ DS )2 . Then we have
X Z
X Z
∗
∗
a (f ) =
aλ (y)fλ (y)dy, a(f ) =
λ=1,2
R3
λ=1,2
aλ (y)fλ (y)dy,
R3
in the sense of quadratic forms.
The number operator N is defined by
(N Φ)(n) (k1 , . . . , kn ) = nΦ(k1 , . . . , kn )
(
)
P
for all Φ ∈ D(N ) = Φ ∈ Fs ,
nkΦ(n) k⊗nk=1 L2 (R3 ;C2 ) < ∞ , and it is easy to see that N is
n≥1
self-adjoint on D(N ). In the sense of quadratic forms, N is given by
X Z
N =
a∗λ (k)aλ (k)dk.
λ=1,2
R3
(
Moreover D(N
1/2
)=
)
Φ ∈ Fs ,
P
n
1/2
kΦ
(n)
n≥1
k⊗nk=1 L2 (R3 ;C2 ) < ∞
and we have
(N 1/2 Φ)(n) (k1 , . . . , kn ) = n1/2 Φ(k1 , . . . , kn )
for all Φ ∈ D(N 1/2 ).
As in [LL] we can decompose an element of Fs in a suitable basis. Namely if (fi )i∈N is an
orthonormal basis of L2 (R3 ; C2 ), the vectors of the form
|i1 , p1 ; . . . ; in , pn if := √
1
a∗ (fi1 )p1 . . . a∗ (fin )pn Ω
p1 ! . . . p n !
constitute an orthonormal basis of F(L2 (R3 ; C2 )) (where Ω = (1, 0, 0, . . . ) denotes the vacuum
vector in Fock space). Any Φ ∈ Fs can be written as
X
X
X
Φ=
Φi1 ,p1 ;...;in ,pn |i1 , p1 ; . . . ; in , pn if ,
n≥0 i1 <i2 <···<in p1 ,...,pn
where the term for n = 0 in the sum is a constant times Ω.
6
2.2
Definition of the Hamiltonian
We denote by m1 and q1 the mass and the charge of the electron respectively, and by m2 and q2
the mass and the charge of the nucleus. Moreover x1 and p1 := −i~∇x1 denote the position and
the momentum of the electron, and x2 , p2 are the position and the momentum of the nucleus.
Let
m1 m2
M = m1 + m2 , µ =
.
m1 + m2
Then the variables R, P of the center of mass, and the relative variables r, p are defined by
m1 x1 + m2 x2
, P = p 1 + p2 ,
M
p
p1
p2
r = x1 − x2 ,
=
−
.
µ
m1
m2
R=
Note that
[P, R] = −i~I
and
[p, r] = −i~I.
We assume that ~ = 1 and c = 1 where c is the velocity of light. Thus the Pauli-Fierz Hamiltonian
of the system we consider is given as an operator acting in H by
HUV :=
1
(pj − qj Aj )2 + Hf + V (r) + U (R).
2m
j
j=1,2
X
(1)
Here Aj := (A1j , A2j , A3j ) is the quantized electromagnetic vector potential in the Coulomb
gauge defined for i = 1, 2, 3 by
Z ⊕
i
Aj =
Ai (xj )dX,
R6
where X = (x1 , x2 ) and for x ∈ R
3
Ai (x) = a∗ (hi (x − ·)) + a(hi (x − ·)),
where the coupling function hi = (hi1 , hi2 ) is defined for λ = 1, 2 by
Z
χ
bΛ (k) i
1
p ελ (k)e−ik.y dk.
hiλ (y) =
2π R3
|k|
The vectors ελ used in the last definiton are the orthonormal polarization vectors in the Coulomb
gauge. They are chosen as
(k2 , −k1 , 0)
ε1 (k) = p 2
k1 + k22
,
ε2 (k) =
k
∧ ε1 (k).
|k|
Note that ε1 (k) and ε2 (k) are well-defined and smooth only on R3 \ Oz where Oz is the axis
{(0, 0, k3 ), k3 ∈ R}. But this singularity is not a problem in this paper.
Finally, Λ is the parameter of the ultraviolet cutoff, and χ
bΛ is a real smooth function depending
7
only on |k|, which is equal to 1 in the ball B(0, Λ/2) and which vanishes outside the ball B(0, Λ).
It is well known that Ai (x) is essentially self-adjoint on Fs0 for all x ∈ R3 (see [RS2]), and one
can verify that in the sense of quadratic forms
Z
1 X
χ
bΛ (k)
p ελ (k) b
A(x) =
a∗λ (k)e−ik.x + b
aλ (k)eik.x dk.
2π
3
|k|
λ=1,2 R
The free energy field of the photons, Hf , acts in Fs = Fs (L2 (R3 ))⊗Fs (L2 (R3 )) and is defined
by
Hf := dΓ(ω) ⊗ I + I ⊗ dΓ(ω),
where w(k) = |k|, and where for all Φ ∈ Fs (L2 (R3 )),
(n)
(dΓ(ω)Φ)
n
X
(k1 , . . . , kn ) =
!
ω(ki ) Φ(n) (k1 , . . . , kn ).
i=1
√
The massive photon field Hf (m) will be defined by replacing ω(k) = |k| with ωm (k) = k 2 + m2 ,
m > 0, in the definition of Hf . Then the massive Hamiltonian HUV (m) is HUV with Hf (m)
replaced by Hf .
Hf is essentially self-adjoint on DS ⊗ DS and we have
X Z
Hf =
|k|b
a∗λ (k)b
aλ (k)dk
λ=1,2
R3
in the sense of quadratic forms.
V is the attractive Coulomb potential and is defined by
V (r) = −
C
,
|r|
where C is a positive constant.
Finally, U is a confining potential for which we make the following assumptions:


(i) U ∈ L1loc (R3 ),



 (ii) inf(U (R)) > −∞ and U − is compactly supported,
(H0 )

(iii) P 2 /2M + U has a non-degenerate ground state φ > 0 with energy − e0 < 0,




and there exists γ such that |φ(R)| ≤ γe−|R|/γ .
In the next subsection we precise the relations between domains of self-adjointness (or domains
of quadratic forms) for the operators that we work with in this paper.
8
2.3
Self-adjointness and domains of quadratic forms
Let q be the quadratic form defined by
X 1
1/2
1/2
((pj − qj Aj )Φ), (pj − qj Aj )Ψ) + (Hf Φ, Hf Ψ).
q(Φ, Ψ) :=
2m
j
j=1,2
(2)
Lemma 2.1 q is closed on Q(p21 + p22 ) ∩ Q(Hf ).
Proof First we have to verify that q is well-defined on Q(p21 + p22 ) ∩ Q(Hf ). Lemma A.4 of [GLL]
shows that
Λ
1/2
1/2
(Aj Φ, Aj Φ) ≤ 32πΛ (Hf Φ, Hf Φ) + (Φ, Φ) ,
(3)
8
6
for all Φ ∈ C∞
0 (R ) ⊗ DS .
1/2
6
Since C∞
0 (R ) ⊗ DS is a core for Hf , this proves that Q(Hf ) ⊂ D(Aj ). Hence q is well-defined.
Next let us show that q is closed on Q(p21 + p22 ) ∩ Q(Hf ). By lemma A.5 of [GLL], we have, for
all Φ ∈ Q(p21 + p22 ) ∩ Q(Hf ),
1/2
1/2
(Hf Φ, Hf Φ) ≤ q(Φ, Φ),
X
(pj Φ, pj Φ) ≤ a.q(Φ, Φ) + b(Φ, Φ),
(4)
(5)
j=1,2
where a, b are positive real numbers.
If Φn ∈ Q(p21 + p22 ) ∩ Q(Hf ) is such that Φn → Φ and q(Φn − Φm , Φn − Φm ) → 0, then (4) yields
Φ ∈ Q(Hf ) and (5) yields Φ ∈ Q(p21 ) ∩ Q(p22 ). Hence q is closed.
In addition, we see that q is positive. Thus, there exists a unique self-adjoint operator, that
we call H00 , associated with q. In other words, q = qH00 (where qA denotes the quadratic form
associated with the self-adjoint operator A).
Lemma 2.2 V − U − is relatively bounded with respect to q in the sense of forms, with relative
bound 0.
Proof According to the assumption (H0 ), U − is infinitesimally small with respect to P 2 . Moreover the Coulomb potential V is infinitesimally small with respect to p2 . Thus, V − U − is
infinitesimally small with respect to p21 + p22 since we have
p2
P2
p2
p21
+ 2 =
+
.
2m1
2m2
2M
2µ
Then V − U − is infinitesimally form-bounded with respect to p21 + p22 (see [RS2], theorem X.18).
We conclude with (5).
With the help of this lemma and the KLMN theorem, we define HUV − as the self-adjoint
operator associated with the closed and semi-bounded quadratic form qH V − defined on Q(p21 +
U
9
p22 ) ∩ Q(Hf ) by qH V − = q + qV −U − .
U
Next, we define the Hamiltonian HUV by
HUV := HUV − u U + ,
(6)
that is to say, HUV is the self-adjoint operator associated with the closed and semi-bounded
quadratic form qHUV defined on Q(HUV − ) ∩ Q(U + ) by qHUV = qH V − + qU + .
U
Remark 2.1 One could have defined the Hamiltonian of the system using a Schrödinger repb 0 defined on
resentation of Fs , say L2 (Q, dµ). Namely, it is proved in [H1] that the operator H
2
2
D(p1 + p2 ) ∩ D(Hf ) by
X 1
b 0 :=
H
(pj − qj Aj )2 + Hf
2m
j
j=1,2
is self-adjoint. This result is obtained thanks to FKN and FKI formulae that lead to the following
functional integral representation:
Z
b
(F, e−tH0 G) =
(F (X0 ), Jt (X)G(Xt ))L2 (Q) dX.
M
Here, M = R6 ×P , where (P, db) is a probability measure space associated with the 6-dimensional
Brownian motion {b(t)}t≥0 , and Xt = X + b(t) is the Wiener process on M .
Moreover,
Jt (X) = Ξ∗0 e−iφ0 (K(t)) Ξt ,
where Ξt is the second quantization of ⊕3 ξt . The isometry ξt : ⊕3 L2 (R3 ) → ⊕3 L2 (R4 ) is defined
by
s
−itk0
ω(k)
e
ξc
fb(k).
t f (k, k0 ) = √
ω(k)2 + |k0 |2
π
φ0 (f ) is a Gaussian random process indexed by real f ∈ ⊕3 L2 (R4 ), on a probability measure
space (Q0 , dµ0 ). Finally, K(t) is the stochastic integral
K(t) = ⊕3i=1 q1
Z
t
ξs ρ(· − Xs )db1i (s) + ⊕3i=1 q2
0
Z
t
ξs ρ(· − Xs )db2i (s),
0
p
with ρb(k) = χ
bΛ (k)/ π 2|k| .
b 0 , so that the KatoNext it is proved that V − U − is infinitesimally small with respect to H
V
−
b − := H
b0 + V − U .
Rellich theorem gives a meaning to H
U
b V is defined in the same way as for H V , that is H
b V := H
b V − u U +.
Finally, H
U
U
U
U
10
Let us show here that the two definitions of the Hamiltonian are the same:
bV .
Proposition 2.1 HUV = H
U
Proof This will follow from two lemmata:
Lemma 2.3 Let A be a semi-bounded self-adjoint operator and let B be a self-adjoint operator
that is relatively bounded with respect to A with a relative bound strictly less than 1. Then
A + B = A u B,
that is to say the definitions given by the Kato-Rellich theorem and the KLMN theorem respectively lead to the same operator.
Proof We easily see that qA+B equals qAuB on D(A), and since this domain is a form core for
each of the two closed quadratic forms, we get qA+B = qAuB . Moreover, since A is semi-bounded,
we can see that the two quadratic forms are semi-bounded. This yields A + B = A u B.
b0.
Lemma 2.4 H00 = H
Proof Since qH00 and qHb 0 are positive, it is sufficient to show that these two closed quadratic
0
forms are equal on a domain that is a form core for the two of them. According to [H1],
6
b
C∞
b 0 . Let us show that it is also a form
0 (R ) ⊗ DS is a core for H0 . Thus it is a form core for qH
core for qH00 .
By lemma A.4 of [GLL], we have
i
X 1 h
1/2
1/2
(1 + |qj |)(pj Φ, pj Φ) + (|qj | + qj2 )(32πΛ(Hf Φ, Hf Φ) + 4πΛ2 (Φ, Φ))
qH00 (Φ, Φ) ≤
2mj
j=1,2
1/2
1/2
+ (Hf Φ, Hf Φ),
6
for all Φ ∈ C∞
0 (R ) ⊗ DS .
6
Now if Φ ∈ Q(p21 +p22 )∩Q(Hf ), there is a sequence Φn ∈ C∞
0 (R )⊗DS such that Φn → Φ, pj Φn →
1/2
1/2
pj Φ for j = 1, 2 and Hf Φn → Hf Φ. From the last inequality, we get qH00 (Φ−Φn , Φ−Φn ) → 0.
6
Hence C∞
0 (R ) ⊗ DS is a form core for qH00 .
Now, HUV (m) is defined similarly by HUV (m) := HUV − (m) u U + , so that we have
Q(HUV (m)) = Q(p21 + p22 ) ∩ Q(Hf (m)) ∩ Q(U + ).
√
We note that the inequalities |k| ≤ k 2 + m2 ≤ |k| + m, for all k ∈ R3 , yield
Q(Hf (m)) = Q(Hf ) ∩ Q(N ).
Thus,
Q(HUV (m)) = Q(HUV ) ∩ Q(N ).
11
(7)
2.4
Massive and massless ground state energy
In this subsection we recall (see [GLL], part 5) that the ground state energy of the massive
Hamiltonian HUV (m), m > 0, converges to the ground state energy of the massless one as m goes
to 0. We will denote by E(A) the infimum of the spectrum of any semi-bounded self-adjoint
operator A, so that we have
E(A) =
inf
(φ, Aφ) =
φ∈D(A),kφk=1
inf
φ∈Q(A),kφk=1
qA (φ, φ).
Lemma 2.5 E(HUV (m)) → E(HUV ).
m→0
Proof We sketch the proof (see [GLL] theorem 5.1 for more details). Namely, if m > m0 > 0,
we have Q(HUV (m)) = Q(HUV (m0 )) ⊂ Q(HUV ) by (7). Then
E(HUV ) =
≤
≤
inf
V)
kΨk=1,Ψ∈Q(HU
qHUV (Ψ, Ψ)
inf
qHUV (Ψ, Ψ)
inf
qHUV (m0 ) (Ψ, Ψ)
V (m0 ))
kΨk=1,Ψ∈Q(HU
V (m0 ))
kΨk=1,Ψ∈Q(HU
= E(HUV (m0 )) ≤ · · · ≤ E(HUV (m)).
Thus E(HUV (m)) converges to a limit E ∗ that is greater than E(HUV ) when m goes to 0.
To see that E ∗ ≤ E(HUV ), let ε > 0 and take Ψ0 ∈ Q(HUV ) such that kΨ0 k = 1 and
qHUV (Ψ0 , Ψ0 ) ≤ E(HUV ) + ε.
(n)
If Πn denotes the projector onto Fs := Φ ∈ Fs , Φ(k) = 0 for all k > n , then we can see as in
e 0 := Πn Ψ0 where n0 is chosen such
[GLL] that qHUV (Πn Ψ0 , Πn Ψ0 ) → qHUV (Ψ0 , Ψ0 ). We set Ψ
0
n→∞
that qHUV (Πn0 Ψ0 , Πn0 Ψ0 )/kΠn0 Ψ0 k2 − qHUV (Ψ0 , Ψ0 ) ≤ ε. Then we have
e 0 ∈ Q(HUV ) ∩ Q(N ) = Q(HUV (m))
Ψ
for all m > 0, so that
E(HUV (m)) =
inf
V (m)),kΨk=1
Ψ∈Q(HU
qHUV (m) (Ψ, Ψ)
e 0, Ψ
e 0 )/kΨ
e 0 k2
≤ qHUV (m) (Ψ
e 0, Ψ
e 0 )/kΨ
e 0 k2 + m.qN (Ψ
e 0, Ψ
e 0 )/kΨ
e 0 k2
≤ qHUV (Ψ
≤ qHUV (Ψ0 , Ψ0 ) + ε + m.n0
≤ E(HUV ) + 2ε + m.n0 .
Letting m → 0, next ε → 0, we get the stated result.
Note that the same result holds when HUV is replaced by H0V (respectively HU0 ).
12
3
Binding conditions
As in [GLL], the key step is to define binding conditions under which we are able to prove that
a ground state exists for the Hamiltonian HUV . We define the binding conditions as:
E(HUV ) < E(H0V ),
(i)
E(HUV ) < E(HU0 ).
(ii)
The proof of the condition (i) follows the one in [GLL] (theorem 2.1), whereas the proof of (ii)
is more difficult and needs the localization methods used in [LL].
Remark 3.1 Note that as soon as (i) and (ii) are satisfied, lemma 2.5 yields
min E(H0V (m)), E(HU0 (m)) − E(HUV (m)) ≥ C > 0,
for any suitable constant C and any m small enough.
3.1
Proof of condition (i)
Following [GLL] theroem 2.1, we shall show that E(HUV ) ≤ E(H0V ) − e0 . The point is to find a
normalized state Φ ∈ Q(HUV ) such that qHUV (Φ, Φ) − Φ, E(H0V ) − e0 + ε Φ ≤ 0 (where ε > 0
is fixed).
Let ε > 0 and let F ∈ D(H0V ),kF k = 1 such that (F, H0V F ) < E(H0V ) + ε.
Define the unitary operator Uy for all y ∈ R3 by
Uy = eiy.(p1 +p2 +dΓ(k)) .
Uy acts in H, and if Ψ ∈ H, Ψ := Ψel ⊗ a∗ (f1 )α1 . . . a∗ (fn )αn Ω, then we have
Uy Ψ = Ψel (· + y, · + y) ⊗ a∗ (f1 (· + y))α1 . . . a∗ (fn (· + y))αn Ω.
3
V
Since [H0V , Uy ] = 0 on C∞
0 (R ) ⊗ DS and since this domain is a core for H0 (see [H1]), then, for
all Ψ ∈ D(H0V ), Uy Ψ ∈ D(H0V ) and [H0V , Uy ]Ψ = 0.
Let us note here that, in particular, this translation invariance of H0V is due to the fact that V
itself is translation invariant. But this becomes false if V is replaced by U , so that we will have
to face a difficulty through the proof of condition (ii).
Now, as in [GLL], we would like to choose Φy := φUy F , for a suitable y, as a trial state. First
we have to show that Φy ∈ Q(HUV ). We know that ∃γ > 0 such that φ(R) ≤ γe−|R|/γ for all
3
R ∈ R3 . Thus, Φy ∈ H for all y. Let ξn (R) = ξ(R/n) be a smooth function in C∞
0 (R ) with
∗ 0 ≤ ξ ≤ 1,
∗ ξ = 1 in the ball B(0, 1),
∗ ξ = 0 outside the ball B(0, 2),
13
and let φn := ξn φ. Then Φny := φn Uy F
→ Φy in H, and Φny ∈ Q(HUV ) since φn is a smooth,
n→∞
compactly supported function. Thus, to be able to conclude that Φy ∈ Q(HUV ), we only need to
show that qHUV (Φny , Φny ) is bounded uniformly in n. Since qHUV is semi-bounded from below, we
would like to check that qHUV (Φny , Φny ) is bounded from above. But
qHUV (Φny , Φny ) =
1
((pj − qj Aj )Φny , (pj − qj Aj )Φny ) + (Φny , U Φny )
2m
j
j=1,2
X
+ (Φny , [Hf + V ]Φny ).
The last term of this sum is uniformly bounded from above since V is negative, since φ2n (R) ≤
φ2 (R) ≤ γ 2 e−|R|/2γ and since (Uy F, Hf Uy F ) < ∞ (because Uy F ∈ D(H0V )). As for the other
terms, following [GLL], we can show
X 1
((pj − qj Aj )Φny , (pj − qj Aj )Φny ) + (Φny , U Φny )
2m
j
j=1,2
2
Z
P
+ U φn (R) hUy F (X), Uy F (X)idX
=
φn (R)
2M
R6
X 1 Z
+
φn (R)2 h(pj − qj A(xj ))Uy F (X), (pj − qj A(xj ))Uy F (X)idX
2m
6
j
R
j=1,2
Z
= −e0
φn (R)2 hUy F (X), Uy F (X)idX
R6
Z
1
+
(P 2 ξn )(R)ξn (R)φ(R)2 hUy F (X), Uy F (X)idX
2M R6
Z
1
φ(R)2 P [ξn (R)(P ξn )(R)hUy F (X), Uy F (X)i]dX
−
M R6
X 1 Z
+
φn (R)2 h(pj − qj A(xj ))Uy F (X), (pj − qj A(xj ))Uy F (X)idX.
2m
6
j
R
j=1,2
All of the terms in this last sum are bounded from above, which can be seen using again the
fact that φ(R) ≤ γe−|R|/γ together with (ξn ), (P ξn ), (P 2 ξn ) are uniformly bounded, and Uy F ∈
D(H0V ). Therefore Φy ∈ Q(HUV ) for all y ∈ R3 . In addition, we have
Z
qHUV (Φy , Φy ) = −e0
φ(R)2 hUy F (X), Uy F (X)idX
R6
X 1 Z
+
φ(R)2 h(pj − qj A(xj ))Uy F (X), (pj − qj A(xj ))Uy F (X)idX
2m
6
j
R
j=1,2
+ (Φy , [Hf + V ]Φy ).
The end of the proof is the same as the one in [GLL]. That is we integrate qHUV (Φy , Φy ) in y
over R3 and do the changes of variables x1 + y → x1 , x2 + y → x2 , which leads to
Z
Z
qHUV (Φy , Φy )dy =
Φ(u)2 du[(F, H0V F ) − e0 (F, F )]
R3
R3
= (F, H0V F ) − e0 ,
14
since kφkL2 (R3 ) = 1 and kF kH = 1.
But we assumed (F, H0V F ) < E(H0V ) + ε and we have
Z
(Φy , Φy )dy = kφk2 kF k2 = 1,
R3
so that
Z
R3
qHUV (Φy , Φy ) − [E(HUV ) − e0 + ε](Φy , Φy )dy < 0.
Therefore ∃y0 ∈ R3 , and Φy0 ∈ Q(HUV ), which is necessary 6= 0, such that
qHUV (Φy0 , Φy0 ) − [E(HUV ) − e0 + ε](Φy0 , Φy0 ) < 0.
Then E(HUV ) < E(H0V ) − e0 + ε, and since this inequality is true for all ε > 0, we obtain
E(HUV ) ≤ E(H0V ) − e0 .
3.2
Proof of condition (ii)
As stated above, we can not follow the proof of the previous subsection because the Hamiltonian
HU0 is not translation invariant. Actually, if (Fj ) denotes a minimizing sequence for HU0 we can
consider two possibilities: either a part of the support of Fj lies in a ball with fixed radius, or Fj
is supported outside balls with increasing radius. In other words, consider a state ”close to” the
ground state. Then, the two particles of the system live either not too far from each other, or,
on the contrary, as far as we want from each other. In the first case the proof is easy, whereas in
the second case it is more difficult.
Namely, we shall localize the electronic particles together with the photons in a similar way
e 0 which operate
as the one used in [LL]. This bring us to pay attention to a new Hamiltonian H
U
in L2 (R6 ) ⊗ Fs ⊗ Fs , and whose ground state energy is such that
e U0 ) > E(HUV ),
E(HU0 ) ≥ E(H
which will give the result.
We begin with the simplest case:
Theorem 3.1 Let (Fj ) be a normalized sequence in Q(HU0 ) such that qHU0 (Fj , Fj ) → E(HU0 ).
j→∞
Assume that
Z
Z
∃ρ > 0, ∃a > 0, ∀j,
B(0,ρ)
Then, E(HUV ) ≤ E(HU0 ) − Ca/ρ.
15
R3
kFj (X)k2 dRdr ≥ a.
Proof Since Q(HUV ) = Q(HU0 ), we have Fj ∈ Q(HUV ). Hence it suffices to write
Z
qHUV (Fj , Fj ) = qHU0 (Fj , Fj ) +
V (X)kFj (X)k2 dX
R6
Z
Z
C
kFj (X)k2 dX
≤ qHU0 (Fj , Fj ) −
B(0,ρ) R3 ρ
C
≤ qHU0 (Fj , Fj ) − a.
ρ
We get the result as j → ∞.
Now we have to deal with the second case. As stated above, we need to define a new
e 0 to be the self-adjoint operator
Hamiltonian acting in L2 (R6 ) ⊗ FS ⊗ Fs . Namely, we define H
U
associated with the closed and semi-bounded quadratic form qHe 0 , with domain Q(p21 + p22 ) ∩
U
e f ) ∩ Q(U + ), such that:
Q(H
qHe 0 (Φ, Ψ) =
U
1
([(p1 − q1 A1 ) ⊗ I]Φ, [(p1 − q1 A1 ) ⊗ I]Ψ)
2m1
1
+
([I ⊗ (p2 − q2 A2 )]Φ, [I ⊗ (p2 − q2 A2 )]Ψ)
2m2
e 1/2 Φ, H
e 1/2 Ψ)
+ (H
f
(8)
f
− 1/2
− ((U )
Φ, (U − )1/2 Ψ) + ((U + )1/2 Φ, (U + )1/2 Ψ),
e f := Hf ⊗ I + I ⊗ Hf . Note that in order to show that q e 0 is closed, one
where we have set H
HU
can follow the subsection 2.3.
Then we have:
Theorem 3.2 Let (Fj ) be a normalized sequence in Q(HU0 ) such that qHU0 (Fj , Fj ) → E(HU0 ).
j→∞
Assume that
∀n ∈ N∗ , ∃jn ,
Z
B(0,n)
Z
R3
kFjn (X)k2 dRdr ≤
1
.
n
(9)
e 0 ) ≤ E(H 0 ).
Then E(HUV ) < E(H
U
U
e 0 ) with the localization method of [LL],
Proof Note that in order to prove that E(HUV ) < E(H
U
e 0 ) ≤ E(H 0 ).
we do not need to assume (9). However, we need it to show that E(H
U
U
We begin with the proof of the first inequality. Since the method is quite similar to prove the
second one, we shall not write the details.
e 0 ).
First step: proof of the inequality E(HUV ) < E(H
U
To show this inequality, we follow [LL]. Namely, as in theorem 4.3 of [LL], we would like first to
16
e 0 ) such that:
find a state Ψ in Q(H
U
a)
the electronic part of Ψ is supported in B(y1 , R0 ) × B(y2 , R0 ),
that is to say Ψ(X) = 0 as soon as x1 ∈
/ B(y1 , R0 ) or x2 ∈
/ B(y2 , R0 ),
b)
the photonic part of Ψ is supported in B(y1 , L) × B(y2 , L),
that is to say the photons of the first component of the tensor product Fs ⊗ Fs
live in B(y1 , L), whereas the photons of the second component live in B(y2 , L),
qHe 0 (Ψ, Ψ)
C1
C2
R0
0
U
≤ E(H̃U ) + 2 +
(1 + | ln(ΛR0 )|),
c)
kΨk2
R0
(L − 2R0 )γ Lγ
where R0 > 0 and L > 2R0 are fixed, where C1 and C2 are positive constants, and where γ is
any real number such that 0 < γ < 1.
We start with localizing the electron and the nucleus in balls of radius R0 .
e 0 ) such that
Lemma 3.1 For any fixed R0 > 0, there exists y1 , y2 ∈ R3 and a state Ψ ∈ Q(H
U
the electronic part of Ψ is supported in B(y1 , R0 ) × B(y2 , R0 ) and
qHe 0 (Ψ, Ψ)
U
kΨk2
≤ E(H̃U0 ) +
C1
,
R02
where C1 is a positive constant.
Proof of the lemma
This lemma is proved in [LL], theorem 4.1. However in [LL], the authors have to deal with the
Pauli principle according to which the states in H have to be antisymmetric under the exchange
of the particles labels. Here the electronic particles are distinct so that we do not have to deal
with this problem. Then the proof becomes a bit simpler.
e 0 ), kΨk = 1 be such that
Let Ψ ∈ D(H
U
e U0 Ψ) < E(H
e U0 ) + 1 .
(Ψ, H
R02
3
Let u ∈ C∞
0 (R ) be such that





∗ u = 1 in the ball B(0, 1/2),
∗ u = 0 outside the ball B(0, 1),
∗ 0 ≤ u ≤ 1.
Let us set
u1,y (X) := u(
and note that
Z
R3
u21,y (X)dy =
x1
− y) ,
R0
Z
R3
u2,y0 (X) := u(
u22,y0 (X)dy 0 =
17
Z
R3
x2
− y 0 ),
R0
u2 (z)dz := β > 0.
e 0 ). Then
Define Ψy,y0 := β1 u1,y u2,y0 Ψ; with the definition of u, we easily see that Ψy,y0 ∈ Q(H
U
we have
Z Z
Z
Z
1
u21,y (X)dy
(Ψy,y0 , Ψy,y0 )dydy 0 = 2
u22,y0 (X)dy 0 < Ψ(X), Ψ(X) > dX
β R6 R3
R6
R3
= (Ψ, Ψ)
= 1,
and
qHe 0 (Ψy,y0 , Ψy,y0 ) =
U
1
(Ψ, |∇x1 u1,y |2 u22,y0 + |∇x2 u2,y0 |2 u21,y Ψ)
2
β
1
e U0 Ψ)
+ 2 Re(u21,y u22,y0 Ψ, H
β
(10)
(here Re(z) denotes the real part of the complex number z).
We compute in one hand
Z
(Ψ, |∇x1 u1,y |2 u22,y0 + |∇x2 u2,y0 |2 u21,y Ψ)dydy 0
6
R
Z Z
1
x1
=β
< Ψ(X), 2 |(∇u)(
− y)|2 Ψ(X) > dydX
R
R
6
3
0
R
0
Z RZ
1
x2
< Ψ(X), 2 |(∇u)(
+β
− y 0 )|2 Ψ(X) > dy 0 dX
R0
R0
R6 R3
2βC0
,
=
R02
R
where C0 = R3 |∇u(z)|2 dz > 0, and in the other hand
Z
e U0 Ψ)dydy 0 = β 2 (Ψ, H
e U0 Ψ).
(u21,y u22,y0 Ψ, H
R6
Thus (10) leads to
Z 2C0
1
e U0 ) Ψy,y0 ) dydy 0
qHe 0 (Ψy,y0 , Ψy,y0 ) − (Ψy,y0 ,
+
+
E(
H
U
βR02
R02
R6
e 0 Ψ) − E(H
e0 ) − 1
= (Ψ, H
U
U
R02
< 0.
Therefore ∃(y1 , y2 ) ∈ R6 such that
e 0 ) + Cste (Ψy ,y , Ψy ,y ),
qHe 0 (Ψy1 ,y2 , Ψy1 ,y2 ) < E(H
U
1 2
1 2
U
R02
and in particular, Ψy1 ,y2 is 6= 0 (here we have set Cste = 1 + 2C0 β).
e 0 ).
Back to the proof of the inequality E(HUV ) < E(H
U
Now, we have to localize the photons around the nucleus and the electron. We do not write the
18
details of the proof here but sketch only it; we refer once again to [LL], lemma 4.3.
e 0 is seen as an operator acting in
First, replacing the Laplacian by the Dirichlet Laplacian, H
U
e 0 ) has a
L2 (B(y1 , R0 ) × B(y2 , R0 ); Fs ⊗ Fs ). We can show that this operator (that we call H
U,D
ground state ΦD , so that in particular
0
e U,D
e U0 ) + Cste ,
(ΦD , H
ΦD ) ≤ qHe 0 (Ψy1 ,y2 , Ψy1 ,y2 ) ≤ E(H
U
R02
where Ψy1 ,y2 is the (normalized) state given by lemma 3.1, which satisfies the Dirichlet boundary
conditions by construction.
e 0 is defined in the same way as H
e 0 in (8); the only modification
Note that the Hamiltonian H
U,D
U
0
1
e
is that the domain of the quadratic form associated with H
is
H
0 (B(y1 , R0 ) × B(y2 , R0 ); Fs ⊗
U,D
+
1
6
e
e
Fs ) ∩ Q(Hf (m)) ∩ Q(U ) instead of H (R ; Fs ⊗ Fs ) ∩ Q(Hf (m)) ∩ Q(U + ). In particular if we
e 0 ).
set ΦD (X) = 0 outside B(y1 , R0 ) × B(y2 , R0 ), then ΦD ∈ Q(H
U
Therefore we would like to localize the photons in the state ΦD .
Recall from section 2.1 that any state Ψ ∈ L2 (R6 ; Fs ⊗ Fs ) can be written as Ψ : X 7→ Ψ(X)
with
Ψ(X) =
X
X
n≥0 i1 <i2 <···<in
n0 ≥0 i0 <i0 <···<i0 0
1
2
n
X
p1 ,...,pn
p01 ,...,p0 0
n
Ψ
i1 ,p1 ;...;in ,pn
i01 ,p01 ;...;i0 0 ,p0 0
n
n
where
|i1 , p1 ; . . . ; in , pn if = √
(X)|i1 , p1 ; . . . ; in , pn if ⊗ |i01 , p01 ; . . . ; i0n0 , p0n0 if ,
1
a∗ (fi1 )p1 . . . a∗ (fin )pn Ω,
p1 ! . . . p n !
and where (fi ) is an orthonormal basis of L2 (R3 ; C2 ).
Then the operator JL will be defined by
JL (|i1 , p1 ; . . . ; in , pn if ⊗ |i01 , p01 ; . . . ; i0n0 , p0n0 if )
0
0
1
1
p 0
=√
a∗ (h1 fi1 )p1 . . . a∗ (h1 fin )pn Ω ⊗ a∗ (h2 fi01 )p1 . . . a∗ (h2 fi0n0 )pn0 Ω,
0
p 1 ! . . . p n ! p1 ! . . . p n 0 !
3
where the functions h1 , h2 ∈ C∞
0 (R ) are defined by


 ∗ 0 ≤ h1 ≤ 1 and 0 ≤ h2 ≤ 1,
∗ h1 = 1 in the ball B(y1 , L/2) and h2 = 1 in the ball B(y2 , L/2),


∗ h1 = 0 outside the ball B(y1 , L) and h2 = 0 outside the ball B(y2 , L).
In other words, h1 localize photons next to the particle x1 (which lives in B(y1 , R0 )) and h2
localize photons next to the particle x2 (which lives in B(y2 , R0 )).
e 0 ), we easily see that Ψ0 ∈ Q(H
e 0 ).
Next we set Ψ0 := JL ΦD /kJL ΦD k2 . Since ΦD ∈ Q(H
U
U
Following [LL], theorem 4.3, we can show that
qHe 0 (JL ΦD , JL ΦD )
U
kJL ΦD k2
C2
e U0 ) + C1 +
≤ E(H
R02
(L − 2R0 )γ
19
R0
Lγ
(1 + | ln(ΛR0 )|),
(11)
for any 0 < γ < 1 and where C1 , C2 > 0.
e 0 to simplify the proof of the
Note that we can use here some invariance of the Hamiltonian H
U
last inequality. Namely, if we set
m2
Tt = eit M
(p1 +dΓ(k))
m1
⊗ e−it M
(p2 +dΓ(k))
,
e 0 ):
we can see that for all t ∈ R3 and all Φ ∈ Q(H
U
qHe 0 (Tt Φ, Tt Φ) = qHe 0 (Φ, Φ).
U
(12)
U
In other words, if one translates the electron (with its cloud of photons) and the nucleus (with
its cloud of photons) without moving the position of the center of mass, one does not modify the
energy of the state under consideration.
We take t = 3L − (y1 − y2 ) and we replace Ψ0 by Ψ0 := Tt JL ΦD /kJL ΦD k2 , so that the new
state Ψ0 has the same properties as the previous one, except that in the new state a distance L
separate the two balls where the particles live. Thus we do not have to pay any attention to the
fact that the balls may overlap or not (as it is done in [LL], lemma 6.1).
Finally we would like to find a state Ξ ∈ Q(HU0 ) whose energy in HU0 would be sufficiently
close to qHe 0 (Ψ0 , Ψ0 ). Then the term (Ξ, V Ξ) would be the main term in qHUV (Ξ, Ξ) and we would
U
be able to conclude.
We shall apply formulae of the type (2.24) of [LL], so that it is convenient to replace A(xi )2
e 0 . We
with the normal-ordered : A(xi )2 : in the definitions of the Hamiltonians HUV and H
U
V
0
e
write the ”normal-ordered Hamiltonians” as : HU : and : HU : respectively. We easily see that
e 0 ) − E(H V ).
e 0 :) − E(: H V :) = E(H
E(: H
U
U
U
U
Now, let (fk ) be an orthonormal basis of L2 (B(y1 , L); C2 ) and let (gl ) be an orthonormal basis
of L2 (B(y2 , L); C2 ). We know that
o
n
1
∗
p1
∗
pn
• |i1 , p1 ; . . . ; in , pn if = √p !...p
a
(f
)
.
.
.
a
(f
)
Ω,
n
∈
N,
i
,
p
∈
N
is an orthonori
i
k
k
1
n
1
n!
2
2
mal basis of Fs (L (B(y1 , L); C )),
1
p0n0
∗
p01
∗
0
0
0
0
0
a
(g
)
.
.
.
a
(g
)
Ω,
n
∈
N,
i
,
p
∈
N
is an or• and |i01 , p01 ; . . . ; i0n0 , p0n0 ig = √p0 !...p
i1
in0
0 !
k k
1
n0
thonormal basis of Fs (L2 (B(y2 , L); C2 )).
So we can write Ψ0 as
X
X
Ψ0 (X) =
n≥0 i1 <i2 <···<in
n0 ≥0 i0 <i0 <···<i0 0
1
2
n
where Ψ
i1 ,p1 ;...;in ,pn
i01 ,p01 ;...;i0 0 ,p0 0
n
n
X
Ψ
p1 ,...,pn
p01 ,...,p0 0
n
i1 ,p1 ;...;in ,pn
i01 ,p01 ;...;i0 0 ,p0 0
n
n
(X)|i1 , p1 ; . . . ; in , pn if ⊗ |i01 , p01 ; . . . ; i0n0 , p0n0 ig ,
∈ L2 (R6 ), and
X
X
n≥0 i1 <i2 <···<in
n0 ≥0 i0 <i0 <···<i0 0
1
2
n
X
p1 ,...,pn
p01 ,...,p0 0
n
Z
R
2
Ψ i1 ,p1 ;...;in ,pn (X) dX = 1.
0
0
0
0
i ,p ;...;i
,p
6
1
20
1
n0
n0
Pick an orthonormal basis {el } of L2 (R3 ) in H2 (R3 ). Thus, we can also write Ψ
Ψ
i1 ,p1 ;...;in ,pn
i01 ,p01 ;...;i0 0 ,p0 0
n
n
(X) =
X
i1 ,p1 ;...;in ,pn
i01 ,p01 ;...;i0 0 ,p0 0
n
n
as
0
Ψl,li1 ,p1 ;...;in ,pn el (x1 )el0 (x2 ).
i01 ,p01 ;...;i0 0 ,p0 0
n
n
l≥0
l0 ≥0
Then we can compute:
q:He 0 : (Ψ0 , Ψ0 )
U
0
1 X X l,l0
0 ,q 0 ;...;j 0 ,q 0 ) ×
Ψ i1 ,p1 ;...;in ,pn Ψm,m
=
j1 ,q1 ;...;jo ,qo δl0 m0 δ(i01 ,p01 ;...;i0 0 ,p0 0 )(j1
1
n
n
o0 o0
0
0
0
0
0
0
2m1
i1 ,p1 ;...;i 0 ,p 0
j1 ,q1 ;...;j 0 0 ,q 0 0
n
n
o
o
n,i,p,l o,j,q,m
Z
el (x1 )|i1 , p1 ; . . . ; in , pn if , : (p1 − q1 A(x1 ))2 : em (x1 )|j1 , q1 ; . . . ; jn , qn if dx1
R3
0
1 X X l,l0
Ψ i1 ,p1 ;...;in ,pn Ψm,m
+
j1 ,q1 ;...;jo ,qo δlm δ(i1 ,p1 ;...;in ,pn )(j1 ,q1 ;...;jo ,qo ) ×
0 ,q 0 ;...;j 0 ,q 0
2m2
i01 ,p01 ;...;i0 0 ,p0 0
j1
1
n
n
o0 o0
n,i,p,l o,j,q,m
Z
el0 (x2 )|i01 , p01 ; . . . ; i0n0 , p0n0 ig , : (p2 − q2 A(x2 ))2 : em0 (x2 )|j10 , q10 ; . . . ; jn0 0 , qn0 0 ig dx2
R3
+
X
X
n,i,p,l o,j,q,m
0
0
0 ,q 0 ;...;j 0 ,q 0 ) ×
Ψl,li1 ,p1 ;...;in ,pn Ψm,m
j1 ,q1 ;...;jo ,qo δlm δl0 m0 δ(i01 ,p01 ;...;i0 0 ,p0 0 )(j1
0
0
1
i01 ,p01 ;...;i0 0 ,p0 0
n
n
n
0 ,q 0 ;...;j 0 ,q 0
j1
1
o0 o0
n
o
o
h|i1 , p1 ; . . . ; in , pn if , Hf |j1 , q1 ; . . . ; jn , qn if i
X X l,l0
0
Ψ i1 ,p1 ;...;in ,pn Ψm,m
+
j1 ,q1 ;...;jo ,qo δlm δl0 m0 δ(i1 ,p1 ;...;in ,pn )(j1 ,q1 ;...;jo ,qo ) ×
n,i,p,l o,j,q,m
i01 ,p01 ;...;i0 0 ,p0 0
n
n
0 ,q 0 ;...;j 0 ,q 0
j1
1
o0 o0
h|i01 , p01 ; . . . ; i0n0 , p0n0 ig , Hf |j10 , q10 ; . . . ; jn0 0 , qn0 0 ig i
X X l,l0
0
0 ,q 0 ;...;j 0 ,q 0 ) ×
+
Ψ i1 ,p1 ;...;in ,pn Ψm,m
j1 ,q1 ;...;jo ,qo δ(i1 ,p1 ;...;in ,pn )(j1 ,q1 ;...;jo ,qo ) δ(i01 ,p01 ;...;i0 0 ,p0 0 )(j1
0
0
1
n,i,p,l o,j,q,m
i01 ,p01 ;...;i0 0 ,p0 0
n
n
n
0 ,q 0 ;...;j 0 ,q 0
j1
1
o0 o0
n
o
Z
U (R)el (x1 )el0 (x2 )em (x1 )em0 (x2 )dx1 dx2 .
R6
Here δ denotes the Kronecker symbol.
Now, we define the state Ξ ∈ L2 (R6 ; Fs ) as
X
X
X
ˆ 01 , p01 ; . . . ; i0n0 , p0n0 ig ,
Ξ(X) =
Ψ i1 ,p1 ;...;in ,pn (X)|i1 , p1 ; . . . ; in , pn if ⊗|i
n≥0 i1 <i2 <···<in
n0 ≥0 i0 <i0 <···<i0 0
1
2
n
p1 ,...,pn
p01 ,...,p0 0
n
i01 ,p01 ;...;i0 0 ,p0 0
n
n
with
ˆ 01 , p01 ; . . . ; i0n0 , p0n0 ig
|i1 , p1 ; . . . ; in , pn if ⊗|i
0
0
1
p
a∗ (fi1 )p1 . . . a∗ (fin )pn a∗ (gi01 )p1 . . . a∗ (gi0n0 )pn0 Ω.
=√
p1 ! . . . pn ! p01 . . . p0n0 !
In particular we can see that Ξ ∈ Q(HUV ) and that kΞk = kΨ0 k = 1.
Thus, applying formulae of the type (2.24) of [LL] to the states
ˆ 01 , p01 ; . . . ; i0n0 , p0n0 ig ,
|i1 , p1 ; . . . ; in , pn if ⊗|i
21
o
we get
q:HUV : (Ξ, Ξ) = [1] + [2] + [3] + [4] + [5],
with
[1] =
0
1 X X l,l0
0 ,q 0 ;...;j 0 ,q 0 ) ×
Ψ i1 ,p1 ;...;in ,pn Ψm,m
j1 ,q1 ;...;jo ,qo δl0 m0 δ(i01 ,p01 ;...;i0 0 ,p0 0 )(j1
1
n
n
o0 o0
0 ,q 0 ;...;j 0 ,q 0
2m1
j1
i01 ,p01 ;...;i0 0 ,p0 0
1
o0 o0
n
n
n,i,p,l o,j,q,m
Z
el (x1 )|i1 , p1 ; . . . ; in , pn if , : (p1 − q1 A(x1 ))2 : em (x1 )|j1 , q1 ; . . . ; jn , qn if dx1
R3
0
1 X X l,l0
+
Ψ i1 ,p1 ;...;in ,pn Ψm,m
j1 ,q1 ;...;jo ,qo δlm δ(i1 ,p1 ;...;in ,pn )(j1 ,q1 ;...;jo ,qo ) ×
0
0
0
0
0
0
2m2
j1 ,q1 ;...;j 0 0 ,q 0 0
i1 ,p1 ;...;i 0 ,p 0
o
o
n
n
n,i,p,l o,j,q,m
Z
el0 (x2 )|i01 , p01 ; . . . ; i0n0 , p0n0 ig , : (p2 − q2 A(x2 ))2 : em0 (x2 )|j10 , q10 ; . . . ; jn0 0 , qn0 0 ig dx2
R3
+
X
X
n,i,p,l o,j,q,m
0
0
0 ,q 0 ;...;j 0 ,q 0 ) ×
Ψl,li1 ,p1 ;...;in ,pn Ψm,m
j1 ,q1 ;...;jo ,qo δlm δl0 m0 δ(i01 ,p01 ;...;i0 0 ,p0 0 )(j1
0
0
1
i01 ,p01 ;...;i0 0 ,p0 0
n
n
n
0 ,q 0 ;...;j 0 ,q 0
j1
1
o0 o0
n
o
o
h|i1 , p1 ; . . . ; in , pn if , Hf |j1 , q1 ; . . . ; jn , qn if i
X X l,l0
0
Ψ i1 ,p1 ;...;in ,pn Ψm,m
+
j1 ,q1 ;...;jo ,qo δlm δl0 m0 δ(i1 ,p1 ;...;in ,pn )(j1 ,q1 ;...;jo ,qo ) ×
n,i,p,l o,j,q,m
i01 ,p01 ;...;i0 0 ,p0 0
n
n
0 ,q 0 ;...;j 0 ,q 0
j1
1
o0 o0
h|i01 , p01 ; . . . ; i0n0 , p0n0 ig , Hf |j10 , q10 ; . . . ; jn0 0 , qn0 0 ig i
X X l,l0
0
0 ,q 0 ;...;j 0 ,q 0 ) ×
Ψ i1 ,p1 ;...;in ,pn Ψm,m
+
j1 ,q1 ;...;jo ,qo δ(i1 ,p1 ;...;in ,pn )(j1 ,q1 ;...;jo ,qo ) δ(i01 ,p01 ;...;i0 0 ,p0 0 )(j1
0
0
1
n,i,p,l o,j,q,m
i01 ,p01 ;...;i0 0 ,p0 0
n
n
n
0 ,q 0 ;...;j 0 ,q 0
j1
1
o0 o0
n
o
o
Z
U (R)el (x1 )el0 (x2 )em (x1 )em0 (x2 )dx1 dx2
R6
+
X
X
n,i,p,l o,j,q,m
0
0
0 ,q 0 ;...;j 0 ,q 0 ) ×
Ψl,li1 ,p1 ;...;in ,pn Ψm,m
j1 ,q1 ;...;jo ,qo δ(i1 ,p1 ;...;in ,pn )(j1 ,q1 ;...;jo ,qo ) δ(i01 ,p01 ;...;i0 0 ,p0 0 )(j1
0
0
1
i01 ,p01 ;...;i0 0 ,p0 0
n
n
n
0 ,q 0 ;...;j 0 ,q 0
j1
1
o0 o0
n
Z
V (r)el (x1 )el0 (x2 )em (x1 )em0 (x2 )dx1 dx2 ,
R6
[2] = −
0
q1 X X l,l0
Ψ i1 ,p1 ;...;in ,pn Ψm,m
j1 ,q1 ;...;jo ,qo δl0 m0 δ(i1 ,p1 ;...;in ,pn )(j1 ,q1 ;...;jo ,qo ) ×
0 ,q 0 ;...;j 0 ,q 0
m1
i01 ,p01 ;...;i0 0 ,p0 0
j1
1
n
n
o0 o0
n,i,p,l o,j,q,m
Z
el (x1 )(p1 em )(x1 )h|i01 , p01 ; . . . ; i0n0 , p0n0 ig , A(x1 )|j10 , q10 ; . . . ; jo0 0 , qo0 0 ig idx1
R3
0
q2 X X l,l0
0 ,q 0 ;...;j 0 ,q 0 ) ×
−
Ψ i1 ,p1 ;...;in ,pn Ψm,m
j1 ,q1 ;...;jo ,qo δlm δ(i01 ,p01 ;...;i0 0 ,p0 0 )(j1
1
n
n
o0 o0
0 ,q 0 ;...;j 0 ,q 0
m2
i01 ,p01 ;...;i0 0 ,p0 0
j1
1
o0 o0
n
n
n,i,p,l o,j,q,m
Z
el0 (x2 )(p2 em0 )(x2 )h|i1 , p1 ; . . . ; in , pn if , A(x2 )|j1 , q1 ; . . . ; jo , qo if idx2 ,
R3
22
o
o
[3] =
0
q12 X X l,l0
Ψ i1 ,p1 ;...;in ,pn Ψm,m
j1 ,q1 ;...;jo ,qo δl0 m0 δ(i1 ,p1 ;...;in ,pn )(j1 ,q1 ;...;jo ,qo ) ×
0
0
0
0
0
0
2m1
i1 ,p1 ;...;i 0 ,p 0
j1 ,q1 ;...;j 0 0 ,q 0 0
n
n
o
o
n,i,p,l o,j,q,m
Z
el (x1 )em (x1 )h|i01 , p01 ; . . . ; i0n0 , p0n0 ig , : A(x1 )2 : |j10 , q10 ; . . . ; jo0 0 , qo0 0 ig idx1
R3
0
q 2 X X l,l0
0 ,q 0 ;...;j 0 ,q 0 ) ×
+ 2
Ψ i1 ,p1 ;...;in ,pn Ψm,m
j1 ,q1 ;...;jo ,qo δlm δ(i01 ,p01 ;...;i0 0 ,p0 0 )(j1
1
n
n
o0 o0
0 ,q 0 ;...;j 0 ,q 0
2m2
i01 ,p01 ;...;i0 0 ,p0 0
j1
1
n
n
o0 o0
n,i,p,l o,j,q,m
Z
el0 (x2 )em0 (x2 )h|i1 , p1 ; . . . ; in , pn if , : A(x2 )2 : |j1 , q1 ; . . . ; jo , qo if idx2 ,
R3
[4] =
0
q12 X X l,l0
Ψ i1 ,p1 ;...;in ,pn Ψm,m
j1 ,q1 ;...;jo ,qo δl0 m0 ×
0
0
0
0
0
0
m1
i1 ,p1 ;...;i 0 ,p 0
j1 ,q1 ;...;j 0 0 ,q 0 0
n
n
o
o
n,i,p,l o,j,q,m
Z
el (x1 )em (x1 )h|i1 , p1 ; . . . ; in , pn if , A(x1 )|j1 , q1 ; . . . ; jo , qo if i
R3
× h|i01 , p01 ; . . . ; i0n0 , p0n0 ig , A(x1 )|j10 , q10 ; . . . ; jo0 0 , qo0 0 ig idx1
+
0
q22 X X l,l0
Ψ i1 ,p1 ;...;in ,pn Ψm,m
j1 ,q1 ;...;jo ,qo δlm ×
0 ,q 0 ;...;j 0 ,q 0
m2
i01 ,p01 ;...;i0 0 ,p0 0
j1
1
n
n
o0 o0
n,i,p,l o,j,q,m
Z
el0 (x2 )em0 (x2 )h|i1 , p1 ; . . . ; in , pn if , A(x2 )|j1 , q1 ; . . . ; jo , qo if i
R3
× h|i01 , p01 ; . . . ; i0n0 , p0n0 ig , A(x2 )|j10 , q10 ; . . . ; jo0 0 , qo0 0 ig idx2 ,
[5] =
X
0
X
i01 ,p01 ;...;i0 0 ,p0 0
n
n
n,i,p,l o,j,q,m
"
XZ
λ
0
Ψl,li1 ,p1 ;...;in ,pn Ψm,m
j1 ,q1 ;...;jo ,qo δlm δl0 m0 ×
0 ,q 0 ;...;j 0 ,q 0
j1
1
o0 o0
|k|hb
aλ (k)|i1 , p1 ; . . . ; in , pn if , |j1 , q1 ; . . . ; jo , qo if i
R3
× h|i01 , p01 ; . . . ; i0n0 , p0n0 ig , b
aλ (k)|j10 , q10 ; . . . ; jo0 0 , qo0 0 ig idk
+
XZ
λ
R3
|k|h|i1 , p1 ; . . . ; in , pn if , b
aλ (k)|j1 , q1 ; . . . ; jo , qo if i
#
× hb
aλ (k)|i01 , p01 ; . . . ; i0n0 , p0n0 ig , |j10 , q10 ; . . . ; jo0 0 , qo0 0 ig idk .
We see that [1] equals q:He 0 : (Ψ0 , Ψ0 ) − ((−V )1/2 Ψ0 , (−V )1/2 Ψ0 ).
U
Moreover, [3] is less than Cste/L2γ according to [LL], lemma 5.6.
23
e where
Finally we can assume [2] + [4] + [5] ≤ 0 because if we replace Ξ by Ξ
X X l,l0
X
X
e
Ψ i1 ,p1 ;...;in ,pn ×
Ξ(X)
=
n≥0 i1 <i2 <···<in
n0 ≥0 i0 <i0 <···<i0 0
1
2
n
p1 ,...,pn
p01 ,...,p0 0
n
i01 ,p01 ;...;i0 0 ,p0 0
n
n
l≥0
l0 ≥0
(−)
ˆ 01 , p01 ; . . . ; i0n0 , p0n0 ig ,
el (x1 )el0 (x2 )|i1 , p1 ; . . . ; in , pn if ⊗|i
(13)
with
(−)
|i1 , p1 ; . . . ; in , pn if
=√
1
(−a∗ (fi1 ))p1 . . . (−a∗ (fin ))pn Ω,
p1 ! . . . p n !
then [1] and [3] do not change whereas [2],[4] and [5] are replaced by their opposite terms.
Now in the state Ψ0 (and also in the state Ξ), the particle x1 is localized in B(y1 , R0 ) ⊂
B(y1 , L), whereas x2 is localized in B(y2 , R0 ) ⊂ B(y2 , L), and we have chosen y1 , y2 such that
the distance between the two balls B(y1 , L) and B(y2 , L) is L. Therefore Ψ0 is supported in
{|r| ≤ 5L}, which yields
−((−V )1/2 Ψ0 , (−V )1/2 Ψ0 ) ≤ −
C
C
(Ψ0 , Ψ0 ) = − .
5L
5L
Here (12) is crucial because otherwise the balls B(y1 , L) and B(y2 , L) could be far away from
each other and we could not estimate ((−V )1/2 Ψ0 , (−V )1/2 Ψ0 ).
To conclude, (11) yields
C1
C2
C
R0
C3
qHUV (Ξ, Ξ) ≤ E(H̃U0 ) + 2 +
,
(1 + | ln(ΛR0 )|) + 2γ −
γ
γ
R0
(L − 2R0 )
L
L
5L
for all γ < 1, all R0 > 0 and all L > 2R0 .
We choose γ such that 34 < γ < 1, and R0 = Lα with 12 < α < 2γ − 1. Then, for L large enough,
C/L becomes the dominant term in the last inequality, that is to say
R0
C
C1
C2
C3
+
(1 + | ln(ΛR0 )|) + 2γ −
< 0.
2
γ
γ
R0
(L − 2R0 )
L
L
5L
Thus
e U0 )
E(HUV ) < E(H
and the proof is complete.
e 0 ) ≤ E(H 0 ).
Second step: proof of the inequality E(H
U
U
The proof uses again the localization methods of [LL] and we only sketch it. Let us note yet
that the localization errors need not to be estimate as precisely as in the previous step. We only
need to know that these corrections can be made as small as we want.
24
Recall that the assumption (9) tells us that there exists a normalized minimizing sequence for
HU0 , (Fj ), which verifies:
Z
Z
1
∗
∀n ∈ N , ∃jn ,
kFjn (X)k2 dRdr ≤ .
n
B(0,n) R3
Let τn and

























νn be functions defined by
∗ τn (r) = 1 if |r| ≤ n − 12
∗ νn (r) = 1 if |r| ≥ n
∗ τn (r) = τ ( |r|−(n−1)
r) and νn (r) = ν( |r|−(n−1)
r) if n −
|r|
|r|
1
2
≤ |r| ≤ n,
where τ and ν are defined on 1/2 ≤ |r| ≤ 1 and are independant on n
∗ 0 ≤ νn ≤ 1 and 0 ≤ τn ≤ 1
∗ νn , τn ∈ C∞ (R3 )
∗ νn2 + τn2 = 1.
Then we have τn Fjn , νn Fjn ∈ Q(HU0 ) and
qHU0 (Fjn , Fjn ) = qHU0 (τn Fjn , τn Fjn ) + qHU0 (νn Fjn , νn Fjn )
− (Fjn , (|∇τn |2 + |∇νn |2 )Fjn ),
with
2
Z
2
(Fjn , (|∇τn | + |∇νn | )Fjn ) =
≤
2
2
Z
kFjn (X)k2 dRdr
(|∇τ (r)| + |∇ν(r)| )
{n− 12 ≤|r|≤n}
R3
Cste
,
n
so that
qHU0 (νn Fjn , νn Fjn ) − E(HU0 )(νn Fjn , νn Fjn ) ≤ qHU0 (Fjn , Fjn ) − E(HU0 ) +
Since
1 ≥ kνn Fjn k2 ≥
Z
B(0,n)c
Z
R3
kFjn (X)k2 dRdr ≥ 1 −
Cste
.
n
(14)
1
,
n
this shows that νn Fjn /kνn Fjn k is a normalized minimizing sequence for HU0 .
Then, we note again νn Fjn = νn Fjn /kνn Fjn k, and we localize the particles in this state. More
precisely, we pick R0 > 0 and L > 0 such that L − 2R0 > 0. Then there exists n0 such that for
all n ≥ n0 , νn Fjn (X) = 0 on {X, |r| ≤ 3L}. Next, with the help of [LL], starting with νn Fjn (for
n large enough), we can construct a normalized state Ξn in Q(HU0 ) such that
a)
the electronic part of Ξn is supported in B(y1 , R0 ) × B(y2 , R0 ),
the photonic part of Ξn is supported in B(y1 , L) ∪ B(y2 , L),
C2
R0
C1
c) qHU0 (Ξn , Ξn ) ≤ E(HU0 ) + 2 +
(1 + | ln(ΛR0 )|),
γ
R0
(L − 2R0 )
Lγ
b)
25
where C1 and C2 are positive constants, and where γ is any real number such that 0 < γ < 1.
In addition, the distance between the balls B(y1 , L) and B(y2 , L) is at least L by construction.
The proof to get this result is close to the one of the first step, that is we localize first the
nucleus and the electron in the state νn Fjn . Next, we replace the Laplacian with the Dirichlet
Laplacian, which defines a new Hamiltonian that has a ground state. Finally, we localize the
photons around the electron and the nucleus in this ground state. Note that the operator JL
that allows us to localize the photons is defined here by:
JL a∗ (hi1 )p1 . . . a∗ (hin )pn Ω := a∗ (hhi1 )p1 . . . a∗ (hhin )pn Ω,
3
where 0 ≤ h ≤ 1 is a function in C∞
0 (R ) that is equal to 1 on B(y1 , L/2) ∪ B(y2 , L/2) and that
is equal to 0 outside B(y1 , L) ∪ B(y2 , L).
Thus, we have h = h|B(y1 ,L) + h|B(y2 ,L) , and we can write Ξn as
Ξn (X) =
X
0
0
Ξni1 ,p1 ;...;ik ,pk (X)a∗ (fi1 )p1 . . . a∗ (fik )pk a∗ (gi01 )p1 . . . a∗ (gi0k0 )pk0 Ω,
i01 ,p01 ;...;i0 0 ,p0 0
k
k
where fij is supported in B(y1 , L) and gi0j0 is supported in B(y2 , L). In other words, all the
factors a∗ (h|B(y1 ,L) f ) are put on the left whereas the factors a∗ (h|B(y2 ,L) f ) are put on the right.
Now, we can define Ψn in L2 (R6 ; Fs ⊗ Fs ) by
Ψn (X) :=
X
0
0
Ξni1 ,p1 ;...;ik ,pk (X)a∗ (fi1 )p1 . . . a∗ (fik )pk Ω ⊗ a∗ (gi01 )p1 . . . a∗ (gi0k0 )pk0 Ω.
i01 ,p01 ;...;i0 0 ,p0 0
k
k
The same computations as the one of the previous step yield
qHe 0 (Ψn , Ψn ) ≤ E(HU0 ) + ε
U
for all n large enough, where ε depends on R0 , L and γ but can be made as small as we want.
e n in order
Note that, contrary to what we did in (13), it is useless to replace Ψn with a state Ψ
to eliminate some terms, since these terms are small when R0 and L are large.
e 0 ) ≤ E(H 0 ) + ε, where ε can be made as small as we want. Thus the proof
This shows that E(H
U
U
is complete.
4
Existence of a ground state for HUV
In this section we shall prove the existence of a ground state for the Hamiltonian HUV . The
proof follows the one in [GLL]. Namely, the existence of a ground state Φm is proved first for
the massive Hamiltonian HUV (m). Next it is shown that Φm decays exponentially in X, so that
Theorems 6.1 and 6.3 of [GLL] concerning kb
aλ (k)Φm k and k∇k b
aλ (k)Φm k follow. Finally the
Rellich-Kondrachov theorem shows that the weak limit of Φm (when m → 0) is a ground state
for HUV .
26
4.1
Existence of a ground state for HUV (m)
As in [GLL], the proof is divided into two steps: the first step is to find a sufficient condition in
order to get the existence of a ground state. Namely, it is sufficient to show that for all normalized sequence (Ψj ) ∈ Q(HUV (m)) which converges weakly to 0 and such that qHUV (m) (Ψj , Ψj ) is
uniformly bounded, we have
lim inf qHUV (m) (Ψj , Ψj ) > E(HUV (m)).
j→∞
(15)
The second step is to prove that the condition (15) is satisfied. This follows again from the
localization methods of [GLL], with some slight modifications.
Theorem 4.1 For all m > 0 small enough, ∃Φm ∈ D(HUV (m)) such that kΦm k = 1 and
HUV (m)Φm = E(HUV (m))Φm .
Proof First step
Assume that (15) is satisfied and let us show that a ground state exists for HUV (m).
Let (Φj ) ∈ Q(HUV (m)) be a normalized sequence such that
qHUV (m) (Φj , Φj ) → E(HUV (m)).
j→∞
1/2 j
Since (Φj ) and ( HUV (m) − E(HUV (m))
Φ ) are bounded sequences, they converge weakly
along some subsequences to limits denoted by Φm and Φ0m respectively. These subsequences are
1/2 j
still denoted by (Φj ) and ( HUV (m) − E(HUV (m))
Φ ). Then we have
1/2 j
1/2
(φ, HUV (m) − E(HUV (m))
Φ ) = ( HUV (m) − E(HUV (m))
φ, Φj ),
for all φ ∈ Q(HUV (m)). When j → ∞, this leads to
1/2
(φ, Φ0m ) = ( HUV (m) − E(HUV (m))
φ, Φm ).
1/2
Therefore Φm ∈ Q(HUV (m)) (and HUV (m) − E(HUV (m))
Φm = Φ0m ).
1/2 j
j
j
j
Then, setting Ψ = Φ −Φm as in [GLL], we have Ψ * 0 and HUV (m) − E(HUV (m))
Ψ * 0,
so that
0 = lim qHUV (m) (Φj , Φj ) − E(HUV (m))(Φj , Φj )
j→∞
1/2
1/2
= lim ( HUV (m) − E(HUV (m))
(Φm + Ψj ), HUV (m) − E(HUV (m))
(Φm + Ψj ))
j→∞
1/2
1/2
= ( HUV (m) − E(HUV (m))
Φm , HUV (m) − E(HUV (m))
Φm )
V
1/2 j
1/2 j
+ lim ( HU (m) − E(HUV (m))
Ψ , HUV (m) − E(HUV (m))
Ψ ).
j→∞
1/2
1/2 j
Thus HUV (m) − E(HUV (m))
Φm = 0 and lim k HUV (m) − E(HUV (m))
Ψ k = 0. Together
j→∞
with (15), this leads to Ψj → 0 strongly, so that kΦm k = 1.
27
Finally kΦm k = 1, Φm ∈ D(HUV (m)) and HUV (m)Φm = E(HUV (m))Φm .
Second step
Let (Ψj ) ∈ Q(HUV (m)) be a normalized sequence which converges weakly to 0 and such that
qHUV (m) (Ψj , Ψj ) is uniformly bounded. Let us show that
lim inf qHUV (m) (Ψj , Ψj ) > E(HUV (m)).
j→∞
Let φ1 , φ2 , φ3 ∈ C∞ (R6 )


 ∗
∗


∗


∗



 ∗

∗



 ∗


 ∗
∗


∗
be such that
φ1 = 1 on the set X ∈ R6 , |r| ≤ 1, |R| ≤ 1 ,
φ1 = 0 on X ∈ R6 , |r| ≥ 2 ∪ X ∈ R6 , |R| ≥ 2 ,
0 ≤ φ1 ≤ 1,
φ2 (X) = φ2 (r),
φ2 = 1 on X ∈ R6 , |r| ≥ 2 ,
φ2 = 0 on X ∈ R6 , |r| ≤ 1 ,
0 ≤ φ2 ≤ 1,
φ3 = 1 on X ∈ R6 , |r| ≤ 1, |R| ≥ 2 ,
φ3 = 0 on X ∈ R6 , |r| ≥ 2 ∪ X ∈ R6 , |R| ≤ 1 ,
0 ≤ φ3 ≤ 1,
and φ21 + φ22 + φ23 = 1.
Moreover we set φi,T (X) = φi (X/T ) for all T > 0 and i = 1, 2, 3. Then, φi,T Ψj ∈ Q(HUV (m)),
and we can show
X
X
2
qHUV (m) (Ψj , Ψj ) =
qHUV (m) (φi,T Ψj , φi,T Ψj ) −
(Ψj , |∇φi,T | Ψj ),
(16)
i=1,2,3
i=1,2,3
where ∇ is the gradient vector in R6 .
Note that |∇φi | ≤ Ci where Ci is a positive constant. Therefore
X
Cste
2
−
(Ψj , |∇φi,T | Ψj ) ≥ − 2 .
T
i=1,2,3
(17)
e V (m) is defined in
Now, let us estimate qHUV (m) (φ1,T Ψj , φ1,T Ψj ). Here, the Hamiltonian H
U
the same way as in (8), as an operator acting in L2 (R6 ; Fs ⊗ Fs ), to be the self-adjoint operator
e f (m)):
associated with the quadratic form with domain Q(p21 + p22 ) ∩ Q(U + ) ∩ Q(H
qHe V (m) (Φ, Ψ) =
U
1
([(p1 − q1 A1 ) ⊗ I]Φ, [(p1 − q1 A1 ) ⊗ I]Ψ)
2m1
1
+
([(p2 − q2 A2 ) ⊗ I]Φ, [(p2 − q2 A2 ) ⊗ I]Ψ)
2m2
e f (m)1/2 Φ, H
e f (m)1/2 Ψ) − ((−V )1/2 Φ, (−V )1/2 Ψ)
+ (H
− ((U − )1/2 Φ, (U − )1/2 Ψ) + ((U + )1/2 Φ, (U + )1/2 Ψ),
28
e f (m) := Hf (m) ⊗ I + I ⊗ Hf (m).
where we have set H
∞
Note that, on C0 (R6 ) ⊗ DS ⊗ DS , we have
e UV (m) = HUV (m) ⊗ I + I ⊗ Hf (m).
H
Since Hf (m) ≥ m.I − m.PΩ , where PΩ denotes the projector onto the subspace spanned by Ω,
we get
qHe V (m) ≥ E(HUV (m)) + m − m.I ⊗ PΩ .
(18)
U
In addition, we define the unitary operator UP from Fs into Fs ⊗ Fs by
UP a∗ (h)UP∗ := a∗ (j1 h) ⊗ I + I ⊗ a∗ (j2 h),
6
for all h ∈ L2 (R3 ; C2 ), and where j1 , j2 ∈ C∞
0 (R ) are such that


∗ j1 = 1 in the ball B(0, 1),



 ∗ j = 0 outside the ball B(0, 2),
1

∗ 0 ≤ j1 ≤ 1,



 ∗ j 2 + j 2 = 1.
1
2
Then, following [GLL], lemma A.1, we can show that
qHUV (m) (φ1,T Ψj , φ1,T Ψj ) = qHe V (m) (UP φ1,T Ψj , UP φ1,T Ψj ) + ν(m, P, T ),
U
where ν(m, P, T ) is such that for all fixed m, T , ν(m, P, T ) → 0.
P →∞
Thus (18) leads to
qHUV (m) (φ1,T Ψj , φ1,T Ψj ) ≥ [E(HUV (m)) + m](φ1,T Ψj , φ1,T Ψj ) + ν(m, P, T )
− m(UP φ1,T Ψj , I ⊗ PΩ UP φ1,T Ψj )
= [E(HUV (m)) + m](φ1,T Ψj , φ1,T Ψj ) + ν(m, P, T )
(19)
− ν 0 (m, P, T, j),
with ν 0 (m, P, T, j) := m(UP φ1,T Ψj , I ⊗ PΩ UP φ1,T Ψj ).
Lemma A.3 in [GLL] tells us that for all fixed m, P, T , lim inf (ν 0 (m, P, T, j)) = 0. The point to
j→∞
get this result is that φ1,T is compactly supported, so that the operator
−1/2

φ1,T Γ(j1 ) 1 +
X
p2j + Hf (m)
j=1,2
is compact, where Γ(j1 ) is defined by Γ(j1 )a∗ (f1 ) . . . a∗ (fn )Ω := a∗ (j1 f1 ) . . . a∗ (j1 fn )Ω.
Next, let us estimate qHUV (m) (φ2,T Ψj , φ2,T Ψj ). We have
qHUV (m) (φ2,T Ψj , φ2,T Ψj ) = qHU0 (m) (φ2,T Ψj , φ2,T Ψj ) − ((−V )1/2 φ2,T Ψj , (−V )1/2 φ2,T Ψj ).
29
Since φ2,T is supported in X ∈ R6 , |r| ≥ T , we get
C
qHUV (m) (φ2,T Ψj , φ2,T Ψj ) ≥ E(HU0 (m) −
(φ2,T Ψj , φ2,T Ψj ).
T
(20)
Finally, let us estimate qHUV (m) (φ3,T Ψj , φ3,T Ψj ). We have
qHUV (m) (φ3,T Ψj , φ3,T Ψj ) = qH0V (m) (φ3,T Ψj , φ3,T Ψj )
+ ((U + )1/2 φ3,T Ψj , (U + )1/2 φ3,T Ψj )
− ((U − )1/2 φ3,T Ψj , (U − )1/2 φ3,T Ψj ).
But φ3,T is supported in X ∈ R6 , |R| ≥ T and we know by (H0 ) that U − is compactly supported. So (U − )1/2 φ3,T = 0 for any T large enough. Therefore
qHUV (m) (φ3,T Ψj , φ3,T Ψj ) ≥ E(H0V (m))(φ3,T Ψj , φ3,T Ψj ),
(21)
for any T large enough.
Then, (16) with the inequalities (17), (19), (20), (21) leads to
qHUV (m) (Ψj , Ψj ) ≥ min E(HUV (m)) + m, E(HU0 (m)), E(H0V (m))
+ ν(m, P, T ) + ν 0 (m, P, T, j) −
Cste
C
− 2 ,
T
T
for any T large enough.
Let ε > 0 and pick T0 large enough such that −C/T0 − Cste/T02 ≥ −ε. Next, pick P0 such that
|ν(m, P0 , T0 )| ≤ ε. Then lim inf (ν 0 (m, P0 , T0 , j)) = 0 for any m small enough, which yields
j→∞
lim inf qHUV (m) (Ψj , Ψj ) ≥ min E(HUV (m)) + m, E(HU0 (m)), E(H0V (m)) − 2ε,
j→∞
for all ε > 0 and any m small enough. Thus,
lim inf qHUV (m) (Ψj , Ψj )
j→∞
≥ min E(HUV (m)) + m, E(HU0 (m)), E(H0V (m))
= E(HUV (m)) + min m, E(HU0 (m)) − E(HUV (m)), E(H0V (m)) − E(HUV (m))
> E(HUV (m)),
for any m small enough (see the remark 3.1 above).
Thus the proof is complete.
4.2
Exponential decay of the ground state Φm
In order to prove the exponential localization of Φm , we follow [GLL], lemma 6.2, with some
modifications. More precisely, we would like to show that k exp(β|X|)Φm k (where β is a suitable
constant) is bounded by a constant which does not depend on m. In [GLL], the bound depended
on m. Here, the proof is simpler, and we do not need to follow [G].
30
Lemma 4.1 Let Φm be a normalized ground state for HUV (m).
Then for all β > 0 such that 0 < β 2 < min(E(H0V ) − E(HUV ), E(HU0 ) − E(HUV )), we have
2
β|X|
Φm ≤ C0 ,
e
H
for any m small enough. Here, C0 is a positive constant which does not depend on m.
Proof For i = 1, 2, 3, φi,T denotes the function defined in the previous subsection. Moreover,
we set
q
φ1,T = φ22,T + φ23,T ,
2
that is φ1,T = 1 − φ21,T .
We have
2 2 2
β|X|
Φm = φ1,T eβ|X| Φm + φ1,T eβ|X| Φm ,
e
and since φ1,T is compactly supported, the first of this two terms is bounded by a positive
constant C1 that does not depend on m.
We set GT := φ1,T exp(fε ) where fε is defined for all ε > 0 by
fε (X) :=
β|X|
.
1 + ε|X|
Note that fε and |∇fε | are bounded functions. Thus, GT Φm ∈ Q(HUV (m)), and using the fact
that Φm is a ground state for HUV (m), we can show
qHUV (m) (GT Φm , GT Φm ) − E(HUV (m))kGT Φm k2 = (Φm , |∇GT |2 Φm ).
But we can compute
|∇GT |2 = |∇φ1,T |2 e2fε + 2(∇φ1,T .∇fε )efε GT + |∇fε |2 G2T .
Therefore
qHUV (m) (GT Φm , GT Φm ) − E(HUV (m))kGT Φm k2 − (GT Φm , |∇fε |2 GT Φm )
= (Φm , |∇φ1,T |2 e2fε + 2(∇φ1,T .∇fε )efε GT Φm )
(22)
≤ C2 ,
where C2 is a positive constant which depends on T but not on m or ε. Here, we used the fact
that ∇φ1,T is compactly supported.
In addition, we note that φ1,T /2 .φ1,T = 0. Thus
X
qHUV (m) (GT Φm , GT Φm ) =
qHUV (m) (GT Φm , φ2i,T /2 GT Φm )
i=2,3
=
X
qHUV (m) (φi,T /2 GT Φm , φi,T /2 GT Φm )
i=2,3
−
X
(GT Φm , |∇φi,T /2 |2 GT Φm ).
i=2,3
31
Now, as in the previous subsection we have
2C
qHUV (m) (φ2,T /2 GT Φm , φ2,T /2 GT Φm ) ≥ E(H0V (m)) −
kφ2,T /2 GT Φm k2 ,
T
and
qHUV (m) (φ3,T /2 GT Φm , φ3,T /2 GT Φm ) ≥ E(HU0 (m))kφ3,T /2 GT Φm k2 ,
for any T large enough. This leads to
qHUV (m) (GT Φm , GT Φm ) ≥ min E(H0V (m)), E(HU0 (m)) kGT Φm k2 −
Cste
2C
+ 2
T
T
kGT Φm k2 ,
for any T large enough. Since |∇fε | ≤ β, we get
qHUV (m) (GT Φm , GT Φm ) − E(HUV (m))kGT Φm k2 − (GT Φm , |∇fε |2 GT Φm )
≥ min E(H0V (m)) − E(HUV (m)), E(HU0 (m)) − E(HUV (m)) kGT Φm k2 − β 2 kGT Φm k2
2C
Cste
−
+ 2 kGT Φm k2 .
T
T
Thus, remark 3.1 leads to
qHUV (m) (GT Φm , GT Φm ) − E(HUV (m))kGT Φm k2 − (GT Φm , |∇fε |2 GT Φm )
1
2C
Cste
V
V
0
V
2
2
≥
min E(H0 ) − E(HU ), E(HU ) − E(HU ) − β kGT Φm k −
+ 2 kGT Φm k2 ,
2
T
T
for any m small enough.
Therefore, we can choose T0 to be such that
qHUV (m) (GT0 Φm , GT0 Φm ) − E(HUV (m))kGT0 Φm k2 − (GT0 Φm , |∇fε |2 GT0 Φm )
1
min E(H0V ) − E(HUV ), E(HU0 ) − E(HUV ) − β 2 kGT0 Φm k2 ,
≥
4
(23)
for any m small enough.
(22) and (23) yields
kGT0 Φm k2 ≤
4C2
:= C3 ,
min E(H0V ) − E(HUV ), E(HU0 ) − E(HUV ) − β 2
for any m small enough and any ε > 0. Thus, as ε → 0, we get
2
φ1,T0 eβ|X| Φm ≤ C3 ,
for any m small enough.
So the proof is complete with C0 := C1 + C3 .
32
4.3
Convergence of the ground state Φm when m → 0
The end of the proof of the existence of a ground state for HUV follows step by step the one in
[GLL]. Namely, it is shown that kb
aλ (k)Φm k and k∇k b
aλ (k)Φm k are bounded for almost every
k, with bounds that do not depend on m. Next the Rellich-Kondrachov theorem leads to the
conclusion. We only give the results.
Theorem 4.2 Let Φm be a normalized ground state for HUV (m), where m > 0 is small. Then,
for almost every k ∈ R3 , we have
kb
aλ (k)Φm k ≤ CΛ (|q1 | + |q2 |)
χ
bΛ (k)
k|X|Φm k.
|k|1/2
(24)
Moreover, for almost every k ∈ R3 such that |k| < Λ/2 and (k1 , k2 ) 6= (0, 0), we have
k∇k b
aλ (k)Φm k ≤ CΛ0 (|q1 | + |q2 |)
|k|1/2
1
p
k|X|Φm k.
k12 + k22
(25)
Here Cλ and Cλ0 are constants that depend on Λ, m1 , m2 , but not on m.
Remark 4.1
1. Note that Φm ∈ Q(N ) by (7). Then (24) means
qN (Φm , Φm ) ≤
CΛ2
2
Z
(|q1 | + |q2 |)
R3
χ
b2Λ (k)
dkk|X|Φm k2 .
|k|
The meaning of (25) is given in the appendix B of [GLL].
2. A key step to obtain (24) and (25) is to use the following gauge transformation: the unitary
operator T is defined by
Z ⊕
P
T =
T (X)dX with T (X) = e−i j=1,2 qj xj .A(0) .
R6
Then we have bbλ (k, X) := T (X)b
aλ (k)T ∗ (X) = b
aλ (k) − iwλ (k, X), with
X
bΛ (k)
1 χ
ελ (k)
qj xj .
1/2
2π |k|
j=1,2
wλ (k, X) =
Hence the transformed Hamiltonian is
e UV (m) := T HUV (m)T ∗ =
H
ej =
with A
R⊕
R6
ej (X)dX, H
e f (m) =
A
ej (X) = A(xj ) − A(0)
A
,
1
ej )2 + H
e f (m) + U + V,
(pj − qj A
2m
j
j=1,2
X
R⊕
e f (m)(X)dX, and
H
X Z
e
Hf (m)(X) =
ωm (k)bb∗λ (k, X)bbλ (k, X)dk.
R6
λ=1,2
33
R3
3. Theorem 4.2 together with lemma 4.1 show that kb
aλ (k)Φm k and k∇k b
aλ (k)Φm k are uniformly bounded for small m.
Now let (mj ) be a sequence that decays to 0 and such that (24) and (25) are satisfied for all
j. We can suppose that Φmj converges weakly to a limit Φ when j goes to ∞. Let us show that
Φ ∈ D(HUV ).
Since, for all j, Q(HUV (mj )) ⊂ Q(HUV ) by (7), we can write
2
1/2
V
Φmj = qHUV (Φmj , Φmj ) − E(HUV )
HU − E(HUV )
≤ qHUV (mj ) (φmj , φmj ) − E(HUV )
= E(HUV (mj )) − E(HUV ) → 0.
j→∞
Thus, for all ψ ∈ Q(HUV ), we have
1/2
1/2
( HUV − E(HUV )
ψ, Φ) = lim ( HUV − E(HUV )
ψ, Φmj )
j→∞
1/2
= lim (ψ, HUV − E(HUV )
Φmj ) = 0.
j→∞
1/2
Φ = 0. This yields
Therefore, Φ ∈ Q(HUV ) and HUV − E(HUV )
Φ ∈ D(HUV )
HUV Φ = E(HUV )Φ.
and
(26)
Then, in the same way as in theorem 7.1 of [GLL], (26), lemma 4.1 and theorem 4.2 lead to
Theorem 4.3 Φmj converges strongly to Φ, so that kΦk = 1 and Φ is a ground state for HUV .
Remark 4.2 With the help of the functional integral representation of remark 2.1, we can prove
V
that the ground state of HUV is non-degenerate. Indeed, it is shown in [H2] that ν −1 e−tHU ν is
positivity improving as an operator acting on L2 (R6 × Q), where L2 (Q) denotes a Schrödinger
representation of F, and where ν is a unitary operator from L2 (Q) to F.
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35