Note on the Retarded van der Waals Potential within the Dipole Approximation
We examine the dipole approximated Pauli-Fierz Hamiltonians of the nonrelativistic QED. We assume that the Coulomb potential of the nuclei, together with the Coulomb interaction between the electrons, can be approximated by harmonic potentials. By an exact diagonalization method, we prove that the b...
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| description | We examine the dipole approximated Pauli-Fierz Hamiltonians of the nonrelativistic QED. We assume that the Coulomb potential of the nuclei, together with the Coulomb interaction between the electrons, can be approximated by harmonic potentials. By an exact diagonalization method, we prove that the binding energy of the two hydrogen atoms behaves as R⁻⁷, provided that the distance between the atoms R is sufficiently large. We employ Feynman's representation of the quantized radiation fields, which enables us to diagonalize Hamiltonians rigorously. Our result supports the famous conjecture by Casimir and Polder.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 036, 34 pages
Note on the Retarded van der Waals Potential
within the Dipole Approximation
Tadahiro MIYAO
Department of Mathematics, Hokkaido University, Sapporo, Japan
E-mail: miyao@math.sci.hokudai.ac.jp
Received February 27, 2019, in final form April 14, 2020; Published online April 26, 2020
https://doi.org/10.3842/SIGMA.2020.036
Abstract. We examine the dipole approximated Pauli–Fierz Hamiltonians of the non-
relativistic QED. We assume that the Coulomb potential of the nuclei together with the
Coulomb interaction between the electrons can be approximated by harmonic potentials.
By an exact diagonalization method, we prove that the binding energy of the two hydrogen
atoms behaves as R−7, provided that the distance between atoms R is sufficiently large. We
employ the Feynman’s representation of the quantized radiation fields which enables us to
diagonalize Hamiltonians, rigorously. Our result supports the famous conjecture by Casimir
and Polder.
Key words: retarded van der Waals potential; non-relativistic QED; Pauli–Fierz Hamilto-
nian; dipole approximation
2020 Mathematics Subject Classification: 81V10; 81V55; 47A75
1 Introduction
London was the first to explain attractive interactions between neutral atoms or molecules by
applying quantum mechanics [16]. Nowadays, the attractive forces are called the van der Waals–
London forces, and are described by the potential energy decaying as R−6 for R sufficiently
large.1 Here, R denotes the distance between two atoms or molecules. It is recognized that
these forces come from the quantum fluctuations of the charges inside the atoms. Because even
a simple hydrogen atom displays a fluctuating dipole, the van der Waals–London forces are
ubiquitous and therefore very fundamental.
Casimir and Polder took the interactions between electrons and the quantized radiation fields
into consideration and perfomed the fourth order perturbative computations [6]. They found
that the finiteness of the speed of light weakens the correlation between nearby dipoles and
causes the attractive potential between atoms to behave as
VCP(R) ∼= − 23
4π
(
1
2π
)2 1
R7
αAαB, R� 1, (1.1)
where αA and αB are the static polarizability of the atoms. The potential VCP is called the
Casimir–Polder potential or the retarded van der Waals potential. For reviews, see, e.g., [5, 11,
13, 18, 19]. Although this result is plausible, Casimir–Polder’s arguments are heuristic, and lack
mathematical rigor.
1More precisely, if one takes the interactions between electrons and the quantized Maxwell field according to
non-relativistic QED into account, the R−6 behavior is true for the near-field region (very vaguely “sufficiently
large R but not too large”, and discussed on [7, p. 157] and [21]), but for the far-field region (where “retardation
effects become important”) the presented results show a R−7 behavior. In the approximation where the quantum
fluctuations of the Maxwell field are ignored, only the electrostatic Coulomb interaction remains. In this case,
the binding energy behaves as R−6 provided that R is sufficently large. This R−6 behavior is well-understood,
mathematically [1, 2, 15, 22].
mailto:miyao@math.sci.hokudai.ac.jp
https://doi.org/10.3842/SIGMA.2020.036
2 T. Miyao
There are few rigorous results concerning the Casimir–Polder potential; In [20, 21], Miyao and
Spohn gave a path integral formula for VCP and applied it to computing the second cumulant.
Under the assumption that all of higher order cumulants behave as O
(
R−9
)
and their coefficients
are small enough to control, they rigorously refound that VCP behaves as R−7 as R → ∞.
Although this assumption appears to be plausible, to prove it is extremely hard. Therefore, to
give a mathematical foundation of the Casimir–Polder potential is an open problem even today.
In the present paper, we will examine the Pauli–Fierz model under the following assumptions
[24, equations (13.127) and (13.123)]:
(C.1) the dipole approxiamtion (see (2.2));
(C.2) the electrons are strongly bound around each nucleus (see (2.3) and (2.4)).
The dipole approximation (C.1) is widely accepted as a convenient procedure in the community
of the nonrelativistic QED [24]. The assumption (C.2) is often useful when we study the low
energy behavior of the system. Under the assumptions, we prove that the binding energy
for two hydrogen atoms actually behaves as R−7. In the context of the Born–Oppenheimer
approximation, this indicates that the effective potential between two hydrogen atoms behaves
as R−7 too. This result supports our assumptions for the model without dipole approximation,
and is expected to become a starting point for study of the non-approximated model. Our proof
relies on the fact that the dipole approximated Hamiltoninas can be diagonalized by applying
Feynman’s representation of the quantized radiation fields [8]. It has been believed that the
dipole approximated model also exhibits R−7 behavior by the forth order perturbation theory.
However, the arguments concering the error terms are completely missing. Indeed, this part is
tacitly assumed to be trivial in literatures. In this paper, we actually perform systematic error
estimates which are far from trivial.
In mathematical physics, it is known that rigorous studies of the Pauli–Fierz Hamiltonian
require an extra care due to the infamous infrared problem [4, 10, 24]. Fortunetely, within the
assumptions (C.1) and (C.2), we can control the problem relatively easily.
Before we proceed, we have additional remarks. In his Ph.D. Thesis [12], Koppen studied the
retarded van der Waals potential; he examined the Pauli–Fierz model with the dipole approxi-
mation (C.1), but the condition (C.2) is not assumed in [12]. In contrast to the present study,
he imposed the infrared cutoff σ on the Hamiltonian in order to apply the naive perturbation
theory and obtained an expansion formula for the binding energy: Eσ(R) =
∞∑
i=0
eiV σ
i (R). Then
he removed the infrared cutoff from each term: Vi(R) := lim
σ→+0
V σ
i (R). Finally, he proved that
some Vi(R) satisfies (1.1). His observation could be regareded as a nice starting point of mathe-
matical analysis of the retarded van der Waals potential, however, there are still some problems
to be considered. For example, the magnetic contributions to the −R−7 decay are completely
overlooked. In addition, in the mathematical study of the Pauli–Fierz model, it is well-known
that to prove that lim
σ→+0
Eσ(R) =
∞∑
i=0
ei lim
σ→+0
V σ
i (R) is very hard problem, the aforementioned
infrared problem.
Our contributions are
• to provide a minimal QED model which can rigorously explain the Casimir–Polder poten-
tial by a relatively simple and easy way;
• to perform systematic error estimates without the infrared cutoff.
In this way, the present paper and the thesis [12] are complementary to each other.
Since the electrons obey Fermi–Dirac statistics, the wave functions of the two-electron system
belong to (H∧H)⊗F
(
L2
(
R3×{1, 2}
))
, where H = L2
(
R3
)
⊗C2, the Hilbert space with spin 1/2,
Note on the Retarded van der Waals Potential within the Dipole Approximation 3
the symbol ∧ indicates the anti-symmetric tensor product and F
(
L2
(
R3 × {1, 2}
))
is the Fock
space over L2
(
R3×{1, 2}
)
. Usually, the ground state of this system is a spin singlet. Thus, the
spatial part of the ground state is symmetric and we can end up with minimizing the energy in
an unrestricted manner on
(
L2
(
R3
)
⊗L2
(
R3
))
⊗F
(
L2
(
R3×{1, 2}
))
. For this reason, we perform
our analysis on
(
L2
(
R3
)
⊗ L2
(
R3
))
⊗ F
(
L2
(
R3 × {1, 2}
))
.2 However, it should be mentioned
that our observation here can not be extended to general N -electron systems, directly.
In fairness, we mention the following two difficulties of the assumptions (C.1) and (C.2). For
details, see discussions in Section 9.
• The condition (C.2) breaks the indistinguishability of the electrons.
• Under the conditions (C.1) and (C.2), we cannot reproduce the exact cancellation of the
term with R−6 decay (the van der Waals–London potential) by the contribution from the
quantized Maxwell field. Note that this cancellation is known to be fundamental to explain
the retarded van der Waals potential [20, 21].
The present paper is organized as follows. In Section 2, we introduce the dipole approximated
Pauli–Fierz Hamiltonian and state the main result. In Section 3, we switch to the Feynman
representation of the quantized radiation fields. This representation enables us to diagonalize
the Hamiltonians as we will see in the following sections. Further, we introduce a canonical
transformation which induces the quantized displacement fields in the Hamiltonians in Section 4.
Section 5 is devoted to the finite volume approximation, which is a standard method in the study
of the quantum field theory [3, 9]. Then we diagonalize the Hamiltonians in Sections 6 and 7.
In Section 8, we give a proof of the main theorem. Section 9 is devoted to the discussions of the
approximations (C.1) and (C.2). In Appendices A, B and C, we collect various auxiliary results
which are needed in the main sections.
2 Main result
Let us consider a single hydrogen atom with an infinitely heavy nucleus located at the origin 0.
The nonrelativistic QED Hamiltonian for this system is given by
H1e =
1
2
(
−i∇− eA(x)
)2 − e2V (x) +Hf .
The nucleus has charge e > 0, and the electron has charge −e. We assume that the charge
distribution % satisfies the following properties:
(A.1) % is normalized:
∫
R3 dx %(x) = 1.
(A.2) %(x) = %(−x). Thus the Fourier transformation %̂ is real.
(A.3) %̂ is rotation invariant, %̂(k) = %̂rad(|k|), of rapid decrease and smooth.
The smeared Coulomb potential V is given by
V (x) =
∫
R3
dk %̂(k)2|k|−2e−ik·x.
The photon annihilation operator is denoted by a(k, λ). As usual, this operator satisfies the
standard commutation relation:
[a(k, λ), a(k′, λ′)∗] = δλλ′δ(k − k′).
2Or we could simply say that one considers the “distinguishable particles”, see Section 9 for detail.
4 T. Miyao4 T. Miyao
Electron 1
Electron 2
x1
x2
Nucleus 1 Nucleus 2r
Figure 1.
The quantized vector potential A(x) is defined by
A(x) =
∑
λ=1,2
∫
R3
dk
%̂(k)√
2|k|
ε(k, λ)
(
e−ik·xa(k, λ)∗ + eik·xa(k, λ)
)
, (2.4)
where ε(k, λ) = (ε1(k, λ), ε2(k, λ), ε3(k, λ)), λ = 1, 2 are polarization vectors. For concreteness,
we choose as
ε(k, 1) =
(k2,−k1, 0)√
k2
1 + k2
2
, ε(k, 2) =
k
|k| ∧ ε(k, 1). (2.5)
Note that A(x) is essentially self-adjoint. We will denote its closure by the same symbol. The
field energy Hf is given by
Hf =
∑
λ=1,2
∫
R3
dk |k|a(k, λ)∗a(k, λ). (2.6)
The operator H1e acts in the Hilbert space L2(R3)⊗F(L2(R3
k×{1, 2})), where F(h) is the bosonic
Fock space over h: F(h) =
⊕∞
n=0 h
⊗sn. Here, ⊗s indicates the symmetric tensor product.
To examine the Casimir-Polder potential, we consider two hydrogen atoms, one located at the
origin and the other at r = (0, 0, R) with R > 0. For computational convenience, we define the
position of the second electron relative to r, see Figure 1. Then the two-electron Hamiltonian
reads
H2e =
1
2
(
− i∇1 − eA(x1)
)2 − e2V (x1) +
1
2
(
− i∇2 − eA(x2 + r)
)2 − e2V (x2)
+ e2VR(x1, x2) +Hf (2.7)
with
VR(x1, x2) = −V (x1 − r)− V (x2 + r) + V (r) + V (r + x2 − x1) (2.8)
=
∫
R3
dk %̂(k)2|k|−2(1− e−ik·x1)(1− eik·x2)eik·r. (2.9)
The operator H2e acts in L2(R3
x1)⊗ L2(R3
x2)⊗ F(L2(R3
k × {1, 2})).
The dipole approximation (C. 1) means the following replacement:
A(x1) ; A(0), A(x2 + r) ; A(r). (2.10)
Figure 1.
The quantized vector potential A(x) is defined by
A(x) =
∑
λ=1,2
∫
R3
dk
%̂(k)√
2|k|
ε(k, λ)
(
e−ik·xa(k, λ)∗ + eik·xa(k, λ)
)
,
where ε(k, λ) = (ε1(k, λ), ε2(k, λ), ε3(k, λ)), λ = 1, 2 are polarization vectors. For concreteness,
we choose as
ε(k, 1) =
(k2,−k1, 0)√
k2
1 + k2
2
, ε(k, 2) =
k
|k| ∧ ε(k, 1). (2.1)
Note that A(x) is essentially self-adjoint. We will denote its closure by the same symbol. The
field energy Hf is given by
Hf =
∑
λ=1,2
∫
R3
dk |k|a(k, λ)∗a(k, λ).
The operator H1e acts in the Hilbert space L2
(
R3
)
⊗ F
(
L2
(
R3
k × {1, 2}
))
, where F(h) is the
bosonic Fock space over h: F(h) =
∞⊕
n=0
h⊗sn. Here, ⊗s indicates the symmetric tensor product.
To examine the Casimir–Polder potential, we consider two hydrogen atoms, one located at
the origin and the other at r = (0, 0, R) with R > 0. For computational convenience, we define
the position of the second electron relative to r, see Fig. 1. Then the two-electron Hamiltonian
reads
H2e =
1
2
(
−i∇1 − eA(x1)
)2 − e2V (x1) +
1
2
(
−i∇2 − eA(x2 + r)
)2 − e2V (x2)
+ e2VR(x1, x2) +Hf
with
VR(x1, x2) = −V (x1 − r)− V (x2 + r) + V (r) + V (r + x2 − x1)
=
∫
R3
dk %̂(k)2|k|−2
(
1− e−ik·x1)(1− eik·x2)eik·r.
The operator H2e acts in L2
(
R3
x1
)
⊗ L2
(
R3
x2
)
⊗ F
(
L2
(
R3
k × {1, 2}
))
.
The dipole approximation (C.1) means the following replacement:
A(x1) ; A(0), A(x2 + r) ; A(r). (2.2)
Note on the Retarded van der Waals Potential within the Dipole Approximation 5
By the assumption (C.2), we can take x1 and x2 sufficiently small. Therefore, we assume that
the Coulomb potential of the nuclei together with the Coulomb interaction between the electrons
can be approximated by harmonic potentials. Then one has
V (xj) ' −
1
2
ν2
0x
2
j + const (2.3)
with ν2
0 = 1
3
∫
dk %̂(k)2 and
VR(x1, x2) '
∫
dk %̂(k)2eik·r(x1 · k̂
)(
x2 · k̂
)
(2.4)
with k̂ = k/|k|. Hence, we arrive at
HD1e =
1
2
(
−i∇− eA(0)
)2
+
1
2
e2ν2
0x
2 +Hf
and
HD2e =
1
2
(
−i∇1 − eA(0)
)2
+
1
2
e2ν2
0x
2
1 +
1
2
(
−i∇2 − eA(r)
)2
+
1
2
e2ν2
0x
2
2
+ e2
∫
R3
dk %̂(k)2eik·r(x1 · k̂)(x2 · k̂) +Hf .
Note that HD1e and HD2e are self-adjoint and bounded from below [14], because the cross-term∫
dk %̂(k)2eik·r(x1 · k̂
)(
x2 · k̂
)
becomes very small provided that R is large enough. As for physical
discussions of the approximation above, see Section 9 in detail.
In what follows, we assume an additional condition:
(A.4) We regard ν0 as a parameter. Thus, ν0 is independent of %.
Hence, there are three parameters e, R and ν0 in our models.
Theorem 2.1. Let E(R) = inf spec(HD2e) and let E = inf spec(HD1e), where spec(X) indicates
the spectrum of a linear operator X. Let
c∞ = max
{
√
2e
∥∥|k|−1%̂
∥∥
L2 ,
‖|k|%̂‖L2√
2eν2
0
,
‖%̂‖L2
ν0
}
.
Choose e and ν0 such that c∞ < 1/2, 1 ≤
√
2eν0 and
√
2e‖%̂‖L2 < 1. Then one has
lim
R→∞
R7(2E − E(R)) =
23
4π
(
1
2π
)2(1
4
αE,at
)2
,
where αE,at = ν−2
0 .
Remark 2.2.
1. The constant αE,at is the dipole moment of a decoupled atom, i.e.,
αE,at =
2
3
〈
ψat|x · (hat − 3eν0/2)−1xψat
〉
, (2.5)
where hat = −1
2∆ +
e2ν20
2 x2 and ψat is the ground state of hat. Note that xjψat is or-
thogonal to ψat: 〈ψat|xjψat〉 = 0. Thus, the vectors (hat − 3eν0/2)−1xjψat in (2.5) are
mathematically meaningful.
6 T. Miyao
2. The restrictions of the parameters in Theorem 2.1 come from technical reasons: As we will
see in the later sections, these are needed in order to control the perturbative expansions
for E and E(R).
Example 2.3. Let η ∈ S
(
R3
)
, the Schwartz space. Suppose that η satisfies the following:
• η(0) = (2π)−3/2;
• η(k) is real-valued;
• η(k) = ηrad(|k|).
For given ξ > 0, we define % by
%̂(k) = η(ξk).
Then % satisfies (A.1)–(A.3). In addition, since
‖%̂‖L2 ∝ ξ−3/2, ‖|k|%̂‖L2 ∝ ξ−5/2, ‖|k|−1%̂‖L2 ∝ ξ−1/2,
the all assumptions in Theorem 2.1 are fulfilled, provided that ξ is large enough. Note that
a typical choice of η is η(k) = (2π)−3/2e−|k|
2
.
3 Feynman Hamiltonians
3.1 Preliminaries
To prove our main result, let us introduce Feynman Hamiltonians of the nonrelativistic QED [8].
These Hamiltonians can be diagonalized readily as we will see in Sections 6 and 7.
First, remark the following identification:
L2
(
R3
)
= L2
e
(
R3
)
⊕ L2
o
(
R3
)
,
where
L2
e
(
R3
)
=
{
f ∈ L2
(
R3
)
| f(−x) = f(x) a.e.
}
,
L2
o
(
R3
)
=
{
f ∈ L2
(
R3
)
| f(−x) = −f(x) a.e.
}
.
For notational convenience, we denote by εj(·, λ) the multiplication operator by the func-
tion εj(·, λ).
We begin with the following lemma.
Lemma 3.1. Let
H1 =
3⋃
j=1
ran
(
εj(·, 1) � L2
e
(
R3
))
, H2 =
3⋃
j=1
ran
(
εj(·, 2) � L2
e
(
R3
))
,
H3 =
3⋃
j=1
ran
(
εj(·, 1) � L2
o
(
R3
))
, H4 =
3⋃
j=1
ran
(
εj(·, 2) � L2
o
(
R3
))
,
where A � X indicates the restriction of A to X. Then H1, H2, H3 and H4 are subspaces of L2
(
R3
)
.
Note on the Retarded van der Waals Potential within the Dipole Approximation 7
Proof. Let D =
{
k ∈ R3 | k1 6= 0, k2 6= 0, k3 6= 0
}
. Trivially, εj(k, λ) is well-defined on D.
In addition, ε1(k, 1)−1, ε2(k, 1)−1, ε1(k, 2)−1, ε2(k, 2)−1 and ε3(k, 2)−1 are well-defined on D.3
Let C0(D) be the set of continuous functions on D of compact support. Because the Lebesgue
measure of Dc, the complement of D, is equal to zero, C0(D) is dense in L2
(
R3
)
. Thus, it holds
that
ran
(
εj(·, 1) � L2
e
(
R3
))
= ran
(
εj(·, 1) � C0,e(D)
)
, j = 1, 2, (3.1)
where C0,e(D) = {f ∈ C0(D) | f(−k) = f(k)}.
Let F,G ∈ H1. Then there exist i, j ∈ {1, 2} such that F ∈ ran
(
εj(·, 1) � L2
e
(
R3
))
and
G ∈ ran
(
εi(·, 1) � L2
e
(
R3
))
. By (3.1), there exist approximating sequences (Fn) ⊂ ran
(
εj(·, 1) �
C0,e(D)
)
and (Gn) ⊂ ran
(
εi(·, 1) � C0,e(D)
)
such that ‖F − Fn‖ → 0 and ‖G − Gn‖ → 0 as
n→ 0. Hence, for each α, β ∈ C, it holds that
αFn + βGn → αF + βG as n→∞. (3.2)
Note that we can write Fn = εj(·, 1)fn and Gn = εi(·, 1)gn with fn, gn ∈ C0,e(D). Thus, we have
Gn = εj(·, 1)g′n, where g′n = εj(·, 1)−1εi(·, 1)gn. Because εj(k, 1)−1εi(k, 1) is an even function
on D, we see that g′n ∈ C0,e(D). Accordingly, αFn + βGn = εj(·, 1)(αfn + βg′n) ∈ ran
(
εj(·, 1) �
C0,e(D)
)
. Combining this, (3.1) and (3.2), we conclude that αF + βG ∈ H1, in particular, H1 is
a subspace of L2
(
R3
)
. By similar arguments, we can prove that H2,H3 and H4 are subspaces
of L2
(
R3
)
. �
Lemma 3.2. We have the following identifications:
L2
(
R3 × {1, 2}
)
= L2
(
R3
)
⊕ L2
(
R3
)
= H1 ⊕ H2 ⊕ H3 ⊕ H4. (3.3)
Proof. The first identification in (3.3) is trivial. In what follows, we will concentrate on the
proof of the second identification.
Note that the multiplication operator εj(·, 1)−1 (j = 1, 2) is self-adjoint and dom
(
εj(·, 1)−1
)
is dense in L2
(
R3
)
. Because ran(εj(·, 1)) ⊇ dom
(
εj(·, 1)−1
)
⊇ C0(D) for j = 1, 2, we obtain
ran(εj(·, 1)) = L2
(
R3
)
. For each f ∈ L2
(
R3
)
, we set fe(k) = 1
2(f(k) + f(−k)) and fo(k) =
1
2(f(k)− f(−k)). Because εj(k, 1)2 is an even function, we have 〈εj(·, 1)fe|εj(·, 1)fo〉 = 0 for all
f ∈ L2
(
R3
)
and j = 1, 2, which implies that ran
(
εj(·, 1) � L2
e
(
R3
))
⊥ ran
(
εj(·, 1) � L2
o
(
R3
))
.
Since
εj(·, 1)f︸ ︷︷ ︸
∈ran(εj(·,1))
= εj(·, 1)fe︸ ︷︷ ︸
∈ran(εj(·,1)�L2
e(R3))
+ εj(·, 1)fo︸ ︷︷ ︸
∈ran(εj(·,1)�L2
o(R3))
, f ∈ L2
(
R3
)
,
we conclude that
L2
(
R3
)
= ran
(
εj(·, 1) � L2
e
(
R3
))
⊕ ran
(
εj(·, 1) � L2
o
(
R3
))
for j = 1, 2.
For i, i′ ∈ {1, 2}, we set µ
(1)
ii′ (k) = εi(k, 1)εi′(k, 1). Because µ
(1)
ii′ (k) is an even function, we see
that, for each f ∈ L2
e
(
R3
)
and g ∈ L2
o
(
R3
)
,
〈εi(·, 1)f |εi′(·, 1)g〉 =
〈
f |µ(1)
ii′ g
〉
= 0.
Therefore, H1 ⊥ H3 holds. Because H1⊕H3 ⊇ ran
(
εj(·, 1) � L2
e
(
R3
))
⊕ran
(
εj(·, 1) � L2
o
(
R3
))
, we
finally arrive at L2
(
R3
)
= H1 ⊕ H3. By arguments similar to the above, we get that L2
(
R3
)
=
H2 ⊕ H4. �
We will construct a useful identification between F
(
L2
(
R3
)
⊕ L2
(
R3
))
and
4⊗
λ=1
F(Hλ) in
Section 3.4. For this purpose, we recall some basic definitions in Sections 3.2 and 3.3.
3These facts immediately follow from (2.1). Here, note that ε(k, 2) is written as ε(k, 2) =
(
k1k3, k2k3,−k21 −
k22
)/
|k|
√
k21 + k22.
8 T. Miyao
3.2 Second quantized operators in F
(
L2
(
R3
)
⊕L2
(
R3
))
Let a(f1⊕ f2) be the annihilation operator acting in F
(
L2(R3×{1, 2})) = F(L2
(
R3
)
⊕L2
(
R3
))
.
As usual, we express this operator as
a(f1 ⊕ f2) =
∑
λ=1,2
∫
R3
dk f∗λ(k)a(k, λ).
The Fock vacuum in F
(
L2
(
R3
)
⊕ L2
(
R3
))
is denoted by Ψ0. Let F be a real-valued function
on R3 which is finite almost everywhere. The multiplication operator by F is also written as F .
The second quantization of F ⊕ F is then given by
dΓ(F ⊕ F ) = 0⊕
[ ∞⊕
n=1
n∑
j=1
1⊗ · · · ⊗ (F ⊕ F )︸ ︷︷ ︸
jth
⊗ · · · ⊗ 1
]
.
Needless to say, dΓ(F⊕F ) acts in F
(
L2
(
R3
)
⊕L2
(
R3
))
. It is known that dΓ(F⊕F ) is essentially
self-adjoint on a dense subspace
{
Ψ = {Ψn}∞n=0 |Ψn ∈ (dom(F )⊕ dom(F ))�n,∃N ∈ N s.t. Ψm = 0 ∀m > N
}
,
where � indicates the algebraic tensor product. We will denote the closure of dΓ(F ⊕F ) by the
same symbol. Symbolically, we express dΓ(F ⊕ F ) as
dΓ(F ⊕ F ) =
∑
λ=1,2
∫
R3
dk F (k)a(k, λ)∗a(k, λ).
3.3 Second quantized operators in
4⊗⊗⊗
λ=1
F(Hλ)
Let aλ(fλ) be the annihilation operator on F(Hλ). We employ the following identifications:
a1(f1) = a1(f1)⊗1l⊗1l⊗1l, a2(f2) = 1l⊗a2(f2)⊗1l⊗1l and so on. Thus, aλ(fλ) can be regarded
as a linear operator acting in the Hilbert space
4⊗
λ=1
F(Hλ). Let F be a real-valued function on R3.
Suppose that F is even: F (−k) = F (k) a.e.. dΓλ(F ) denotes the second quantization of F which
acts in F(Hλ). As before, we can also regard dΓλ(F ) as a linear operator acting in
4⊗
λ=1
F(Hλ).
The Fock vacuum in F(Hλ) is denoted by Ψλ. We will freely use the following notations:
aλ(f) =
∫
R3
dk f(k)∗a(k, λ), f ∈ Hλ,
dΓλ(F ) =
∫
R3
dk F (k)a∗(k, λ)a(k, λ), λ = 1, 2, 3, 4.
3.4 Identifications between F
(
L2
(
R3
)
⊕L2
(
R3
))
and
4⊗⊗⊗
λ=1
F(Hλ)
For each f = (f1, f2) ∈ L2
(
R3
)
⊕ L2
(
R3
)
, we set
bij(f) = a
(
εi(·, 1)f1 ⊕ εj(·, 2)f2
)
,
cij(f) = a1(εi(·, 1)f1,e) + a2(εj(·, 2)f2,e)− ia3(εi(·, 1)f1,o)− ia4(εj(·, 2)f2,o), (3.4)
where fe(k) = (f(k) + f(−k))/2 and fo(k) = (f(k)− f(−k))/2. Let Ψ0 be the Fock vacuum in
F
(
L2
(
R3
)
⊕ L2
(
R3
))
: Ψ0 = 1⊕ 0⊕ 0⊕ · · · .
Note on the Retarded van der Waals Potential within the Dipole Approximation 9
Lemma 3.3. We define a linear operator V : F(L2
(
R3
)
⊕ L2
(
R3
)
)→
4⊗
λ=1
F(Hλ) by
VΨ0 =
4⊗
λ=1
Ψλ,
V
[
N∏
`=1
bi`j`(f`)
∗
]
Ψ0 =
[
N∏
`=1
ci`j`(f`)
∗
]
4⊗
λ=1
Ψλ
for each f1, . . . ,fN ∈ L2
(
R3
)
⊕ L2
(
R3
)
and N ∈ N. Then V can be extended to the unitary
operator. In what follows, we denote the extension by the same symbol. Then we have
V bij(f)V −1 = cij(f) (3.5)
for each f ∈ L2
(
R3
)
⊕ L2
(
R3
)
and i, j ∈ {1, 2, 3}, where the bar indicates the closure of the
operator.
Proof. For i, i′ ∈ {1, 2, 3}, we set
µ
(1)
ii′ (k) = εi(k, 1)εi′(k, 1), µ
(2)
ii′ (k) = εi(k, 2)εi′(k, 2).
For f ,f ′ ∈ L2
(
R3
)
⊕ L2
(
R3
)
and i, j, i′, j′ ∈ {1, 2, 3}, define
D(f |f ′)ij;i′j′ =
〈
f1|µ(1)
ii′ f
′
1
〉
+
〈
f2|µ(2)
ii′ f
′
2
〉
.
First, we prove that
{
bij(f)|f ∈ L2
(
R3
)
⊕L2
(
R3
)
, i, j ∈ {1, 2, 3}
}
and
{
cij(f)|f ∈ L2
(
R3
)
⊕
L2
(
R3
)
, i, j ∈ {1, 2, 3}
}
satisfy the similar commutations relations, that is,
[bij(f), bi′j′(f
′)∗] = D(f |f ′)ij;i′j′ , [bij(f), bi′j′(f
′)] = 0
and
[cij(f), ci′j′(f
′)∗] = D(f |f ′)ij;i′j′ , [cij(f), ci′j′(f
′)] = 0.
To see this, note that µ
(1)
ii′ (k) and µ
(2)
ii′ (k) are even functions. Thus,
〈
fe|µ(1)
ii′ go
〉
= 0 =
〈
fe|µ(2)
ii′ go
〉
, f, g ∈ L2
(
R3
)
.
Accordingly, we have
[cij(f), ci′j′(f
′)∗] =
〈
f1,e|µ(1)
ii′ f
′
1,e
〉
+
〈
f2,e|µ(2)
ii′ f
′
2,e
〉
+
〈
f1,o|µ(1)
ii′ f
′
1,o
〉
+
〈
f2,0|µ(2)
ii′ f
′
2,o
〉
= D(f |f ′)ij;i′j′ .
To check other commutation relations are easy.
Using the above fact, we readily confirm that
〈[
N∏
`=1
bi`j`(f`)
∗
]
Ψ0
∣∣∣∣∣
[
N ′∏
`=1
bi′`j
′
`
(f ′`)
∗
]
Ψ0
〉
=
〈[
N∏
`=1
ci`j`(f`)
∗
]
4⊗
λ=1
Ψλ
∣∣∣∣∣
[
N ′∏
`=1
ci′`j
′
`
(f ′`)
∗
]
4⊗
λ=1
Ψλ
〉
for every f1, . . .fN ,f
′
1, . . . ,f
′
N ′ ∈ L2
(
R3
)
⊕ L2
(
R3
)
and N,N ′ ∈ N. From (3.3), it follows that
the subspace spanned by the set of vectors
{[ N∏
`=1
bi`j`(f`)
∗
]
Ψ0
}
is dense in F
(
L2
(
R3
)
⊕L2
(
R3
))
and the subspace spanned by the set of vectors
{[ N∏
`=1
ci`j`(f`)
∗
] 4⊗
λ=1
Ψλ
}
is dense in
4⊗
λ=1
Hλ.
Hence, V can be extended to the unitary operator. To check (3.5) is easy. �
10 T. Miyao
Lemma 3.4. Let F be a real-valued even function on R3. Assume that F is continuous. Then
we obtain
V dΓ(F ⊕ F )V −1 =
4∑
λ=1
dΓλ(F ).
Proof. For readers’ convenience, we will provide a sketch of the proof. We will continue to use
the notations in the proof of Lemma 3.3. Set
Bij(f1, . . . ,fN ) =
[
N∏
`=1
bi`j`(f`)
∗
]
Ψ0,
Cij(f1, . . . ,fN ) =
[
N∏
`=1
ci`j`(f`)
∗
]
4⊗
λ=1
Ψλ
for f1, . . . ,fN ∈ L2
(
R3
)
⊕ L2
(
R3
)
, i = (i1, . . . , iN ), j = (j1, . . . , jN ) ∈ {1, 2, 3}N and N ∈ N.
We define dense subspaces of F
(
L2
(
R3
)
⊕ L2
(
R3
))
and
4⊗
λ=1
Hλ by
V1 = Lin
{
Bij(f1, . . . ,fN )
∣∣f1, . . . ,fN ∈ C0
(
R3
)
⊕C0
(
R3
)
, i, j∈{1, 2, 3}N , N ∈ N
}
,
V2 = Lin
{
Cij(f1, . . . ,fN )
∣∣f1, . . . ,fN ∈ C0
(
R3
)
⊕C0
(
R3
)
, i, j∈{1, 2, 3}N , N ∈ N
}
, (3.6)
where Lin(S) indicates the linear span of S. As is well-known, dΓ(F ⊕ F ) and
4∑
λ=1
dΓλ(F ) are
essentially self-adjoint on V1 and V2, respectively. We readily confirm that
dΓ(F ⊕ F )Bij(f1, . . . ,fN ) =
N∑
α=1
Bij(f1, . . . , F ⊕ Ffα, . . . ,fN ),
4∑
λ=1
dΓλ(F )Cij(f1, . . . ,fN ) =
N∑
α=1
Cij(f1, . . . , F ⊕ Ffα, . . . ,fN ).
Therefore, by Lemma 3.3, we obtain
V dΓ(F ⊕ F )V −1 =
4∑
λ=1
dΓλ(F ) on V2.
This concludes the proof of Lemma 3.4. �
3.5 Definition of the Feynman Hamiltonians
In this subsection, we introduce the Feynman Hamiltonians. To this end, let
A (x) =
∫
R3
dk
%̂(k)√
2|k|
{
ε(k, 1)
(
a(k, 1) cos(k · x) + a(k, 3) sin(k · x)
)
+ ε(k, 2)
(
a(k, 2) cos(k · x) + a(k, 4) sin(k · x)
)
+ h.c.
}
,
H0 =
4∑
λ=1
∫
R3
dk |k|a(k, λ)∗a(k, λ).
Note on the Retarded van der Waals Potential within the Dipole Approximation 11
Here, h.c. denotes the hermite conjugates of the preceeding terms. Note that A (x) is essentially
self-adjoint on V2 defined by (3.6). We denote its closure by the same symbol. By (3.4) and (3.5),
we have the following:
V A(x)V −1 = A (x), V HfV
−1 = H0.
Now we define the two-electron Feynman Hamiltonian HF2e by
HF2e =
1
2
(
−i∇1 − eA (0)
)2
+
1
2
e2ν2
0x
2
1 +
1
2
(
−i∇2 − eA (r)
)2
+
1
2
e2ν2
0x
2
2
+ e2
∫
R3
dk %̂(k)2eik·r(x1 · k̂
)(
x2 · k̂
)
+H0.
Remark thatHF2e acts in L2
(
R3
x1
)
⊗L2
(
R3
x2
)
⊗
( 4⊗
λ=1
F(Hλ)
)
and is bounded from below, provided
that R is sufficiently large.
The following proposition plays an important role in the present paper.
Proposition 3.5. If R is large enough, V HD2eV
−1 = HF2e.
Proof. Apply Lemmas 3.3 and 3.4. �
As for the one-electron Feynman Hamiltonian, we obtain the following.
Proposition 3.6. Let
HF1e =
1
2
(
−i∇− eA (0)
)2
+
1
2
e2ν2
0x
2 +H0.
We have V HD1eV
−1 = HF1e.
In Remark 7.2, we will explain why the Feynman Hamiltonians are useful.
4 Canonical transformations
Let U be a unitary operator on L2
(
R3
x1
)
⊗ L2
(
R3
x2
)
⊗
( 4⊗
λ=1
F(Hλ)
)
defined by
U = exp{iex1 ·A (0) + iex2 ·A (r)}.
Then one readily confirms that
U∗(−i∇1)U = −i∇1 + eA (0), U∗(−i∇2)U = −i∇2 + eA (r)
and
U∗a(k, λ)U =
a(k, λ) + ie
%̂(k)√
2|k|
ε(k, λ) · (x1 + x2 cos(k · r)) for λ = 1, 2,
a(k, λ) + ie
%̂(k)√
2|k|
ε(k, λ− 2) · x2 sin(k · r) for λ = 3, 4.
Here, we used the following fact:
eTa(k, λ)e−T = a(k, λ) +G(k, λ),
12 T. Miyao
where T =
4∑
λ=1
{aλ(G(·, λ))− a(G(·, λ))∗}∗∗, G(·, λ) ∈ Hλ. Hence, we arrive at4
U∗HF2eU = −1
2
∆1 +
1
2
e2ν2x2
1 −
1
2
∆2 +
1
2
e2ν2x2
2 + ex1 · E(0) + ex2 · E(r) +H0
+ e2
∫
R3
dk %̂(k)2 cos(k · r)(x1 · x2), (4.2)
where ν2 = 2ν2
0 and
E(x) = i
∫
R3
dk
√
|k|
2
%̂(k)
{
ε(k, 1)
(
cos(k · x)a(k, 1) + sin(k · x)a(k, 3)
)
+ ε(k, 2)
(
cos(k · x)a(k, 2) + sin(k · x)a(k, 4)
)
− h.c.
}
.
Let Nf be the number operator defined by Nf =
4∑
λ=1
dΓλ(1). Applying the “Fourier transfor-
mation” e−iπNf/2 in the Fock space,5 we obtain that
H = eiπNf/2U∗HF2eUe−iπNf/2
= −1
2
∆1 +
1
2
e2ν2x2
1 −
1
2
∆2 +
1
2
e2ν2x2
2 + ex1 · Ê(0) + ex2 · Ê(r) +H0
+ e2
∫
R3
dk %̂(k)2 cos(k · r)(x1 · x2), (4.3)
where
Ê(x) =
∫
R3
dk |k|%̂(k)
{
ε(k, 1)
(
cos(k · x)q(k, 1) + sin(k · x)q(k, 3)
)
+ ε(k, 2)
(
cos(k · x)q(k, 2) + sin(k · x)q(k, 4)
)}
(4.4)
and
q(k, λ) =
1√
2|k|
(
a(k, λ) + a(k, λ)∗
)
.
Since due to the assumption (A.3) the last term in (4.3) gives a rapidly decreasing contribution
as a function of R to the ground state energy, we ignore this term from now on.
Finally, we define
K = −1
2
∆ +
1
2
e2ν2x2 + ex · Ê(0) +H0.
By an argument similar to the construction of U , we can construct a unitary operator u on
L2
(
R3
)
⊗
( 4⊗
λ=1
F(Hλ)
)
such that K = eiπNf/2uHF1eu
−1e−iπNf/2.
4The reason why the last term in the right-hand side of (4.2) appears is as follows. After performing the unitary
transformation, we see that U∗HF2eU contains the term concerning
(
x1 · k̂
)(
x2 · k̂
)
and (ε(k, λ) · x1)(ε(k, λ) · x2),
which is given by
e2
∫
R3
dk %̂(k)2 cos(k · r)
{(
x1 · k̂
)(
x2 · k̂
)
+
∑
λ=1,2
(ε(k, λ) · x1)(ε(k, λ) · x2)
}
. (4.1)
Here, we used the fact that
∫
dk %̂(k)2 sin(k · r)
(
x1 · k̂
)(
x2 · k̂
)
= 0. By applying the basic property∑
λ=1,2
|ε(k, λ)〉〈ε(k, λ)| = 1l3 − |k̂〉〈k̂|, we conclude that (4.1) is equal to e2
∫
R3 dk %̂(k)2 cos(k · r)(x1 · x2).
5Let π(k, λ) = − i√
2
(a(k, λ) − a(k, λ)∗) and φ(k, λ) = 1√
2
(a(k, λ) + a(k, λ)∗). We can confirm that
[π(k, λ), φ(k′, λ′)] = −iδλλ′δ(k − k′). Recalling the fact [−id/dx, x] = −i, π(k, λ) and φ(k, λ) can be re-
garded as a multiplication operator and a differential operator, respectively. Now, we readily check that
eiπNf/2π(k, λ)e−iπNf/2 = φ(k, λ) holds, which corresponds to the relation FxF−1 = −id/dx, where F is the
Fourier transformation on L2(R). This similarity is a reason why we refer to the unitary operator eiπNf/2 as the
Fourier transformation.
Note on the Retarded van der Waals Potential within the Dipole Approximation 13
5 Lattice approximated Hamiltonians
In order to exactly compute the ground state energies of H and K, we will first introduce
the lattice approximation of Hamiltonians. As we will see in later sections, the approximated
Hamiltonians can be regarded as Hamiltonians of finite dimensional harmonic oscillator, which
are exactly solvable.
For each Λ > 0, let χΛ be an ultraviolet cutoff function given by χΛ(k) = 1 if |k| ≤ Λ,
χΛ(k) = 0 otherwise. We define a linear operator ÊΛ(x) by replacing %̂(k) with %̂(k)χΛ(k) in
the definition of Ê(x), i.e., the equation (4.4). We also define H0,Λ by
H0,Λ =
4∑
λ=1
∫
R3
dk |k|χΛ(k)a(k, λ)∗a(k, λ).
The Hamiltonians with a cutoff Λ are defined by
HΛ = −1
2
∆1 +
1
2
e2ν2x2
1 −
1
2
∆2 +
1
2
e2ν2x2
2 + ex1 · ÊΛ(0) + ex2 · ÊΛ(r) +H0,Λ,
KΛ = −1
2
∆ +
1
2
e2ν2x2 + ex · ÊΛ(0) +H0,Λ.
We readily see that HΛ and KΛ respectively converge to H and K in the norm resolvent sense
as Λ→∞.
Let M be the (momentum) lattice with a cutoff Λ, namely,
M =
{
l ∈ (2πZ/L)3
∣∣ |li| ≤ 2πΛ, i = 1, 2, 3
}
\{0}.
For later use, we label the elements of M as
M = {k1, . . . , kN}.
Then the lattice approximated Hamiltonians are defined by
HL,Λ = −1
2
∆1 +
1
2
e2ν2x2
1 −
1
2
∆2 +
1
2
e2ν2x2
2 + ex1 · ÊL,Λ(0) + ex2 · ÊL,Λ(r) +H0,L,Λ,
KL,Λ = −1
2
∆ +
1
2
e2ν2x2 + ex · ÊL,Λ(0) +H0,L,Λ,
where
ÊL,Λ(x) =
(
2π
L
)3/2 ∑
k∈M
|k|%̂(k)
{
ε(k, 1)
(
cos(k · x)q(k, 1) + sin(k · x)q(k, 3)
)
+ ε(k, 2)
(
cos(k · x)q(k, 2) + sin(k · x)q(k, 4)
)}
, (5.1)
H0,L,Λ =
1
2
4∑
λ=1
∑
k∈M
(
p(k, λ)2 + |k|2q(k, λ)2
)
− 2
∑
k∈M
|k|
with p(k, λ) = 1
i
√
|k|
2 (a(k, λ)− a(k, λ)∗). The lattice approximated operators act in the Hilbert
space L2
(
R3
x1
)
⊗L2
(
R3
x2
)
⊗
( 4⊗
λ=1
F(HL,Λ,λ)
)
or L2
(
R3
)
⊗
( 4⊗
λ=1
F(HL,Λ,λ)
)
, where HL,Λ,λ = `2∗(M)
∩ Hλ. Here, `2∗(M) is the `2(M) equipped with a modified norm
‖f‖∗ =
(
2π
L
)3/4
(∑
k∈M
|f(k)|2
)1/2
,
14 T. Miyao
and we regard `2∗(M) as a closed subspace of L2
(
R3
)
. Note that p(k, λ) and q(k, λ) are essentially
self-adjoint on the finite particle subspace of
4⊗
λ=1
F(HL,Λ,λ). In what follows, we denote their
closures by same symbols, respectively. q(k, λ) and p(k, λ) is a canonical pair of the photonic
displacement coordinate and its conjugate momentum satisfying the standard commutation
relations:
[p(k, λ), q(k′, λ′)] = −iδkk′δλλ′ ,
[p(k, λ), p(k′, λ′)] = 0 = [q(k, λ), q(k′, λ′)].
Recall the identification F(C) = L2(R). Using this, we can naturally embed
4⊗
λ=1
F(HL,Λ,λ) into
( 4⊗
λ=1
L2
(
R3
))⊗#M
. In addition, p(k, λ) and q(k, λ) can be regarded as the differential and
multiplication operators, respectively.
The following proposition is a basis for our computation.
Proposition 5.1. For each z ∈ C\R, one has
lim
Λ→∞
lim
L→∞
(HL,Λ − z)−1 = (H − z)−1,
lim
Λ→∞
lim
L→∞
(KL,Λ − z)−1 = (K − z)−1
in the operator norm topology.
Proof. See, e.g., [3, 9]. �
6 Diagonalization I: One-electron Hamiltonian
In this section, we diagonalize the one-electron Hamiltonian KL,Λ. To this end, let
Fx(k, 1) =
(
2π
L
)3/2
|k|%̂(k) cos(k · x), Fx(k, 2) =
(
2π
L
)3/2
|k|%̂(k) cos(k · x),
Fx(k, 3) =
(
2π
L
)3/2
|k|%̂(k) sin(k · x), Fx(k, 4) =
(
2π
L
)3/2
|k|%̂(k) sin(k · x).
We define a linear operator T(x) from `2(M × {1, . . . , 4}) to C3 by
T(x)f =
4∑
λ=1
∑
k∈M
|ε(k, λ)〉Fx(k, λ)f(k, λ)
for each f = {f(k, λ) | k ∈M,λ ∈ {1, . . . , 4}} ∈ `2(M ×{1, . . . , 4}). Here, we used the following
notation: ε(k, 3) := ε(k, 1) and ε(k, 4) := ε(k, 2). The adjoint of T(x) is denoted by T∗(x). Note
that
(T∗(x)a)(k, λ) = 〈ε(k, λ) | a〉3Fx(k, λ), a ∈ C3,
where 〈·|·〉3 stands for the inner product in C3.
Using the above notations, the interaction term x·ÊL,Λ(r) in KL,Λ is expressed as x·ÊL,Λ(r) =
〈T(r)q |x〉3 = 〈x |T(r)q〉3, where q = {q(k, λ) | k ∈ M,λ ∈ {1, . . . , 4}}. On the other hand, the
field energy can be represented by
H0,L,Λ =
1
2
(
p2 + 〈q|S0q〉
)
− 1
2
tr
[√
S0
]
,
Note on the Retarded van der Waals Potential within the Dipole Approximation 15
where p = {p(k, λ) | k ∈M,λ ∈ {1, . . . , 4}} and
S0 =
|k1|21l4 O
|k2|21l4
. . .
O |kN |21l4
.
Hence, KL,Λ can be rewritten as
KL,Λ = −1
2
∆ +
1
2
e2ν2x2 + e〈x |T(r)q〉3 +
1
2
(
p2 + 〈q |S0q〉
)
− 1
2
tr
[√
S0
]
. (6.1)
By setting φ = (x, q) and π = (−i∇,p), one sees that
KL,Λ =
1
2
(
〈π|π〉+ 〈φ |ωφ〉
)
− 1
2
tr
[√
ω0
]
+
3
2
eν,
where
ω = ω0 +Q, ω0 =
(
e2ν2 0
0 S0
)
, Q = e
(
0 T(r)
T∗(r) 0
)
.
The following lemma is a basic input.
Lemma 6.1. If 1 ≤
√
2eν0 and
√
2e‖%̂‖∗ < 1, then ω ≥ 0.
Proof. By (A.3), we have ‖T(r)f‖ ≤
√
2‖%̂‖∗
∥∥S1/2
0 f
∥∥ for all f ∈ `2(M × {1, . . . , 4}). Hence,
for all ϕ = (a,f) ∈ C3 ⊕ `2(M × {1, . . . , 4}), we have, by the Schwarz inequality,
|〈ϕ|Qϕ〉| ≤ 2
√
2e‖%̂‖∗‖a‖3
∥∥S1/2
0 f
∥∥
≤
√
2e‖%̂‖∗
(
‖a‖23 +
∥∥S1/2
0 f
∥∥2)
≤
√
2e‖%̂‖∗〈ϕ|ω0ϕ〉,
provided that 1 ≤ e2ν2. This concludes the proof of Lemma 6.1. �
Therefore, the ground state energy of KL,Λ is given by the following formula.
Proposition 6.2. Let EL,Λ = inf spec(KL,Λ). If 1 ≤
√
2eν0 and
√
2e‖%̂‖∗ < 1, then one has
EL,Λ =
1
2
tr
[√
ω −√ω0
]
+
3
2
eν.
Proof. We provide a sketch of the proof. First, we diagonalize ω as
ω = U−1 diag(λ1, . . . , λ4N+3)U,
where U is a unitary matrix and λ1, . . . , λ4N+3 are positive eigenvalues of ω. By setting φ̃ = Uφ
and π̃ = Uπ, we can express KL,Λ as
KL,Λ =
1
2
〈
π̃|π̃
〉
+
1
2
〈
φ̃|diag(λ1, . . . , λ4N+3)φ̃
〉
− 1
2
tr[
√
ω0] +
3
2
eν. (6.2)
Because π̃j and φ̃j satisfy the Weyl relation: eitπ̃ieisφ̃j = eistδijeisφ̃jeitπ̃i , the von Neumann’s
uniqueness theorem [23, Theorem VIII.14] tells us that there is a unitary operator τ : L2
(
R4N+3
)
→ L2
(
R4N+3
)
such that τ φ̃jτ
−1 = xj and τ π̃jτ
−1 = −i∂/∂xj . Therefore, the right-hand side
16 T. Miyao
of (6.2) can be regarded as a Hamiltonian for 4N + 3-dimensional harmonic oscillator. Since the
lowest eigenvalue of the Hamiltonian −1
2∆2
j +
λj
2 x
2
j is equal to
√
λj/2, we obtain that
EL,Λ =
1
2
4N+3∑
j=1
√
λj −
1
2
tr
[√
ω0
]
+
3
2
eν =
1
2
tr
[√
ω −√ω0
]
+
3
2
eν.
This finishes the proof of Proposition 6.2. �
Applying the elementary fact
1
π
∫ ∞
−∞
ds
a
s2 + a
=
√
a, (6.3)
we have that
EL,Λ =
1
2π
∫ ∞
−∞
ds tr
[
ω
(
s2 + ω
)−1 − ω0
(
s2 + ω0
)−1]
+
3
2
eν
=
1
2π
∞∑
j=1
(−1)j
∫ ∞
−∞
ds tr
[
ω0
(
s2 + ω0
)−1{
Q
(
s2 + ω0
)−1}j]
+
1
2π
∞∑
j=1
(−1)j+1
∫ ∞
−∞
ds tr
[{
Q
(
s2 + ω0
)−1}j]
+
3
2
eν
=
1
2π
∞∑
j=1
(−1)j+1
∫ ∞
−∞
ds s2 tr
[(
s2 + ω0
)−1{
Q
(
s2 + ω0
)−1}j]
+
3
2
eν. (6.4)
Since Q is off-diagonal, (6.4) becomes
EL,Λ = − 1
2π
∞∑
j=1
∫ ∞
−∞
ds s2 tr
[(
s2 + ω0
)−1
Q(s)2j
]
+
3
2
eν, (6.5)
where Q(s) =
(
s2 +ω0
)−1/2
Q
(
s2 +ω0
)−1/2
. In what follows, we will examine the convergence of
the right-hand side of (6.5). As we will see, this series absolutely converges and (6.5) is rigorously
justified if ν0 is large enough.
We begin with the following basic lemma.
Lemma 6.3. We have the following
∥∥T(x)
(
s2 + S0
)−1/2∥∥ ≤
√
2‖%̂‖∗,
∥∥(s2 + S0
)−1/2T∗(x)
∥∥ ≤
√
2‖%̂‖∗
for all x ∈ R3, where ‖f‖∗ =
√(
2π
L
)3 ∑
k∈M
|f(k)|2 for each f ∈ `2(M).
Proof. For each f ∈ `2(M × {1, . . . , 4}), we have, by (A.3),
∥∥T(x)
(
s2 + S0
)−1/2
f
∥∥2
= 〈f |T∗s(x)Ts(x)f〉
=
〈
f |
(
s2 + S0
)−1/2M(x, x)
(
s2 + S0
)−1/2
f
〉
≤
∣∣∣∣∣
∑
k,λ
Fx(k, λ)
(
s2 + k2
)1/2 f(k, λ)
∣∣∣∣∣
2
≤
∥∥(s2 + k2
)−1/2
Fx
∥∥2‖f‖2.
Because
∥∥(s2 + k2
)−1/2
Fx
∥∥2 ≤ 2‖%̂‖2∗, we conclude that
∥∥T(x)
(
s2 + S0
)−1/2∥∥ ≤
√
2‖%̂‖∗. �
Note on the Retarded van der Waals Potential within the Dipole Approximation 17
Lemma 6.4. Let
D(s) = 2e2s2
(
s2 + e2ν2
)−1
×
{(
s2 + e2ν2
)−1∥∥(s2 + |k|2
)−1/2|k|%̂
∥∥2
∗ +
∥∥(s2 + |k|2
)−1|k|%̂
∥∥2
∗
}
. (6.6)
Then we have the following:
(i) For all s ∈ R,
s2 tr
[(
s2 + ω0
)−1
Q(s)2n
]
≤
(√
2
ν
‖%̂‖∗
)2n−2
D(s).
(ii) Let a =
(√
2
ν ‖%̂‖∗
)2
. If a < 1, then we have
∞∑
n=1
s2 tr
[(
s2 + ω0
)−1
Q(s)2n
]
≤ 1
1− aD(s).
Remark that lim
L→∞
a ≤
(√
2
ν ‖%̂‖L2
)2 ≤ c2
∞ < 1/4 holds for all Λ > 0 by the assumption in
Theorem 2.1. Thus, the condition a < 1 is satisfied provided that L is sufficiently large.
(iii) D(s) ∈ L1(R) and
1
2π
∫
R
dsD(s) ≤ e
ν
‖%̂‖2∗. (6.7)
Proof. We set Ts(r) = T(r)
(
s2 + S0
)−1/2
. First, consider the case where n = 1. Because
Q(s)2 = e2
(
s2 + e2ν2
)−1
(
Ts(r)T∗s(r) 0
0 T∗s(r)Ts(r)
)
, (6.8)
we obtain that
s2 tr
[(
s2 + ω0
)−1
Q(s)2
]
= e2s2
(
s2 + e2ν2
)−2
tr
[
Ts(r)T∗s(r)
]
+ e2s2
(
s2 + e2ν2
)−1
tr
[(
s2 + S0
)−1T∗s(r)Ts(r)
]
.
By (A.3) and (A.4), we have
tr
[
Ts(r)T∗s(r)
]
≤ 2
(
2π
L
)3 ∑
k∈M
(
s2 + |k|2
)−1|k|2|%̂(k)|2,
tr
[(
s2 + S0
)−1T∗s(r)Ts(r)
]
≤ 2
(
2π
L
)3 ∑
k∈M
(
s2 + |k|2
)−2|k|2|%̂(k)|2.
Thus, we get (i) for n = 1.
To prove the assertion for n ≥ 2, we remark that ‖Q(s)‖ ≤
√
2
ν ‖%̂‖∗, which immediately
follows from Lemma 6.3 and (6.8). Thus, by using the fact Q(s)2n ≤ ‖Q(s)‖2n−2Q(s)2, we have
tr
[(
s2 + ω0
)−1/2
Q(s)2n
(
s2 + ω0
)−1/2]≤ ‖Q(s)‖2n−2tr
[(
s2 + ω0
)−1/2
Q(s)2
(
s2 + ω0
)−1/2]
.
Applying the result for n = 1, we get the desired result for n ≥ 2. (ii) immediately follows
from (i).
18 T. Miyao
By using the formula (C.1) with a = b = e2ν2 and c = |k|2, we see that
1
2π
∫ ∞
−∞
ds 2e2s2
(
s2 + e2ν2
)−2∥∥(s2 + |k|2
)−1/2|k|%̂
∥∥2
∗
=
e2
2π
(
2π
L
)3 ∑
k∈M
2π
2eν
|k|2|%̂(k)|2
(eν + |k|)2
≤ e
2ν
‖%̂‖2∗.
Similarly, by using the formula (C.1) with a = e2ν2 and b = c = |k|2, we obtain
1
2π
∫ ∞
−∞
ds 2e2s2
(
s2 + e2ν2
)−1∥∥(s2 + |k|2
)−1|k|%̂
∥∥2
∗
=
e2
2π
(
2π
L
)3 ∑
k∈M
2π
2|k|(|k|+ eν)2
|k|2|%̂(k)|2 ≤ e
2ν
‖%̂‖2∗.
Inserting these into (6.6), we obtain the assertion (iii). �
Corollary 6.5. The right-hand side of (6.5) absolutely converges, provided that
√
2‖%̂‖∗ < ν,
1 ≤
√
2eν0 and
√
2e‖%̂‖∗ < 1. In addition, to exchange the series with the integral in (6.5)
(or (6.4)) can be justified.
7 Diagonalization II: Two-electron Hamiltonian
Next we will diagonalize HL,Λ. This is actually possible because we employ the Feynman Hamil-
tonian, see Remark 7.2 for details. By an argument similar to that of the proof of (6.1), HL,Λ
can be expressed as
HL,Λ = −1
2
∆1 +
1
2
e2ν2x2
1 −
1
2
∆2 +
1
2
e2ν2x2
2 + e〈T(0)q|x1〉3 + e〈T(r)q|x2〉3
+
1
2
(
p2 + 〈q|S0q〉
)
− 1
2
tr
[√
S0
]
.
By setting Φ = (x1, x2, q) and Π = (−i∇1,−i∇2,p), we have that
HL,Λ =
1
2
(
〈Π|Π〉+ 〈Φ|ΩΦ〉
)
− 1
2
tr
[√
Ω0
]
+ 3eν,
where
Ω = Ω0 +Q1 +Q2,
Ω0 =
e2ν2 0 0
0 e2ν2 0
0 0 S0
, Q1 = e
0 0 T(0)
0 0 0
T∗(0) 0 0
, Q2 = e
0 0 0
0 0 T(r)
0 T∗(r) 0
.
By an argument similar to that in the proof of Proposition 6.2, we get the following useful
formula.
Proposition 7.1. Let EL,Λ(R) = inf spec(HL,Λ). If 1 ≤
√
2eν0 and
√
2e‖%̂‖∗ < 1, then Ω ≥ 0
and
EL,Λ(R) =
1
2
tr
[√
Ω−
√
Ω0
]
+ 3eν.
Note on the Retarded van der Waals Potential within the Dipole Approximation 19
Remark 7.2 (Why are the Feynman Hamiltonians helpful?). From the expression (5.1), we see
that ÊL,Λ(x) can be written as a sum of multiplication operators q(k, λ). As we already knew,
this fact is a key to the diagonalization of HL,Λ. In contrast to the Feynman Hamiltonians, in
the standard representation, ÊL,Λ(x) corresponds to the following operator:
(
2π
L
)3/2 ∑
λ=1,2
∑
k∈M
%̂(k)ε(k, λ)
{
cos(k · x)|k|q(k, λ) + sin(k · x)|k|−1p(k, λ)
}
. (7.1)
In (7.1), both multiplication and differential operators appear, provided that x 6= 0. At first
glance, it appears that diagonalizing the Hamiltonians in this representation requires extra
efforts.
Moreover, it can be readily seen that, by (6.3),
EL,Λ(R) =
1
2π
∞∑
j=1
(−1)j+1
∫ ∞
−∞
ds s2 tr
[(
s2 + Ω0
)−1{
(Q1 +Q2)
(
s2 + Ω0
)−1}j]
+ 3eν. (7.2)
To examine this formal series, let us introduce the following notation:
〈O1O2 · · ·On〉 =
1
2π
∫ ∞
−∞
ds s2 tr
[(
s2 + Ω0
)−1
O1(s)O2(s) · · ·On(s)
]
,
where Oi(s) =
(
s2 + Ω0
)−1/2
Oi
(
s2 + Ω0
)−1/2
. Then (7.2) can be expressed as
EL,Λ(R) =
∞∑
j=1
(−1)j+1〈(Q1 +Q2) · · · (Q1 +Q2)︸ ︷︷ ︸
j
〉+ 3eν
=
∞∑
j=1
(−1)j+1〈Q1 · · ·Q1︸ ︷︷ ︸
j
〉+
∞∑
j=1
(−1)j+1〈Q2 · · ·Q2︸ ︷︷ ︸
j
〉
+
∞∑
j=1
∑
i1,...,ij∈{1,2}
{i1,...,ij}6={1,1,...,1},{2,2,...,2}
(−1)j+1〈Qi1 · · ·Qij 〉+ 3eν. (7.3)
Since Q1 and Q2 are off-diagonal, we have
EL,Λ(R) = −
∞∑
j=1
〈Q1 · · ·Q1︸ ︷︷ ︸
2j
〉 −
∞∑
j=1
〈Q2 · · ·Q2︸ ︷︷ ︸
2j
〉
−
∞∑
j=1
∑
i1,...,i2j∈{1,2}
{i1,...,i2j}6={1,1,...,1},{2,2,...,2}
〈Qi1 · · ·Qi2j 〉+ 3eν.
On the other hand, we remark that, by Corollary 6.5,
EL,Λ = −
∞∑
j=1
〈Q1 · · ·Q1︸ ︷︷ ︸
2j
〉+
3
2
eν = −
∞∑
j=1
〈Q2 · · ·Q2︸ ︷︷ ︸
2j
〉+
3
2
eν,
provided that
√
2‖%̂‖∗ < ν, 1 ≤
√
2eν0 and
√
2e‖%̂‖∗ < 1. Thus, we formally arrive at the
following formula:
EL,Λ(R)− 2EL,Λ = −
∞∑
j=1
∑
i1,...,i2j∈{1,2}
{i1,...,i2j}6={1,1,...,1},{2,2,...,2}
〈Qi1 · · ·Qi2j 〉. (7.4)
20 T. Miyao
Our next task is to prove the convergence of the right-hand side of (7.4). For this purpose, we
need some preliminaries. Let
I2j =
{
I = {i1, . . . , i2j}, i1, . . . , i2j ∈ {1, 2}
∣∣ I 6= {1, 1, . . . , 1}, {2, 2, . . . , 2}
}
.
For each I ∈ I2j , we set |I| = i1 + i2 + · · ·+ i2j . Furthermore, we use the following notation:
QI = Qi1Qi2 · · ·Qi2j , I = {i1, . . . , i2j} ∈ I2j .
Lemma 7.3. Let I ∈ I2j. If |I| is an odd number, then 〈QI〉 = 0.
Proof. Note that
(
s2 +Ω0
)−1
, Q1(s)2 and Q2(s)2 are diagonal operators, while Q1(s)Q2(s) and
Q2(s)Q1(s) are off-diagonal operators, see Appendix A. Hence, if |I| is an odd number, then(
s2 + Ω0
)−1
Qi1(s) · · ·Qi2j (s) is an off-diagonal operator. Accordingly,
tr
[(
s2 + Ω0
)−1
Qi1(s) · · ·Qi2j (s)
]
= 0.
This concludes the proof of Lemma 7.3. �
Let I(e)
2j = {I ∈ I2j | |I| is even}. By Lemma 7.3, we have
the r.h.s. of (7.4) = −
∞∑
j=1
∑
I∈I(e)2j
〈QI〉. (7.5)
Lemma 7.4. For each s ∈ R and I = {i1, . . . , i2j} ∈ I(e)
2j , we set
QI(s) = Qi1(s) · · ·Qi2j (s)
and
EI(s) = s2 tr
[(
s2 + ω
)−1
Q∗I\{i1}(s)QI\{i1}(s)
]
,
where Q∗I\{i1}(s) =
(
QI\{i1}(s)
)∗
. For all R > 0, we have the following:
(i) For each s ∈ R and I ∈ I(e)
2j ,
s2
∣∣ tr
[(
s2 + ω
)−1
QI(s)
]∣∣ ≤ D(s)1/2EI(s)
1/2, (7.6)
where D(s) is given by (6.6).
(ii) Recall that a is defined by a =
(√
2
ν ‖%̂‖∗
)2
. If a < 1/4, then
∞∑
j=1
∑
I∈I(e)2j
EI(s)
1/2 ≤ D(s)1/2 4
1− 4a
. (7.7)
Thus, D(s)1/2
∞∑
j=1
∑
I∈I(e)2j
EI(s)
1/2 ∈ L1(R) and
∞∑
j=1
∑
I∈I(e)2j
|〈QI〉| ≤
e
ν
‖%̂‖2∗
4
1− 4a
.
Note that as we mentioned in Lemma 6.4, the condition a < 1/4 is satisfied provided that L
is large enough.
Note on the Retarded van der Waals Potential within the Dipole Approximation 21
Proof. For notational simplicity, we set G =
(
s2 +ω0
)−1
. By the Schwarz inequality | tr[A∗B]|
≤ tr[A∗A]1/2 tr[B∗B]1/2, we obtain
∣∣ tr
[
G1/2Qi1(s) · · ·Qi2j (s)G1/2
]∣∣ ≤ tr
[
G1/2Qi1(s)Qi1(s)G1/2
]1/2
× tr
[
G1/2Qi2j (s)Qi2j−1(s) · · ·Qi2(s)Qi2(s) · · ·Qi2j (s)G1/2
]1/2
,
which implies that
s2
∣∣ tr
[(
s2 + ω
)−1
QI(s)
]∣∣ ≤
{
s2 tr
[(
s2 + ω
)−1
Qi1(s)Qi1(s)
]}1/2
EI(s)
1/2.
Because
s2 tr
[(
s2 + ω
)−1
Qi(s)Qi(s)
]
≤ D(s), (7.8)
we conclude (i).
From Lemma 6.3 and (6.8), we obtain that
‖Qi(s)‖ ≤
√
2
ν
‖%̂‖∗.
Hence, EI(s) ≤ a2j−2s2 tr
[(
s2 + ω
)−1
Qi2j (s)Qi2j (s)
]
≤ a2j−2D(s) by (7.8). Therefore, we
obtain (7.7).
One observes that
∞∑
j=1
∑
I∈I(e)2j
s2
∣∣ tr
[(
s2 + ω
)−1
QI(s)
]∣∣ ≤
(7.6)
∞∑
j=1
∑
I∈I(e)2j
D(s)1/2EI(s)
1/2 ≤
∞∑
j=1
22jaj−1D(s)
≤
(7.7)
4
1− 4a
D(s).
In the second inequality, we have used the fact that #I(e)
2j ≤ 22j Accordingly, we get
∞∑
j=1
∑
I∈I(e)2j
|〈QI〉| ≤
4
1− 4a
1
2π
∫
R
dsD(s) ≤ 4
1− 4a
e
ν
‖%̂‖2∗
by (6.7). �
Corollary 7.5. If
√
2‖%̂‖∗ < ν, 1 ≤
√
2eν0 and
√
2e‖%̂‖∗ < 1, then the r.h.s. of (7.4) converges
absolutely for every R > 0. In addition, to exchange the series with the integral, i.e., 〈· · ·〉
in (7.4) (or (7.3)) can be justified.
8 Proof of Theorem 2.1
For each I ∈ I(e)
2j , #I indicates the cardinality of I. Notice that #I is different from |I| =
i1 + · · ·+ i2j .
8.1 Analysis of 〈QI〉 with #I = 2
We claim that
〈Q1Q2〉 = 〈Q2Q1〉 = 0. (8.1)
To see this, let I = {1, 2} or {2, 1}. Trivially, |I| = 1 + 2 = 3. By Lemma 7.3, we conclude (8.1).
22 T. Miyao
8.2 Analysis of 〈QI〉 with #I = 4
In this subsection, we will examine the following terms:
∑
i1,...,i4∈{1,2}
{i1,...,i4}6={1,1,1,1},{2,2,2,2}
〈Qi1Qi2Qi3Qi4〉 = A + B,
where
A = 〈Q1Q1Q2Q2〉+ 〈Q2Q2Q1Q1〉 (8.2)
and
B = 〈Q1Q2Q1Q2〉+ 〈Q2Q1Q2Q1〉+ 〈Q2Q1Q1Q2〉+ 〈Q1Q2Q2Q1〉. (8.3)
In Appendix B, we will prove the following lemmas.
Lemma 8.1. We have
lim
R→∞
lim
Λ→∞
lim
L→∞
R7A =
23
4π
(
1
2π
)2(1
4
αE,at
)2
.
Lemma 8.2. We have
lim
R→∞
lim
Λ→∞
lim
L→∞
R9〈Q1Q2Q2Q1〉 = lim
R→∞
lim
Λ→∞
lim
L→∞
R9〈Q2Q1Q1Q2〉 =
g
e2ν6
,
where g is a constant independent of e, ν0 and R. Moreover, 〈Q1Q2Q1Q2〉 = 〈Q2Q1Q2Q1〉 = 0.
Thus, lim
R→∞
lim
Λ→∞
lim
L→∞
R9B = 2g/e2ν6.
8.3 Analysis of 〈QI〉 with #I ≥ 6
Let I = {i1, . . . , i2j} ∈ I(e)
2j . We will examine the following two cases, separately.
Case 1: There exists a unique number ` ∈ {1, 2, . . . , 2j − 1} such that i` + i`+1 = 3.
Case 2: There exist at least two numbers m,n ∈ {1, 2, . . . , 2j − 1} such that im + im+1 =
in + in+1 = 3.
Example 8.3. For readers’ convenience, we provide some examples below:
Case 1: I =
{
1, 1,
i3+i4=3︷︸︸︷
1, 2 , 2, 2, 2, 2
}
,
{
1, 1, 1, 1,
i5+i6=3︷︸︸︷
1, 2 , 2, 2
}
.
Case 2: I =
{
1, 1,
i3+i4=3︷︸︸︷
1, 2 , 2, 2, 2,
i8+i9=3︷︸︸︷
2, 1 , 1
}
,
{
1, 1, 1, 1, 1,
i6+i7=3︷︸︸︷
1, 2 ,
i8+i9=3︷︸︸︷
2, 1 , 1
}
.
8.3.1 Case 1
In Appendix B, we will prove the following lemma.
Lemma 8.4. Assume that I satisfies the condition in Case 1. If R is sufficiently large, then we
have
lim
Λ→∞
lim
L→∞
|〈QI〉| ≤ R−9α2
E,at
(‖%̂‖2L2
3ν2
)#I/2−2
C,
where C is a positive number independent of e, I, R and ν0.
Note on the Retarded van der Waals Potential within the Dipole Approximation 23
8.3.2 Case 2
The purpose here is to prove Lemma 8.6 below. To this end, we begin with the following lemma.
Lemma 8.5. Let G =
(
s2 + Ω0
)−1
. For each j ∈ {1, 2}, we have
‖QjG‖ ≤ D(%̂),
where D(%̂) = max
{√
2e
∥∥|k|−1%̂
∥∥
∗,
√
2
eν2
‖|k|%̂‖∗
}
.
Proof. By (A.3) and (A.4), we readily show that
∥∥T(r)
(
s2 + S0
)−1∥∥ ≤
√
2
∥∥|k|−1%̂
∥∥
∗,
∥∥(s2 + e2ν2
)−1T∗(r)
∥∥ ≤
√
2
e2ν2
‖|k|%̂‖∗.
This concludes the proof of Lemma 8.5. �
Lemma 8.6. Let j ≥ 3. For each I ∈ I(e)
2j satisfying the condition in Case 2, we have
|〈QI〉| ≤ c2j−4
L 〈Q2Q1Q1Q2〉,
where cL = max
{
D(%̂),
√
2
ν ‖%̂‖∗
}
.
Proof. By the assumption in the condition Case 2, there exist at least two numbers m,n ∈
{1, 2, . . . , 2j − 1} such that im + im+1 = in + in+1 = 3. Hence, I can be decomposed as
I = A ∪ {im, im+1} ∪ B ∪ {in, in+1} ∪ C. Without loss of generality, we may assume that
{im, im+1} = {in, in+1} = {1, 2}. Thus,
〈QI〉 = 〈QAQ1Q2QBQ1Q2QC〉.
Let QI(s) = Qi1(s)Qi2(s) · · ·Qi2j (s). By the Schwarz inequality, we have
∣∣ tr
[
G1/2QI(s)G
1/2
]∣∣ ≤ Φ
1/2
1 Φ
1/2
2 , (8.4)
where
Φ1 = tr
[
G1/2QA(s)Q1(s)Q2(s)Q2(s)Q1(s)Q∗A(s)G1/2
]
,
Φ2 = tr
[
G1/2Q∗C(s)Q2(s)Q1(s)Q∗B(s)QB(s)Q1(s)Q2(s)QC(s)G1/2
]
.
First, we estimate Φ1. By the cyclic property of the trace, we have
Φ1 = tr
[
Q2(s)Q1(s)Q∗A(s)G1/2G1/2QA(s)Q1(s)Q2(s)
]
= tr
[
Q2(s)Q1(s)G1/2G−1/2Q∗A(s)G1/2G1/2QA(s)G−1/2G1/2Q1(s)Q2(s)
]
. (8.5)
Because
G1/2QA(s)G−1/2 = (GQa1)(GQa2) · · · (GQa#A),
where A = {a1, a2, . . . , a#A}, we have, by Lemma 8.5,
∥∥G1/2QA(s)G−1/2
∥∥ ≤ D(%̂)#A.
Thus, by (8.5) and the cyclic property of the trace,
Φ1 ≤ D(%̂)2#A tr
[
G1/2Q1(s)Q2(s)Q2(s)Q1(s)G1/2
]
. (8.6)
24 T. Miyao
As for Φ2, we have
Φ2 ≤ ‖QB(s)‖2 tr
[
G1/2Q∗C(s)Q2(s)Q1(s)Q1(s)Q2(s)QC(s)G1/2
]
.
By an argument similar to the one in the proof of (8.6), one obtains that
tr
[
G1/2Q∗C(s)Q2(s)Q1(s)Q1(s)Q2(s)QC(s)G1/2
]
≤ D(%̂)2#C tr
[
G1/2Q2(s)Q1(s)Q1(s)Q2(s)G1/2
]
. (8.7)
By using the fact ‖QB(s)‖ ≤
(√
2
ν ‖%̂‖∗
)2#B
and (8.7), we have
Φ2 ≤
(√
2
ν
‖%̂‖∗
)2#B
D(%̂)2#C tr
[
G1/2Q2(s)Q1(s)Q1(s)Q2(s)G1/2
]
. (8.8)
Combining (8.4), (8.6) and (8.8), we arrive at
|〈QI〉| ≤
(√
2
ν
‖%̂‖∗
)#B
D(%̂)#A+#C〈Q1Q2Q2Q1〉1/2〈Q2Q1Q1Q2〉1/2
≤ c2j−4
L 〈Q1Q2Q2Q1〉1/2〈Q2Q1Q1Q2〉1/2.
Because 〈Q1Q2Q2Q1〉 = 〈Q2Q1Q1Q2〉, we obtain the desired result. �
8.4 Completion of the proof of Theorem 2.1
First, remark that lim
Λ→∞
lim
L→∞
EL,Λ = E and lim
Λ→∞
lim
L→∞
EL,Λ(R) = E(R) by Proposition 5.1. We
divide I(e)
2j as I(e)
2j = I(e)
2j,1 ∪ I
(e)
2j,2, where
I(e)
2j,α =
{
I ∈ I(e)
2j | I satisfies the condition in Case α
}
, α = 1, 2.
Note that #I(e)
2j,1 = 2j − 1 and #I(e)
2j,2 ≤ 22j .
By (7.5) and (8.1), one obtains that
2EL,Λ − EL,Λ(R) = A + B +
∑
j≥3
∑
I∈I(e)2j,1
〈QI〉+
∑
j≥3
∑
I∈I(e)2j,2
〈QI〉,
where A and B are defined by (8.2) and (8.3), respectively. Therefore,
∣∣R7
{
2EL,Λ − EL,Λ(R)−A
}∣∣ ≤ R7B +
∑
j≥3
∑
I∈I(e)2j,1
R7|〈QI〉|+
∑
j≥3
∑
I∈I(e)2j,2
R7|〈QI〉|. (8.9)
We will estimate the three terms in the right-hand side of (8.9). By Lemma 8.2, we can easily
control the first term. As for the second term, by Lemma 8.4, we have
lim
Λ→∞
lim
L→∞
∑
j≥3
∑
I∈I(e)2j,1
R7|〈QI〉| ≤
Lemma 8.4
R−2
∑
j≥3
(
#I(e)
2j,1
)
Cα2
E,at
(‖%̂‖2L2
3ν2
)j−2
= R−2Cα2
E,at
∑
j≥3
(2j − 1)
(‖%̂‖2L2
3ν2
)j−2
. (8.10)
Note on the Retarded van der Waals Potential within the Dipole Approximation 25
Note that because ‖%̂‖2L2/3ν
2 < 1, the right-hand side of (8.10) converges. On the other hand,
using Lemma 8.6, one obtains that
∑
j≥3
∑
I∈I(e)2j,2
R7|〈QI〉| ≤
Lemma 8.6
∑
j≥3
(
#I(e)
2j,2
)
c2j−4
L R7〈Q2Q1Q1Q2〉
≤
∑
j≥3
22jc2j−4
L R7〈Q2Q1Q1Q2〉. (8.11)
Note that because lim
L→∞
cL = c∞ < 1/2, the right-hand side of (8.11) converges, provided that L
is sufficiently large.
Combining (8.9), (8.10) and (8.11), and using Lemma 8.2, we finally arrive at
lim
R→∞
∣∣∣∣∣R
7
{
2E − E(R)− 23
4π
(
1
2π
)2(1
4
αE,at
)2
}∣∣∣∣∣
≤
2 + c−4
∞
∑
j≥3
(2c∞)2j
lim
R→∞
lim
Λ→∞
lim
L→∞
R7〈Q1Q2Q2Q1〉
+ lim
R→∞
R−2Cα2
E,at
∑
j≥3
(2j − 1)
(‖%̂‖2L2
3ν2
)2j−2
= 0.
This concludes the proof of Theorem 2.1.
9 Discussions
9.1 Indistinguishability of the electrons
The original Hamiltonian H2e has the indistinguishability of the electrons, i.e., the Hamiltonian
is unchanged under the exchange of x1 ↔ x2 + r. In contrast to this, the approximated Hamil-
tonian HD2e breaks the indistinguishability. Nevertheless, the Hamiltonian HD2e does explain
the Casimi–Polder potential as we show in Theorem 2.1. The distinguishability comes from the
assumptions (C.1) and (C.2). However, to justify the assumptions is still open.
One way to avoid the unjustified derivation of HD2e is to directly start with the Hamilto-
nian H given by (4.3) without the last term, which can for instance be directly taken from [24,
equation (13.127)] and then extended to the two-particle case. Alternatively and equivalently,
the many-particle case is presented, e.g., in [17, Section 4]. If we start from this form, the
necessary assumptions are stated as follows:
• We assume distinguishability of the two electrons by localizing electron 1 at 0, such that
electron 1 experiences the field Ê(0), while electron 2 is localized at r and hence experiences
the field Ê(r).
• We discard all self-interaction terms and approximate the atomic Coulomb potential by
a harmonic potential.
In this manner, we can construct a minimal QED model which describes the Casimir–Polder
potential. Note that, since the particle 1 and 2 only communicate via the photon field, and due
to distinguishability, the actual choice of coordinate systems is insubstantial such that we can
choose for particle 2 a coordinate system that is centered at r.
26 T. Miyao
9.2 Cancellation mechanism of the van der Waals–London force
As we performed in [20], the attractive R−7 decay (the retarded van der Waals potential) appears
due to the exact cancellation of the terms with R−6 decay (the van der Waals–London potential)
originating from VR by the contribution from the quantized radiation field. Note that the
conditions (A.1)–(A.3) are assumed in [20] as well, but (C.1) and (C.2) are not. As we saw
in the present paper, this kind of the cancellation mechanism cannot be reproduced under
the conditions (C.1), (C.2) and (A.1)–(A.4). In this sense, our assumptions, especially (C.1)
and (C.2) would be unphysical.
In many literatures, the retardation on the van der Waals potential is examined under the
condition (C.1) alone. In these studies, the cancellation of the terms with R−6 decay is pre-
supposed and only the 4-th order perturbation theory is performed without estimating higher
order terms.6 As far as we know, to examine the exact cancellation mechanism under only
the condition (C.1) is still unsolved. This problem could be a key to achieving mathematically
complete understanding of the retarded van der Waals potential.
A Useful formulas
In this appendix, we give a list of useful formulas. Let Ts(x) = T(x)
(
s2 + S0
)−1/2
. First, we
give some formulas for Qi:
Q1(s) = e
(
s2 + e2ν2
)−1/2
0 0 Ts(0)
0 0 0
T∗s(0) 0 0
,
Q2(s) = e
(
s2 + e2ν2
)−1/2
0 0 0
0 0 Ts(r)
0 T∗s(r) 0
,
Q1(s)2 = e2
(
s2 + e2ν2
)−1
Ts(0)T∗s(0) 0 0
0 0 0
0 0 T∗s(0)Ts(0)
, (A.1)
Q2(s)2 = e2
(
s2 + e2ν2
)−1
0 0 0
0 Ts(r)T∗s(r) 0
0 0 T∗s(r)Ts(r)
, (A.2)
Q1(s)Q2(s) = e2
(
s2 + e2ν2
)−1
0 Ts(0)T∗s(r) 0
0 0 0
0 0 0
,
Q2(s)Q1(s) = e2
(
s2 + e2ν2
)−1
0 0 0
Ts(r)T∗s(0) 0 0
0 0 0
.
Let M(r, r′) be a linear operator on `2(M × {1, . . . , 4}) defined by
(M(r, r′)f)(k, λ) =
4∑
λ′=1
∑
k′∈M
Mk,λ;k′,λ′(r, r
′)f(k′, λ′), f ∈ `2(M × {1, . . . , 4}),
Mk,λ;k′,λ′(r, r
′) = 〈ε(k, λ)|ε(k′, λ′)〉3Fr(k, λ)Fr′(k
′, λ′).
The following formulas are readily checked:
T∗s(r)Ts(r′) =
(
s2 + S0
)−1/2M(r, r′)
(
s2 + S0
)−1/2
, (A.3)
6A kind of weak cancellation mechanics is discussed in [12] by imposing the infrared cutoff.
Note on the Retarded van der Waals Potential within the Dipole Approximation 27
Ts(r)T∗s(r′) =
4∑
λ=1
∑
k∈M
|ε(k, λ)〉〈ε(k, λ)|
(
s2 + k2
)−1
Fr(k, λ)Fr′(k, λ). (A.4)
Note that T∗s(r)Ts(r′) is a map from `2(M ×{1, . . . , 4}) to `2(M ×{1, . . . , 4}), while Ts(r)T∗s(r′)
is a map from C3 to C3.
B Numerical computations
B.1 Proof of Lemma 8.1
We will extend the methods in [20, 21]. By (A.1) and (A.2), we have
Q1(s)2Q2(s)2 = e4
(
s2 + e2ν2
)−2
0 0 0
0 0 0
0 0 T∗s(0)Ts(0)T∗s(r)Ts(r)
,
which implies that
〈Q1Q1Q2Q2〉 =
e4
2π
∫
R
ds
s2
(
s2 + e2ν2
)2 tr
[
T∗s(0)Ts(0)T∗s(r)Ts(r)
]
.
By (A.3) and the fact
∑
λ1,λ2=1,2
(〈ε(k1, λ1)|ε(k2, λ2)〉3)2 = 1 +
(
k̂1 · k̂2
)2
with k̂ = k/|k|, we have
tr
[
T∗s(0)Ts(0)T∗s(r)Ts(r)
]
= tr
[(
s2 + Ω0
)−2M(0, 0)
(
s2 + Ω0
)−1M(r, r)
]
=
∑
k1,λ1
∑
k2,λ2
(
s2 + k2
1
)−2(
s2 + k2
2
)−1
(〈ε(k1, λ1)|ε(k2, λ2)〉3)2
× k2
1k
2
2%̂(k1)2%̂(k2)2 cos(k1 · r) cos(k2 · r) (B.1)
=
∑
k1
∑
k2
(
s2 + k2
1
)−2(
s2 + k2
2
)−1(
1 +
(
k̂1 · k̂2
)2)
k2
1k
2
2%̂(k1)2%̂(k2)2 cos(k1 · r) cos(k2 · r).
By using the assumption (A.3), we have
the r.h.s. of (B.1)
=
∑
k1
∑
k2
(
s2 + k2
1
)−2(
s2 + k2
2
)−1(
1 +
(
k̂1 · k̂2
)2)
k2
1k
2
2%̂(k1)2%̂(k2)2 cos((k1 + k2) · r).
Hence, we arrive at
〈Q1Q1Q2Q2〉 =
e4
2
(
2π
L
)6 ∑
k1,k2∈M
|k1|2|k2|2
(
1 +
(
k̂1 · k̂2
)2)
%̂2(k1)%̂2(k2) cos{(k1 + k2) · r}
× I2,2,1
(
e2ν2; |k1|2; |k2|2
)
where
Ina,nb,nc(a; b; c) =
1
π
∫ ∞
−∞
ds
s2
(
s2 + a
)na(s2 + b
)nb(s2 + c
)nc . (B.2)
Thus, we obtain that
lim
Λ→∞
lim
L→∞
〈Q1Q1Q2Q2〉 =
e4
2
∫
R3×R3
dk1dk2|k1|2|k2|2
(
1 +
(
k̂1 · k̂2
)2)
%̂2(k1)%̂2(k2)
28 T. Miyao
× cos{(k1 + k2) · r}I2,2,1
(
e2ν2; |k1|2; |k2|2
)
. (B.3)
By scalings Rk1 ; k1 and Rk2 ; k2, we have
the r.h.s. of (B.3) = R−10 e
4
2
∫
dk1dk2 |k1|2|k2|2
(
1 +
(
k̂1 · k̂2
)2)
%̂2(k1/R)%̂2(k2/R)ei(k1+k2)·n̂
× I2,2,1
(
e2ν2; |k1|2/R2; |k2|2/R2
)
, (B.4)
where n̂ = r/R = (0, 0, 1). Let us switch to spherical coordinates (r, ϕ, θ) by
k̂ = (Y cosϕ, Y sinϕ,X), X = cos θ, Y = sin θ.
Clearly X2 + Y 2 = 1. Then we have
k̂1 · k̂2 = cos(ϕ1 − ϕ2)Y1Y2 +X1X2
and hence, by taking the symmetry between r1 and r2 variables into consideration, we obtain
the r.h.s. of (B.4) = R−10 e
4
2
∫ ∞
0
dr1
∫ ∞
0
dr2
∫ 1
−1
dX1
∫ 1
−1
dX2 S(X1, X2)
× r4
1r
4
2eir1X1eir2X2I
(
e2ν2; r2
1/R
2; r2
2/R
2
)
%̂2
rad(r1/R)%̂2
rad(r2/R),
where
S(X1, X2) =
∫ 2π
0
dϕ1
∫ 2π
0
dϕ2
{
1 +
(
cos(ϕ1 − ϕ2)Y1Y2 +X1X2
)2}
= 6π2 − 2π2
(
X2
1 +X2
2
)
+ 6π2X2
1X
2
2 . (B.5)
and
I(a; b; c) =
1
2
{
I2,2,1(a; b; c) + I2,1,2(a; b; c)
}
.
By (C.2) and (C.3), we decompose I(a; b; c) as
I(a; b; c) = Ire(a; b; c) + Iir(a; b; c),
where
Ire(a; b; c) =
I1,1,1(a; b; c)
8
√
abc
(
1
A
+
1
C
)
,
Iir(a; b; c) =
I1,1,1(a; b; c)
8
√
ab
1
A
(
2
A
+
1
C
)
+
I1,1,1(a; b; c)
8
√
ac
1
C
(
2
C
+
1
A
)
with A =
√
a+
√
b, B =
√
b+
√
c and C =
√
c+
√
a. First, we compute the contribution from
the term Ire. By the formula
Ire
(
e2ν2; r2
1/R
2; r2
2/R
2
)
=
R3
8eνr1r2(eν + r1/R)(eν + r2/R)
(
1
eν + r1/R
+
1
eν + r2/R
)∫ ∞
0
dt e−t(r1+r2),
the contribution can be expressed as
e3
16ν
R−7
∫ ∞
0
dt
∫ ∞
0
dr1
∫ ∞
0
dr2
(
1
eν + r1/R
+
1
eν + r2/R
)
Note on the Retarded van der Waals Potential within the Dipole Approximation 29
×
{
3
2
[1]r1 [1]r2 − [1]r1 [X2]r2 +
3
2
[X2]r1 [X2]r2
}
, (B.6)
where
[A(X)]r(t) = (2π)
r3e−tr%̂2
rad(r/R)
eν + r/R
∫ 1
−1
dX eirXA(X).
For readers’ convenience, we will explain how to compute the integral (B.6). Let ϕ(x) be the
Fourier transformation of %̂2
rad(r): ϕ(x) = (2π)−1/2
∫
R e−irx%̂2
rad(r)dr. Here, we extend %̂2
rad
to a function on R by %̂2
rad(−r) := %̂2
rad(r) for r > 0. Note that ϕ(x) decays rapidly by the
assumption (A.3). By the convolution theorem in the Fourier analysis, we have
∫ ∞
0
dr [1]r =
∫ ∞
0
ds
∫
R
dx (1 + x/R)e−seν
12(t+ s/R)2 − 4(1 + x/R)2
{
(t+ s/R)2 + (1 + x/R)2
}3ϕ(x) (B.7)
and
∫ ∞
0
dr
[1]r
eν + r/R
=
∫ ∞
0
ds
∫
R
dx (1 + x/R)se−seν
12(t+ s/R)2 − 4(1 + x/R)2
{
(t+ s/R)2 + (1 + x/R)2
}3ϕ(x).
[Here, we explain how we derive (B.7). First, we observe that
∫ ∞
0
dr[1]r = 4π
∫ ∞
0
dr
r2e−tr%̂2
rad(r/R)
eν + r/R
sin r
= 4π
∫ ∞
0
ds e−seν Im
∫
R
dr 1+(r)r2e−(t+s/R)r
︸ ︷︷ ︸
=:f(r)
%̂2
rad(r/R)eir,
where 1+(r) = 1 if r > 0, 1+(r) = 0 if r ≤ 0. By the convolution theorem
(
(2π)1/2
(
ĝĥ
)∨
= g∗h
)
,
we have
(2π)1/2 Im
∫
R
dr f(r)%̂2
rad(r/R)eirx =
(
ϕR ∗ Im f̌
)
(x),
where ϕR(x) = Rϕ(Rx) and (g ∗ h)(x) =
∫
R g(y)h(x− y)dy. Because
Im f̌(x) = (2π)−1/2 6x(t+ s/R)2 − 2x3
{
(t+ s/R)2 + x2
}3 ,
we get (B.7).] Hence, by the dominated convergence theorem, we obtain
lim
R→∞
e3
16ν
∫ ∞
0
dt
∫ ∞
0
dr1
∫ ∞
0
dr2
(
1
eν + r1/R
+
1
eν + r2/R
)
[1]r1 [1]r2
=
π
8ν4
%̂4(0)
∫ ∞
0
dt
(
−4 + 12t2
)2
(
t2 + 1
)6 .
Similarly, we obtain that
lim
R→∞
e3
16ν
∫ ∞
0
dt
∫ ∞
0
dr1
∫ ∞
0
dr2
(
1
eν + r1/R
+
1
eν + r2/R
)
[1]r1
[
X2
]
r2
=
π
8ν4
%̂4(0)
∫ ∞
0
dt
(
−4 + 12t2
)(
−12 + 4t2
)
(
t2 + 1
)6
30 T. Miyao
and
lim
R→∞
e3
16ν
∫ ∞
0
dt
∫ ∞
0
dr1
∫ ∞
0
dr2
(
1
eν + r1/R
+
1
eν + r2/R
)[
X2
]
r1
[
X2
]
r2
=
π
8ν4
%̂4(0)
∫ ∞
0
dt
(
−12 + 4t2
)2
(
t2 + 1
)6 .
Summarizing the above results, we arrive at
e4
2
∫ ∞
0
dr1
∫ ∞
0
dr2
∫ 1
−1
dX1
∫ 1
−1
dX2 S(X1, X2)r4
1r
4
2eir1X1eir2X2
× Ire
(
e2ν2; r2
1/R
2; r2
2/R
2
)
%̂2
rad(r1/R)%̂2
rad(r2/R) =
23π3
2ν4
%̂4(0)R−7 + o
(
R−7
)
,
where we used the following formula in [20]:
∫ ∞
0
dt
{
3
2
A(t)2 −A(t)B(t) +
3
2
B(t)2
}
= 92π3
with
A(t) =
−4 + 12t2
(
1 + t2
)3 and B(t) =
4
(
−3 + t2
)
(
1 + t2
)3 .
As for the contribution from Iir, we have, by an argument similar to that of the computation
concerning with Ire,
e4
2
∫ ∞
0
dr1
∫ ∞
0
dr2
∫ 1
−1
dX1
∫ 1
−1
dX2 S(X1, X2) r4
1r
4
2eir1X1eir2X2
× Iir
(
e2ν2; r2
1/R
2; r2
2/R
2
)
%̂2
rad(r1/R)%̂2
rad(r2/R) = const ·R−9 + o
(
R−9
)
.
To summarize, we obtain that
lim
R→∞
lim
Λ→∞
lim
L→∞
R7〈Q1Q1Q2Q2〉 =
23
8π
(
1
2π
)2(1
4
αE,at
)2
.
Similarly, we get
lim
R→∞
lim
Λ→∞
lim
L→∞
R7〈Q2Q2Q1Q1〉 =
23
8π
(
1
2π
)2(1
4
αE,at
)2
.
This concludes the proof of Lemma 8.1.
B.2 Proof of Lemma 8.2
We readily see that 〈Q1Q2Q1Q2〉 = 〈Q2Q1Q2Q1〉 = 0 by the formulas in Appendix A. In
what follows, we evaluate 〈Q1Q2Q2Q1〉 and 〈Q2Q1Q1Q2〉. Because the argument here is almost
pallarel to the proof of Lemma 8.1, we provide a sketch only. As before, we have
lim
Λ→∞
lim
L→∞
〈Q2Q1Q1Q2〉 =
e4
2π
∫
dk1dk2
(
1 +
(
k̂1 · k̂2
)2)|k1|2|k2|2%̂2(k1)%̂2(k2)
× cos(k1 · r) cos(k2 · r)I3,1,1
(
e2ν2; |k1|2; |k2|2
)
= R−10 e
4
2
∫ ∞
0
dr1
∫ ∞
0
dr2
∫ 1
−1
dX1
∫ 1
−1
dX2 S(X1, X2)r4
1r
4
2%̂
2
rad(r1/R)%̂2
rad(r2/R)
Note on the Retarded van der Waals Potential within the Dipole Approximation 31
× cos(r1X1) cos(r2X2)I3,1,1
(
e2ν2; r2
1/R
2; r2
2/R
2
)
. (B.8)
Remark the following formula:
I3,1,1
(
e2ν2; r2
1/R
2; r2
2/R
2
)
= (eν)−6 R
r1 + r2
+ o(R),
which follows from (C.4). Inserting this into (B.8), we formally obtain that
lim
Λ→∞
lim
L→∞
〈Q2Q1Q1Q2〉 =
g
e2ν6
R−9 + o
(
R−9
)
.
To justify this rough argument, we carefully have to treat the oscillatory integral as we did in
the proof of Lemma 8.1. Similarly, we see that lim
Λ→∞
lim
L→∞
〈Q1Q2Q2Q1〉 = g
e2ν6
R−9 + o
(
R−9
)
.
B.3 Proof of Lemma 8.4
In this case, there exist two numbers m,n ∈ N with m+ n ≥ 3 such that 〈QI〉 =
〈
Q2m
1 Q2n
2
〉
or
〈QI〉 =
〈
Q2m
2 Q2n
1
〉
. We will study the case where 〈QI〉 =
〈
Q2m
1 Q2n
2
〉
only. By using the formulas
in Appendix A, one obtains that
lim
Λ→∞
lim
L→∞
〈
Q2m
1 Q2n
2
〉
= e2(m+n)
∑
λ1,...,λm+n=1,2
∫
dk1 · · · dkm+n Im+n,2,1m+n−1
(
e2ν2; |k1|2; . . . ; |km+n|2
)
×
m+n∏
j=1
|kj |2%̂2(kj)
〈ε1|ε2〉〈ε2|ε3〉 · · · 〈εm+n|ε1〉 cos(km · r) cos(km+1 · r), (B.9)
where εj = ε(kj , λj) and Im+n,2,1m+n−1(a0; . . . ; am+n) = Im+n,2,1, . . . , 1︸ ︷︷ ︸
m+n−1
(a0; . . . ; am+n) with
In0,n1,...,nk(a0; a1; . . . ; ak) =
1
π
∫ ∞
−∞
ds
s2
k∏
j=0
(s2 + aj)nj
.
By scalings Rkm ; km and Rkm+1 ; km+1, we get
the r.h.s. of (B.9) = e2(m+n)R−10
∑
λ1,...,λm+n=1,2
∫
dk1 · · · dkm+n
× Im+n,2,1m+n−1
(
e2ν2; |k1|2; . . . ; |km−1|2; |km|2/R2; |km+1|2/R2; |km+2|2; . . . ; |km+n|2
)
×
[ ∏
j 6=m,m+1
|kj |2%̂2(kj)
]
〈ε1|ε2〉〈ε2|ε3〉 · · · 〈εm+n|ε1〉
× |km|2|km+1|2%̂2(km/R)%̂2(km+1/R) cos(km · n̂) cos(km+1 · n̂). (B.10)
Switching to the polar coordinates as we did in the proof of Lemma 8.1, we see that
the r.h.s. of (B.10) = e2(m+n)R−9
∑
λ1,...,λm+n=1,2
m+n∏
j=1
∫
drj
∫
dXj
∫
dϕj
[
1
π
∫ ∞
−∞
ds s2
]
×FR(r1, . . . , rm−1, rm+2, . . . , rm+n; s)GR(rm, rm+1; s)
× 〈ε1|ε2〉〈ε2|ε3〉 · · · 〈εm+n|ε1〉 cos(rmXm) cos(rm+1Xm+1), (B.11)
32 T. Miyao
where
FR =
(
e2ν2 +R−2s2
)−(m+n)
[ ∏
j 6=m,m+1
r4
j %̂
2
rad(rj)
(
r2
j +R−2s2
)−1
]
,
GR = r4
mr
4
m+1%̂
2
rad(rm/R)%̂2
rad(rm+1/R)
(
s2 + r2
m
)−1(
s2 + r2
m+1
)−1
.
Next, we will perform Xj- and ϕj-integrations for j 6= m,m + 1. For this purpose, we remark
that
∑
λ=1,2
∫ 1
−1
dX
∫ 2π
0
dϕ |ε(k, λ)〉〈ε(k, λ)| = 4π
3
1l3,
where 1l3 is the identity matrix acting in C3. Using this and the fact that
∑
λm,λm+1=1,2
(〈εm|εm+1〉3)2 = 1 +
(
k̂m · k̂m+1
)2
,
we get
the r.h.s. of (B.11) = e2(m+n)R−9
(
4π
3
)m+n−2 [ 1
π
∫ ∞
−∞
dss2
]
∏
j 6=m,m+1
∫
drj
FR
×
[∫
drmdrm+1
∫
dXmdXm+1
]
S(Xm, Xm+1)GR cos(rmXm) cos(rm+1Xm+1), (B.12)
where S(Xm, Xm+1) is defined by (B.5). Because
FR ≤
(
e2ν2
)−(m+n)
[ ∏
j 6=m,m+1
r2
j %̂
2
rad(rj)
]
,
we obtain that
the r.h.s. of (B.12) ≤ R−9
(‖%̂‖2L2
3ν2
)m+n−2
ν−4 1
π
∫ ∞
−∞
ds s2
∫
drmdrm+1
×
∫
dXmdXm+1 GRS(Xm, Xm+1) cos(rmXm) cos(rm+1Xm+1).(B.13)
Here, we used the fact that the factor
∫
drm
∫
drm+1
∫
dXmdXm+1[· · · ]
in the r.h.s. of (B.13) is positive for all s ≥ 0. Using the elementary formula
1
π
∫ ∞
−∞
ds
s2
(
s2 + r2
m
)(
s2 + r2
m+1
) =
1
rm + rm+1
=
∫ ∞
0
dt e−t(rm+rm+1),
we have
the r.h.s. of (B.13) = R−9
(‖%̂‖2L2
3ν2
)m+n−2
× ν−4
∫ ∞
0
dt
{
3
2
[[1]][[1]]− [[1]]
[[
X2
]]
+
3
2
[[
X2
]][[
X2
]]}
, (B.14)
Note on the Retarded van der Waals Potential within the Dipole Approximation 33
where
[[A(X)]](t) = (2π)
∫ ∞
0
dr r4e−tr%̂2
rad(r/R)
∫ 1
−1
dXeirXA(X).
We can compute [[1]] and [[X2]] as
[[1]] =
∫
R
dx
−24t3 + 24t(1 + x/R)2
{
t2 + (1 + x/R)2
}4 ϕ(x),
[[X2]] =
∫
R
dxFt(1 + x/R)ϕ(x),
where
Ft(a) = 8
(1− a)t5 − 2a
(
a2 + a− 3
)
t3 − a3
(
a2 + 3a+ 6
)
t
(
t2 + a2
)4 .
Since ϕ(x) decays rapidly, we readily see that the integral in (B.14) is uniformly bounded
provided that R is sufficiently large.
C Basic properties of Ina,nb,nc
(a; b; c)
Here, we will give a list of basic properties of Ina,nb,nc(a; b; c) defined by (B.2).
The following result is easily checked:
I1,1,1(a; b; c) =
1
ABC
, (C.1)
where A =
√
a+
√
b, B =
√
b+
√
c and C =
√
c+
√
a. Using this, we have
I2,2,1(a; b; c) =
1
4
√
ab
I1,1,1(a; b; c)
{
2
A2
+
1
AC
+
1
AB
+
1
BC
}
, (C.2)
I2,1,2(a; b; c) =
1
4
√
ac
I1,1,1(a; b; c)
{
2
C2
+
1
AC
+
1
BC
+
1
AB
}
(C.3)
and
I3,1,1(a; b; c) =
1
8a
I1,1,1(a; b; c)
{
2
A2
+
2
C2
+
2
AC
+
1√
aA
+
1√
aC
}
. (C.4)
Acknowledgements
The original idea of the present paper comes from an unpublished sketch by Herbert Spohn.
I would like to thank the kind referees for very helpful comments. The discussions in Section 9
heavily rely on their comments. This work was partially supported by KAKENHI 18K03315.
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1 Introduction
2 Main result
3 Feynman Hamiltonians
3.1 Preliminaries
3.2 Second quantized operators in F(to.L2(to.R3)to. L2(to.R3)to.)to.
3.3 Second quantized operators in =14F(H)
3.4 Identifications between F(to.L2(to.R3)to.L2(to.R3)to.)to. and =14 F(H)
3.5 Definition of the Feynman Hamiltonians
4 Canonical transformations
5 Lattice approximated Hamiltonians
6 Diagonalization I: One-electron Hamiltonian
7 Diagonalization II: Two-electron Hamiltonian
8 Proof of Theorem 2.1
8.1 Analysis of "426830A QI"526930B with #I=2
8.2 Analysis of "426830A QI"526930B with # I=4
8.3 Analysis of "426830A QI"526930B with # I6
8.3.1 Case 1
8.3.2 Case 2
8.4 Completion of the proof of Theorem 2.1
9 Discussions
9.1 Indistinguishability of the electrons
9.2 Cancellation mechanism of the van der Waals–London force
A Useful formulas
B Numerical computations
B.1 Proof of Lemma 8.1
B.2 Proof of Lemma 8.2
B.3 Proof of Lemma 8.4
C Basic properties of Ina, nb, nc(a; b; c)
References
|
| id | nasplib_isofts_kiev_ua-123456789-210714 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-17T12:04:33Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Miyao, Tadahiro 2025-12-15T15:29:32Z 2020 Note on the Retarded van der Waals Potential within the Dipole Approximation. Tadahiro Miyao. SIGMA 16 (2020), 036, 34 pages 1815-0659 2020 Mathematics Subject Classification: 81V10;81V55;47A75 arXiv:1902.05207 https://nasplib.isofts.kiev.ua/handle/123456789/210714 https://doi.org/10.3842/SIGMA.2020.036 We examine the dipole approximated Pauli-Fierz Hamiltonians of the nonrelativistic QED. We assume that the Coulomb potential of the nuclei, together with the Coulomb interaction between the electrons, can be approximated by harmonic potentials. By an exact diagonalization method, we prove that the binding energy of the two hydrogen atoms behaves as R⁻⁷, provided that the distance between the atoms R is sufficiently large. We employ Feynman's representation of the quantized radiation fields, which enables us to diagonalize Hamiltonians rigorously. Our result supports the famous conjecture by Casimir and Polder. The original idea of the present paper comes from an unpublished sketch by Herbert Spohn. I would like to thank the kind referees for their very helpful comments. The discussions in Section 9 heavily rely on their comments. This work was partially supported by KAKENHI 18K03315. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Note on the Retarded van der Waals Potential within the Dipole Approximation Article published earlier |
| spellingShingle | Note on the Retarded van der Waals Potential within the Dipole Approximation Miyao, Tadahiro |
| title | Note on the Retarded van der Waals Potential within the Dipole Approximation |
| title_full | Note on the Retarded van der Waals Potential within the Dipole Approximation |
| title_fullStr | Note on the Retarded van der Waals Potential within the Dipole Approximation |
| title_full_unstemmed | Note on the Retarded van der Waals Potential within the Dipole Approximation |
| title_short | Note on the Retarded van der Waals Potential within the Dipole Approximation |
| title_sort | note on the retarded van der waals potential within the dipole approximation |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210714 |
| work_keys_str_mv | AT miyaotadahiro noteontheretardedvanderwaalspotentialwithinthedipoleapproximation |