Spectrum and states of the BCS Hamiltonian with sources
We consider the BCS Hamiltonian with sources, as proposed by Bogolyubov and Bogolyubov, Jr. We prove that the eigenvectors and eigenvalues of the BCS Hamiltonian with sources can be exactly determined in the thermodynamic limit. Earlier, Bogolyubov proved that the energies per volume of the BCS Hami...
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irk-123456789-1647522020-02-11T01:26:10Z Spectrum and states of the BCS Hamiltonian with sources Petrina, D.Ya. Статті We consider the BCS Hamiltonian with sources, as proposed by Bogolyubov and Bogolyubov, Jr. We prove that the eigenvectors and eigenvalues of the BCS Hamiltonian with sources can be exactly determined in the thermodynamic limit. Earlier, Bogolyubov proved that the energies per volume of the BCS Hamiltonian with sources and the approximating Hamiltonian coincide in the thermodynamic limit. Розглянуто БКШ гамільтоніан з джерелами, який був запропонований Боголюбовим та Боголюбовим (мол.). Доведено, що власні вектори та власні значення БКШ гамільтоніана з джерелами можна визначити точно в термодинамiчнiй границі. Раніше Боголюбовим було встановлено, що питомі енергії БКШ та апроксимуючого гамільтоніанів збігаються в термодинамічній границі. 2008 Article Spectrum and states of the BCS Hamiltonian with sources / D.Ya. Petrina // Український математичний журнал. — 2008. — Т. 60, № 9. — С. 1243–1269. — Бібліогр.: 13 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/164752 517.9+531.19 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Petrina, D.Ya. Spectrum and states of the BCS Hamiltonian with sources Український математичний журнал |
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We consider the BCS Hamiltonian with sources, as proposed by Bogolyubov and Bogolyubov, Jr. We prove that the eigenvectors and eigenvalues of the BCS Hamiltonian with sources can be exactly determined in the thermodynamic limit. Earlier, Bogolyubov proved that the energies per volume of the BCS Hamiltonian with sources and the approximating Hamiltonian coincide in the thermodynamic limit. |
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Petrina, D.Ya. |
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Petrina, D.Ya. |
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Petrina, D.Ya. |
| title |
Spectrum and states of the BCS Hamiltonian with sources |
| title_short |
Spectrum and states of the BCS Hamiltonian with sources |
| title_full |
Spectrum and states of the BCS Hamiltonian with sources |
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Spectrum and states of the BCS Hamiltonian with sources |
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Spectrum and states of the BCS Hamiltonian with sources |
| title_sort |
spectrum and states of the bcs hamiltonian with sources |
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Інститут математики НАН України |
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2008 |
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Статті |
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http://dspace.nbuv.gov.ua/handle/123456789/164752 |
| citation_txt |
Spectrum and states of the BCS Hamiltonian with sources / D.Ya. Petrina // Український математичний журнал. — 2008. — Т. 60, № 9. — С. 1243–1269. — Бібліогр.: 13 назв. — англ. |
| series |
Український математичний журнал |
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AT petrinadya spectrumandstatesofthebcshamiltonianwithsources |
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2025-07-14T17:20:37Z |
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1837643740244606976 |
| fulltext |
UDC 517.9+531.19
D. Ya. Petrina (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
SPECTRUM AND STATES OF THE BCS HAMILTONIAN
WITH SOURCES
СПЕКТР ТА СТАНИ БКШ ГАМIЛЬТОНIАНА З ДЖЕРЕЛАМИ
We consider the BCS Hamiltonian with the sources as it has been proposed by Bogolyubov and
Bogolyubov (jr.). We prove that the eigenvectors and eigenvalues of the BCS Hamiltonian with the
sources can be determined exactly in the thermodynamic limit. Earlier, Bogolyubov proved that the
energies per volume of the BCS Hamiltonian with sources and the approximating Hamiltonian coincide
in the thermodynamic limit.
Розглянуто БКШ гамiльтонiан з джерелами, який був запропонований Боголюбовим та Боголюбовим
(мол.). Доведено, що власнi вектори та власнi значення БКШ гамiльтонiана з джерелами можна
визначити точно в термодинамiчнiй границi. Ранiше Боголюбовим було встановлено, що питомi
енергiї БКШ та апроксимуючого гамiльтонiанiв збiгаються в термодинамiчнiй границi.
Introduction. In the series of papers [1 – 7] and book [8], we have investigated the
spectrum and eigenfunctions of the BCS Hamiltonian [9]
HΛ =
∑
p̄
(
p2
2m
− µ
)
a+
p̄ ap̄ +
g
V
∑
p1,p2
vp1vp2a
+
p1
a+
−p1
a−p2ap2 (1)
in a finite cube Λ with periodic boundary conditions. Here, p is discrete momenta p =
=
2π
L
(n1, n2, n3), ni ⊂ Z, i = 1, 2, 3. By p̄ we denote momenta p and spin σ = ±1
2
,
p =
(
p, +
1
2
)
, −p =
(
−p,−1
2
)
, g is a coupling constant, vp is the potential, µ is the
chemical potential, V = L3 is the volume of the cube Λ, and L is the length of the edge
of the cube.
In the special subspace of pairs we have represented the Hamiltonian HΛ as sum
of two operators AΛ and BΛ. The spectrum and eigenfunctions of the operator AΛ can
be determined exactly in subspaces of arbitrary n pairs, and pairs do not interact. The
operator BΛ describes the interaction of pairs and it tends to zero as the volume V tends
to infinity for an arbitrary finite number n of pairs. The operator BΛ can be considered as
a perturbation of the operator AΛ, and the spectrum of the operator HΛ is a perturbation
of the spectrum of the operator AΛ and can be determined asymptotically exactly as the
volume V tends to infinity.
It is not a surprise that the Hamiltonian HΛ has eigenfunctions in the subspace of
n pairs because the operator of the number of particles commutes with the Hamiltonian
HΛ. It is a great surprise that this phenomenon has not been recognized earlier.
We have discovered a new branch of the spectrum and eigenfunctions of the BCS
Hamiltonian HΛ that differs from the well-known spectrum corresponding to the BCS
ground state and its excitations. This new branch of eigenfunctions consists of an
arbitrary number n = 1, 2, . . . of pairs in ground state and excitation of these pairs with
an arbitrary orbital momenta l = 0, 1, . . . and a continuous energy divided by a nonzero
gap from the energy of the pairs in ground state.
c© D. YA. PETRINA, 2008
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 9 1243
1244 D. YA. PETRINA
From the physical point of view, the first main difference between the classical BCS
branch of spectrum and eigenfunctions and the new branch of spectrum and eigenfunc-
tions consists of the following: In the excitations of the BCS ground states, a certain
number of pairs are replaced by the same number of electrons. In the excitations of the
new ground state, a certain number of pairs are replaced by the same number of pairs
with orbital momenta and continuous energy divided from the energy of pairs in the
ground state by a nonzero gap.
The same situation is true for the excitations. Namely, an arbitrary number n of
excited pairs of the second branch is the eigenfunction of the BCS Hamiltonian, while
excited n pairs of the first branch are not eigenfunction of the BCS Hamiltonian.
The second main difference consists of the following: the BCS ground state is a
coherent vector of the same pairs, as well as the ground state of the new branch. But
each pair and an arbitrary number of pairs of the new branch are eigenvectors of the
BCS Hamiltonian in the thermodynamic limit when an arbitrary number of pairs of the
BCS ground state are not eigenfunctions of the BCS Hamiltonian. It is interesting to
mention that Cooper in his famous paper [10], in fact, investigated the BCS Hamiltonian
in the subspace of one pair and promised to investigate the spectrum in the subspace
of an arbitrary number n of pairs. We solved this problem in the series of our papers
[1 – 7] and the book [8].
In the given paper, we consider the BCS Hamiltonian with sources, as it has been
proposed by Bogolyubov [11] and Bogolyubov (jr.) [12]. Namely, we consider the
Hamiltonian
HΛ,ν =
∑
p̄
(
p2
2m
− µ
)
a+
p̄ ap̄ +
g
V
∑
p1,p2
vp1vp2a
+
p1
a+
−p1
a−p2ap2+
+ν
∑
p
vpa
+
p a+
−p + ν
∑
p
vpa−pap, (2)
where ν > 0 is parameter and ν
∑
p
vpa
+
p a+
−p and ν
∑
p
vpa−pap are the operators of
sources.
Bogolyubov had explained [13] that sources are introduced into the original BCS
Hamiltonian in order to choose the proper solution (eigenvectors). In the final results
one should put ν = 0. It will be shown that the Hamiltonian with sources (2) has
only one branch of the spectrum, namely the ground state and its excitations discovered
by Bardeen, Cooper, and Schrieffer. The eigenvectors and their eigenvalues depend
continuously on the parameter ν, but at ν = 0 they coincide with the corresponding
BCS eigenvectors and their eigenvalues, but not with the second branch. This means
that the BCS Hamiltonian remembers perturbation by sources, which are unbounded
operators.
The BCS Hamiltonian with sources does not have eigenfunctions in the subspace
of n pairs because the operators of sources connect the subspaces of n + 1 and n − 1
pairs with the subspace of n pairs. Therefore, it is necessary to consider the eigenvalue
problem in the entire subspace of pairs. We showed that the BCS Hamiltonian with
sources (2) can be represented in it as the sum of two operators AΛ,ν and BΛ, where the
operator BΛ tends to zero in the thermodynamic limit and can be considered as a pertur-
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 9
SPECTRUM AND STATES OF THE BCS HAMILTONIAN WITH SOURCES 1245
bation. The operator AΛ,ν coincides in the thermodynamic limit with the approximating
Hamiltonian on the ground and excited BCS states
Ha
Λ,ν =
∑
p̄
(
p2
2m
− µ
)
a+
p̄ ap̄ + (cΛ + ν)
∑
p
vpa
+
p a+
−p+
+(cΛ + ν)
∑
p
vpa
+
p a−pap − g−1c2
vV I, (3)
where cΛ is a certain constant and I is the identity operator. As is known, the spectrum
and eigenvectors of the approximating Hamiltonian can be determined exactly.
Thus, we have proved that the eigenvectors and eigenvalues of the BCS Hamilto-
nian can be determined exactly in the thermodynamics limit. Earlier, Bogolyubov [11]
proved that the energies per volume of the BCS Hamiltonian with sources (2) and of the
approximating Hamiltonian coincide in the thermodynamic limit. His result was very
unique information about the spectrum of the BCS Hamiltonian.
1. BCS Hamiltonian with sources in the subspace of pairs. 1.1. Action of the
BCS Hamiltonian with sources in the subspace of pairs. Consider the BCS Hamilto-
nian with sources in a finite cube Λ of volume V
HΛ,ν =
∑
p̄
(
p2
2m
− µ
)
a+
p̄ ap̄ +
g
V
∑
p1,p2
vp1vp2a
+
p1
a+
−p1
a−p2ap2 +
+ ν
∑
p
vpa
+
p a+
−p + ν
∑
p
vpa−pap, (1.1)
where ν > 0 is the same parameter, which characterizes the last two operators in
HΛ,ν known as sources. Note that the summation is carried out over momenta p =
=
2π
4
(n1, n2, n3), where L is the length of the edge of the cube Λ and ni are integer
numbers i = 1, 2, 3, p̄ = (p,±1). The original BCS Hamiltonian is obtained for ν = 0,
i.e., HΛ,ν |ν=0= HΛ.
The sources have been introduced by Bogolyubov [11] “in order to choose a proper
solution for eigenvalue problem”.
We introduce the subspace of pairs HP
f =
∞∑
n=0
1
n!
∑
k1 6=...6=kn
fn(k1, . . . , kn)a+
k1
a+
−k1
. . . a+
kn
a+
−kn
|0〉 =
=
∞∑
n=0
1
n!
∑
k1,...,kn
fn(k1, . . . , kn)a+
k1
a+
−k1
. . . a+
kn
a+
−kn
|0〉, (1.2)
where fn(k1, . . . , kn) are the wave functions of n pairs of electrons with opposite mo-
menta and spins and are symmetric, |0〉 is the vacuum state, and the summation is carried
out over all k1 6= . . . 6= kn. We suppose that the functions fn(k1, . . . , kn) are defined
for all momenta k1, . . . , kn, but not only for k1 6= . . . 6= kn. In the last expression on
the right-hand side of (1.2) we add terms with equal momenta. These terms are equal to
zero.
In the subspace of pairs HP we introduce the following scalar product of two ele-
ments f and g:
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 9
1246 D. YA. PETRINA
(f, g)′ =
∞∑
n=0
1
n!
∑
k1 6=...6=kn
fn(k1, . . . , kn)gn(k1, . . . , kn) =
=
∞∑
n=0
∑′
k1 6=... 6=kn
fn(k1, . . . , kn)gn(k1, . . . , kn), (1.3)
where
∑′
k1 6=... 6=kn
means that the summation is carried out over all k1 6= . . . 6= kn, and
the points k1 6= . . . 6= kn that differ only by permutations are identified. The norm is
defined as follows: ‖f‖′ = {(f, f)′} 1
2 .
We will also use the scalar product of two elements f and g defined as follows:
(f, g) =
∞∑
n=0
1
n!
∑
k1,...,kn
fn(k1, . . . , kn)gn(k1, . . . , kn), (1.4)
where the summation is carried out over all k1, . . . , kn, including momenta that coincide.
We have the norm ‖f‖ = (f, f)
1
2 .
In order to perform the thermodynamic limit we need the following scalar products
and norms:
(f, g)′V =
∞∑
n=0
1
V n
∑′
k1 6=...6=kn
fn(k1, . . . , kn)gn(k1, . . . , kn) =
=
∞∑
n=0
1
n!
1
V n
∑
k1 6=...6=kn
fn(k1, . . . , kn)gn(k1, . . . , kn), (1.5)
‖f‖′V =
{
(f, f)′V
} 1
2
and
(f, g)V =
∞∑
n=0
1
n!
1
V n
∑
k1,...,kn
fn(k1, . . . , kn)gn(k1, . . . , kn), (1.6)
‖f‖V =
{
(f, f)V
} 1
2 .
We suppose that the potential vp satisfies the following conditions: it has support
on compact D, it is continuous for p ∈ D and
1
V
∑
p
|vp|2 = ‖v‖2 < ∞, where
supp∈D |vp| = v < ∞ uniformly with respect to Λ(V ). The wave functions of n pairs
fn(k1, . . . , kn) are supposed to have support in the domain D with respect to all the
momenta k1, . . . , kn.
Repeating the calculation performed in our previous paper [6] with obvious modifi-
cations connected with sources, one obtains for HΛ,νf the following expression:
(HΛ,νf)n(k1, . . . , kn) =
(
2k2
1
2m
+ . . . +
2k2
n
2m
− 2µn
)
fn(k1, . . . , kn) +
+
g
V
n∑
i=1
vki
vpfn
(
k1, . . . ,
i
p
, . . . , kn
)
+
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 9
SPECTRUM AND STATES OF THE BCS HAMILTONIAN WITH SOURCES 1247
+ ν
n∑
i=1
vkifn−1(k1, . . . ,
i
∨, . . . , kn) + ν
∑
p6=(k)n
vpfn+1(p, k1, . . . , kn) −
− g
V
n∑
i=1
n∑
1=j 6=i
vkivkj fn
(
k1, . . . ,
i
kj
, . . . , kn
)
, (1.7)
(k)n = (k1, . . . , kn), n ≥ 1,
where
(
k1, . . . ,
i
p
, . . . , kn
)
≡ (k1, . . . , ki−1, p, ki+1 . . . , kn) and (k1, . . . ,
i
∨, . . . , kn) ≡
≡ (k1, . . . , ki−1, ki+1, . . . , kn).
Note that expression (1.7) was derived, according to the Fermi statistics, for all
momenta such that k1 6= . . . 6= kn. Due to the fact that functions fn(k1, . . . , kn) are
defined for all k1, . . . , kn, but not only for k1 6= . . . 6= kn, we will suppose that
expression (1.7) is also true for all k1, . . . , kn. For example, equating in expression
HΛ,νf coefficients of the same products of the operators of creation pairs, including
operators with equal momenta, one obtains expression (1.7) for all k1, . . . , kn including
equal momenta. In what follows we will consider both expressions when we investigate
the eigenvalue and eigenvector problem for the Hamiltonian HΛ,ν .
Note that both expressions coincide in the thermodynamic limit as V → ∞ (Λ ↗
↗ R3). This will be shown later in subsection 3.1.
Denote by AΛ,ν and BΛ the following operators:
(AΛ,νf)n(k1, . . . , kn) =
(
2k2
1
2m
+ . . . +
2k2
n
2m
− 2µn
)
fn(k1, . . . , kn) +
+
g
V
n∑
i=1
vki
vpfn
(
k1, . . . ,
i
p
, . . . , kn
)
+
+ ν
n∑
i=1
vkifn−1(k1, . . . ,
i
∨, . . . , kn) + ν
∑
p6=(k)n
vpfn+1(p, k1, . . . , kn),
(1.8)
(BΛf)n(k1, . . . , kn) = − g
V
n∑
i=1
n∑
1=j 6=i
vki
vkj
fn
(
k1, . . . ,
i
kj
, . . . , kn
)
.
In what follows, we will consider the operators AΛ,ν and BΛ defined by formulas
(1.4), for k1 6= . . . 6= kn and as well for all k1, . . . , kn.
Now we want to attract attention to some special properties of the operator HΛ,V ,
namely, to the operator AΛ,ν . If ν = 0, then the subspaces of n pairs are invariant
with respect to HΛ, AΛ = AΛ,0, and BΛ, and we have considered and investigated
the eigenvalue problem in the subspaces. One can see from (1.8) that the subspaces of
n pairs are not invariant with respect to HΛ,ν and AΛ,ν because (AΛ,νf)n(k1, . . . , kn)
contains terms with fn−1 and fn+1. Therefore, we can consider the eigenvalue problems
for HΛ,ν and AΛ,ν only in the entire subspace of pairs.
Prior to the investigation of this problem, we define the domain of definition of the
Hamiltonian HΛ,ν and the operators AΛ,ν and BΛ. If follows from formulas (3.3) and
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 9
1248 D. YA. PETRINA
(3.4) of our paper [3] that the operator AΛ = AΛ,0 is well defined and bounded in the
subspaces HP
n of n pairs if the potential vp1vp2 satisfies the above described conditions.
The operator BΛ, according to estimates (3.9′) and (3.9′′) in [3], is a bounded operator
in HP
n with scalar product (1.4), ‖B‖ ≤ g|v||v|
V
1
2
n(n− 1), and its norm tends to zero as
V →∞ for arbitrary fixed n.
It is interesting to mention that Cooper in his famous paper [10], in fact, investi-
gated the BCS Hamiltonian in the subspace of one pair and promised to investigate the
spectrum in the subspace of an arbitrary number n of pairs. We solved this problem in
the series of our papers [1 – 7] and the book [8]. Thus, to define the operator HΛ,ν it is
sufficient to estimate the norms of the operators of sources.
1.2. Estimates of the norms of the operators of sources. Consider the operators
of the sources ν
∑
p
vpa
+
p a+
−p, and ν
∑
p
vpa−pap. It follows from (1.7) that(
ν
∑
p
vpa
+
p a+
−pf)n(k1, . . . , kn
)
= ν
n∑
i=1
vki
fn−1
(
k1, . . . ,
i
∨, . . . , kn
)
,
(
ν
∑
p
vpa−papf)n(k1, . . . , kn
)
= ν
∑
p6=(k)n
vpfn+1(p, k1, . . . , kn), (1.9)
ν > 0.
It is easy to obtain the following equalities:
1
n!
∑
k1,...,kn
|vk1fn−1(k2, . . . , kn)|2 =
=
1
n!
V
1
V
∑
k1
vk2
1
∑
k1,...,kn
|fn−1(k2, . . . , kn)|2 = V
1
n
‖v‖2‖fn−1‖2,
1
n!
∑
k1,...,kn
∣∣∣∣∣∣
∑
p6=(k)n
vpfn+1(p, k1, . . . , kn)
∣∣∣∣∣∣
2
≤
≤ 1
n!
V
1
V
∑
p
|vp|2
∑
p,k1,...,kn
∣∣fn+1(p, k1, . . . , kn)
∣∣2 = V (n + 1)‖v‖2‖fn+1‖2.
Note that we have used the norm connected with the scalar product (1.4).
From these inequalities one gets∥∥∥∥∥∥ν
(∑
p
vpa
+
p a+
−pf
)
n
∥∥∥∥∥∥ ≤ n
1
2 V
1
2 ν‖v‖‖fn−1‖,
∥∥∥∥∥∥ν
(∑
p
vpa−papf
)
n
∥∥∥∥∥∥ ≤ (n + 1)
1
2 V
1
2 ν‖v‖‖fn+1‖
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 9
SPECTRUM AND STATES OF THE BCS HAMILTONIAN WITH SOURCES 1249
and this means that the sources are unbounded operators when V → ∞ even in the
subspace of n pairs HP
n .
Finally, one has
∥∥∥∥∥ν∑
p
vpa
+
p a+
−pf
∥∥∥∥∥ ≤ V
1
2 ν‖v‖
∞∑
n=0
(n + 1)
1
2 ‖fn‖,
∥∥∥∥∥ν∑
p
vpa−papf
∥∥∥∥∥ ≤ V
1
2 ν‖v‖
∞∑
n=0
(n + 1)
1
2 ‖fn+1‖.
(1.10)
Note that summation with respect to n is carried out from 0 to N, where N is the
number of quasimomenta
2π
L
(n1, n2, n3) which lie in the domain D and N tends to
infinity as V →∞.
It follows from (1.10) that the operators of sources become unbounded in the ther-
modynamic limit as V →∞ in the whole subspace of pairs HP .
Now estimate the norms of the operators of sources using the norm of subspace of
pairs (1.6) connected with the volume V. One has
1
n!
1
V n
∑
k1,...,kn
∣∣vk1fn−1(k2, . . . , kn)
∣∣2 =
=
1
n!
1
V
∑
k1
vk2
1
1
V n−1
∑
k1,...,kn
∣∣fn−1(k2, . . . , kn)
∣∣2 = ‖v‖2V
1
n
‖fn−1‖2V ,
1
n!
1
V n
∑
k1,...,kn
∣∣∣∣∣∣
∑
p 6=(kn)
vpfn+1(p, k1, . . . , kn)
∣∣∣∣∣∣
2
≤
≤ 1
n!
V 2 1
V
∑
p
v2
p
1
V n+1
∑
p,k1,...,kn
∣∣fn+1(p, k1, . . . , kn)
∣∣2 =
= (n + 1)V 2‖v‖2V
(
‖fn+1‖V
)2
.
From these inequalities one gets
∥∥∥∥∥ν
(∑
p
vpa
+
p a+
−pf
)
n
∥∥∥∥∥
V
≤ n
1
2 ν‖v‖V ‖fn−1‖V ,
∥∥∥∥∥ν
(∑
p
vpa−papf
)
n
∥∥∥∥∥
V
≤ (n + 1)
1
2 νV ‖v‖V ‖fn+1‖V .
Finally one has
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 9
1250 D. YA. PETRINA∥∥∥∥∥ν∑
p
vpa
+
p a+
−pf
∥∥∥∥∥
V
≤ ν‖v‖V
∞∑
n=0
(n + 1)
1
2 ‖fn‖V ,
∥∥∥∥∥ν∑
p
vpa−papf
∥∥∥∥∥
V
≤ V ν‖v‖V
∞∑
n=0
(n + 1)
1
2 ‖fn+1‖V .
(1.11)
It follows from (1.11) that the operator of sources becomes unbounded as V → ∞
because the volume V is present in the second inequality in (1.11).
The unboundedness of the operators of sources has an important consequence.
Namely, this means that even for arbitrary small parameter ν the operators of sources
cannot be considered as the small perturbation of the operator HΛ = HΛ,ν |ν=0. We will
show later that the eigenvalues and eigenvectors of the operators HΛ,ν , which will be
calculated exactly in the thermodynamic limit (as V →∞), do not coincide when ν = 0
with the corresponding eigenvalues and eigenvectors of the operator HΛ (as V →∞).
2. Representation of HΛ,ν through the approximating Hamiltonian on coherent
states of pairs. 2.1. Hamiltonian HΛ,ν on coherent states of pairs. Consider the state
of pairs (1.2) with wave functions of n pairs which are equal to the product of wave
functions of one pair, namely
f =
∞∑
n=0
1
n!
∑
k1 6=...6=kn
f1(k1) . . . f1(kn)a+
k1
a+
−k1
. . . a+
kn
a+
−kn
|0〉 =
=
∞∑
n=0
1
n!
∑
k1
f1(k1)a+
k1
a+
−k1
. . .
∑
kn
f1(kn)a+
kn
a+
−kn
|0〉. (2.1)
Then one has the following identity:
1
n!
∑
k1 6=...6=kn
g
V
n∑
i=1
vki
∑
p
vpf1(p)f1(k1) . . .
. . .
i
∨ . . . f1(kn)a+
k1
a+
−k1
. . .
i
∨ . . . a+
kn
a+
−kn
|0〉 =
=
cΛ
n!
n∑
i=1
∑
k1 6=...6=kn
vki
f1(k1) . . .
i
∨ . . . f1(kn)a+
k1
a+
−k1
. . .
i
∨ . . . a+
kn
a+
−kn
|0〉 =
=
cΛ
n!
n∑
i=1
∑
k1,...,kn
vki
f1(k1) . . .
i
∨ . . . f1(kn)a+
k1
a+
−k1
. . . a+
ki
a+
−ki
. . . a+
kn
a+
−kn
|0〉, (2.2)
where
a+
k1
a+
−k1
. . .
i
∨ . . . a+
kn
a+
−kn
≡ a+
k1
a+
−k1
. . . a+
ki−1
a+
−ki−1
a+
ki+1
a+
−ki+1
. . . a+
kn
a+
−kn
and
cΛ =
g
V
∑
p
vpf1(p). (2.3)
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SPECTRUM AND STATES OF THE BCS HAMILTONIAN WITH SOURCES 1251
Note that in (2.2) we added the terms with equal momenta which are equal to zero.
Denote by AI
Λ the part of the operator AΛ,ν , that describes the interaction of two
particles with opposite momenta and spin. It is defined by the second term in (1.7). It
follows from (2.2) that the operator AI
Λ on the state (2.1) is defined as follows:
AI
Λf = cΛ
∞∑
n=1
1
n!
n∑
i=1
∑
k1 6=... 6=kn
f(k1)a+
k1
a+
−k1
. . . vkia
+
ki
a+
−ki
. . . f(kn)a+
kn
a+
−kn
|0〉 =
= cΛ
∞∑
n=1
1
n!
n∑
i=1
∑
k1
f(k1)a+
k1
a+
−k1
. . .
∑
ki
vki
a+
ki
a+
−ki
. . .
. . .
∑
kn
f(kn)a+
kn
a+
−kn
|0〉. (2.4)
Note that in (2.4) only the terms with k1 6= . . . 6= ki 6= . . . 6= kn are different from zero.
Consider the operator
A+
Λ = cΛ
∑
k
vka+
k a+
−k, (2.5)
where cΛ is defined according to (2.3). By direct calculation and by analogy with the
operator ν
∑
p
vpa
+
p a+
−p one obtains
A+
Λf = cΛ
∞∑
n=0
1
n!
∑
k
vka+
k a+
−k
∑
k1
f1(k1)a+
k1
a+
−k1
. . .
∑
kn
f1(kn)a+
kn
a+
−kn
|0〉 =
= cΛ
∞∑
n=0
1
(n + 1)!
n+1∑
i=1
∑
k1
f1(k1)a+
k1
a+
−k1
. . .
∑
ki
vki
a+
ki
a+
−ki
. . .
. . .
∑
kn+1
f(kn+1)a+
kn+1
a+
−kn+1
|0〉 =
= cΛ
∞∑
n=1
1
(n)!
n∑
i=1
∑
k1
f1(k1)a+
k1
a+
−k1
. . .
∑
ki
vkia
+
ki
a+
−ki
. . .
. . .
∑
kn
f1(kn)a+
kn
a+
−kn
|0〉. (2.6)
Comparing (2.4) with (2.6) one concludes that
AI
Λf = A+
Λf. (2.7)
on coherent states of pairs (2.1).
Taking into account the last equality and formulas (1.7), (1.8), one can represent the
Hamiltonian HΛ,ν , on coherent states of pairs (2.1) as follows:
HΛ,νf =
(∑
p̄
(
p2
2m
− µ
)
a+
p̄ a+
−p̄ + cΛ
∑
p
vpa
+
p a+
−p +
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 9
1252 D. YA. PETRINA
+ ν
∑
p
vpa
+
p a+
−p + +ν
∑
p
vpa−pap + B
)
f =
=
(∑
p̄
(
p2
2m
− µ
)
a+
p̄ a−p̄ + cΛ
∑
p
vpa
+
p a+
−p + cΛ
∑
p
vpa−pap +
+ ν
∑
p
vpa
+
p a+
−p + ν
∑
p
vpa−pap
)
f + Bf − cΛ
∑
p
vpa−papf =
= Ha
Λ,νf + Bf − cΛ
∑
p
vpa−papf, (2.8)
where
Ha
Λ,ν =
∑
p̄
(
p2
2m
− µ
)
a+
p̄ ap̄ + (cΛ + ν)
∑
p
vpa
+
p a+
−p + (cΛ + ν)
∑
p
vpa−pap
(2.9)
is the approximating Hamiltonian introduced by Bogolyubov. (In formula (2.8) we
added and subtracted the operator cΛ
∑
p
vpa−pap.) Note that the constant cΛ (2.3) is
defined by the function f1(p) and varies with f(p).
2.2. Eigenvalue and eigenvector problem for the approximating Hamiltonian
Ha
Λ,ν . It is well known that the operator Ha
Λ,ν can be diagonal. Namely introduce the
following operators:
αk = ukak + wka+
−k, α+
k = uka+
k + wka−k,
α−k = uka−k − wka+
k , α+
−k = uka+
−k − wkak,
uk = (2)−
1
2
[
1 + εk(ε2
k + (cΛ + ν)2v2
k)−
1
2
] 1
2
, (2.10)
wk = (2)−
1
2
[
1− εk(ε2
k + (cΛ + ν)2v2
k)−
1
2
] 1
2
,
εk =
k2
2m
− µ, Ek =
(
ε2
k + (cΛ + ν)2v2
k
) 1
2 .
Note that uk = 1, wk = 0 for k /∈ D and we consider transformation (2.10) only for
k ∈ D.
The approximating Hamiltonian Ha
Λ,ν (2.9) can be represented through the operators
αk, α+
k , α−k, α+
−k in the following diagonal form:
Ha
Λ,ν =
∑
p̄
Epα
+
p̄ αp̄ +
∑
p
[
εp − (ε2
p + (cΛ + ν)2v2
p)
1
2
]
, (2.11)
for p /∈ D,
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SPECTRUM AND STATES OF THE BCS HAMILTONIAN WITH SOURCES 1253
Ha
Λ,ν =
∑
p̄
(
p2
2m
− µ
)
a+
p̄ ap̄.
Denote by Φa
0 the following state:
Φa
0 =
∏
k
(1 + fa(k)a+
k a+
−k)|0〉,
fa(k) =
−wk
uk
. (2.12)
It is easy to check by direct calculation that the state Φa
0 is the vacuum state for the
operators αk, α+
k , α−k, α+
−k. Indeed,
αpΦa
0 = 0, α−pΦa
0 = 0, (2.13)
because
αpΦa
0 = (wp − wp)α+
−p
∏
k 6=p
(1 + fa(k)a+
k a+
−k)|0〉 = 0,
α−pΦa
0 = (−wp + wp)α+
p
∏
k 6=p
(1 + fa(k)a+
k a+
−k)|0〉 = 0.
Therefore
Ha
Λ,νΦa
0 =
∑
P⊂D
[
εp − (ε2
p + (cΛ + ν)2v2
p)
1
2
]
Φa
0 , (2.14)
and this means that the vacuum state Φa
0 is the eigenvector of Ha
Λ,ν with the eigenvalue∑
P⊂D
[
εp −
(
ε2
p + (cΛ + ν)2v2
p
) 1
2
]
.
It follows from (2.10) that uk > 0 because vk 6= 0,
∣∣εk− (ε2
k +(cΛ +ν)2v2
k)−
1
2
∣∣ < 1
in D and, thus, the function fa(k) is uniformly bounded in D, i.e., |fa(k)| < f. The
state Φa
0 (2.12) can be represented as follows:
Φa
0 =
∞∑
n=0
∑′
k1 6=...6=kn
fa(k1) . . . fa(kn)a+
k1
a+
−k1
. . . a+
kn
a+
−kn
|0〉 =
=
∞∑
n=0
1
n!
∑
k1 6=...6=kn
fa(k1) . . . fa(kn)a+
k1
a+
−k1
. . . a+
kn
a+
−kn
|0〉 ≡
=
∞∑
n=0
1
n!
∑
k1,...,kn
fa(k1) . . . fa(kn)a+
k1
a+
−k1
. . . a+
kn
a+
−kn
|0〉. (2.15)
In the last expression we added terms equal to zero with equal momenta. It has been
proved in our paper [3] that on the state (2.12) with |fa(k)| < f the operator B can be
estimates as follows: ∣∣(Φa
0 ,BΦa
0)′V
∣∣ < |g|v2
V
α2f4eαf2,
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1254 D. YA. PETRINA
∣∣(BΦa
0 ,BΦa
0)′V
∣∣ ≤ gv4
V 2
(
6α4f8eαf2 + 2α2f4 + α3f6
)
, (2.16)
α =
8π
3(2π)2
[2m(µ + w)]
3
2 .
Then from (2.8), (2.15) one gets
lim
V→∞
∣∣∣∣∣∣
(
Φa
0 ,
[
HΛ,ν −Ha
Λ,ν − cΛ
∑
p
vpa−pap
]
Φa
0
)′
V
∣∣∣∣∣∣ = lim
V→∞
∣∣(Φa
0 ,BΦa
0)′V
∣∣ = 0,
(2.17)
lim
V→∞
∣∣∣∣∣
([
HΛ,ν −Ha
Λ,ν − cΛ
∑
p
vpa−pap
]
Φa
0 ,
[
HΛ,ν −Ha
Λ,ν−
−cΛ
∑
p
vpa−pap
]
Φa
0
)′
V
∣∣∣∣∣∣ = lim
V→∞
(BΦa
0 ,BΦa
0)′V = 0,
lim
V→∞
∥∥∥∥∥
[
HΛ,ν −Ha
Λ,ν − cΛ
∑
p
vpa−pap
]
Φa
0
∥∥∥∥∥
′
V
= 0. (2.18)
Thus we note that inequalities (2.16), (2.17) are true not only for coherent states of
pairs, but for arbitrary f ∈ HP with wave functions of n pairs fn(k1, . . . , kn) satisfying
conditions sup
(k)n
∣∣fn(k1, . . . , kn)
∣∣ < fn, f < ∞, see [3] formulas (5.8), (5.11). We al-
ready know that the state Φa
0 is the eigenvector of the operator Ha
Λ,ν with the eigenvalue∑
P⊂D
[
εp − (ε2
p + (cΛ + ν)2v2
p)
]
(see (2.14)).
Now consider the operator c
∑
p
vpa−pap on the state Φa
0 . One has
cΛ
∑
p
vpa−papΦa
0 =
= cΛ
∞∑
n=0
1
n!
∑
k1 6=...6=kn
∑
p6=(k)n
vpf
a(p)fa(k1) . . . fa(kn)a+
k1
a+
−k1
. . . a+
kn
a+
−kn
|0〉.
Further one has
lim
V→∞
1
V
(
cΛ
∑
p
vpa−papΦa
0 ,Φa
0
)′
V
=
= lim
V→∞
1
V
cΛ
∑
p6=(k)n
vpf
a(p)
∞∑
n=0
1
n!
1
V n
∑
k1 6=...6=kn
|fa(k1)|2 . . . |fa(kn)|2 =
= g−1c2
∑
n=0
1
n!
(∫
|fa(k)|2dk
)n
= g−1c2e
∫
|fa(k)|2dk
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SPECTRUM AND STATES OF THE BCS HAMILTONIAN WITH SOURCES 1255
because
lim
V→∞
1
V
∑
p6=(k)n
vpf
a(p) = g−1c,
lim
V→∞
1
V n
∑
k1 6=...6=kn
|fa(k1)|2 . . . |fa(kn)|2 =
(∫
|fa(k)|2dk
)n
,
n ≥ 1, lim
V→∞
cΛ = c,
lim
V→∞
(Φa
0 ,Φa
0)′V = e
∫
|fa(k)|2dk.
We have taken into account that in the integral sums one can neglect the hyperplanes
p = k1, . . . , p = kn in the first sum and the hyperplanes k1 = k2, . . . , kn−1 = kn in
the second sums. (We have omitted some details of proof. It will be presented in more
complicated situation, connected with excited states of Φa
0 , in Section 4.)
If f is a state (2.1) with an arbitrary wave function f(k) of one pair, then one has
lim
V→∞
1
V
(
cΛ
∑
p
vpa−papΦa
0 , f
)′
V
=
= lim
V→∞
1
V
cΛ
∞∑
n=0
1
n!
1
V n
∑
p6=(k)n
vpf
a(p)
∑
k1 6=... 6=kn
fa(k1)f(k1) . . . fa(kn)f(kn) =
= c2e
∫
fa(k)f(k)dk.
The last formula can be interpreted as a proof that lim
V→∞
1
V
cΛ
∑
p
vpa−papΦa
0 = c2Φa
0
in the weak sense on coherent states.
It is obvious that
lim
V→∞
1
V 2
(
cΛ
∑
p
vpa−papΦa
0 , cΛ
∑
p
vpa−papΦa
0
)′
V
= c4e
∫
|fa(k)|2dk.
Thus, in the above described sense, the operator cΛ
∑
p
vpa−pap on the state Φa
0 tends
in the thermodynamic limit to the operator of multiplication by g−1c2V
lim
V→∞
cΛ
∑
p
vpa−papΦa
0 = c2V̄ Φa
0 , (2.19)
where V̄ denotes the infinite volume of the entire three-dimensional Euclidean space.
The state Φa
0 is the eigenvector of the operator Ha
Λ,ν with eigenvalue
∑
p⊂D
[
εp −
− (ε2
p + (cΛ + ν)2v2
p)
1
2
]
. If follows from (2.14) and (2.15) that the same state Φa
0 is the
eigenvector of the operator Ha
Λ,ν − cΛ
∑
p
vpa−pap in the thermodynamic limit with
the eigenvalue
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1256 D. YA. PETRINA
lim
V→∞
∑
p⊂D
[
εp − (ε2
p + (cΛ + ν)2v2
p)
1
2
]
− c2
ΛV
=
= V̄
[∫
(ε(p)− (ε(p)2 + (c + ν)2v(p)2)
1
2 )− c2
]
.
3. BCS Hamiltonian for infinite volume (in the entire Euclidean space). 3.1. Op-
erator AΛ in the thermodynamic limit. Define the operator AΛ for one pair (one puts
ν = 0 in Aν)
(AΛf1)(k1) =
(
2k2
1
2m
− 2µ
)
f1(k) +
g
V
∑
p
vk1vpf1(p)
and denote it by H2,Λ
(H2,Λf1)(k1) =
(
2k2
1
2m
− 2µ
)
f1(k) +
g
V
∑
p
vk1vpf1(p). (3.1)
In our previous paper [6] we have completely investigated the eigenvalue and eigenvec-
tor problem for the operator H2,Λ with finite and infinite cube Λ.
Now we want to explain in what sense the operator H2,Λ converges to H2,R3 ≡ H2.
For this purpose, we restrict ourselves to a potential vk with the following compact
support D
(
k
∣∣∣∣ k2
2m
−µ
∣∣∣∣ < ω
)
, ω > 0. Consider a continuous function f1(k) with support
D and defined for arbitrary k ∈ D, but not only for quasidiscrete k =
2π
L
(n1, n2, n3),
ni ∈ Z, i = 1, 2, 3. The norm of the function f1(k) with these quasidiscrete k is defined
as follows:
‖f1‖Λ =
{
1
V
∑
k
|f1(k)2|
} 1
2
, V =
L3
(2π)3
. (3.2)
The norm of the function f1(k) with continuous momenta k is defined as follows:
‖f‖R3 = ‖f‖ =
{∫
|f1(k)|2dk
} 1
2
. (3.3)
It is obvious that ‖f‖Λ tends to ‖f‖ as V → ∞ (L → ∞) because the expression
1
V
∑
k
|f1(k)|2 is the Riemann sum which converges to
∫ ∣∣f1(k)
∣∣2dk, i.e.,
lim
V→∞
1
V
∑
k
∣∣f1(k)
∣∣2 =
∫
|f1(k)|2dk = ‖f1‖. (3.4)
For each of terms in (H2,Λf1)(k) we have
lim
V→∞
1
V
∑
k
∣∣∣∣(2k2
2m
− 2µ
)
f1(k)
∣∣∣∣2 =
∫ ∣∣∣∣(2k2
2m
− 2µ
)
f1(k)
∣∣∣∣2 dk,
lim
V→∞
1
V
∑
k
∣∣∣∣∣ 1V ∑
p
v(k)vpf1(p)
∣∣∣∣∣
2
=
∫
|v(k)|2dk|
∫
v(p)f1(p)dp|2.
(3.5)
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SPECTRUM AND STATES OF THE BCS HAMILTONIAN WITH SOURCES 1257
It is obvious that all the integrals are convergent due to continuity of f1(k), v(k) and the
fact that they have compact support D. (We denote by v(k) potential vk with continuous
momenta k.)
Define the operator H2 for infinite Λ = R3 as follows:
(H2f1)(k1)
(
2k2
1
2m
− 2µ
)
f1(k1) + v(k1)
∫
v(p)f1(p)dp. (3.6)
From the above obtained equalities (3.2) – (3.5) one concludes that
lim
V→∞
‖H2,Λf1‖Λ = ‖H2f1‖ (3.7)
and in this sense the operator H2,Λ converges to the operator H2 as V →∞(Λ → R3).
The operator AΛ = AΛ,ν |ν=0 is represented through the operators H2,Λ on wave
functions of n pairs as follows:
(AΛfn)(k1, . . . , kn) =
=
(
[H2,Λ ⊗ I . . .⊗ I + . . . + I ⊗ I . . .⊗H2,Λ]fn)(k1, . . . , kn
)
, n ≥ 1, (3.8)
where I is the identity operator.
Define the following norm for finite and infinite Λ
‖fn‖Λ =
{
1
V n
∑
k1,...,kn
|fn(k1, . . . , kn)|2
}1
2
,
‖fn‖Λ=R3 = ‖fn‖ =
{∫
|fn(k1, . . . , kn)|2dk1 . . . dkn
}1
2
(3.9)
for continuous functions with compact support D with respect to each of variables
k1, . . . , kn. It is obvious that
lim
V→∞
‖fn‖Λ = ‖fn‖. (3.10)
As a simple consequence one obtains that
lim
V→∞
‖AΛfn‖Λ = ‖Afn‖. (3.11)
if the operator A is defined as follows:
(Afn)(k1, . . . , kn) =
=
(
[H2 ⊗ I . . .⊗ I + . . . + I ⊗ I . . .⊗H2]fn)(k1, . . . , kn
)
=
=
n∑
i=1
[(
2k2
1
2m
− 2µ
)
+ v(ki)g
∫
v(p)fn
(
k1, . . . ,
i
p
, . . . kn
)
dp
]
, n ≥ 1. (3.12)
The BCS Hamiltonian HΛ = HΛ,ν = 0 on states of pairs f (1.2) is represented accord-
ing to (1.7), (1.8) as the sum of two operators AΛ and BΛ
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1258 D. YA. PETRINA
HΛf = AΛf + BΛf. (3.13)
Taking into account estimates (2.16), (2.17) and formulas (3.9) – (3.11) one can conclude
that for finite elements f (1.2) (elements f with fn(k1, . . . , kn) = 0 for n > n0, n0 is
finite number, and
∣∣fn(k1, . . . , kn)
∣∣ < fn, f < ∞)
lim
V→∞
‖HΛf‖′V = lim
V→∞
‖AΛf‖′V = ‖Af‖. (3.14)
Note that we need a finite element f to have ‖Af‖ < ∞.
In this sense the BCS Hamiltonian is defined directly for infinite volume as follows:
Hf = Af,
(3.15)
(Hf)n(k1, . . . , kn) = (H2 ⊗ I . . .⊗ I + . . . + I ⊗ I . . .⊗H2)fn(k1, . . . , kn).
The last expression for the BCS Hamiltonian, directly for infinite volume, had been
obtained in our paper [1] in 1970.
Remark 3.1. By a slight modification of formulas (3.8), (3.9) from our paper [1]
one can obtain the following estimate for the operator B in the space HP with norm
‖f‖V (1.6)
‖Bfn‖V ≤ g
|V |‖V ‖
V
1
2
n(n− 1)‖f‖V . (3.16)
From this estimate one can conclude that lim
V→∞
‖Bf‖V = 0 for finite f ∈ HP and the
estimate sup
(k)n
|fn(k1, . . . , kn)| ≤ fn, f < ∞ is not imposed on fn(k1, . . . , kn). We have
formula
lim
V→∞
‖HΛf‖ = ‖Af‖ = ‖Hf‖. (3.17)
Formulas (3.13) – (3.16) show in what sense lim
V→∞
HΛ = H.
3.2. BCS and approximating Hamiltonians for infinite volume [1, 2]. Now we
show to prove that the BCS Hamiltonian for infinite volume represented by the operator
A (3.14) coincides with the approximating Hamiltonian on coherent states of pairs. It
can be done in full analogy with calculation performed in the Section 1.
Denote by AI the part of operator A that describes the interaction of particles with
opposite momenta and spin. It is defined as follows:
(AIf)n(k1, . . . , kn) =
n∑
i=1
v(ki)g
∫
v(p)fn
(
k1, . . . ,
i
p
, . . . kn
)
dp.
The operator AI coincides with the operator A+ = c
∫
v(k)a+(k)a(−k)dk on
coherent states of pairs
f =
∞∑
n=0
1
n!
∫
f(k1)a+(k1)a+(−k1)dk1 . . .
∫
f(kn)a+(kn)a+(−kn)dkn|0〉 (3.18)
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SPECTRUM AND STATES OF THE BCS HAMILTONIAN WITH SOURCES 1259
with c = g
∫
v(p)f(p)dp. (For the creation and annihilation operators with continuous
momenta k we use the notation a+(k), a(k).) Indeed,
AIf = c
∞∑
n=1
1
n!
n∑
i=1
∫
f(k1)a+(k1)a+(−k1)dk1 . . .
. . .
∫
v(ki)a+(ki)a+(−ki)dki . . .
∫
f(kn)a+(kn)a+(−kn)dkn,
A+f = c
∞∑
n=1
A
n!
n∑
i=1
∫
f(k1)a+(k1)a+(−k1)dk1 . . .
. . .
∫
v(ki)a+(ki)a+(−ki)dki . . .
∫
f(kn)a+(kn)a+(−kn)dkn
and thus
AIf = A+f. (3.19)
This means that the BCS Hamiltonian H =
∑∞
n=0
⊗
∑n
i=1
I⊗ . . .⊗ i
H2
⊗ . . .⊗ I
on the coherent states f = e
∫
f(k)a+(k)a+(−k)dk coincides with the operator
H =
∫ (
k2
2m
− µ
)
a+(k̄)a(k̄)dk + c
∫
v(k)a+(k)a+(−k)dk. (3.20)
Now introduce the approximating Hamiltonian
Happr = H + c
∫
v(k)a(−k)a(k)dk − c
∫
v(k)a(−k)a(k)dk =
=
{∫ (
k2
2m
− µ
)
a+(k̄)a(k̄)dk + c
∫
v(k)a+(k)a+(−k)dk +
+ c
∫
v(k)a(−k)a(k)dk
}
− c
∫
v(k)a(−k)a(k)dk. (3.21)
Note that the operator c
∫
v(k)a(−k)a(k)dk on the coherent states of pairs f (3.17) is
defined as follows:
c
∫
v(k)a(−k)a(k)dkf = g−1c2V̄ f,
1
V
c
∫
v(k)a(−k)a(k)dkf = c2f.
(3.22)
We use the formula δ(0) = V̄ .
Note that the constant c = g
∫
v(k)f(k)dk is the same for all n, because in inte-
gral
∫
v(p)f(p)dp one can neglect behavior of the integrand v(k)f(k) in the points
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1260 D. YA. PETRINA
p = k1, . . . , kn. According to the Fermi statistics, the operator c
∫
v(p)a(−p)a(p)dp
annihilates only two operators a+(p)a+(−p) from one pair with momenta p and (−p).
Finally, on coherent states f one obtains
Happrf =
(∫ (
k2
2m
− µ
)
a+(k̄)a(k̄)dk + c
∫
v(k)a+(k)a+(−k)dk +
+c
∫
v(k)a(−k)a(k)dk − c2g−1V̄ I
)
where I is the identity operator.
In this sense, for infinite volume, the BCS Hamiltonian H coincides with the ap-
proximating Hamiltonian Happr on coherent states.
The coherent state Φa
0 = e
∫
fa
0 (k)a+(k)a+(−k)dk|0〉 is the eigenvector of the approxi-
mating Hamiltonian as well as of the BCS Hamiltonian with the following eigenvalue:
V̄
∫ [
ε(p)− (ε(p)2 + c2 + v(p)2)
]
dp, c =
∫
v(p)fa
0 (p)dp.
4. Hamiltonian HΛ,ν on excited states. 4.1. Excited states. Denote by f(p)l,(g)m
=
= fp1,...,pl;q1,...,qm
the following state:
f(p)l,(q)m
= a+
p̄1
. . . a+
p̄l
a+
q1
a+
−q1
. . . a+
qm
a+
−qm
f =
= a+
p̄1
. . . a+
p̄l
a+
q1
a+
−q1
. . . a+
qm
a+
−qm
∞∑
n=0
1
n!
×
×
∑
k1 6=...6=kn
fn(k1, . . . , kn)a+
k1
a+
−k1
. . . a+
kn
a+
−kn
|0〉
where f was defined by (1.2) and f ∈ HP
V . We say that f(p)l,(q)m
is an excited state
with momenta p1, . . . , pl of l particles and momenta (q1,−q1), . . . , (qm,−qm) of m
pairs. We assume that any two momenta (pi, pj) do not coincide with some pairs of
momenta from the set (k1,−k1, . . . , kn,−kn) = (k)n, n = 1, 2, . . . , ki ⊂ D or from
the set (q1,−q1, . . . , qm,−qm) = (q)m, qj ⊂ D, but some pi can coincide with some
momenta from the sets (k)n. If some momenta from (p)l coincide with some momenta
from (q)m, then f(p)l,(q)m
= 0.
We consider f(p)l,(q)m
as an element of HF with respect to momenta (p)l, (q)m and
as an element of HP
V with respect to momenta (k)n, n = 1, 2, . . . , i.e., f(p)l,(q)m
∈
∈ HF ⊗HP
V . We will use the notation (f(p)l,(q)m
, g(p)l,(q)m
)′V for the scalar product of
two elements f(p)l,(q)m
and g(p)l,(q)m
from HF ⊗HP
V .
The scalar product of two sequences f(p)l,(q)m
and g(p)l,(q)m
is equal to the following
expression:
(f(p)l,(q)m
, g(p)l,(q)m
)′V =
=
∞∑
n=0
1
V n
∑′
(k)n 6=(p)l 6=(q)m
fn(k1, . . . , kn)gn(k1, . . . , kn) =
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SPECTRUM AND STATES OF THE BCS HAMILTONIAN WITH SOURCES 1261
=
∞∑
n=0
1
n!
1
V n
∑′
(k)n 6=(p)l 6=(q)m
fn(k1, . . . , kn)gn(k1, . . . , kn). (4.1)
Note that
∑′
(k)n 6=(p)l 6=(q)m
means that one considers only k1 6= k2 6= . . . 6= kn
and identifies (k)n that differs only by permutation. Consider the action of HΛ,ν on
f(p)l,(q)m
and for the sake of simplicity we consider only operators a+
p with spins H.
By analogy with (1.3) one obtains
HΛ,νf(p)l,(q)m
=
=
∞∑
n=0
1
n!
∑
k1,...,kn
{[
l∑
i=1
(
p2
i
2m
− µ
)
+
m∑
i=1
(
2q2
j
2m
− 2µ
)
+
n∑
i=1
(
2k2
i
2m
− 2µ
)]
×
× fn(k1, . . . , kn) +
g
V
n∑
i=1
[∑
p
vki
vpfn
(
k1, . . . ,
1
p
, . . . , kn
)
−
−
n∑
1=j 6=i
vkivkj fn
(
k1, . . . ,
i
kj
, . . . , kn
)
−
l∑
j=1
vkivpj fn
(
k1, . . . ,
i
pj
, . . . , kn
)
−
−
m∑
j=1
vki
vqj
fn
(
k1, . . . ,
i
qj
, . . . , kn
)]}
×
× a+
p1
. . . a+
pl
a+
q1
a+
−q1
. . . a+
qm
a+
−qm
a+
k1
a+
−k1
. . . a+
kn
a+
−kn
|0〉 +
+
g
V
m∑
j=1
∑
kn+1
vkn+1vqj
fn(k1, . . . , kn) ×
× a+
p1
. . . a+
pl
a+
q1
a+
−q1
. . .
j
∨ . . . a+
qm
a+
−qm
a+
k1
a+
−k1
. . . a+
kn
a+
−kn
a+
kn+1
a+
−kn+1
|0〉 +
+ ν
n∑
i=1
vki
fn−1(k1, . . . ,
i
∨ . . . , kn)a+
p1
. . . a+
pl
a+
q1
a+
−q1
. . .
. . . a+
qm
a+
−qm
a+
k1
a+
−k1
. . . a+
kn
a+
−kn
|0〉 +
+ ν
∑
p6=(k)n 6=(p)l 6=(q)m
vpfn+1(p, k1, . . . , kn)a+
p1
. . . a+
pl
a+
q1
a+
−q1
. . .
. . . a+
qm
a+
−qm
a+
k1
a+
−k1
. . . a+
kn
a+
−kn
|0〉 +
+ ν
m∑
j=1
vqj
fn(k1, . . . , kn)a+
p1
. . . a+
pl
a+
q1
a+
−q1
. . .
j
∨ . . .−qm
+a+
k1
a+
−k1
. . . a+
kn
a+
−kn
|0〉.
(4.2)
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1262 D. YA. PETRINA
This yields
HΛ,νf(p)l,(q)m
=
=
[
l∑
i=1
(
p2
i
2m
− µ
)
I +
m∑
i=1
(
2q2
j
2m
− 2µ
)
I
]
f(p)l,(q)m
+
+(AΛ + B(p)l,(q)m
)f(p)l,(q)m
+ C(p)l, (q)mf(p)l,(q)m
+
+
[
ν
∑
p
vpa
+
p a+
−p + ν
∑
p
vpa−pap
]
f(p)l,(q)m
, (4.3)
where AΛ is defined by the third and fourth term, the operator B(p)l,(q)m
is defined by
the fifth, sixth and seventh terms, the operator C(p)l,(q)m
is defined by the eighth term, the
operator ν
∑
p
vpa
+
p a+
−p is defined by the ninth term and the operator ν
∑
p
vpa−pap
is defined by the last two terms, I is the identity operator.
4.2. Estimates for the operators B(p)l,(q)m
, C(p)l,(q)m
. We restrict ourselves to
fn(k1, . . . , kn) such that
∣∣fn(k1, . . . , kn)
∣∣ ≤ fn. For the operator B(p)l,(q)m
one has
(see detail in [4])
∣∣(f(p)l,(q)m
,B(p)l,(q)m
f(p)l,(q)m
)′V
∣∣ ≤ |g|v2
V
∞∑
n=2
Nn
V n
n(n + l + m− 1)
n!
f2n ≤
≤ |g|v2(l + m + 1)
V
∞∑
n=2
α2f2n
(n− 2)!
≤ |g|v2(l + m + 1)
V
α2f4eαf2,
(
‖B(p)l,(q)m
f(p)l,(q)m
‖′V
)2
=
(
B(p)l,(q)m
f(p)l,(q)m
,B(p)l,(q)m
f(p)l,(q)m
)
≤
≤ g2v4
V 2
∞∑
n=2
Nn
V n
(n(n + l + m− 1))2
n!
f2n ≤ g2v4
V 2
∞∑
n=2
αn(n(n + l + m− 1))2
n!
f2n.
It is obvious that the last series is convergent.
For the operator C(p)l,(q)m
one has∣∣(f(p)l,(q)m
, C(p)l,(q)m
f(p)l,(q)m
)′V
∣∣ ≤
≤
∞∑
n=0
Nn
V n
∑
k1 6=...6=kn
∣∣fn(k1, . . . , kn)
∣∣2 |g|
V
m∑
j=1
|Vgj ,qj
| ≤
≤
∞∑
n=0
Nn
V n
1
n!
f2n v2|g|m
V
|g|
V
mv2eαf2,
(
‖C(p)l,(q)m
f(p)l,(q)m
‖′V
)2 =
(
C(p)l,(q)m
f(p)l,(q)m
, C(p)l,(q)m
f(p)l,(q)m
)′
V
≤
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SPECTRUM AND STATES OF THE BCS HAMILTONIAN WITH SOURCES 1263
≤
∞∑
n=0
1
V n
∑′
k1 6=...6=kn
g2
V 2
m∑
i=1
∑
kn+1
|Vkn+1,qi
|2 +
g2
V 2
m∑
i6=j=1
|Vqi,qi
||Vqj ,qj
|
×
×|fn(k1, . . . , kn)|2 ≤
≤
∞∑
n=0
Nn
V n
f2n
(
g2
V 2
αv4 +
g2
V 2
m2v4
)
≤ g2eαf2
V
(
αv4m +
m2v4
V
)
.
Thus, the norms of the operators B(p)l,(q)m
, C(p)l,(q)m
on elements f(p)l,(q)m
tend to
zero as V tends to infinity.
By analogy with Section 2 we have the following representation of the operator
HΛ,ν through the operator Ha
Λ,ν on the states f(p)l,(q)m
HΛ,νf(p)l,(q)m
=
=
(∑
p̄
(
p2
2m
− µ
)
a+
p̄ ap̄ + cΛ
∑
p
vpa
+
p a+
−p + ν
∑
p
vpa
+
p a+
−p + ν
∑
p
vpa−pap +
+ B(p)l,(q)m
+ C(p)l,(q)m
)
f(p)l,(q)m
=
=
(∑
p̄
(
p2
2m
− µ
)
a+
p̄ ap̄ + (cΛ + ν)
∑
p
vpa
+
p a+
−p +
+ (cΛ + ν)
∑
p
vpa−pap + B(p)l,(q)m
+ C(p)l,(q)m
− cΛ
∑
p
vpa−pap
)
f(p)l,(q)m
=
=
(
Ha
Λ,ν + B(p)l,(q)m
+ C(p)l,(q)m
− cΛ
∑
p
vpa−pap
)
f(p)l,(q)m
.
From the above obtained estimates for the operators B(p)l,(q)m
, C(p)l,(q)m
one can prove
the following theorem.
Theorem 4.1. On the states of excited pairs f(p)l,(q)m
, one has
lim
V→∞
∥∥∥∥∥(HΛ,ν −Ha
Λ,ν − cΛ
∑
p
vpa−pap)f(p)l,(q)m
∥∥∥∥∥
′
V
=
= lim
V→∞
∥∥∥(B(p)l,(q)m
+ C(p)l,(q)m
)f(p)l,(q)m
∥∥∥′
V
= 0.
It is well known that the states f(p)l,(q)m
with f = Φa
0 and the operators
α+
q1
α+
−q1
. . . α+
qm
α+
−qm
instead of the operators a+
q1
a+
−q1
. . . a+
qm
a+
−qm
, where
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1264 D. YA. PETRINA
Φa
0 =
∞∑
n=0
1
n!
∑
k1 6=...6=kn
fa
0 (k1) . . . fa
0 (kn)a+
k1
a+
−k1
. . . a+
kn
a+
−kn
|0〉 =
=
∞∑
n=0
1
n!
∑
k1,...,kn
fa
0 (k1) . . . fa
0 (kn)a+
k1
a+
−k1
. . . a+
kn
a+
−kn
|0〉,
are the eigenvectors of the approximating Hamiltonian Ha
Λ,ν with eigenvalues
l∑
i=1
Epi + 2
m∑
i=1
Eqi +
∑
p
[
εp −
√
ε2
p + (c + ν)2v2
p
]
. (4.4)
From the estimates established above one obtains the following theorem.
Theorem 4.2. If the states of pairs f = (1, 0, f1(k1), . . . , fn(k1, . . . , kn), . . .)
satisfy the conditions ∣∣fn(k1, . . . , kn)
∣∣ ≤ fn, f < ∞, n > 1,
uniformly with respect to V and have supports in Dn, then the expressions∣∣∣∣∣
[
HΛ −AΛ −
l∑
i=1
(
p2
i
2m
− µ
)
I −
m∑
i=1
(
g2
i
2m
− µ
)
I−
−ν
∑
p
vpa
+
p a+
−p − ν
∑
p
vpa−pap
]
f(p)l,(q)m
∣∣∣∣∣
′
V
tend to zero as V →∞ for arbitrary fixed l and m.
4.3. Approximating Hamiltonian on the states f(p)l,(q)m
. In this subsection, we
restrict ourselves to the state of pairs such that
f =
(
1, 0, f0
1 (k1), f0
1 (k2), . . . , f0
1 (kn), . . .
)
,
∣∣f0
1 (k)
∣∣ ≤ f < ∞
and define on these f(p)l,(q)m
the following approximating Hamiltonian:
Ha
Λ,ν =
∑
p̄
(
p2
2m
− µ
)
a+
p̄ ap̄ + (cΛ + ν)
∑
p
vpa
+
p a−p+
+(cΛ + ν)
∑
p
vpa−pap − g−1c2
ΛV,
cΛ =
g
V
∑
p
vpf
0
1 (p).
By analogy with the calculations performed in Section 2, we have
AIf(p)l,(q)m
= cΛ
∑
p
vpa
+
p a+
−pf(p)l,(q)m
,
cΛ
∑
p
vpa−papf(p)l,(q)m
= cΛ
∞∑
n=1
1
(n− 1)!
∑
k1
f0
1 (k1)a+
k1
a+
−k1
. . .
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SPECTRUM AND STATES OF THE BCS HAMILTONIAN WITH SOURCES 1265
. . .
∑
kn−1
f0
1 (kn − 1)a+
kn−1a
+
−kn−1
∑
p6=(k)n−1,p6=(p)l,p6=(g)m
vpf
0
1 (p)a+
p1
. . .
. . . a+
pl
a+
q1
a+
−q1
. . . a+
qm
a+
−qm
|0〉+ cΛ
m∑
i=1
vqi
f
(p)l,(
i
∨
q)m
,
where (
i
∨
q)m ≡ (q1, . . . ,
i
∨, . . . , qm).
4.4. Operator cΛ
∑
p
vpa−pap on excited states. Now consider in detail the
operator cΛ
∑
p
vpa−pap on the states f(p)l,(q)m
. One obtains
cΛ
∑
p
vpa−papf(p)l,(q)m
=
= g−1c2
ΛV f(p)l,(q)m
+ cΛB1
(p)l,(q)m
f(p)l,(q)m
+ cΛ
m∑
i=1
vqi
f
(p)l,(
i
∨
q)m
,
where the operator B1
(p)l,(q)m
is defined as follows:
B1
(p)l,(q)m
f(p)l,(q)m
=
= −cΛa+
p1
. . . a+
pl
a+
q1
a+
−q1
. . . a+
qm
a+
−qm
∞∑
n=1
1
n!
∑
k1
fa
0 (k1)a+
k1
a+
−k1
. . .
. . .
∑
kn
fa
0 (kn)a+
kn
a+
−kn
|0〉
∑
k=(k)n,k=(p)l,k=(q)m|k1 6=k2 6=kn
vkfa
0 (k).
One obtains the following estimates [4]:
1
V
∣∣(f(p)l,(q)m
,B1
(p)l,(q)m
f(p)l,(q)m
)′V
∣∣ ≤ 1
V
v(l + m + 1)αf3eαf2
,
1
V
(
B1
(p)l,(q)m
f(p)l,(q)m
,B1
(p)l,(q)m
f(p)l,(q)m
f(p)l,(q)m
)′
V
≤
≤ 1
V
v2(l + m + 1)2(αf4 + 2α2f6eαf2
),
1
V
∣∣∣∣∣
(
m∑
i=1
vqi
f
(p)l,(
i
∨
q)m
,
m∑
i=1
vqi
f(p)l,(q)m
)′
V
∣∣∣∣∣ ≤ 1
V
v2m2eαf2
.
From these estimates one obtains the following theorem.
Theorem 4.3. The expressions
1
V
(
f(p)l,(q)m
, (HΛ −Ha
Λ,ν − g−1c2
ΛV − cΛB1
(p)l,(q)m
)f(p)l,(q)m
−
− cΛ
m∑
i=1
vqi
f
(p)l,(
i
∨
q)m
)′
V
,
(4.5)
1
V
∥∥∥∥∥
[
(HΛ −Ha
Λ,ν − g−1cΛV − cΛB1
(p)l,(q)m
)f(p)l,(q)m
−
− cΛ
m∑
i=1
vqif
(p)l,(
i
∨
q)m
]∥∥∥∥∥
′
V
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 9
1266 D. YA. PETRINA
tend to zero in the thermodynamic limit (as V → ∞) for arbitrary fixed numbers l
and m.
Note that the numbers l and m can tend to infinity together with V but in such a
way that (l + m)/V tends to zero as V →∞. In this sense the states f(p)l,(q)m
are the
eigenvectors of the Hamiltonian HΛ in the thermodynamic limit. If m = 0, then the
states f(p)l are the eigenvectors of the Hamiltonian HΛ in the following sense:
lim
V→∞
∥∥∥∥∥(HΛ −Ha
Λ,ν − cΛ
∑
p
vpa−pap)f(p)l
∥∥∥∥∥ = 0. (4.6)
In this case the operator cΛ
∑
p
vpa−pap on the state f(p)l
can be considered as the
operator of multiplication in the thermodynamic limit
cΛ
∑
p
vpa−papf(p)l
→V→∞V g−1c2
Λf(p)l
, lim
V→∞
cΛ = c. (4.7)
It follows from the relation
lim
V→∞
1
V
∑
p6=(k)m
p6=(p)l
vpf
a
0 (p) =
∫
v(p)fa
0 (p)dp = c.
Thus we use the factor
1
V
in expression (4.5) mainly to neglect the operator
cΛ
∑m
i=1
vqi
f
(p)l,(
i
∨
q)m
.
Now we give the proof of formula (1). The expression cΛ
∑
k
vka−kakΦ0 can be
estimated as follows: The sum
A−Φ0 = cΛ
∑
k
vka−kakΦ0 = cΛ
∑
k
vkf0
1 (k)|0〉+
+ cΛ
∞∑
n=2
1
n!
n∑
i=1
∑
k1
f0
1 (k1)a+
k1
a+
−k1
. . .
∑
ki 6=k1...
ki 6=kn
vkif
0
1 (ki) . . .
∑
kn
f0
1 (kn)a+
kn
a+
−kn
|0〉,
should be divided into two parts 0 ≤ n ≤ n0 and n0 < n < ∞, and the number n0 is
of order V δ, where 0 < δ < 1, i.e., n0 ≤ V δ. Then one has
lim
V→∞
1
V 2
(A−Φ0,A−Φ0)′ = lim
V→∞
1
V 2
c2
Λ
(∑
k
vkf0
1 (k)
)2
+
+ lim
V→∞
c2
Λ
n0∑
n=2
1
V n−1
∑′
k1 6=...6=kn
∣∣∣f0
1 (k1)|2 . . .
i
∨ . . . |f0
1 (kn)
∣∣∣2×
×
(
1
V
∑
ki 6=k1,...,ki 6=kn
vki
f0
1 (ki)
)2
+
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SPECTRUM AND STATES OF THE BCS HAMILTONIAN WITH SOURCES 1267
+ lim
V→∞
c2
Λ
∞∑
n=n0+1
1
V n−1
∑′
k1 6=...6=kn
∣∣∣f0
1 (k1)|2 . . .
i
∨ . . . |f0
1 (kn)
∣∣∣2×
×
(
1
V
∑
ki 6=k1,...,ki 6=kn
vkif
0
1 (ki)
)2
.
For n ≤ n0 one has
lim
V→∞
1
V
∑
ki 6=k1,...,ki 6=kn
vki
f0
1 (ki) =
= lim
V→∞
1
V
(∑
(k)i
vkif
0
1 (ki)−
∑
ki 6=k1,...,ki 6=kn
vkif
0
1 (ki)
)
=
= lim
V→∞
1
V
∑
ki
vkif
0
1 (ki) =
∫
v(ki)f0
1 (ki)dki = c,
because
lim
V→∞
1
V
∣∣∣∣∣ ∑
ki 6=k1,...,ki 6=kn
vki
f0
1 (ki)
∣∣∣∣∣ ≤ 1n
V
vf ≤ lim
V→∞
V δ
V
vf = 0
for n < n0 < V δ, 0 < δ < 1. For the sum over n ≥ n0 + 1 one has the following
estimate:
N∑
n=n0+1
1
V n−1
∑′
k1 6=...6=kn
∣∣f0
1 (k1)
∣∣2 . . .
i
∨ . . .
∣∣f0
1 (kn)
∣∣2∣∣∣∣∣ 1V ∑
ki 6=k1,...,ki 6=kn
vki
f0
1 (ki)
∣∣∣∣∣
2
≤
≤
N∑
n=n0+1
1
V n−1
Nn−1
(n− 1)!
f2nv2n2 ≤
∞∑
n=n0+1
αn−1f2nv2 n2
(n− 1)!
.
The last series is convergent and tends to zero as V →∞ because n0. Thus,
lim
V→∞
1
V 2
(A−Φ0, A
−Φ0)′V = g−2c4 lim
V→∞
(Φ0,Φ0)′V
where lim
V→∞
(Φ0,Φ0)′V =
∑∞
n=0
1
n!
∑
k1 6=...6=kn
|fa
0 (k1)|2 . . .
i
∨ . . . |fa
0 (kn)|2 =
= e
∫
|fa
0 (k)|2dk. Using an analogous calculation one can show that
lim
V→∞
1
V
(Φ, A−Φ0)′V = g−1c2 lim
V→∞
(Φ, Φ0)′V
for an arbitrary coherent state
Φ = exp
(∑
k
fka+
k a+
−k
)
|0〉, |fk| < f < ∞.
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 9
1268 D. YA. PETRINA
In the above described sense the operator
1
V
A− on Φ0 in the thermodynamic limit is
equal to the operator of multiplication by constant g−1c2.
By analogous calculation one can show that
lim
V→∞
cΛ
∑
p
vpa−papΦ(p)l,(q)m
=
= cΛ lim
V→∞
[
a+
p1
. . . a+
pl
a+
q1
a+
−q1
. . . a+
qm
a+
−qm
∞∑
n=0
1
n!
∑
k1
fa
0 (k1)a+
k1
a+
−k1
. . .
. . .
∑
kn
fa
0 (kn)a+
kn
a+
−kn
|0〉
∑
k 6=(k)n, k 6=(p)l,k 6=(q)m
vkfa
0 (k) + cΛ
m∑
i=1
vqiΦ
(p)l,(
i
∨
q)m
]
=
= g−1c2V Φ(p)l,(q)m
+ c
m∑
i=1
vqiΦ
(p)l,(
i
∨
q)m
.
Remark 4.1. The factor
1
V
in expressions (4.5) could be replaced by factor
1
V δ
where δ is arbitrary number 0 < δ < 1 for arbitrary fixed m.
Remark 4.2. Consider the excitation created by the operators α+
k α+
−k or αkα−k.
It is easy to check that
α+
p Φ0 =
√
1 + (fa(p))2a+
p
∏
k 6=p
(1 + fa(k)a+
k a+
−k)|0〉,
α+
−pΦ0 =
√
1 + (fa(p))2a+
−p
∏
k 6=p
(1 + fa(k)a+
k a+
−k)|0〉,
α+
q α+
−qΦ0 = (−fa(q) + a+
q a+
−q)
∏
k 6=q
(1 + fa(k)a+
k a+
−k)|0〉.
It follows from the obtained formulas that the excitations
ϕ(p)l,(q)m
= α+
p1
. . . α+
pl
α+
q1
α+
−q1
. . . α+
qm
α+
−qm
Φa
0
can be expressed via certain linear combinations of excitations f(p)l,(q)i
, i = 0, 1, . . . ,m,
where expressions
√
1 + (fa(pi))2, i = 1, . . . , l, fa(qj), j = 1, . . . ,m, should be
considered as coefficients independent on momenta (k)n. All results obtained above
about coincidence in the thermodynamic limit of the model BCS and approximating
Hamiltonians on the excitations f(p)l,(q)m
also hold for the excitations ϕ(p)l,(q)m
.
1. Petrina D. Ya. On Hamiltonians of quantum statistics and a model Hamiltonian in the theory of
superconductivity // Teor. i Mat. Fiz. – 1970. – 4, № 3. – P. 394 – 411.
2. Petrina D. Ya., Yatsyshin V. P. On model Hamiltonian in the theory of superconductivity // Ibid. –
1972. – 10, № 2. – P. 283 – 299.
3. Petrina D. Ya. Spectrum and states of BCS Hamiltonian in finite domain. I. Spectrum // Ukr. Mat.
Zh. – 2000. – 52, № 5. – P. 667 – 690.
4. Petrina D. Ya. Spectrum and states of the BCS Hamiltonian in finite domains. II. Spectra of excita-
tions // Ibid. – 2001. – 53, № 8. – P. 1080 – 1101.
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 9
SPECTRUM AND STATES OF THE BCS HAMILTONIAN WITH SOURCES 1269
5. Petrina D. Ya. Spectrum and states of the BCS Hamiltonian in finite domain. III. The BCS Hamil-
tonian with mean-field interaction // Ibid. – 2002. – 54, № 11. – P. 1486 – 1504.
6. Petrina D. Ya. Model BCS Hamiltonian and approximating Hamiltonian for an infinite volume. IV.
Two branches of their common spectra and states // Ibid. – 2003. – 55, № 2. – P. 174 – 197.
7. Petrina D. Ya. New second branch of spectrum of the BCS Hamiltonian and “pseudo-gap” // Ibid.
– 2005. – 57, № 11. – P. 1508 – 1534.
8. Petrina D. Ya. Mathematical foundations of quantum statistical mechanics. – Dordrecht: Kluwer
Acad. Publ., 1995. – 461 p.
9. Bardeen J., Cooper L. N., Schrieffer J. R. Theory of superconductivity // Phys. Rev. – 1957. – 108.
– P. 1175 – 1204.
10. Cooper L. N. Bound electron pairs in a degenerate Fermi gas // Ibid. – 1956. – 104. – P. 1189 – 1190.
11. Bogolyubov N. N. On the model Hamiltonian in the theory of superconductivity // Selected Paper
of N. N. Bogolyubov. – Kiev: Naukova Dumka, 1970. – 3. – P. 110 – 173.
12. Bogolyubov N. N. (jr.) A method of investigations of model Hamiltonian. – Moscow: Nauka, 1974.
– 178 p.
13. Bogolyubov N. N., Zubarev D. N., Tserkovnikov Yu. A. Asymptotically exact solution for model
Hamiltonian of theory of superconductivity // Selected Papers of N. N. Bogolyubov. – Kiev: Naukova
Dumka, 1960. – 3. – P. 98 – 109.
Received 28.04.07
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