New second branch of the spectrum of the BCS Hamiltonian and a “pseudogap”
The BCS Hamiltonian of superconductivity has the second branch of eigenvalues and eigenvectors. It consists of wave functions of pairs of electrons in ground and excited states. The continuous spectrum of excited pairs is separated by a nonzero gap from the point of the discrete spectrum that corres...
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irk-123456789-1658992020-02-18T01:28:41Z New second branch of the spectrum of the BCS Hamiltonian and a “pseudogap” Petrina, D.Ya. Статті The BCS Hamiltonian of superconductivity has the second branch of eigenvalues and eigenvectors. It consists of wave functions of pairs of electrons in ground and excited states. The continuous spectrum of excited pairs is separated by a nonzero gap from the point of the discrete spectrum that corresponds to the pair in the ground state. The corresponding grand partition function and free energy are exactly calculated. This implies that, for low temperatures, the system is in the condensate of pairs in the ground state. The sequence of correlation functions is exactly calculated in the thermodynamic limit, and it coincides with the corresponding sequence of the system with approximating Hamiltonian. The gap in the spectrum of excitations depends continuously on temperature and is different from zero above the critical temperature corresponding to the first branch of the spectrum. In our opinion, this fact explains the phenomenon of “pseudogap.” Гамільтоніан БКШ теорії надпровідності має другу вітку власних значень та власних векторів. Ця вітка складається з хвильових функцій пар електронів в основному та збуджених станах. Неперервний спектр збуджених пар відділений відмінною від нуля щілиною від точки дискретного спектра, що відповідає парі в основному стані. Відповідна велика статистична сума та вільна енергія вирахувані точно. Звідси випливає, що при низьких температурах система є в конденсаті пар в основному стані. Послідовність кореляційних функцій вирахувана точно у термодинамічній границі і збігається з відповідною послідовністю системи з апроксимуючим гамільтоніаном. Щілина в спектрі збуджень залежить неперервно від температури і є відмінною від нуля і на відрізку вище критичної температури, що відповідає першій вітці спектра. На думку автора, цей факт пояснює феномен „псевдощілини". 2005 Article New second branch of the spectrum of the BCS Hamiltonian and a “pseudogap” / D.Ya. Petrina // Український математичний журнал. — 2005. — Т. 57, № 11. — С. 1508–1533. — Бібліогр.: 17 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165899 517.9 + 531.19 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Petrina, D.Ya. New second branch of the spectrum of the BCS Hamiltonian and a “pseudogap” Український математичний журнал |
| description |
The BCS Hamiltonian of superconductivity has the second branch of eigenvalues and eigenvectors. It consists of wave functions of pairs of electrons in ground and excited states. The continuous spectrum of excited pairs is separated by a nonzero gap from the point of the discrete spectrum that corresponds to the pair in the ground state. The corresponding grand partition function and free energy are exactly calculated. This implies that, for low temperatures, the system is in the condensate of pairs in the ground state. The sequence of correlation functions is exactly calculated in the thermodynamic limit, and it coincides with the corresponding sequence of the system with approximating Hamiltonian. The gap in the spectrum of excitations depends continuously on temperature and is different from zero above the critical temperature corresponding to the first branch of the spectrum. In our opinion, this fact explains the phenomenon of “pseudogap.” |
| format |
Article |
| author |
Petrina, D.Ya. |
| author_facet |
Petrina, D.Ya. |
| author_sort |
Petrina, D.Ya. |
| title |
New second branch of the spectrum of the BCS Hamiltonian and a “pseudogap” |
| title_short |
New second branch of the spectrum of the BCS Hamiltonian and a “pseudogap” |
| title_full |
New second branch of the spectrum of the BCS Hamiltonian and a “pseudogap” |
| title_fullStr |
New second branch of the spectrum of the BCS Hamiltonian and a “pseudogap” |
| title_full_unstemmed |
New second branch of the spectrum of the BCS Hamiltonian and a “pseudogap” |
| title_sort |
new second branch of the spectrum of the bcs hamiltonian and a “pseudogap” |
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Інститут математики НАН України |
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2005 |
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Статті |
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| citation_txt |
New second branch of the spectrum of the BCS Hamiltonian and a “pseudogap” / D.Ya. Petrina // Український математичний журнал. — 2005. — Т. 57, № 11. — С. 1508–1533. — Бібліогр.: 17 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT petrinadya newsecondbranchofthespectrumofthebcshamiltonianandapseudogap |
| first_indexed |
2025-07-14T20:20:11Z |
| last_indexed |
2025-07-14T20:20:11Z |
| _version_ |
1837655042024275968 |
| fulltext |
UDC 517.9+531.19
D. Ya. Petrina (Inst. Math. Nat. Acad. Sci. Ukraine, Kiev)
NEW SECOND BRANCH OF SPECTRA
OF THE BCS HAMILTONIAN AND “PSEUDOGAP”∗
NOVA DRUHA VITKA SPEKTRA HAMIL\TONIANA BKÍ
TA “PSEVDOWILYNA”
The BCS Hamiltonian of superconductivity has the second branch of eigenvalues and eigenvectors. It consists
from wave functions of pairs of electrons in ground and excited states. The continuous spectra of excited pairs
is divided by different from zero gap from the point of discrete spectra corresponding to the pair in ground
state. The corresponding grand partition function and free energy is exactly calculated. It follows from it that
for low temperatures system is in condensate of pairs in ground state. The sequence of correlation functions
is exactly calculated in the thermodynamic limit and it coincides with corresponding sequence of system with
approximating Hamiltonian. The gap in spectra of excitations depends continuously on temperature and is
different from zero above the critical temperature corresponding to the first branch of spectra. It seems to us that
this fact explains the phenomena of “pseudogap”.
Hamil\tonian BKÍ teori] nadprovidnosti ma[ druhu vitku vlasnyx znaçen\ ta vlasnyx vektoriv. Cq
vitka sklada[t\sq z xvyl\ovyx funkcij par elektroniv v osnovnomu ta zbudΩenyx stanax. Nepererv-
nyj spektr zbudΩenyx par viddilenyj vidminnog vid nulq wilynog vid toçky dyskretnoho spektra,
wo vidpovida[ pari v osnovnomu stani. Vidpovidna velyka statystyçna suma ta vil\na enerhiq vyraxu-
vani toçno. Zvidsy vyplyva[, wo pry nyz\kyx temperaturax systema [ v kondensati par v osnovnomu
stani. Poslidovnist\ korelqcijnyx funkcij vyraxuvana toçno u termodynamiçnij hranyci i zbiha[t\sq
z vidpovidnog poslidovnistg systemy z aproksymugçym hamil\tonianom. Wilyna v spektri zbudΩen\
zaleΩyt\ neperervno vid temperatury i [ vidminnog vid nulq i na vidrizku vywe krytyçno] temperatury,
wo vidpovida[ perßij vitci spektra. Na dumku avtora, cej fakt poqsng[ fenomen ,,psevdowilyny”.
Introduction. In the series of papers [1 – 7] we investigated the eigenvalues and eigen-
vectors of the BCS Hamiltonian for system of electrons in finite cube Λ with periodic
boundary condition and in the entire space R3. It was shown that the BCS Hamilto-
nian has two branches of eigenvalues and eigenvectors — the first is well known ground
state and its excitations with corresponding eigenvalues discovered by Bardeen, Cooper
and Schrieffer [8]. Bogolyubov [9] showed that the mean energies per volume of ground
states of the BCS Hamiltonian and of the approximating Hamiltonian coincide in the
thermodynamic limit as Λ → R3 in some sense.
Recently we showed that the mean energies per volume of all the excited states of
the both Hamiltonians coincide in the thermodynamic limit, and the BCS Hamiltonian
and the approximating Hamiltonian coincide as the quadratic form in the thermodynamic
limit [5].
The second branch of eigenvalues and eigenvectors has been discovered by author
first directly for infinite system [6] and recently for finite system in cube Λ [1 – 4]. If
was shown that the eigenvalues determined in the cube Λ tend to the corresponding
eigenvalues determined in the entire R3 in the thermodynamic limit. For the second
branch of the eigenvectors mean energies per volume of the ground and the excited states
of the BCS and approximating Hamiltonians also coincide in the thermodynamic limit.
On this second branch of the eigenvectors the BCS and the approximating Hamiltonians
also coincide as the quadratic forms, in the thermodynamic limit.
Describe shortly the second branch of eigenvectors and eigenvalues directly for in-
finite system in R3. Consider the Hamiltonian H2 for wave function of one pair of
∗ The INTAS Grant 2000-15 is gratefully acknowledged.
c© D. Ya. PETRINA, 2005
1508 ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
NEW SECOND BRANCH OF SPECTRA OF THE BCS HAMILTONIAN AND “PSEUDOGAP” 1509
electrons with opposite momenta and spin
H2f(k) =
(
2k2
2m
− 2µ
)
f(k) + gv(k)
∫
v(p)f(p)dp, (1)
where m is mass of electron, µ — chemical potential, v(k)v(p) separated potential,
g — coupling constant. Denote by f0(k) the eigenvector with lowest eigenvalue E0.
f0(k) and E0 satisfy equations(
2k2
2m
− 2µ
)
f0(k) + c0v(k) = E0f0(k), c0 = g
∫
v(p)f0(p)dp, (2)
1 = g
∫
v2(p)dp
E0 −
2p2
2m
+ 2µ
.
For certain potential v(k) = v(|k|) with support in layer
∣∣∣∣ k2
2m
− µ
∣∣∣∣ < ω, ω > 0,
equation (2) for eigenvalue E0 has unique solution E0 < −2ω with gap |E0 + 2ω| =
= ∆ �= 0 and corresponding eigenvector (normalized to unity)
f0(k) =
v(k)
E0 −
2p2
2m
+ 2µ
∫
v(p)2dp(
E0 −
2p2
2m
+ 2µ
)2
− 1
2
. (3)
E0 is discrete eigenvalue. These results has been obtained by Cooper [10] and Yam-
aguchi [11].
We consider the operator H2 on functions with support in layer
∣∣∣∣ k2
2m
− µ
∣∣∣∣ < ω,
ω > 0. The Hamiltonian H2 has also eigenvectors corresponding to the continuous
spectra −2ω < E − 2µ < 2ω. Namely, some of these eigenvectors fE(k) with eigen-
value E − 2µ are orthogonal to v(k), i.e.,
∫
v(p)fE(p)dp = 0, and satisfy equations(
2k2
2m
− 2µ
)
fE(k) = (E − 2µ)fE(k), E =
p2
m
. (4)
Eigenvectors fE(k) can be represented as superposition of the following eigenfunctions:
fE,nl(k) =
(m|k|
2
)1
2 δ(|k| − |p|)
|k|2 Ynl(θ, ϕ), |n| + l ≥ 1, (5)
where Ynl(θ, ϕ) is spherical function.
For l = 0 we have
fE,00(k) =
(
m|k|
2
)1
2 δ(|k| − |p|)
|k|2 +
gv(|k|)c1
k2
m
− E − iε
,
c1 =
(
m|p|
2
) 1
2
4πv(|p|)
1 − 4πg
∫
v2(|p′|)|p′|2d|p′|
p′2
m
− E − iε
−1
. (6)
One supposes lim
ε→0
in (6). Eigenvalue E − 2µ is degenerated 2l + 1 times because
for fixed l number n takes value n = 0,±1, . . . ,±l. Eigenvalue E − 2µ corresponds
to excited pair with eigenvectors fE,nl(k).
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
1510 D. YA. PETRINA
The 2n-particle Hamiltonian or the Hamiltonian Hn for n pairs is defined as fol-
lows:
Hn = H2 ⊗ I ⊗ . . .⊗ I + . . .+ I ⊗ . . .⊗ I ⊗H2, n > 2, (7)
and it has eigenfunctions proportional to
f0(k1) . . . f0(ks)fE1(ks+1) . . . fEn−s
(kn) (8)
with eigenvalues
sE0 + E1 − 2µ+ · · · + En−s − 2µ, −2ω + 2µ ≤ Ei ≤ 2ω + 2µ, 1 ≤ i ≤ n− s.
Note that formulae (7) is crucial point in our paper. If defines the BCS Hamiltonian
in subspace of n pairs and has been discovered in our paper [6]. If means that BCS
Hamiltonian is identical to the Hamiltonians Hn(7) in subspace of n pairs and general
Hilbert space of translation invariant functions [6]. If follows from (7) that the Hamilto-
nian H2(1) introduced by Cooper [10] in connection with the theory of superconductivity
for wave function of one pair defines the BCS Hamiltonian for arbitrary numbers of n
pairs by formulae (7).
Note that we obtained wave functions of excited pairs (5), (6) with arbitrary angular
momenta l ≥ 0.
The Hamiltonians Hn, n ≥ 2, coincide with the BCS Hamiltonian [8]
H =
∫ (
p2
2m
− µ
)
a+(p̄)a(p̄)dp̄+
+
g(2π)3
V
∫
v(p)v(p′)a+(p)a+(−p)a(−p′)a(p′)dpdp′ (9)
on eigenfunctions (8) (description of notation used in (9) will be given in Section 1).
These facts have been discovered in our papers [1 – 6]. This means that n wave func-
tions of pairs of electrons with opposite momenta and spins in ground or excited states
are eigenvectors of the BCS Hamiltonian. The interaction of the BCS Hamiltonian is
only cause to create bound state of pairs of electrons, but pairs do not interact between
themselves.
Define the ground state φ0 as coherent state of pairs with wave functions f0(k) (3)
φ0 = e
∫
f0(k)a+(k)a+(−k)dk |0〉
and the BCS ground state
φa
0 = e
∫
fa
0 (k)a+(k)a+(−k)dk |0〉,
where
fa
0 (k) = −
(
(ε2(k) + c2v2(k))
1
2 − ε(k)
)1
2
(
(ε2(k) + c2v2(k))
1
2 + ε(k)
)− 1
2
,
ε(k) =
k2
2m
− µ.
The constant c is defined from condition of minimum of (φa
0 , Hφ
a
0).
The main difference between the ground state discovered by Bardeen, Cooper and
Schrieffer φa
0 and the second ground state φ0 consists in the following. The BCS ground
state φa
0 is determined from condition of minimum of mean energy (φa
0 , Hφ
a
0) for the all
coherent state of pairs φa
0 , but the second ground state φ0 is determined from condition
of minimum of mean energy (φn
0 , Hφ
n
0 ) for the all states φn
0 of n pairs of the coherent
state φ0 of pairs.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
NEW SECOND BRANCH OF SPECTRA OF THE BCS HAMILTONIAN AND “PSEUDOGAP” 1511
For these second branch of spectra we calculated exactly grand partition function
Ξ = e(2π)−3V e−βE0 exp
{
(2π)−3V
2ω∫
−2ω
e−β( 2k2
2m −2µ)
[
α
(
k2
m
)
+
+
∞∑
l=1
(2l + 1)e−β1(l+1)l
]
d
(
k2
m
− 2µ
) }
, (10)
where β is inverse temperature, β1 =
β
I
and I is inertia momenta and α
(
k2
m
)
is
some function which will be defined later. Free energy per volume is equal to
− 1
β
lim
V →∞
1
V
ln Ξ = − (2π)−3
β
(
e−βE0+
+
2ω∫
−2ω
e−β( 2k2
2m −2µ)
[
α
(
k2
m
)
+
∞∑
l=0
(2l + 1)e−β1(l+1)l
]
d
(
k2
m
− 2µ
) )
. (11)
It follows from (11) that for low temperature (β → ∞) and due to the gap in spectra of
the Hamiltonian H2 system exhibits condensation of pairs in ground state
− 1
β
lim
V →∞
1
V
ln Ξ ≈ − (2π)−3
β
e−βE0 .
We also proved that the correlation functions associated with the second branch of
eigenvalues and eigenvectors (8) of the BCS Hamiltonian coincide with the correlation
functions associated with the following approximating Hamiltonian [9]:
Happr =
∫
a+(k̄)
(
k2
2m
− µ
)
a(k̄)dk̄+
+c
∫
v(k)a+(k)a+(−k)dk + c
∫
v(k)a(−k)a(k)dk + C(c)V, (12)
where constant c is defined as follows:
c =
∫
v(p)
f0(p)e−βE0 +
2ω+2µ∫
−2ω+2µ
fE,00(p)e−β(E−2µ)α(E)dE
dp (13)
and constant C(c) is determined from condition of coincidence of grand partition func-
tions (10) of the BCS (9) and the approximating (12) Hamiltonians.
Stress that constant c is defined directly by formulae (13), it is different from zero for
arbitrary 0 ≤ β < ∞ and depends on β continuously. Recall that in the approximating
Hamiltonian that corresponds to the first branch of spectra constant c is defined from
condition of minimum of the free energy with respect to c. The condition of minimum is
reduced to certain nonlinear equation that has nontrivial solution for the temperature T
less than some critical Tc. In the case of the second branch of eigenvalues and eigenvec-
tors constant c is different from zero for all the temperatures.
If seems to us that this fact explains the phenomena of “pseudogap”. Indeed, the
eigenvalues of the approximating Hamiltonian (12) that correspond to n-particle excita-
tion with momenta p1, . . . , pn are defined through formulae
E(k1) + . . .+ E(kn), E(k1) =
√(
k2
2m
− µ
)2
+ c2v2(k),
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
1512 D. YA. PETRINA
where c2v2(k) characterizes the gap in spectra and according to (13) the gap is different
from zero for all the temperatures.
1. The model BCS Hamiltonian. 1.1. Equation for ground state. Consider the
model BCS Hamiltonian [8] for infinite cube Λ = R3
H =
∫ (
p2
2m
− µ
)
a+(p̄)a(p̄)dp̄+
+
g(2π)3
V
∫
v(p)v(p′)a+(p)a+(−p)a(−p′)a(p′)dpdp′ = H0 +HI , (1.1)
where V = V (R3) is the volume of the three-dimensional space R3, g is coupling
constant, p̄ denote momenta p and spin σ = ±1, dp̄ means integration with respect to
p and summation with respect to σ = ±1, p = (p, 1), −p = (−p,−1) a+(p̄), a(p̄)
are the operators of creation and annihilation of electrons with momenta p and spin σ.
The model Hamiltonian (1.1) has a rigorous meaning in the Hilbert space of transla-
tion-invariant functions and its spectra has been investigated in detail [4, 6]. We present
a short review of these results. We consider the Hamiltonian H (1.1) on functions with
support in layer
∣∣∣∣ k2
2m
− µ
∣∣∣∣ < ω, ω > 0.
Let us consider the following coherent state:
Φ0 = e
∫
f0(k)a+(k)a+(−k)dk |0〉 =
=
∞∑
r=0
1
r!
∫
f0(k1) . . . f0(kr) ×
×a+(k1)a+(−k1) . . . a+(kr)a+(−kr)dk1 . . . dkr |0〉 =
∞∑
r=0
1
r!
Φr
0 (1.2)
and determine the normalized to unity function f0(k) from condition that each Φr
0 is an
eigenvector of H with the lowest eigenvalue. From these conditions we obtain
r∑
i=1
(2k2
i
2m
− 2µ
)
f0(k1) . . . f0(kr) +
+
n∑
i=1
g
∫
v(k)f0(k)dkf0(k1) . . .
i
v(ki)
. . . f0(kr) =
= rE0f0(k1) . . . f0(kr), HΦr
0 = rE0Φr
0. (1.3)
Obtaining (1.3) we used the identity
(2π)3
V
δ(0) = 1, and the fact that, according to
the Fermi statistics, in Φ0 pairs with the same momenta are absent. By using the method
of separation of variables one concludes that f0(k) is solution of the equation(
2k2
2m
− 2µ
)
f0(k) + c0v(k) = E0f0(k),
c0 = g
∫
v(k)f0(k)dk,
(1.4)
and eigenvalue E0 is solution of the equation
1 = g
∫
v2(p)
E0 −
2p2
2m
+ 2µ
dp. (1.5)
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
NEW SECOND BRANCH OF SPECTRA OF THE BCS HAMILTONIAN AND “PSEUDOGAP” 1513
From (1.4) one gets normalized to unity f0(k)
f0(k) =
v(k)
E0 −
2k2
2m
+ 2µ
∫
v2(p)dp(
E0 −
2p2
2m
+ 2µ
)2
− 1
2
.
Let proceed to investigation of equation (1.5).
1.2. Equation for discrete eigenvalue. We have the following equation for eigen-
value (1.5):
1 = g
∫
v2(p)
E −
(
2p2
2m
− 2µ
)dp,
where v(p) is different from zero in layer −ω + µ ≤ p2
2m
≤ ω + µ and for the sake
of simplicity we put v2(p) =
α
|p| , α > 0. By using spherical system of coordinate
equation is performed to the following form:
1 = 2πgαm
∫
−ω+µ< p2
2m <ω+µ
2|p|
m
d|p|
E −
(
2p2
2m
− 2µ
) =
= ga
2ω∫
−2ω
dx
E − x
= |g|a
2ω∫
−2ω
dx
x− E
, (1.6)
where a = 2παm > 0.
We calculate the last integral in (1.6) supposing that E < −2ω, and obtain
1 = |g|a ln
2ω − E
−2ω − E
. (1.7)
This equation has the unique solution
E0 = 2ω
1 + e
1
a|g|
1 − e
1
a|g|
< −2ω < 0 (1.8)
that is eigenvalue corresponding to the following normalized to unity eigenfunction
f0(k) =
v(k)
E0 −
2k2
2m
+ 2µ
∫
v2(p)(
E0 −
2p2
2m
+ 2µ
)2 dp
− 1
2
. (1.9)
Consider equation (1.6) for −2ω < E < 2ω, i.e., 2ω−E > 0, −2ω−E < 0. The func-
tion ln
2ω − E
−2ω − E
is holomorphic function and for its negative argument
2ω − E
−2ω − E
< 0
equation is defined as follows:
1 = ln
2ω − E
2ω + E
+ iπ,
2ω − E
2ω + E
> 0.
If means that equation (1.7) has not solution with −2ω < E < 2ω.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
1514 D. YA. PETRINA
Show that equation (1.7) has also not solution with E > 2ω. Indeed in this case
2ω − E
−2ω − E
=
E − 2ω
E + 2ω
> 0
and equation (1.7)
1 = |g|a ln
E − 2ω
E + 2ω
has solution
E = 2ω
1 + e
1
a |g|
1 − e
1
a |g| < −2ω < 0.
But this contradicts our assumption that E > 2ω.
Now show that in general case equation
1 = g
∫
v2(p)
E − 2p2
2m
+ 2µ
dp = ϕ(E)
can have only finite number of real solutions −2ω < E < 2ω. Indeed, the right-hand
side of equation ϕ(E) is holomorphic function with respect to E in complex plane
outside the interval ImE = 0, −2ω < ReE < 2ω. For potential v(p) that is holomor-
phic function in a neighborhood of this interval the function ϕ(E) has boundary value
on this interval and is here a holomorphic function. (This easy follows from method of
holomorphic continuation by using deformation of the contour of integration.) Therefore
ϕ(E) can take the value 1 only in finite number of points on the interval ImE = 0,
−2ω < ReE < 2ω.
In what follows we will consider only the case v2(p) =
α
|p| , and in this case equation
(1.6) has unique solution (1.8) E0 < −2ω.
1.3. Eigenfunction of continuous spectra. Consider equation for eigenfunctions
corresponding to continuous spectra
H2fE(k) =
(
2k2
2m
− 2µ
)
f(k) + gv(k)
∫
v(p′)f(p′)dp̄′ = (E − 2µ)f(k), (1.10)
−2ω + 2µ < E < 2ω + 2µ.
This equation is equivalent to the following integral equation [11]:
fE(k) =
(m|k|
2
) 1
2
δ(k − p) +
gv(k)
2k2
2m
− E − iε
∫
v(p′)f(p′)dp′,
2p2
2m
= E, ε > 0.
(1.11)
Represent fE(k) and δ(k − p) as follows:
fE(k) =
∞∑
l=0
l∑
n=−l
ΨE,l(|k|)Yln(k̂)Y ∗
ln(p̂) =
=
∞∑
l=0
ΨE,l(|k|)4π(2l + 1)Pl(k̂ · p̂), |k̂| = 1, |p̂| = 1,
δ(k − p) =
∑
ln
1
k2
δ(|k| − |p|)Yln(k̂)Y ∗
ln(p̂) =
=
∞∑
l=0
1
k2
δ(|k| − |p|)4π(2l + 1)Pl(k̂ · p̂).
(1.12)
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NEW SECOND BRANCH OF SPECTRA OF THE BCS HAMILTONIAN AND “PSEUDOGAP” 1515
There Yln(k̂) is normalized spherical function, k̂ = (sin θ cosϕ, sin θ sinϕ, cos θ)
Yln(k̂) = einϕP
|n|
l (cos θ)Nln,
Nln =
(
(l − |n|)!(2l + 1)
(l + |n|)!4π
)1
2
,
P
|n|
l (ξ) = (1 − ξ2)
|n|
2
d|n|
dξ|n|
Pl(ξ),
Pl(ξ) is Legendre polinom.
Note that
∑∞
l=0
∑l
n=−l
Yln(k̂)Y ∗
ln(p̂) is δ-function δ(k̂, p̂) on sphere and if ϕ(k̂)
is arbitrary smooth function then∫ ∞∑
l=0
l∑
n=−l
Yln(k̂)Y ∗
ln(p̂)ϕ(p̂)dp̂ = ϕ(k̂),
where dp̂ is the element of unit sphere |p̂| = 1. Substituting expressions (1.12) into
(1.11) and using orthogonality Y ∗
ln(p̂′) to v(p′), l + |n| ≥ 1 one obtains
ΨE,0(|k|) =
(
m|k|
2
)1
2 δ(|k| − |p|)
k2
+
gv(k)c1
2k2
2m
− E − iε
,
c1 =
(
m|p|
2
)1
2
v(p)4π
1 − 4πg
∫
v2(p′)p′2dp′
2p′2
2m
− E − iε
−1
,
ΨE,l(k) =
(m|k|
2
)1
2 δ(|k| − |p|)
k2
.
Denote as above by Ek =
k2
m
, Ep =
p2
m
and use the following obvious formula:
δ(Ek − Ep) = mδ(k2 − p2) =
m
2|k|δ(|k| − |p|). (1.13)
It follows from (1.12) and (1.13) that
ΨE,l(|k|) =
( |m|k
2
)1
2
k2
δ(|k| − |p|) =
(
4
Ekm3
)1
4
δ(Ek − Ep), l ≥ 1. (1.14)
The eigenfunction of the operator H2 that corresponds to continuous eigenvalue
−2ω ≤ E − 2µ ≤ 2ω and orbital momenta l ≥ 1 can be represented follows:
fE,l(k, p) = ΨE,l(|k|)
l∑
n=−l
Yln(k̂)Y ∗
ln(p̂) =
=
(
4
Ekm3
)1
4
δ(Ek − Ep)
l∑
n=−l
Yln(k̂)Y ∗
ln(p̂) =
=
(
4
Ekm3
)1
4
δ(Ek − Ep)
1
4π
(2l + 1)Pl(k̂ · p̂). (1.15)
1.4. Some formulaes. Now calculate the following expression by using commutation
relations:
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1516 D. YA. PETRINA
〈0 |
∫
dp̂
∫
dk1
E2∫
E1
dEp1
(
4
Ek1m
3
)1
4
δ(Ek1 − Ep1)×
×
l∑
n1=−l
Y ∗
ln1
(k̂1)Yln1(p̂)a(−k1)a(k1)×
×
∫
dk2
E2∫
E1
dEp2
(
4
Ek2m
3
)1
4
δ(Ek2 − Ep2)×
×
l∑
n2=−l
Yln2(k̂2)Y ∗
ln2
(p̂)a+(k2)a(−k2) |0〉 =
= (2π)−3V (2l + 1)(E2 − E1), (1.16)
l ≥ 1, E1 = −2ω + 2µ, E2 = 2ω + 2µ.
If one first performs integration with respect to p̂ using orthogonality of Yln(p̂) then
one obtains the following equivalent representation of (1.16)
〈0 |
∫
dk1
E2∫
E1
dEp1
( 4
Ek1m
3
)1
4
δ(Ek1 − Ep1)a(−k1)a(k1)×
×
∫
dk2
E2∫
E1
dEp2
( 4
Ek2m
3
)1
4
δ(Ek2 − Ep2)×
×
l∑
n=−l
Y ∗
ln(k̂1)Yln(k2)a+(k2)a+(−k2)|0〉 =
= (2π)−3V (2l + 1)(E2 − E1).
The same result will be obtained if one considers more general expression
〈0 |
∫
dk1
E2∫
E1
dEp1
( 4
Ek1m
3
)1
4
δ(Ek1 − Ep1)
l∑
n1=−l
Y ∗
ln1
(k̂1)a(−k1)a(k1)×
×
∫
dk2
E2∫
E1
dEp2
( 4
Ek2m
3
)1
4
δ(Ek2 − Ep2)×
×
l∑
n2=−l
Yln2(k̂2)a+(k2)a+(−k2) |0〉 =
= (2π)−3V (2l + 1)(E2 − E1). (1.17)
We omit the same calculation as in (1.16).
It is obvious that functions
fE,l(k) =
l∑
n=−l
fE,ln(|k|) =
( 4
Ekm3
)1
4
δ(Ek − Ep)
l∑
n=−l
Yln(k̂), l ≥ 1,
fE,ln(k) =
( 4
Ekm3
)1
4
δ(Ek − Ep)Yln(k̂), n = 0,±1, . . . ,±l,
(1.18)
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NEW SECOND BRANCH OF SPECTRA OF THE BCS HAMILTONIAN AND “PSEUDOGAP” 1517
are also eigenfunctions of the operator H2 with continuous eigenvalue −2ω ≤ E ≤ 2ω,
because
∫
v(p′)
l∑
n=−l
Ynl(p̂′)dp′ = 0, l ≥ 1,
and (
2k2
2m
− 2µ
)
fE,l(k) = (Ep − 2µ)fE,l(k), Ep = Ek.
In what follows for the sake of simplicity we will use expression (1.17), (1.18) calcu-
lating different averages.
Remark that we use the factor
(
m|k|
2
) 1
2
in (1.14) in order to have in formulae
(1.16) – (1.18) (2π)−3V (2l + 1)
∫ E2
E1
dE without any factor depending on E [12].
Note that f0(k) has been found by Cooper [10], but he did not calculate eigenfunction
corresponding to continuous spectra. We also used papers of Yamaguchi [11].
1.5. Superposition of eigenfunctions corresponding to continuous spectra. Denote
by fln(k) and fl(k) functions
fln(k) =
E2∫
E1
( 4
Ekm3
)1
4
δ(Ek − E)Yln(k̂)dE, n = 0,±1, . . . ,±l,
fl(k) =
E2∫
E1
fE,l(k)dE =
E2∫
E1
( 4
Ekm3
)1
4
δ(Ek − E)
l∑
n=−l
Yln(k̂)dE =
=
l∑
n=−l
fln(k), l ≥ 1,
f0(k) =
E2∫
E1
( 4
Ekm3
)1
4
δ(Ek − E) +
gv(k)c1
2k2
2m
− E − iε
dE
that is superposition of functions fE,l(k) with respect to E with fixed unit vector p̂.
Note that f0(k) �= f0(k) because f0(k) corresponds to discrete eigenvalue E0 and
f0(k) corresponds to superposition of eigenvalues with continuous eigenvalues −2ω ≤
≤ E − 2µ ≤ 2ω and l = 0. The constant c1 was defined in Subsection 1.3.
Denote by f(k) function
f(k) =
∞∑
l=0
fl(k). (1.19)
Note that function f(k) is given by formal series (1.19) and it contains the wave
functions of excited pairs with arbitrary angular momenta.
2. Ground and excited states and grand partition function. 2.1. Ground and ex-
cited states of infinite system. The ground state Φ0 of infinite system can be represented
as follows:
Φ0 =
(
1, 0,
∫
f0(k1)a+(k1)a+(−k1)dk1 |0〉, 0, . . .
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
1518 D. YA. PETRINA
. . . ,
1
r!
∫
f0(k1)a+(k1)a+(−k1)dk1 . . .
. . .
∫
f0(kr)a+(kr)a+(−kr)dkr |0〉, 0 . . .
)
. (2.1)
The state Φ0 is the coherent state of pair of electrons with opposite momenta and
spin with wave function f0(k) that corresponds to the lowest eigenvalue E0 < 0 of H2.
It can be represented as follows:
Φ0 = exp
(∫
f(k)a+(k)a+(−k)dk
)
|0〉. (2.2)
Φ0 is the ground state of system with the Hamiltonian H, because
H
∫
f0(k1)a+(k1)a+(−k1)dk1 . . .
∫
f0(kr)a+(kr)a+(−kr)dkr |0〉 =
= rE0
∫
f0(k1)a+(k1)a+(−k1)dk1 . . .
∫
f0(kr)a+(kr)a+(−kr)dkr |0〉.
Define the following state:
Φs =
1
s!
∫
f(k1)a+(k1)a+(−k1)dk1 . . .
∫
f(ks)a+(ks)a+(−ks)dksΦ0. (2.3)
The state fE1(k1)a+(k1)a+(−k1) . . . fEs(ks)a+(ks)a+(−ks)Φr
0 is the eigenvector of
H with eigenvalue
(
(E1 − 2µ) + . . .+ (Es − 2µ) + rE0
)
. The state f(k1)a+(k1) ×
×a+(−k1) . . . f(ks)a+(ks)a+(−ks)Φr
0 is superposition of states fE1(k1)a+(k1) ×
×a+(−k1) . . . fEs(ks)Φr
0. Φs is excitation of the ground state Φ0 by s excited pairs
with wave functions f(k). Vectors Φs are orthogonal to Φ0 and themselves and their
linear combinations forms the Hilbert space of states. In this Hilbert space all the averages
will be calculated.
Define the state
Φ =
∞∑
s=0
1
s!
∫
f(k1)a+(k1)a+(−k1) . . .
∫
f(kl)a+(ks)a+(−ks)Φ0 =
= e
∫
f(k)a+(k)a+(−k)dke
∫
f0(k)a+(k)a+(−k)dk |0〉. (2.4)
Φ is excitation of the ground state Φ0 by arbitrary number of pairs with wave functions
f(k). Note that f(k) is orthogonal to f0(k) and therefore excitations are orthogonal to
Φ0. It is easy to construct the operator of creation and annihilation for which Φ0 is the
vacuum (see [2]).
Remark 2.1. Now explain how to obtain Φ0, Φs and Φ for infinite system in R3
from those for finite system in cube Λ with center at the origin of coordinate. For system
situated in Λ the ground state is defined as follows by analogy with the BCS ground
state [8]
ΦΛ
0 =
∏
k
(1 + f0(k)a+
k a
+
(−k)) |0〉 =
(
1, 0,
∑
k
f0(k1)a+
k1
a+
−k1
|0〉, 0, . . .
. . . ,
1
r!
∑
k1 �=... �=kr
f0(k1)a+
k1
a+
−k1
. . . f0(kr)a+
kr
a+
−kr
|0〉, . . .
)
,
ΦΛ
s =
1
s!
∑
k1 �=... �=ks
f(k1)a+
k1
a+
−k1
. . . f(ks)a+
ks
a+
−ks
ΦΛ
0 ,
ΦΛ =
∞∑
s=0
1
s!
∑
k1 �=... �=ks
f(k1)a+
k1
a+
−k1
. . . f(ks)a+
ks
a+
−ks
ΦΛ
0 ,
(2.5)
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NEW SECOND BRANCH OF SPECTRA OF THE BCS HAMILTONIAN AND “PSEUDOGAP” 1519
where k is quasimomenta k =
2π
L
n, n = (n1, n2, n3), ni ∈ Z, and L is length of the
edge of the cube Λ. Note that in (2.5) summation is carried out over all k1 �= . . . �= kr
and k1 �= . . . �= ks. We have taken into account that pairs of electrons cannot have
the same momenta, but they can have the same wave functions because the operators
a+
ki
a+
−ki
, a+
kj
a+
−kj
commute for ki �= kj .
In order to obtain Φ0, Φs, Φ for infinite system in R3 one formally replaces the
operators a+
k , a
+
−k by a+(k), a+(−k) and sums
∑
k
by integrals
∫
dk according to
the formulas
a+(±k) = lim
V →∞
( V
(2π)3
)1
2
a+
±k, lim
V →∞
(2π)3
V
∑
k
f(k) =
∫
f(k)dk.
It is easy to see that ΦΛ
0 , ΦΛ
s , ΦΛ (2.5) become Φ0, Φs, Φ (2.2) – (2.4) in this
limit.
2.2. Grand partition function. Eigenvalues E1 ≤ E ≤ E2 are degenerated by
angular momenta l = 0, 1, 2, . . . . Each l is again degenerated (2l + 1) time. The
eigenfunction fE,l(k) corresponds to energy E and above described angular momenta.
According to law of quantum statistical mechanics it is necessary to take into account
the angular momenta together with energy. Denote by M the operator of angular mo-
mentum. In spherical system of coordinate
M2 =
1
sin θ
∂
∂θ
(
sin θ
∂
∂θ
)
+
1
sin2 θ
∂2
∂ϕ2
, M2Yln(k̂) = (l + 1)lYln(k̂),
k̂ = (cos θ cosϕ, cos θ sinϕ, sin θ).
Denote by
M2
s = M2 ⊗ I ⊗ . . .⊗ I + . . .+ I ⊗ . . .⊗M2 (2.6)
and by
M̂2 =
∞∑
s=1
M2
s . (2.7)
Recall that
M2
s (f(k1) . . . f(ks)) = (M2f(k1)) . . . f(ks) + . . .+ f(k1) . . . (M2f(ks)). (2.8)
Denote by β the inverse temperature, and by β1 =
β
2Il
, where Il is inertia momenta
of pair with angular momenta l.
In statistical averages of considered system at the inverse temperature β one should
use the operator e−β(H + 1
2I M̂
2) = e−βH − β1M̂
2
instead of the operator e−βH.(
Recall that I, β1 =
β
2I
depend on l when the operator M̂2 acts on function with an-
gular momentum l, for the sake of simplicity one omits sign l in β1 and often in I.
)
Consider expression
(Φ, e−βH−β1M̂2
Φ) = 〈0|
∞∑
s1=0
1
s1!
∫
f(k1)a(−k1)a(k1)dk1 . . .
. . .
∫
f(ks1)a(−ks1)a(ks1)dks1×
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1520 D. YA. PETRINA
×
∞∑
r1=0
1
r1!
∫
f0(p1)a(−p1)a(p1)dp1 . . .
. . .
∫
f0(pr1)a(−pr1)a(pr1)dpr1×
×e−βH−β1M2
∞∑
s=0
1
s!
∫
f(k′1)a
+(k′1)a
+(−k′1)dk′1 . . .
. . .
∫
f(k′s)a
+(k′s)a
+(−k′s)dk′s×
×
∞∑
r=0
1
r!
∫
f0(p′1)a
+(p′1)a
+(−p′1)dp′1 . . .
. . .
∫
f0(p′r)a
+(p′r)a
+(−p′r))dpr |0〉. (2.9)
It is obvious that expression (2.9) is grand partition function Ξ because (s + n) -
particle states are invariant with respect to action of the Hamiltonian H and operator
M̂2. Therefore in (Φ, e−βH−β1M̂2
Φ) different from zero contributions belong to equal
s = s1, r = r1, l = l1 and E = E′, but it is the grand partition function. Thus
we have taken into account pairs in ground state and all excited states of pairs with all
energies and angular momenta and therefore
Ξ = (Φ, e−βH−β1M̂2
Φ) = Tre−βH−β1M̂2
. (2.10)
Now calculate Ξ. We begin with calculation of some integrals
〈0 |
∫
f0(k1)a(−k1)a(k1)dk1
∫
f0(k2)a+(k2)a+(−k2)dk2 |0〉 =
= V
∫
f0(k)2dk = V,
∫
f0(k)2dk = 1,
(2.11)
〈0 |
∫
E2∫
E1
dE
(
4
Ekm3
)1
4
δ(Ek − E)Y ∗
ln(k̂1)a(−k1)a(k1)
dk1×
×
∫
f0(k2)a+(k2)a+(−k2)dk2 |0〉 = 0, l + |n| ≥ 1.
In these equalities we used orthogonality f0(k) to Yln(k̂), l + |n| ≥ 1. We will also
use equality (1.16) or equivalent equality (1.17) with fE,l(k) (1.18), and orthogonality
f0(k) and f0(k) that corresponds to different eigenvalues of H2.
It is obvious that
e−βH−β1M̂2 1
s!
(∫
fln(k)a+(k)a+(−k)dk
)s
×
× 1
r!
(∫
f0(k)a+(k)a+(−k)dk
)r
|0〉 =
=
1
s!
( ∫
fln(k1)a+(k1)a+(−k1)dk1 . . .
. . .
∫
fln(ks)a+(ks)a+(−ks)e
s∑
i=1
[−β(
2k2
i
2m −2µ)−β1(l+1)l]
dks
)
×
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NEW SECOND BRANCH OF SPECTRA OF THE BCS HAMILTONIAN AND “PSEUDOGAP” 1521
× 1
r!
( ∫
f0(ks+1)a+(ks+1)a+(−ks+1)dks+1 . . .
. . .
∫
f0(ks+r)a+(ks+r)a+(−ks+r)dks+re
−βE0r
)
|0〉 =
=
1
s!
(∫
fln(k)e−β( 2k2
2m −2µ)e−β1(l+1)la+(k)a+(−k)dk
)s
×
× 1
r!
(∫
f0(k)e−βE0a+(k)a+(−k)dk
)r
|0〉. (2.12)
We used formulae
e−β1M̂2
Yln(k̂) = e−β1(l+1)lYln(k̂), n = 0,±1, . . . ,±l.
If one considers flini(ki) with different lini, i = 1, . . . , s, then one will have in (2.12)
1
s!
(∫
fl1n1(k1)e−β(
2k2
1
2m −2µ)e−β1(l1+1)l1a+(k1)a+(−k1)dk1 . . .
. . .
∫
flsns(ks)e−β(
2k2
s
2m −2µ)e−β1(ls+1)lsa+(ks)a+(−ks)dks
)
.
Recall that spectra of the operator M2 with eigenvalue (l + 1)l is degenerated 2l + 1
times. Using formulas (1.16), (2.11), (2.12) one obtains
Ξ = (Φ, e−βH−β1M̂2
Φ) = 〈0 |
∞∑
s1=0
1
s1!
(∫
f(k)a(−k)a(k)dk
)s1
×
×
∞∑
r1=0
1
r1!
(∫
f0(k)a(−k)a(k)dk
)r1
×
×
∞∑
s=0
1
s!
∫
dk
E2∫
E1
dE
∑
l+|n|≥0
fE,ln(k)e−β(E−2µ)−β1(l+1)la+(k)a+(−k)dk
s
×
×
∞∑
r=0
1
r!
(∫
f0(k)e−βE0a+(k)a+(−k)dk
)r
=
=
∞∑
s=0
1
s!
(2π)−3V
E2∫
E1
e−β(E−2µ)
(
α(E) +
∞∑
l=1
(2l + 1)e−β1(l+1)l
)
dE
s
×
×
∞∑
n=0
1
n!
((2π)−3V e−βE0)n =
= exp
(2π)−3V
E2∫
E1
e−β(E−2µ)
(
α(E) +
∞∑
l=1
(2l + 1)e−β1(l+1)l
)
dE
×
× exp((2π)−3V e−βE0) =
= exp
[
(2π)−3V
2ω∫
−2ω
e−β( k2
m −2µ)
(
α
(k2
m
)
+
∞∑
l=1
(2l + 1)e−β1(l+1)l
)
×
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1522 D. YA. PETRINA
×d
(
k2
m
− 2µ
) ]
exp((2π)−3V e−βE0),
(2.13)
E1 = −2ω + 2µ, E2 = 2ω + 2µ.
Note that factor 2l+1 is connected with degeneracy of eigenvalue (l+1)l by 2l+1
time (n = 0,±1, . . . ,±l).
Remark 2.2. We calculated exactly factor e−β(E − 2µ) only for l ≥ 1 (see (1.16),
(1.17)). Show that for l = 0 we have the factor e−β(E − 2µ)α(E) where function
α(E) will be defined below. If follows from formulae
∫
E2∫
E1
Ψ̄E′,0(|k|)dE′
∫
ΨE,0(|k|)e−β(E−2µ)dE
d|k| =
=
E2∫
E1
dE′
E2∫
E1
dEe−β(E−2µ)
{∫
Ψ̄E′,0(|k|)ΨE,0(|k|)d|k|
}
=
=
E2∫
E1
dE′
E2∫
E1
dEe−β(E−2µ)δ(E′ − E)α(E′) =
E2∫
E1
dEe−β(E−2µ)α(E).
We used condition of orthogonality of eigenvectors ΨE,0(|k|)∫
Ψ̄E′(|k|)ΨE(|k|)d|k| = δ(E′ − E)α(E′).
We can not prove directly that α(E) = 1. Thus we calculated exactly the grand par-
tition function for system of pairs with one ground state with eigenvalue E0 and excited
states with continuous spectra in interval −2ω < E−2µ < 2ω. This system of pairs can
be considered as system of unpenetrated bosons in momentum space, because according
to the Fermi statistics pairs can not occupy the same momenta. We used extensively that
state Φ is the coherent state of pairs in ground and excited states.
Note that obtained grand partition function does not coincide with those for Bose –
Einstein or Fermi – Dirac statistics. It is the grand partition function of system of nonin-
teracting pairs of electrons with opposite momenta and spins. We have taken into account
that pairs of electrons cannot have the same momenta, but they can have the same wave
functions because the operators a+
ki
a+
−ki
, a+
kj
a+
−kj
commute for ki �= kj .
2.3. Definition of energy connected with orbital momenta. Now we proceed to
calculate inertia momenta. Use the following formula [12] (see formulae (2.57))
eik̄r =
4π
kr
(
mk
2
) 1
2 ∑
ln
ilul(kr)Ynl(r̂)Y ∗
ln(k̂) =
=
1
kr
(
mk
2
)1
2 ∑
l=0
ilul(kr)(2l + 1)Pl(r̂ · k̂), |k̂| = 1, |r̂| = 1, (2.14)
where
ul(kr) =
(
1
2
πkr
)1
2
jl+ 1
2
(kr)
and jl+ 1
2
(kr) is the Bessel function, Pl(r̂ · k̂) is the Legendre polinom.
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NEW SECOND BRANCH OF SPECTRA OF THE BCS HAMILTONIAN AND “PSEUDOGAP” 1523
Function ul(kr) has the following asymptotic expression:
ul(z) ≈
z→∞
sin(z − 1
2
πl), ul(z) ≈
z→0
zl+1
(2l + 1)!!
. (2.15)
Recall that
∣∣∣∣4πkr (
mk
2
)
1
2ul(kr)
∣∣∣∣
2
r2 is the density of probability with respect to r for
given l, n. Now define energy connected with orbital momenta l :
(l + 1)l
Il
= (l + 1)l
∞∫
0
mk
2r2
(4π
kr
)2
|ul(kr)|2r2dr =
= (l + 1)l
∞∫
0
(4π
kr
)2mk
2
|ul(kr)|2dr, l ≥ 1. (2.16)
It follows from asymptotic behavior of function ul(kr) (2.14) that integral (2.15) is con-
vergent and inertia momenta Il depend on l. Moreover, Il can be calculated exactly.
We have (see [13, p. 443], §13.42, formulae (1))
(l + 1)l
Il
= (l + 1)l
∞∫
0
(
4π
kr
)2
mk
2
1
2
πkr|jl+ 1
2
(kr)|2dr =
=
(l + 1)l(2π)3m
2
∞∫
0
|jl+ 1
2
(kr)|2
r
dr =
(l + 1)l(2π)3m
2(2l + 1)
. (2.17)
It follows from (2.17) that
1
Il
=
(2π)3m
2(2l + 1)
. (2.18)
This means that inertia momenta
Il =
2(2l + 1)
(2π)3m
is proportional to 2l + 1 and it grows together with l. The series
∞∑
l=1
(2l + 1)e−
β(l+1)l
Il =
∞∑
l=1
(2l + 1)e−
β(l+1)l(2π)3m
2(2l+1)
is absolutely convergent.
2.4. Free energy. As known the free energy per volume is defined as follows:
− 1
β
lim
V →∞
1
V
ln Ξ =
= − (2π)−3
β
e−βE0 +
E2∫
E1
e−β(E−2µ)
(
α(E) +
∞∑
l=1
(2l + 1)e−β1(l+1)l
)
dE
. (2.19)
Consider asymptotic of the free energy in limit of low temperature β → ∞. It is obvious
that due to the fact that E0 < 0 we have for β → ∞
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1524 D. YA. PETRINA
− 1
β
1
V
ln Ξ ∼= − (2π)−3
β
(
e−βE0 +
0∫
−2ω
e−β( k2
m −2µ) ×
×
(
α
(
k2
m
)
+
∞∑
l=1
(2l + 1)e−β1(l+1)l
)
d
(
k2
m
− 2µ
) )
. (2.20)
It follows from (2.20) that for β → ∞ system exists in layer −2ω ≤ 2k2
2m
− 2µ ≤ 0
below Fermi sphere. If one takes into account that E0 is divided from the continuous
spectra by gap ∆ = min
E
(E0 − E) < 0 then
− 1
β
1
V
ln Ξ ∼= − (2π)−3
β
e−βE0
(
1 +
0∫
−2ω
eβ(E0− k2
m +2µ)
(
α
(
k2
m
)
+
+
∞∑
l=1
(2l + 1)e−β1(l+1)l
))
d
(
k2
m
− 2µ
)
≈ e−βE0 . (2.21)
This means that for low temperature
1
V
ln Ξ ∼= e−βE0 and system is in state of pairs of
ground state, i.e., we have condensation of pairs in ground state. May be this phenomena
of condensation explains hypothesis of Schafroth, Butler and Blatt [14]. Note that this
condensation of pairs in ground state is different from Bose – Einstein condensation of
free boson system. There does not exists any critical temperature. Note that we are able
to calculate the grand partition function and free energy per volume directly for infinite
volume. We obtained complete and detailed description of eigenvectors and eigenvalues
of the BCS Hamiltonian and it permits us to prove that the grand partition function is
exponent which depends on volume multiplied by finite expression proportional to the
free energy.
3. Equation for correlation functions and “pseudogap”. 3.1. Equation for corre-
lation functions. Consider the operator a(p̄), a+(p̄) in the Heisenberg representation
a(t, p̄) = ēiHta(p̄)eiHt, a+(t, p̄) = e−iHta+(p̄)eiHt. (3.1)
They satisfy the Heisenberg equations
i
∂a(t, p)
∂t
=
=
(
p2
2m
− µ
)
a(t, p) + v(p)a+(t,−p) (2π)3
V
∫
v(p′)a(t,−p′)a(t, p′)dp′,
−i∂a
+(t, p)
∂t
=
=
(
p2
2m
− µ
)
a+(t, p) +
(2π)3
V
∫
v(p′)a+(t, p′)a+(t,−p′)dp′v(p)a(t,−p).
(3.2)
It is easy to derive equations for a(t,−p), a+(t,−p).
Consider the following correlation functions:
〈a+(t1, p1)a(t2, p2)〉 =
1
Ξ
Tr
(
a+(t1, p1)a(t2, p2)e−βH−β1M̂2
)
,
〈a+(t1, p1)a+(t2, p2)〉 =
1
Ξ
Tr
(
a+(t1, p1)a+(t2, p2)e−βH−β1M̂2
)
,
〈a(t1, p1)a(t2, p2)〉 =
1
Ξ
Tr
(
a(t1, p1)a(t2, p2)e−βH−β1M̂2
)
,
(3.3)
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NEW SECOND BRANCH OF SPECTRA OF THE BCS HAMILTONIAN AND “PSEUDOGAP” 1525
and derive for them equations using the Heisenberg equations (3.1). Begin with the first
correlation function (3.3). One obtains
i
∂
∂t2
〈a+(t1, p1)a(t2, p2)〉 =
(
p2
2
2m
− µ
)
〈a+(t1, p1)a(t2, p2)〉 +
+v(p2)
(2π)3
V
∫
v(p′)〈a+(t1, p1)a+(t2,−p2)a(t2,−p′)a(t2, p′)〉dp′. (3.4)
Note that average 〈a+(t1, p1)a+(t2,−p2)a(t2,−p′)a(t2, p′)〉 is invariant with re-
spect to transformation t1 → t1 + t0, t2 → t2 + t0 with arbitrary t0. For example,
substitute in the average operators e−iH(−t2)a+(t1, p1)eiH(−t2), e−iH(−t2) ×
×a+(t2,−p2)eiH(−t2), e−iH(−t2)a(t2,−p′)eiH(−t2), e−iH(−t2)a(t2, p′)eiH(−t2) in-
stead of the operators a+(t1, p1), a+(t2,−p2), a(t2,−p′), a(t2, p′) and use that av-
erage is invariant with respect to cyclic permutation of operators. Then one obtains the
equality
〈a+(t1, p1)a+(t2,−p2)a(t2,−p′)a(t2, p′)〉 =
= 〈a+(t1 − t2, p1)a+(0, p2)a(0,−p′)a(0, p′)〉, (3.5)
a+(0, p2) = a+(p2), a(0,−p′) = a(−p′), a(0, p′) = a(p′) and substitutes it in (3.4).
Proceed to investigate the second term in the right-hand side of (3.4).
Consider expression
(2π)3
V
∫
v(p′)a(−p′)a(p′)dp′e−βH−β1M̂2
Φ =
=
(2π)3
V
∫
v(p′)a(−p′)a(p′)dp′e−βH−β1M̂2×
×
∞∑
s=0
1
s!
(∫
f(k)a+(k)a+(−k)dk
)s
×
×
∞∑
r=0
1
r!
(∫
f0(k)a+(k)a+(−k)dk
)r
|0〉 =
=
(2π)3
V
∫
v(p′)a(−p′)a(p′)dp′×
×
∞∑
s=0
1
s!
(∫
dk
E2∫
E1
dE
∑
l+|n|≥0
fE,ln(k)×
×e−β(E−2µ)−β1(l+1)la+(k)a+(−k)
)s
×
×
∞∑
r=0
1
r!
(∫
f0(k)e−βE0a+(k)a+(−k)dk
)r
|0〉 =
=
∫
v(p′)
f0(p′)e−βE0 +
E2∫
E1
fE,0(p′)e−β(E−2µ)α(E)dE
dp′×
×
∞∑
s=0
1
s!
( ∫
dk
E2∫
E1
dE
∑
l+|n|≥0
fE,ln(k)×
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1526 D. YA. PETRINA
×e−β(E−2µ)−β1(l+1)la+(k)a+(−k)
)s
×
×
∞∑
r=0
1
r!
(∫
f0(k)e−βE0a+(k)a+(−k)dk
)r
|0〉 =
= ce−βH−β1M̂2
∞∑
s=0
1
s!
∫
dk
2ω∫
−2ω
dE
∑
l+|n|≥1
fE,ln(k)a+(k)a+(−k)
s
×
×
∞∑
r=0
1
r!
(∫
f0(k)a+(k)a+(−k)dk
)r
|0〉 = ce−βH−β1M̂2
Φ,
(3.6)
c =
∫
v(p)
f0(p)e−βE0 +
E2∫
E1
fE,0(p)e−β(E−2µ)α(E)dE
dp.
We used orthogonality of v(k) to fE,ln(k) with l ≥ 1.
Now substitute obtained expression in the second term of the right-hand side of (3.4).
One obtains
v(p2)
Ξ
(2π)3
V
∫
v(p′)(Φ, a+(t1, p1)a+(t2,−p2)a(t2,−p′)a(t2, p′)e−βH−β1M̂2
Φ) =
= c
v(p2)
Ξ
(Φ, a+(t1 − t2, p1)a+(0,−p2)e−βH−β1M̂2
Φ) =
= cv(p2)〈a+(t1, p1), a+(t2, p2)〉.
We used again that averages are invariant with respect to cyclic permutation of operators
and therefore 〈a+(t1 − t2, p1)a+(0, p2)〉 = 〈a+(t1, p1)a+(t2, p2)〉.
Taking the last formula into account one obtains equation (3.4) in the final form
i
∂
∂t2
〈a+(t1, p1)a(t2, p2)〉 =
=
(
p2
2
2m
− µ
)
〈a+(t1, p1)a(t2, p2)〉 + cv(p2)〈a+(t1, p1)a+(t2,−p2)〉. (3.7)
Derive equation for the third correlation function (3.3). One obtains
i
∂
∂t2
〈a(t1, p1)a(t2, p2)〉 =
=
(
p2
2
2m
− µ
)
〈a(t1, p1)a(t2, p2)〉 +
+ v(p2)
(2π)3
V
∫
v(p′)〈a(t1, p1)a+(t2,−p2)a(t2,−p′)a(t2, p′)〉dp′. (3.8)
By using formulae (3.6) in the second term of the right-hand side of (3.8) one represent
equation (3.8) in the following form:
i
∂
∂t2
〈a(t1, p1)a(t2, p2)〉 =
=
(
p2
2
2m
− µ
)
〈a(t1, p1)a(t2, p2)〉 + cv(p2)〈a(t1, p1)a+(t2,−p2)〉. (3.9)
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NEW SECOND BRANCH OF SPECTRA OF THE BCS HAMILTONIAN AND “PSEUDOGAP” 1527
Equation for the second correlation function one obtains by conjugation of the equation,
for 〈a(t1, p1)a(t2, p2)〉
−i ∂
∂t2
〈a+(t2, p2)a+(t1, p1)〉 =
=
( p2
2
2m
− µ
)
〈a+(t2, p2)a+(t1, p1)〉 + c̄v(p2)〈a(t2,−p2)a+(t1, p1)〉. (3.10)
Recall that fE,0(k) is complex function and therefore constant c is complex number.
3.2. Cluster property and equation for Green functions. In the second term of
the right-hand side of (3.4) one is faced with the problem of giving a meaning to the
integral with inverse volume V. The analogical problem has been already solved for
equations for correlation function for models of superconductivity, superfluidity, Huang –
Yang – Luttinger model and other model [15]. We will follow the method used for above
mentioned models.
Namely we suppose that correlation functions satisfy the cluster property. For
example
〈a+(t1, p1)a+(t2, p2)a(t2,−p′)a+(t2, p′)〉 =
= 〈a+(t1, p1)a+(t2, p2)〉〈a(t2,−p′)a(t2, p′)〉 + . . . , (3.11)
where the rest terms consist from sum of all possible two particle correlation functions.
If one substitutes the right-hand side of (3.11) in (3.4) then one obtains
i
∂
∂t2
〈a+(t1, p1)a(t2, p2)〉 =
=
( p2
2
2m
− µ
)
〈a+(t1, p1)a(t2, p2)〉 + cv(p2)〈a+(t1, p1)a+(t2,−p2)〉,
where
c =
(2π)3
V
∫
v(p′)〈a(o,−p′1)a(0, p′)〉dp′. (3.12)
It was used that correlation function 〈a(t2,−p′1)a(t2, p′)〉 does not depend on time
( it depends on difference of times of operators a(t2,−p′) and a(t2, p′)).
One can calculate
(2π)3
V
∫
v(p′)〈a(0,−p′)a(0, p′)〉dp′ exactly as in Subsection 3.1
(Φ,
(2π)3
V
∫
v(p′)a(0,−p′)a(0, p′)dp′e−βH−β1M̂2
Φ)
(Φ, e−βH−β1M̂2Φ)
= c.
Thus we obtained the same result as before in Subsection 3.1.
4. Model and approximating Hamiltonian. 4.1. Model Hamiltonian on state Φ.
Consider model BCS Hamiltonian on the state Φ
HΦ =
[ ∫
a+(k̄)
(
k2
2m
− µ
)
a(k̄)dk̄+
+
(2π)3
V
∫ ∫
v(k)a+(k)a+(−k)v(k′)a(−k′)a(k′)dkdk′
]
×
×e
∫
f(k)a+(k)a+(−k)dke
∫
f0(k)a+(k)a+(−k)dk |0〉 =
=
[∫
a+(k̄)
( k2
2m
− µ
)
a(k̄)dk̄ + c
∫
v(k)a+(k)a+(−k)dk
]
Φ, (4.1)
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1528 D. YA. PETRINA
where
c =
∫
v(p)
f0(p) +
E2∫
E1
dEfE,0(p)
dp. (4.2)
Calculating HΦ one uses that
(2π)3
V
∫
v(k′)a(−k′)a(k′)dk′
∫
f(k)a+(k)a+(−k)dk |0〉 =
=
(2π)3
V
∫
δ(k − k)v(k)f(k)dk |0〉 =
V
V
∫
v(k)
E2∫
E1
fE,0(k)dE
dk
because
∫
fE,l(k)v(k)dk = 0 for l ≥ 1. It was also used that
(2π)3
V
∫
v(k′)a(−k′)a(k′)dk′
∫
f0(k)a+(k)a+(−k)dk |0〉 =
=
(2π)3
V
∫
δ(k − k)v(k)f0(k)dk |0〉 =
V
V
∫
v(k)f0(k)dk |0〉.
Taking into account that
Φ = e
∫
f(k)a+(k)a+(−k)dke
∫
f0(k)a+(k)a+(−k)dk |0〉
one obtains that
(2π)3
V
∫
v(k′)a(−k′)a(k′)dk′Φ = cΦ. (4.3)
4.2. Approximating Hamiltonian on state Φ. Define the approximating Hamiltonian
Ha =
∫
a+(k̄)
(
k2
2m
− µ
)
a(k̄)dk̄+
+c
∫
v(k)a+(k)a+(−k)dk + c̄
∫
v(k)a(−k)a(k)dk − |c|2V, (4.4)
where c is defined according to (4.2). Show that
HΦ = HaΦ. (4.5)
The action HΦ was already calculated (4.1), (4.2). To prove (4.5) it is sufficient to show
that (
c̄
∫
v(k)a(−k)a(k)dk − |c|2V
)
Φ = 0, (4.6)
but it is simple consequence of (4.3).
Remark 4.1. It is obvious that
Φ = e
∫
f(k)a+(k)a+(−k)dke
∫
f0(k)a+(k)a+(−k)dk |0〉 = e
∫
[f(k)+f0(k)]a+(k)a+(−k)dk |0〉.
Include f0(k) in f0(k) and introduce the functions
f̃0(k) = f0(k) + f0(k), f̃(k) =
∞∑
l=1
f l(k),
and define the following vectors:
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NEW SECOND BRANCH OF SPECTRA OF THE BCS HAMILTONIAN AND “PSEUDOGAP” 1529
Φ̃0 = e
∫
f̃0(k)a+(k)a+(−k)dk |0〉,
Φ̃s =
1
s!
∫
f̃(k1)a+(k1)a+(−k1)dk1 . . .
∫
f̃(ks)a+(ks)a+(−ks)dksΦ̃0, s ≥ 0.
The vector Φ can be represented as follows:
Φ =
∞∑
s=0
Φ̃s = e
∫
f̃(k)a+(k)a+(−k)dke
∫
f̃0(k)a+(k)a+(−k)dk |0〉 =
= e
∫
[f̃(k)+f̃0(k)]a+(k)a+(−k)dk |0〉 = e
∫
[f(k)+f0(k)]a+(k)a+(−k)dk |0〉 =
= e
∫
f(k)a+(k)a+(−k)dke
∫
f0(k)a+(k)a+(−k)dk |0〉.
We have proved that HΦ = HaΦ, but we also have HΦ̃s = HaΦ̃s, s ≥ 0, because
(2π)3
V
∫
v(p)a(−p)a(p)dpΦ̃s = cΦ̃s,
∫
v(p)f̃(p)dp = 0,
where as above c =
∫
v(p)[f0(p)+f0(p)]dp, and it was used that
∫
v(p)f̃(p)dp = 0.
This means that H coincides with Ha not only on vectors Φ and Φ̃0 but also on
all the excitations Φ̃s, s ≥ 1, of the state Φ̃0. Excitations Φ̃s are orthogonal to Φ̃0
and to themselves and linear combinations of Φ̃0, Φ̃s, s ≥ 1, form the Hilbert space of
states. On this Hilbert space the BCS and approximating Hamiltonians coincide.
The approximating Hamiltonian has its own branch of eigenvalues and eigenvectors.
It is operator — quadratic form of operators of creations and annihilations and therefore
it can be diagonalized
Ha =
∫
E(k)α+(k̄)α(k̄)dk̄ + C(c)V, (4.7)
where
E(k) =
√
ε(k)2 + c2v2(k), C(c) =
∫ [
ε(k) −
√
ε(k)2 + c2v2(k)
]
dk − c2, (4.8)
ε(k) =
k2
2m
− µ.
The operators α+(k̄), α(k̄) satisfy the same canonical anticommutation relations as
the operators a+(k̄), a(k) and are expressed through the operators a+(k̄), a(k) by the
following formulae [9, 16]:
α+(k) = u(k)a+(k) + w(k)a(−k), α+(−k) = u(k)a+(−k) − w(k)a(k),
α(k) = u(k)a(k) + w(k)a+(−k), α(−k) = u(k)a(−k) − w(k)a(k),
(4.9)
u(k) =
1√
2
(1 + ε(k)(ε2(k) + c2v2(k))−
1
2 )
1
2 ,
w(k) =
1√
2
(1 − ε(k)(ε2(k) + c2v2(k))−
1
2 )
1
2 .
The approximating Hamiltonian Ha (4.4) has the following eigenvectors:
Φa
0 = e
∫
fa
0 (k)a+(k)a+(−k)dk |0〉, (4.10)
where
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1530 D. YA. PETRINA
fa
0 (k) = −
(
(ε2(k) + c2v2(k))
1
2 − ε(k)
) 1
2
(
(ε2(k) + c2v2(k))
1
2 + ε(k)
)− 1
2
. (4.11)
Eigenvector Φa
0 is the vacuum for operators α+(k̄), α(k̄) because α(k̄)Φa
0 = 0,
and eigenvector for Ha
HaΦa
0 = C(c)V Φa
0 . (4.12)
The operator Ha has also eigenvectors α+(k̄1)α . . . α+(k̄n)Φa
0
Haα
+(k̄1) . . . α+(k̄n)Φa
0 = (E(k1) + . . .+ E(kn))α+(k̄1) . . . α+(k̄n)Φa
0 . (4.13)
(Note that in this case the constant c is defined from condition of minimum of the func-
tion C(c) and it is different from that defined according to (4.2).)
These eigenvectors are n-particle excitations of Φa
0 . The eigenvectors (4.11), (4.14)
with k̄i �= −k̄j , (i, j) ⊂ (1, . . . , n) are also eigenvectors of the operator H with the
same eigenvalues (see [8, 9]).
To eigenvectors (4.14) corresponds the following grand partition function
Ξa = e
V
∫ dp
eβE(p)+1
+V C(c)
,
i.e., the grand partition function of free system of fermions with energy E(p). Recall
again that in this subsection the constant c is defined according to (4.2).
Remark 4.2. Note that the wave function of one pair of electrons with opposite mo-
menta and spin fa
0 (k) is not the eigenfunction of the two particle Hamiltonian H2. The
wave function of n ≥ 2 pairs fa
0 (k1) . . . fa
0 (kn) also are not the eigenfunction of the
BCS Hamiltonian. Only the coherent state of pairs Φa
0 is eigenfunction of the BCS
Hamiltonian in the following sense [5, 9]
lim
V →∞
1
V
(
Φa
0,Λ, HΛΦa
0,Λ
)
= lim
V →∞
1
V
(
Φa
0,Λ, Ha,ΛΦa
0,Λ
)
,
where Φa
0,Λ, Ha,Λ, HΛ are the restriction of Φa
0 , Ha, H in a cube Λ. This formula is
also true if one puts excitations (4.14) instead of Φa
0 .
Usually in presentation of the theory of superconductivity authors begin with equa-
tion (2) for the Cooper pair in the ground state f0(k), but later continue with coherent
state (4.11), from condition of minimum of
(
Φa
0,Λ, HΛΦa
0,Λ
)
define fa
0 (k) (4.12) that is
different from the wave function of the original Cooper pair f0(k) and is not eigenfunc-
tion of H2. They also consider excitation of the ground state Φa
0 (see for example [17]).
It is surprise that the eigenvalue problem has not been considered for H in 2n-
particle space in spite of promise made by Cooper in his pioneering paper [10]. This
problem has solved in our papers [1 – 6].
4.3. Approximating Hamiltonian and Green functions. Consider the approximating
Hamiltonian (4.4) but with parameter c defined as in equations for Green functions
c =
∫
v(p)
f0(p)e−βE0 +
E2∫
E1
fE,0(p)e−β(E−2µ)α(E)dE
dp. (4.14)
It is easy to check that equations (3.7) – (3.10) for correlations functions of the BSC
model Hamiltonian H completely coincide with equation for the same correlation func-
tion of the approximating Hamiltonian Ha (4.4) but with the constant (4.15).
We omit almost obvious calculation for arbitrary correlation functions and simple
proof that the equations for them for the BCS model Hamiltonian H coincide with the
corresponding equations for the approximating Hamiltonian Ha.
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NEW SECOND BRANCH OF SPECTRA OF THE BCS HAMILTONIAN AND “PSEUDOGAP” 1531
Remark 4.3. The sequence of correlation functions do not depend on the last fourth
term in (4.4). In our case of the second branch of eigenvalues and eigenvectors the fourth
term should be determined from condition of coincidence of the free energy of the Hamil-
tonian (4.4) (with unknown fourth term) with the free energy (2.19).
4.4. Pseudogap. We have proved that equations for correlation functions of the model
with the BCS Hamiltonian H and those equations of the model with approximating
Hamiltonian Ha with constant c (4.15) coincide. This means that their solutions, i.e.,
correlation functions also coincide if initial data coincide. In this sense model with the
BCS Hamiltonian H and model with the approximating Hamiltonian Ha are thermody-
namically equivalent, i.e., their states, described by correlation functions, coincide.
At first sight we obtained well known result, first estalilished by Bogolyubov [9] for
zero temperature and by Bogolyubov [16] for arbitrary temperatures concerning thermo-
dynamic equivalence of the BCS and approximating Hamiltonians.
In fact there is one fundamental difference. It consists in the following. In our case
for the second branch of eigenvalues and eigenvectors the constant
c =
∫
v(p)
f0(p)e−βE0 +
E2∫
E1
fE,0(p)e−β(E−2µ)α(E)dE
dp
in the approximating Hamiltonian Ha is calculated exactly and it does depend on tem-
perature, and it is different from zero for arbitrary β > 0.
In Bogolyubov’s cases [9, 16] constant c in his approximating Hamiltonian is solu-
tion of nonlinear equation that defines minimum of free energy and it also depends on
temperature. It is different from zero below certain critical temperature Tc and equal to
zero for temperature greater than Tc.
The constant c defines the gap in spectrum of excitation, namely one particle excita-
tion with momentum p has the following energy:
E(p) =
√(
p2
2m
− µ
)
+ |c|2v2(p)
with the gap ∆(p) = |c|2v2(p).
In our case for the second branch of eigenvalues and eigenvectors the constant c
also depends on temperature, the one particle excitation with momenta p has also the
following energy
E(p) =
√(
p2
2m
− µ
)
+ |c|2v2(p)
with the gap ∆(p) = |c|2v2(p), but the gap in our case does not vanish for temper-
ature greater than Tc, i.e., in the BCS model exists “pseudogap” that depends also on
temperature and exists for all the temperatures:
c =
∫
v(p)
f0(p)e−βE0 +
E2∫
E1
fE,0(p)e−β(E−2µ)dE
dp.
Note that for the second branch of eigenvalues and eigenvectors the approximating
Hamiltonian appears in two different cases. In the first one it coincides with BCS Hamil-
tonian H on Φ, (Φ̃0, Φ̃s) and with the constant c (4.2). In the second one it appearers
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1532 D. YA. PETRINA
in the equations for the correlation functions, i.e., equations for correlation functions de-
fined for model with BCS Hamiltonian H coincide with those equations defined for
model with the approximating Hamiltonian Ha, but the constant c does depends on
temperature c =
∫
v(p)
[
f0(p)e−βE0 +
∫ E2
E1
fE,0(p)e−β(E−2µ)dE
]
dp.
4.5. Three phases of BCS model. The model Hamiltonian has two branches of eigen-
values and eigenfunctions. It has been well known ground state Φa
0 (4.11) and its exci-
tations (4.14) and constant c has been defined from condition of minimum of energy of
ground state C(c) per volume (for zero temperature). For different from zero tempera-
ture constant c is defined from condition of minimum of free energy per volume or, that
is the same, from equations for correlation function. In this case constant c depend on
temperature, it is different from zero for temperature less than critical temperature Tc,
T < Tc, and vanish for temperature greater than Tc, T > Tc. This branch of spectra is
associated with superconductivity.
We have showed [1 – 4] that the BCS hamiltonian H has the second branch of eigen-
values and eigenvectors, namely ground state Φ0 (2.2) and excited states Φs (2.3)
(or Φ̃0, Φ̃s (4.7)).
We calculated the grand partition function and correlation functions, that coincide
with correlation functions of the approximating Hamiltonian Ha but for second branch
constant c (4.15) also depends on temperature and it is different from zero for arbitrary
temperature. The gap in spectrum ∆ = |c|2v2(p) is different from zero for arbitrary
temperature and it is known as “pseudogap.”
In fact, the BCS Hamiltonian has also the third branch of spectra, namely spectra of
free system of electrons, that correspond to normal metal.
Summarizing above described three branches of spectra and eigenfunctions one can
say that system with the BCS Hamiltonian can exist in three different phases:
1) superconducting phase with gap different from zero for T < Tc;
2) phase with ”pseudogap” different from zero for all the temperatures;
3) normal phase that corresponds to free system of electrons.
These phases correspond to three branches of spectra and eigenvectors. The first and
third phases have been known, the second phase correspond to the second branch of spec-
tra and eigenvectors has been recently discovered in our papers [1 – 6]. It is possible that
for some temperature T > Tc the system with Hamiltonian H is in the third normal
phase and in this sense “pseudogap” disappears.
Remark 4.4. We calculated the grand partition function and the Green functions tak-
ing into account the all orbitel momente l = 0, 1, 2, . . . . From phisical point of view one
should take into account only even l = 0, 2, 4, . . . .
1. Petrina D. Ya. Spectrum and states of BCS Hamiltonian in finite domain. I. Spectrum // Ukr. Math. J. –
2000. – 52, # 5. – P. 667 – 690.
2. Petrina D. Ya. Spectrum and states of the BCS Hamiltonian in finite domains. II. Spectra of excitations //
Ibid. – 2001. – 53, # 8. – P. 1080 – 1101.
3. Petrina D. Ya. Spectrum and states of the BCS Hamiltonian in finite domain. III. The BCS Hamiltonian
with mean-field interaction // Ibid. – 2002. – 54, # 11. – P. 1486 – 1504.
4. 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.
5. Petrina D. Ya. BCS model Hamiltonian of the theory of superconductivity as a quadratic form // Ibid. –
2004. – 56, # 3. – P. 309 – 338.
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NEW SECOND BRANCH OF SPECTRA OF THE BCS HAMILTONIAN AND “PSEUDOGAP” 1533
6. Petrina D. Ya. On Hamiltonians of quantum statistical mechanics and a model Hamiltonian in the theory
of superconductivity // Teor. i Mat. Fiz. – 1970. – 4, # 3. – P. 394 – 411.
7. Petrina D. Ya., Yatsyshin V. P. On a model Hamiltonian in the theory of superconductivity // Teor. i Mat.
Fiz. – 1972. – 10, # 2. – P. 283 – 299.
8. Bardeen J., Cooper L. N., Schieffer J. R. Theory of superconductivity // Phys. Rev. – 1957. – 108, # 5. –
P. 1175 – 1204.
9. Bogolyubov N. N. On the model Hamiltonian in the theory of superconductivity // Selected papers of
N. N. Bogolyubov. – Kiev: Nauk. Dymka, 1970. – 3. – P. 110 – 173.
10. Cooper L. N. Bound electron pairs in a degenerate Fermi gas // Phys. Rev. – 1956. – 104, # 4. – P. 1189 –
1190.
11. Yamaguchi Y. Two-nucleon problem where the potential in nonlocal but separable. I, II // Ibid. – 1954. –
95, # 6. – P. 1628 – 1634, 1635 – 1643.
12. Newton R. G. Scattering theory of waves and particles. – Moscow: Mir, 1969. – 607 p.
13. Watson G. N. Bessel functions, I. – New York, 1958. – 798 p.
14. Schafroth M. R., Butler S. T., Blatt J. M. Helv. Phys. Acta. – 1957. – 30. – P. 93.
15. Petrina D. Ya. Mathematical foundations of quantum statistical mechanics. – Dordrecht: Kluwer
Acad.Publ., 1995. – 461 p.
16. Bogolyubov N. N. (jr). A method of investigation of model Hamiltonian. – Moscow: Nauka, 1974. – 178 p.
17. Schrieffer J. R. Theory of superconductivity. – Moscow: Nauka, 1970. – 311 p.
Received 10.10.2004
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