Integrable superconductivity and Richardson equations
For the integrable generalized model of superconductivity, the solution of the Richardson equations is studied for a model spectrum. For the case of a narrow band, the solution is presented in terms of generalized Laguerre or Jacobi polynomials. In the asymptotic limit, when the Richardson equations...
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| author | Belokolos, E. D. Бєлоколос, Є. Д. |
| author_facet | Belokolos, E. D. Бєлоколос, Є. Д. |
| author_sort | Belokolos, E. D. |
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| datestamp_date | 2020-03-18T19:51:00Z |
| description | For the integrable generalized model of superconductivity, the solution of the Richardson equations is studied for a model spectrum. For the case of a narrow band, the solution is presented in terms of generalized Laguerre or Jacobi polynomials. In the asymptotic limit, when the Richardson equations are transformed into a singular integral equation, the properties of the integration contour are discussed and the spectral density is calculated. The conditions of appearance of gaps in the spectrum are investigated. |
| first_indexed | 2026-03-24T02:40:05Z |
| format | Article |
| fulltext |
UDC 517.9
E. D. Belokolos (Inst. Magnetism Nat. Acad. Sci. Ukraine, Kyiv)
INTEGRABLE SUPERCONDUCTIVITY
AND RICHARDSON EQUATIONS
INTEHROVNA NADPROVIDNIST|
I RIVNQNNQ RIÇARDSONA
For the integrable generalized model of superconductivity a solution of the Richardson equations for a
spectrum of model is studied. For the case of narrow band the solution is presented in terms of the
generalized Laguerre or Jacobi polynomials. In asymptotic limit, when the Richardson equations are
transformed to an integral singular equation, the properties of an integration contour are discussed and a
spectral density is calculated. Conditions for appearance of gaps in the spectrum are considered.
Dlq intehrovno] uzahal\neno] modeli nadprovidnosti doslidΩeno rozv’qzannq rivnqn\ Riçardsona
dlq spektra modeli. U vypadku vuz\ko] zony rozv’qzok podano v terminax uzahal\nenyx polino-
miv Laherra ta Qkobi. V asymptotyçnomu vypadku, koly rivnqnnq Riçardsona transformugt\sq
v intehral\ne synhulqrne rivnqnnq, z’qsovano vlastyvosti kontura intehruvannq ta rozraxovano
spektral\nu wil\nist\. Rozhlqnuto umovy poqvy wilyn u spektri.
1. Introduction. In 1957 J. Bardeen, L. N. Cooper and J. R. Schrieffer [1] has intro-
duced the BCS Hamiltonian to describe a superconductivity. Later N. N. Bogolubov
et. al. [2] proved the integrability of the BCS Hamiltonian in the thermodynamic limit.
The integrability of the BCS Hamiltonian for a finite number of particles was proved in
1963 by R. W. Richardson [3]. M. Gaudin developed in the 1976 the appropriate ma-
thematical theory and proposed the integrable generalizations of the BCS Hamiltonian
which present an interest for a number of physical problems [4 – 6].
Later the integrable generalizations of the BCS Hamiltonian have been studied
L. Amico, J. Dukelsky, G. Sierra and many others (see, e.g., a review article [7]).
In this paper we study generalizations of the integrable BCS Hamiltonian which is
possible to describe the old and new superconductivity on equal level. The first reason
for this is a fact that both types of superconductivity are based on a concept of the Co-
oper pair of electrons bound by attractive force, i.e., they have same origin, and there-
fore in order to explain new superconductivity we have no necessity in new extravagant
interactions, scenarios and so on, we need just to generalize the well known BCS Ha-
miltonian. The second reason is that the essence of the superconductivity phenomenon
is the coherence property, if we use physical language, which is equivalent in our opi-
nion to the integrability property of Hamiltonian, if we use mathematical language. We
know that the BCS Hamiltonian is integrable.
These reasons lead us to an idea to consider the superconductivity phenomenon by
means of a single integrable Hamiltonian which in one domain of parameters describe
the old superconductivity and in other domain will describe possibly the new supercon-
ductivity.
In this paper we rewiew the integrable pairing Hamiltonians and discuss a solution
of the Richardson equations. A structure of the paper is follows. In Section 2 we pre-
sent the Gaudin construction of quantum integrable Hamiltonians and in Section 3 give
a discussion of the Richardson equations. In Section 4 we show that under special con-
ditions the solutions of the Richardson equations are zeros of the Laguerre or Jacobi
polynomials. The asymptotic expression of the Richardson equations in a form of the
singular integral equation is presented in Section 5. In Section 6 we remind some facts
of a theory of singular integral equations and in Section 7 obtain a solution of the Ri-
chardson singular integral equation in a special case.
2. The integrable pairing Hamiltonian. 2.1. Commuting quadratic operator
forms. The generators S
α
, α = 1, 2, 3, of the algebra sl2 are well known to satisfy
the following relations:
© E. D. BELOKOLOS, 2007
314 ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
INTEGRABLE SUPERCONDUCTIVITY AND RICHARDSON EQUATIONS 315
[ ],S Sα β = i Sεαβγ
γ ,
[ ],S S3 ± = ± S± , [ ],S S+ − = 2 3S , S± = S iS1 2± ,
[ ],Sα S2 = 0, S2 = ( ) ( ) ( )S S S1 2 2 2 3 2+ + .
Now let us take N copies of this algebra,
Sk
α , α ∈{ , , }1 2 3 , k N∈ …{ , , , }1 2 ,
[ ],S Sj k
3 ± = ± δ jk kS± , [ ],S Sj k
+ − = 2 3δ jk kS ,
and construct the quadratic operator forms
Hj = w S Sjk j k
k k j
N
α α α
= ≠
∑
1,
, j N∈ …{ , , , }1 2 .
These operator forms commute with each other,
[ ],H Hj k = 0,
if coefficients of these forms satisfy the equations
w w w w w wij jk ik ki ki ij
α γ β α γ β+ + = 0.
A solution of this equations looks as follows,
wij
1 =
′θ
θ
θ
θ
11
10
10
11
0
0
( )
( )
( )
( )
u
u
ij
ij
, wij
2 =
′θ
θ
θ
θ
11
00
00
11
0
0
( )
( )
( )
( )
u
u
ij
ij
,
wij
3 =
′θ
θ
θ
θ
11
01
01
11
0
0
( )
( )
( )
( )
u
u
ij
ij
, uij = u ui j− ,
where
θab u( ) = θ τab u( ; ) = exp π τ πi n a i n a u b
n
+
+ +
+
∈
∑ 2
2
2 2
2
Z
are theta-functions with characteristics and
′θab( )0 =
d u
du
ab
u
θ ( )
=0
.
Further we impose one more additional commutativity condition,
[ ],S Hj
3 = 0, S3 = Si
i
N
3
1=
∑ ,
which is equivalent to the relations
wij
1 = – wji
1 = wij
2 = – wji
2 .
In this case we can transform the equations for quantities wij
α to that ones,
w wij jk ji ikv v+ = w wik jk ,
where
wij : = wij
1 = wij
2 , vij : = wjk
3 .
These equations have solution
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
316 E. D. BELOKOLOS
v jk = qB q u uj kcoth ( )− ,
wjk =
qB
q u uj ksinh ( )−
,
where parameters B, uj are real and the parameter q may be real or imaginary.
If the parameter q is real then the quantities v jk , wjk are hyperbolic functions, if
the parameter q is imaginary then the quantities v jk , wjk are trigonometric functi-
ons. There are three different types of integrable models depending of values of the
parameter q :
the hyperbolic model at q = 1,
the trigonometric model at q = i,
the rational model at q = 0.
We can write down also the following commuting quadratic operator forms,
Rj = S Hj j
3 − , [ ],R Rj k = 0.
It is possible to consider the operators Rj
, j = 1, 2, … , N, as a complete set of integ-
rals of motions of a Hamiltonian which is an arbitrary function of these operators.
2.2. The integrable pairing Hamiltonian. Let us construct a following integr-
able Hamiltonian:
H = 2
2
2
1
3
ε β α
α
j j
j
j
j
j
j
jR A R S∑ ∑ ∑ ∑+
+
=
( ) ,
where εj
, βj
, j = 1, 2, … , N, and A are some real constants.
If we express the integrals of motion Rj
in terms of operators S
α we transform
our Hamiltonian to the form
H = 2 3 3 3ε j j
j
jk j k
j k
jk j k
j k
S g S S U S S∑ ∑ ∑+ − +–
, ,
,
with the following interaction functions:
gjk =
qB
q u u
j k
j k
( )
sinh ( )
ε ε−
−
, j ≠ k,
Ujk = A qB q u uj k j k− − −( )coth ( )ε ε , j ≠ k,
gjj = – βj
, Ujj = A j+ β ,
βj
=
1
2
qB q u u C
k j
k j k j
≠
∑ − − +( )coth ( )ε ε .
Here parameters A, B, C are arbitrary real constants.
We can express the operators Sk
α in terms of the annihilation and creation opera-
tors of electrons ciσ , ciσ
+ ,
Sj
− = c cj j↓ ↑, Sj
+ = ( )Sj
− † = c c
j j↑ ↓
† † ,
Sj
3 = 1
2
1( )c c c c
j j j j↑ ↑ ↓ ↓+ −† † .
Using these expressions we can present the Hamiltonian in a form of generalized BCS
Hamiltonian
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
INTEGRABLE SUPERCONDUCTIVITY AND RICHARDSON EQUATIONS 317
H =
i
N
i i
i j
N
ij i i i i
i j
N
ij i in g c c c c U n n
= =
↑
+
↓
+
↓ ↑
=
′∑ ∑ ∑+
1 1 1
ε σ σ σ–
, ,
.
2.3. The rational limit. At q → 0 and εj = uj
, j = 1, … , N, we come to the
isotropic case gij = g
, i.e., to the BCS Hamiltonian
HBCS =
i
i i i
i j
i i j j
c c g c c c c
, ,
–
σ
σ σε∑ ∑+
↑
+
↓
+
↓ ↑ =
j
N
j j
j
N
k
N
j kS g S S
= = =
+ −∑ ∑ ∑+
1
3
1 1
2 1
2
ε – .
The BCS Hamiltonian has the commuting integrals of motion
Ri = S gi
k k i
N
i k
i k
3
1
− ⋅
−= ≠
∑
,
S S
ε ε
,
[ ],H RBCS i = 0, [ ],R Rj i = 0.
The number of pairs M and the Hamiltonian HBCS are expressed in terms of these in-
tegrals,
M =
i
L
iR
=
∑ +
1
1
2
,
HBCS =
i
N
i i
i
N
i
i
N
iR g R g
= = =
∑ ∑ ∑+
+
1 1
2
1
22
1
2
ε – S .
2.4. Eigenvalues and eigenstates. It is easy to prove that the joint eigenfunction
of operators H, Rj is of the form [4 – 6]
Ψ =
α αω=
↑ ↓
=
∏ ∑ −1 1
0
M
j j
jj
N c c
u
† †
sinh( )
,
where 0 = ↓ … ↓, , is the vacuum and ωα , α = 1, 2, … , M, satisfy to the ge-
neralized Richardson equations
2
1β β α
β αω ω
= ≠
∑ −
,
coth ( )
M
q q =
l
N
lq q u
B=
∑ − −
1
1
coth ( )ωα .
The eigenvalues E of the Hamiltonian H are
E =
j
N
j
j k
N
jkU ABM B
= =
∑ ∑+ + +
1 1
24 1ε
,
( ) – qB q u
j
N M
j j
= =
∑ ∑ −
1 1α
αε ωcoth ( ).
3. The Richardson equations. Three integrable models are described with the Ri-
chardson equations which we can present in the following unified form:
2
1β β α
β αω ω
= ≠
∑ −
,
coth ( )
M
q q =
l
N
lq q u
G=
∑ − −
1
1
coth ( )ωα .
At the limit q → 0 the Richardson equations are
2
1
1β β α β αω ω= ≠
∑ −,
M
=
l
N
lu G=
∑ −
−
1
1 1
ωα
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
318 E. D. BELOKOLOS
In variables
ξα = exp( )2qωα , ζl = exp( )2qul
the Richardson equations in general case,
2
1β β α
β αω ω
= ≠
∑ −
,
coth ( )
M
q q =
l
N
lq q u
G=
∑ − −
1
1
coth ( )ωα ,
attain the form
2 1
1β β α β αξ ξ= ≠
∑ −,
M
=
l
N
l
L
=
∑ −
−
1
1
ζ ξ ξα α
,
where
L =
1
2 2
1
qG
N
M− + − .
The Richardson equations have solutions ωα , α = 1, … , M, which form a set
symmetric with respect to the real axis.
4. Special exact solutions of the Richardson equations. We can solve the Ri-
chardson equations in a special case of narrow band when the parameters ul , l = 1,
2, … , N, coincide.
4.1. The rational case.
Theorem 4.1. If in the Richardson equations for the rational case
2
1
1β β α β αω ω= ≠
∑ −,
M
=
l
N
lu G=
∑ −
−
1
1 1
ωα
the following conditions are satisfied:
ul = 0, l = 1, 2, … , N,
then the Richardson equations have solutions
ωα = Gxα , α = 1, 2, … , M,
where xα are zeros of the generalized Laguerre polynomial
L xM
N− −1( ) .
If N = 0, 1, … , M – 1, then among ωα there are M – N – 1 positive numbers
and N + 1 numbers ωα = 0. If N ≥ M – 1, then all numbers ωα are complex
pairwise conjugated numbers but one negative number for the odd M.
Proof. Let in the Richardson equations for the rational case
2 1
1β β α α βω ω= ≠
∑ −,
M
+
l
N
lu G=
∑ −
−
1
1 1
ωα
= 0,
the following conditions are satisfied:
ul = 0, l = 1, 2, … , N,
then these equations attain the form
2 1
1β β α α βω ω= ≠
∑ −,
M
–
N
ωα
–
1
G
= 0.
As a result of last equalities the polynomial
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
INTEGRABLE SUPERCONDUCTIVITY AND RICHARDSON EQUATIONS 319
f ( x ) =
β
βω
=
∏ −
1
M
x( )
is solution of the differential equation
x
d f
dx
x
G
N
df
dx
M
G
f
2
2 − +
+ = 0.
The generalized Laguerre polynomial
y ( x ) = L cxn
a ( )
satisfies the differential equation
x
d y
dx
cx a
dy
dx
cny
2
2 1− − − +( ) = 0.
Comparing these differential equations we get
c =
1
G
, a = – N – 1, n = M,
and therefore
f ( x ) = L
x
GM
N− −
1 .
Let 0 ≤ N < M – 1. Then for N = 0, 1, … , M – 1, the generalized Laguerre
polynomial has M – N – 1 positive zeros and one zero of the order N + 1 at x = 0.
Let N ≥ M – 1. Then all zeros of the generalized Laguerre polynomial are comp-
lex pairwise conjugated numbers but one negative zero for the odd M.
The theorem is proved.
4.2. The general case.
Theorem 4.2. If in the Richardson equations for general case
2
1β β α
β αω ω
= ≠
∑ −
,
coth ( )
M
q q =
l
N
lq q u
G=
∑ − −
1
1
coth ( )ωα ,
the following conditions are satisfied:
ul =
ln 2
2q
, l = 1, 2, … , N,
then the Richardson equations have solutions
ωα =
ln( )x
q
α +1
2
, α = 1, 2, … , M,
where xα are zeros of the generalized Jacobi polynomial
P xM
N L− + − +( ), ( )( )1 1 ,
and
L =
1
2 2
1
qG
N
M− + − .
The location of ωβ in complex plane is defined by distribution of zeros xβ o f
the generalized Jacobi polynomials.
Let us assume that q ∈ R . Let us define for the generalized Jacobi polynomial
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
320 E. D. BELOKOLOS
P xn
a b( , )( ) the number N a b1( , ) , N a b2( , ), N a b3( , ) of zeros on the intervals ( – 1,
+ 1 ) , ( – ∞ , – 1 ) , ( 1, + ∞ ) appropriately (see explicit expressions for these quanti-
ties in [8, 9]). If N a b1( , ) + N a b3( , ) = M, then all numbers ωβ are all real, if
N a b1( , ) + N a b3( , ) < M, then some numbers ωβ are complex pairwise conjugated.
Proof. Let for the Richardson equations in general case
2
1β β α
α βω ω
= ≠
∑ −
,
coth ( )
M
q q +
l
N
lq q u
G=
∑ − −
1
1
coth ( )ωα = 0,
we introduce new variables
xα = exp( )2 1qωα − , α = 1, 2, … , M,
ζl = exp( )2qul , l = 1, 2, … , N,
and present these equations as follows:
2
1 1
1 11 1β β α α β α αζ= ≠ =
∑ ∑−
+
− −
−
+,
M
l
N
lx x x
L
x
= 0,
where
L =
1
2 2
1
qG
N
M− + − .
If we assume
ζl = 2, l = 1, 2, … , N,
then the equations attain the form
2
1
1 11β β α α β α α= ≠
∑ −
−
−
−
+,
M
x x
N
x
L
x
= 0.
As a result of last equalities the polynomial
f ( x ) =
β
β
=
∏ −
1
M
x x( ), xβ = exp( )2 1qωβ −
satisfies the differential equation
( ) ( ) ( ) ( )1 12
2
2− + − + +[ ] + − − −x
d f
dx
N L N L x
df
dx
M M N L f = 0.
The generalized Jacobi polynomial
y ( x ) = P xn
a b( , )( )
satisfies the differential equation
( ) ( ) ( ) ( )1 2 12
2
2− + − − + +[ ] + + + +x
d y
dx
b a a b x
dy
dx
n n a b y = 0.
Comparing these equations we get
a = – ( N + 1 ) , b = – ( L + 1 ) , n = M,
and therefore
f ( x ) = P xM
N L− + − +( ), ( )( )1 1
.
The theorem is proved.
We can show than an appearance of complex solutions of the Richardson solutions
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
INTEGRABLE SUPERCONDUCTIVITY AND RICHARDSON EQUATIONS 321
indicates an appearance of a spectral gap [4, 6].
In order to obtain further results we must use some facts of a theory of ordinary
differential equations with polynomial solutions [10, 11].
5. Asymptotic form of the Richardson equations. Let us consider the limit
N, M → ∞ , G → 0
under the condition
lim
N
M
N→∞
= m, lim
N
GN
→∞
= g.
5.1. Rational case. In this limit the Richardson equations in the rational case are
transformed to a singular integral equation
P
r d
Γ
∫
′ ′
′ −
( )ξ ξ
ξ ξ
=
Ω
∫ −
−ρ ε ε
ε ξ
( )d
g
1
, ξ ∈ Γ ,
where the density of states ρ ( ε ) and the density of pairs r ( ξ ) satisfy the conditions
Ω
∫ ρ ε ε( )d =
1
2
,
Γ
∫ r d( )ξ ξ = m,
Γ
∫ ξ ξ ξr d( ) = e.
Here Ω is a support of unperturbed spectrum and Γ is a support of spectrum of pairs.
The integrals of motion look now as follows:
λ ( ε ) = –
1
2
+
−
− ′ ′
− ′
∫ ∫g
r d
P
d
Γ Ω
( ) ( )ξ ξ
ε ξ
ρ ε ε
ε ε
,.
5.2. General case. In asymptotic limit the general Richardson equations
2
1β β α
β α
= ≠
∑ −
,
coth ( )
M
q q e e =
l
N
lq q e
G=
∑ − −
1
1
coth ( )ε α
are transformed to a singular integral equation
P q q r d
Γ
∫ ′ − ′ ′coth ( ) ( )ξ ξ ξ ξ =
Ω
∫ − −q q d
g
coth ( ) ( )ε ξ ρ ε ε 1
,
where the density of states ρ ( ε ) and the density of pairs r ( ξ ) satisfy the conditions
Ω
∫ ρ ε ε( )d =
1
2
,
Γ
∫ r d( )ξ ξ = m,
Γ
∫ ξ ξ ξr d( ) = e.
In new variables
ε = exp( )2qu , ξ = exp( )2qω ,
the above singular integral equation attains the form
P
r d
˜
˜( )
Γ
∫
′ ′
′ −
ξ ξ
ξ ξ
=
˜
˜ ( )
˜
Ω
∫ −
−ρ ε ε
ε ξ
d
g
1
2
,
where
˜ ( )ρ ε = ρ ( u ) = ρ ε1
2q
ln
,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
322 E. D. BELOKOLOS
˜( )r ξ = r ( ω ) = r
q
1
2
lnξ
,
1
2g̃
=
1
2
1
2b
q m− −
.
This integral equation coincide with the similar integral equation for the BCS case
P
r d
Γ
∫
′ ′
′ −
( )ξ ξ
ξ ξ
=
Ω
∫ −
−ρ ε ε
ε ξ
( )d
g
1
2
.
Thus our problem now is to solve the singular integral equation. To this end let us
remind some appropriate facts of such a type equations (see, e.g., [12, 13]).
6. Bounded solution for the Cauchy type integral inversion problem. First of
all let us consider the inversion of the Cauchy type integral
1
π
φ
i
s ds
s t
L
( )
−∫ = f ( t )
, t ∈ L
,
where the contour L is assumed to be of a general form consisting of separate arcs
without common ends, i.e., the contour
L =
Lk
k
p
=1
∪ =
( , )a bk k
k
p
=1
∪ .
We study various possibilities.
6.1. The free term is a function of the Hölder class. The solution of the inversi-
on problem φ ( t ) , which belongs to the class of functions H, i.e., the solution φ ( t )
,
which is bounded in all ends of the contour L, exists only at special conditions.
Theorem 6.1. If a contour L consists of separate arcs and the function f ( t )-∈
∈ H , then the singular integral equation
1
0π
φ
i
t dt
t t
L
( )
−∫ = f ( t0 )
, t0 ∈ L
,
has a solution φ ( t ) ∈ H iff the function f ( t ) satisfy the following conditions:
f t t
R t
dt
k
L
( )
( )∫
, k = 0, 1, … , p – 1, R ( t ) = ( )( )t a t bk k
k
p
− −
=
∏
1
.
6.2. The free term is sum of a function of the Hölder class and a polynomial.
We remind that in generic case there are no solutions which are bounded at all ends of
the contour L consisting of separate arcs. We illustrate this statement in case when
the free term is just a polynomial.
Theorem 6.2. If the singular integral equation
1
0π
φ
i
t dt
t t
L
( )
−∫ = P ( t0 )
, t0 ∈ L
,
with a contour L , consisting of separate arcs, and a polynomial P ( t ) of degree
deg P ( t ) ≤ p – 1 as the free term f ( t ) has a solution φ ( t ) of the class H, then it
is necessary
φ ( t ) = 0, P ( t ) = 0.
Now we prove a following generalization of this statement.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
INTEGRABLE SUPERCONDUCTIVITY AND RICHARDSON EQUATIONS 323
Theorem 6.3. If a contour L consists of separate arcs, the function f ( t ) ∈ H,
and the polynomial P ( t ) has a degree deg P ( t ) ≤ p – 1, then the singular integral
equation
1
0π
φ
i
t dt
t t
L
( )
−∫ = f ( t0 ) + P ( t0 )
, t0 ∈ L
,
has a unique bounded solution φ ( t ) ∈ H, iff the polynomial P ( z ) is of the form
P ( z ) =
1
πi
f t
R t
Q z Q t
z t
dt
L
( )
( )
( ) ( )∫ −
−
,
Q ( z ) = R z O z( ) ( )+ −1 , z → ∞ ,
R ( z ) = ( )( )z a z bi i
i
p
− −
=
∏
1
.
Proof. If a solution of our singular integral equation exists it is to be of the form
φ ( t0 ) =
R t
i
f t dt
R t t t
L
( ) ( )
( ) ( )
0
0π −∫ ,
R ( t ) = ( )( )t a t bk k
k
p
− −
=
∏
1
.
Let us put this solution into the equation. In such a way we ensure that it is indeed so-
lution and obtain in addition the expression for the polynomial P ( t ) .
To prove this statement we introduce two functions
Ψ ( z ) =
R z
i
f t dt
R t t z
L
( ) ( )
( ) ( )2π −∫ ,
Φ ( z ) =
1
2π
φ
i
t dt
t z
L
( )
−∫ .
We have obviously
Ψ Ψ( ) ( )t t+ −− = φ ( t ) ,
Φ Φ( ) ( )t t+ −+ = f ( t ) + P ( t )
.
Therefore we have
Φ ( z ) =
1
2π
φ
i
t dt
t z
L
( )
−∫ =
1
2πi
t dt
t z
L
Ψ( )
˜ −∫ =
=
1
2
1
2
1 1
1 1π πi
R t dt
t z i
f t dt
R t t t
L L
( ) ( )
( ) ( )˜ − −∫ ∫ =
1
2
1
2
1 1
1 1π πi
f t dt
R t i
R t dt
t z t t
L L
( )
( )
( )
( )( )˜
∫ ∫ − −
,
where L̃ is a clock-wise closed contour encircling the contour L. Using the Cauchy
residue theorem we obtain
1
2 1πi
R t dt
t z t t
L
( )
( )( )˜ − −∫ =
R z
t z
Q z Q t
z t
( ) ( ) ( )
1
1
1−
+ −
−
,
where
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
324 E. D. BELOKOLOS
R z( ) = Q z Q z( ) ( )+ −1 .
Here we have taken in account that the point t1 is inside the contour L̃ , and the point
z is outside the contour L̃ . We have taken in account also that the integral vanishes if
z or t1 goes to infinity. Hence we obtain
Φ ( z ) =
R z
i
f t dt
R t t z
P z
L
( ) ( )
( ) ( )
( )
2
1
2
1 1
1 1π −
+∫ ,
P ( z ) =
1
πi
f t
R t
Q z Q t
z t
dt
L
( )
( )
( ) ( )∫ −
−
.
Using the equality
Φ Φ+ −+( ) ( )t t = f t P t( ) ( )+ ,
we arrive to conclusion of the theorem.
6.3. The free term is sum of a function of the Hölder class and a piecewise
constant. As we said earlier, in generic case there is no solution of the inversion
problem which is bounded at ends of the contour of integration. We demonstrate this
statement in case when the free term is just a piecewise constant.
Theorem 6.4. If the singular integral equation
1
0π
φ
i
t dt
t t
L
( )
−∫ = Ck
, t0 ∈ Lk
,
with a contour L , consisting of separate arcs, and a piecewise constant as the free
term f ( t ) , has a solution φ ( t ) of the class H, then it is necessary
φ ( t ) = 0, Ck = 0, k = 1, … , p .
Now we prove a generalization of this statement.
Theorem 6.5. If a contour L consists of separate arcs, the function f ( t ) ∈ H,
and Ck
, k = 1, … , p, are some constants, then the singular integral equation
1
0π
φ
i
t dt
t t
L
( )
−∫ = f ( t0 ) + Ck
, t0 ∈ Lk
,
has a unique bounded solution φ ( t ) ∈ H , wich is of the form
φ ( t0 ) =
R t
i
f t
R t t t
t
ds
R s s t
dt
L
k
k
p
Lk
( ) ( )
( )
( )
( ) ( )
0
0 1 0
1
π
ω∫ ∑ ∫−
+
−
=
,
iff there exist such polynomials ωk ( t ) , k = 1, … , p, of the order deg ωk ( t ) ≤ p – 1
that
Ck =
ωk
L
t f t
R t
dt
( ) ( )
( )∫ , k = 1, … , p .
These polynomials ω k ( t ) are defined by the function f t( ) and the contour L
( more exactly, by the ends ak
, bk and mutual arrangement of arcs Lk ) .
There is another formulation of the above theorem.
Theorem 6.6. If a contour L consists of separate arcs, the function f ( t ) ∈ H
and Ck
, k = 1, … , p, are some constants, then the singular integral equation
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
INTEGRABLE SUPERCONDUCTIVITY AND RICHARDSON EQUATIONS 325
1
0π
φ
i
t dt
t t
L
( )
−∫ = f ( t0 ) + Ck
, t0 ∈ Lk
,
has a unique bounded solution φ ( t ) ∈ H , wich is of the form
φ ( t0 ) =
R t
i
f t dt
R t t t
C
dt
R t t t
L
k
k
p
Lk
( ) ( )
( ) ( ) ( ) ( )
0
0 1 0π −
+
−
∫ ∑ ∫
=
,
iff the Ck satisfy the linear equations
a C Ajk k
k
p
j
=
∑ +
1
= 0, j = 0, 1, … , p – 1,
where
ajk =
t
R t
dt
j
Lk
( )∫ , Aj =
t f t
R t
dt
j
L
( )
( )∫ .
7. The Richardson singular integral equation. Applying the results presented
above to solve the Richardson singular integral equation we should find first of all the
integration contour L in course of asymptotic analysis. This is difficult problem even
for classical polynomials. We shall assume further that the integration contour is de-
fined by this time.
7.1. The inversion problem for the Cauchy type integral with an open arc.
Here we determine for an arc L = ab and a function f ( t ) ∈ H a unique solution bo-
unded in all nodes for the equation
1
0π
φ
i
t
t t
dt
L
( )
−∫ = f ( t ) + C.
Solution is of the form
φ ( t0 ) =
R t
i
f t dt
R t t t
L
( ) ( )
( ) ( )
0
0π −∫ ,
C =
1
πi
f t dt
R t
L
( )
( )∫ , R ( t ) = ( t – a ) ( t – b ) .
7.2. A special case of solution of the Richardson singular integral equation.
Determine for arcs Γ = ab and Ω = cd ∈ R and a function ρ ( ε ) ∈ H a unique
solution bounded in all nodes of the contour L for the equation
r d( )′ ′
′ −∫ ξ ξ
ξ ξ
Γ
=
ρ ε ε
ε ξ
( )d
C
−
+∫
Ω
, ξ ∈ Γ .
Hence we should study at first properties of the function
f ( ξ ) =
ρ ε ε
ε ξ
( )d
C
−
+∫
Ω
, ξ ∈ Γ .
We use further the polynomial
R ( t ) = ( t – a ) ( t – b ) .
In order to obtain the solution we need the following Cauchy residue formulae:
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
326 E. D. BELOKOLOS
1
π εi
dt
R t t( ) ( )−∫
Γ
=
1
2π εi
dt
R t t( ) ( )˜ −∫
Γ
=
1
R( )ε
,
1
π ξ εi
dt
R t t t( ) ( )( )− −∫
Γ
=
1
2π ξ εi
dt
R t t t( ) ( )( )˜ − −∫
Γ
=
1
R( ) ( )ε ε ξ−
.
Here Γ̃ is a clock-wise closed curve encircling the arc Γ. In the last formula we have
taken in account that the point ξ is inside of the contour L̃ , and the point ε is
outside of the contour L̃ .
The solution is
r ( ξ ) =
φ ξ
π
( )
i
=
R
i
f t dt
R t t
( )
( )
( )
( ) ( )
ξ
π ξ2 −∫
Γ
=
R
i
dt
R t t
d
t
( )
( ) ( ) ( )
( )ξ
π ξ
ερ ε
ε2 − −∫ ∫
Γ Ω
=
=
R
i
d
dt
R t t t
( )
( )
( )
( ) ( )( )
ξ
π
ερ ε
ξ ε2
Ω Γ
∫ ∫ − −
=
R
i
d
R
( ) ( )
( ) ( )
ξ
π
ρ ε ε
ε ε ξ−∫
Ω
.
The constant is
C =
1
πi
f t dt
R t
( )
( )
Γ
∫ =
1
π
ρ ε ε
εi
dt
R t
d
t( )
( )
Γ Ω
∫ ∫ −
=
1
π
ερ ε
εi
d
dt
R t t
( )
( ) ( )
Ω Γ
∫ ∫ −
=
ρ ε ε
ε
( )
( )
d
R
Ω
∫ .
8. Conclusion. Thus we have studied a number of the integrable quantum models
and discussed different methods to solve the appropriate Richardson equations. For the
case of narrow band a solution of the Richardson equations is presented in terms of ze-
ros of the generalized Laguerre or Jacobi polynomials. We have also formulated the
conditions for appearance of gaps in the spectrum, i.e., an appearance of complex solu-
tion of the Richardson equations. In asymptotic limit, when the Richardson equations
are transformed to an integral singular equation, we have studied possible solutions and
their relations to a spectral density.
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SSSR. – 1957. – 117, # 5. – S.-788 – 791.
3. Richardson R. W. A restricted class of exact eigenstates of the pairing-force Hamiltonian // Phys.
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P. 1087 – 1098.
5. Gaudin M. La fonction d’onde de Bethe. – Paris: Masson, 1983. – 352 p.
6. Gaudin M. Modèles exactement résolus // CEA Saclay-Service Phys. Théor. Edit. Phys. – 1995.
7. Dukelsky J., Pittel S., Sierra G. Exactly solvable Richardson – Gaudin models for many-body
quantum systems // nucl-ph/0405011.
8. Hilbert D. Über die Discriminante der im Endlichen abbrechenden hypergeometrischen Reihe // J.
Math. – 1888. – 103. – S. 337 – 345.
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10. Hille E. Ordinary differential equations in the complex domain. – New York: John Wiley & Sons,
1976.
11. Ince E. L. Ordinary differential equations. – New York: Dover Publ., 1956.
12. Haxov F. D. Kraev¥e zadaçy. – M.: Nauka, 1977. – 640 s.
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Received 09.10.2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
|
| id | umjimathkievua-article-3309 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:40:05Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c6/bb4ca1cb63881f6c782f0a8f42ce68c6.pdf |
| spelling | umjimathkievua-article-33092020-03-18T19:51:00Z Integrable superconductivity and Richardson equations Інтегровна надпровідність і рівняння Річардсона Belokolos, E. D. Бєлоколос, Є. Д. For the integrable generalized model of superconductivity, the solution of the Richardson equations is studied for a model spectrum. For the case of a narrow band, the solution is presented in terms of generalized Laguerre or Jacobi polynomials. In the asymptotic limit, when the Richardson equations are transformed into a singular integral equation, the properties of the integration contour are discussed and the spectral density is calculated. The conditions of appearance of gaps in the spectrum are investigated. Для інтегровної узагальненої моделі надпровідності досліджено розв'язання рівнянь Річардсона для спектра моделі. У випадку вузької зони розв'язок подано в термінах узагальнених поліномів Лагерра та Якобі. В асимптотичному випадку, коли рівняння Річардсона трансформуються в інтегральне сингулярне рівняння, з'ясовано властивості контура інтегрування та розраховано спектральну щільність. Розглянуто умови появи щілин у спектрі. Institute of Mathematics, NAS of Ukraine 2007-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3309 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 3 (2007); 314–326 Український математичний журнал; Том 59 № 3 (2007); 314–326 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3309/3363 https://umj.imath.kiev.ua/index.php/umj/article/view/3309/3364 Copyright (c) 2007 Belokolos E. D. |
| spellingShingle | Belokolos, E. D. Бєлоколос, Є. Д. Integrable superconductivity and Richardson equations |
| title | Integrable superconductivity and Richardson equations |
| title_alt | Інтегровна надпровідність і рівняння Річардсона |
| title_full | Integrable superconductivity and Richardson equations |
| title_fullStr | Integrable superconductivity and Richardson equations |
| title_full_unstemmed | Integrable superconductivity and Richardson equations |
| title_short | Integrable superconductivity and Richardson equations |
| title_sort | integrable superconductivity and richardson equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3309 |
| work_keys_str_mv | AT belokolosed integrablesuperconductivityandrichardsonequations AT bêlokolosêd integrablesuperconductivityandrichardsonequations AT belokolosed íntegrovnanadprovídnístʹírívnânnâríčardsona AT bêlokolosêd íntegrovnanadprovídnístʹírívnânnâríčardsona |