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7232019 Fano-An Existence Proof 1968
httpslidepdfcomreaderfullfano-an-existence-proof-1968 16
C o m m u n
m a t h P h y s 10 274mdash27 9 (1968)
An Existence
Proof
for the
G ap E q u a t i o n
in the Superconductivity
Theory
P B ILLARD and G
F A N O
C e n t r e de P hys iq u e Th eo r iq u e et D e p a r te m e n t
d e P h y siq u e M a t h e m a t iq u e d U n i ve r sit e d Aix M arseille
Eece i ved M ay
28 1968
Abstract An existen ce th eo rem for th e gap e q u a t io n in t h e s u pe rco n d u c t i vit y
t h e o r y is
given
as a consequence of the Schauder 983085 Tychon off th eo rem Suffic ien t
condi t ions on t h e k e rn e l are
given
which in su re the e xis tenc e of a so l u t io n am o n gst
a par t icu lar c lass of co n t i n u o u s fu n c t io n s The case of a pos i t ive kernel is s tudied
in
deta i l
1 Introduction
F o r
a non relativistic many983085fermion system the existence of a
superfluid or superconducting state is related to the appearence of
n o n trivial solutions in a non linear integral equation called the
gap equation
Various approximation methods for finding the solution of the gap
equation
have been devised [1 2 3] which give rise to a linearization
of the equation All these methods produce solutions with the same non983085
analytic behaviour for small values of the interaction strength A neces983085
sary condition for the appearence of non trivial solutions has been
given
a long time ago by
C O O P E R M I L L S
and
SESSLEE983085
[4] (see also ref 1) The
convergence of an iterative procedure has been proved under certain
conditions
by
K I T A M U R A
[5] Fixed point theorems were
first
used by
O D E H
[6] We prove here an existence theorem under entirely different
assumptions which cover many cases of physical interest We make use
of the Schauder983085Tychonoff theorem which
allows
us to find a solution
amongst a particular class of continuous functions
2 The Existence Theorem
Let us consider the gap equation in its simplest form (ie the equation
for the spherically symmetrical solutions at zero temperature)
oo
= K(k h)
φ
dk
r
(1)
J ] (amp
2
mdash I )
2
+ φ (kY
Su ppo r t ed by c o n t ra c t D G R S T N o
6700
976
7232019 Fano-An Existence Proof 1968
httpslidepdfcomreaderfullfano-an-existence-proof-1968 26
G ap
Equ at ion 275
a n d m a k e t h e
following
h y p o t h e s e s o n t h e ke r n e l
K (k k )
K is a m e a s u r a b l e r e a l va l u e d b o u n d e d fu n c t io n o n R + x j β
+
le t
M gt
0 b e a b o u n d su c h t h a t
Kk k )
^
M
fo r
every
x +
( I )
[There
exists a compact interval =
k
ξ
1
^
k lt
f
2
C
R
+
( Π )
jfi lt 1 lt pound2 such
that
K(k k ) ^ 0 for (k k ) ζ l x J
[There
exist
three positive numbers a A ε ( 0 lt a lt A ε gt 0)
|such
that
th e
following
inequalities hold
ϊ) j dk
gt 1 +
ε
for
k ζ I
( I Π
a
) |g(t
gt
fe
) |983085 p= ^ ^ lt 4 g for
J V(^
2
983085 I )
2
+ A
2
~ ~
A
R + 983085 I
n ]
τ
( I Π
8
)
Γ |Z(Jfc ifc)l
1
= d fe
^ 1 for a l l
k ζ R +
j ) ( ^
2
983085 I )
2
+ ^ 4
2
C
( I V ) T h e r e
exists
a n L gt 0 su c h t h a t
f K(k
v
k)
983085 Z ( amp
2
k)983085
7
==ά ===rdk
l K V J
V 2gt I y
(
^
2
_
1 ) 2 +
^
2
fo r every (k
v
k
2
) ζ R + x JR +
F o r
t h e r e m a i n d e r of t h i s se c t io n w e will c o n s i d e r o n l y k e r n e l s v e r i 983085
fying c o n d it io n s ( I ) ( I V)
Definition 1 L e t ^ ( J R
+
) be t h e sp a c e of all c o n t in u o u s n u m e r ic a l
f u n c t i o n s o n R
+
w it h t h e t o p o lo g y of u n ifo r m c o n ve r ge n c e o n c o m p a c t s
$F R
+
) i s a F r e c h e t s p a c e
W e c o n s i d e r n o w t h e
following
s u b s e t o f
J f = ^Γ (li I
2 J
laquo
3
^ L ) = 1 ζ ^ ( B + )
real
valued
= sup|(jfc)| ^ 4inf (i) ^ α λ ( )= su p
I t i s
straightforward
t o
prove
th e
following
proposition
Proposition 1 J Γ is a convex compact
subset
of ^
r
(R+) and 0 $ X
Furthermore we have
Proposition 2
T he application T
X
983085 gt
amp (R
+
) defined for every
(T(f))(k)=
fκ kk)983085=
fk
]
983085 dyen
is a continuous mapping of J f ^ίo J^
7232019 Fano-An Existence Proof 1968
httpslidepdfcomreaderfullfano-an-existence-proof-1968 36
276
P
BILLARD
and G
F A N O
Proof By (I ) (T (f))
(k)
is defined for every
k ζ R
+
because
mdash for every
k ζ R
+
](amp
2
983085 I )
2
983085
and
Tf)
is real valued because of (I )
By (IV) and th e condition foo ^
A
we have
2
T( f ) (h) 983085 T(f) (k
2
) ltZ f K(k
v
V) 983085 K(k
2
k) y^j
^
L k
plusmn
983085 k
2
fo r (k
1
k
2
) ζR+ x
Therefore λ (ίΓ ()) ^ L which implies in particular
T(f)
t h e r m o r e
for every
k ζ R
+
we have by (IΠ
3
)
= j=
k
2
Fur983085
Consequently
||5
r r
( ) | |
0 O
^ ^4
F o r
amp ξ we have by (I I)
inf (amp) α
( I I I
2
) and the inequality
~ L gt
jy I jz(h h
f
tj ) lυ hi
mdash I
J fυ tίgt )
R+983085 I
y = J = = d k gt
α (l + β ) 983085 α β = α so inf
T(f) (k) a
Therefore T (
JΓ ) C Ct
I t still remains to prove the continuity of
T
As Jf
C lt^R
+
)
is a metrizable space in order to prove the continuity
of
T
on Jf it is sufficient to show that from
in
it follows that Γ (
n
) 983085 ^z^r ^( ) in ^
In order to see that this is the case lets
Ά x
an arbitrary number
η gt
0 and write
τ ( f
n
) (k ) 983085 Til)
(k)
)
r f i f
0
7232019 Fano-An Existence Proof 1968
httpslidepdfcomreaderfullfano-an-existence-proof-1968 46
G ap
Equation 277
If k
x
is chosen large enough so that
CO
Γ
l
2
983085
2
3 lts2
~
1
we have J
2
^ 9830854983085 independently of
k ζ R
+
and
n k
x
being fixed by this
c o n d i t i o n
there is uniform convergence of
t o zero on the interval [0 fej when n^oo This follows from the
inequality
bull |
Λ
V)
As Jj ^ M f |^
n
( i ) | ^^ J th^re exists an entire w
0
such that for n ^ w
0
o
J
x
^ 983085 | independently of Therefore w ^
0
=Φ [ Γ (
n
) 983085 ^( )||oo ^
η
which proves Proposition 2
Theorem Eq (1) admits at least one solution φ ζ Jf (Therefore in
particular
φ = 0 j
Proof The theorem follows immediately from Propositions 1 and 2
by applying the Schauder983085 Tychonoff theorem [7]
BemarJc
Condition IV holds if the following condition is verified
[There
exists
N gt 0 such that for every
fixed
h
r
ζ R
+
the function
( V )
K
k
Kvk) = Kh k) k
ζ Λ +) verifies
λ (K
r
)
^
N
This happens in particular if for every fixed k ζ R
+
the function
K
k
gt is continuous on R
+
difFerentiable on jR+ except at most for a de983085
numerable set of points of R
+
and the absolute value of this derivative
is majorized by N
I n general condition
I I I j
can be satisfied with a sufficiently small
a gt 0 and condition I I I
3
can be satisfied with a sufficiently large A gt 0
I n
order to produce a large clase of kernels
fulfilling
all the conditions
it is then sufficient to consider kernels which vanish sufficiently fast
outside of x (in order to
verify
condition I I I
2
) and which are suf983085
ficiently regular (in order to
verify
condition IV)
3
The Case of a
Positive
Kernel
If K(k k
f
) gt 0 for every (k
k
r
) ζR
+
x J+ one is tempted to put
= R
+
because the inequality I I I
2
is then automatically satisfied
7232019 Fano-An Existence Proof 1968
httpslidepdfcomreaderfullfano-an-existence-proof-1968 56
278 P B I L L A R D and G
However since in all reasonable physical cases lim
K (k k
f
)
= 0 it is not
β gtoo
possible in general to find in a such that the inequality 11^ written with
= R
+
is satisfied
I n order to avoid this difficulty we consider in the place of Jf the
following subset of JF (R
+
)
X
==
X
1
(a
A L) = ζ ^(R+) real valued gt 0
IIU = sup () lt A
f(k )
gt aK(k 1) (f) pound iλ
IceuroR+ )
nd we m ke the following hypotheses on the kernel
K(k k )
(Γ ) K is a measurable bounded function gt 0 on R
+
x R
+
let M
be a bound such that Kk k
f
) ^ M for every
(k k) ζR+ x β +
(II) There exists
a n α gt 0
such that the following inequality holds
r K V)KVl)
d yen
gt
ι
forall
k R+
K(h
9
1)
](A
2
983085 I )
2
+ aKψ I )
2
K(h
9
1)
](A
2
983085 I)
2
+ aKψ I)
2
( I I I ) There exists an L gt 0 such that
[Kk
1
k)983085K(k
2
k)983085=
A
983085dk
r
lt Lk
λ
983085 k
2
J
ι κ x
κ 2 n
V(k
2
983085 I)
2
+ A
2
~
for every (k
v
k
2
) ζ R
+
x R+
where A is a positive number
verifying
the inequalities
M f
l
=rdk
ltgt 1
AgtaK(ll)
I t
is straightforward to prove that propositions 1 and 2 as well as
the existence theorem hold equally well if we replace
Jf
by JΓ and we
take into account the new hypotheses (I )
( I I ) ( I I I )
on the kernel In
particular
if ζ Jf we have making use of the inequality
( I I )
gt f
Kkk
f
)~j983085
α Z (
^
1
gt
^ gt
aK(k)
~ R+ l^
2
983085
1
)
2
+
aK(V
I)
2
~
I
2
Example Let us consider the kernel
oo
k)=983085 ^j983085
dr e983085
ur
sπ ikr sinkr
o
k
2
[ α
2
+ (A + F )
2
] [α
2
+ (k 983085 k
7232019 Fano-An Existence Proof 1968
httpslidepdfcomreaderfullfano-an-existence-proof-1968 66
Gap Equation 279
This kernel arises naturally in physical situations (see ref 2) it
corresponds to an a tt ract ive two body po ten tial of the form
V(r) =
Ve~
ocr
i
r being the interpart icle distance I t is easy to verify tha t
K(kk) ^
2F
dK(kk)
dk
8 7 1
lt sup 1mdash983085
for
every (k k) ζ R+
x
R+
and therefore the kernel verifies the conditions (I) (III) (see the pre983085
ceding Remark) It is also
immediate
to see that the function
= mi
is continuous and strictly positive for
k gt
0
Therefore choosing
a
sufficiently small in order that
fD k )
j(amp
2
983085 I )
2
+
a
2
K(k
9
I )
2
also co ndition (I I ) is verified and the existence theorem applies
Acknowledgements
One of us
G
F)
would like
t o
thank P rofessor
D K A ST L E R
for
the
hospitality received
at the
University
of
Aix983085 Marseille
and
theD GRST
for financial support
References
1 BOGOLIUBOV N N V V TOLMACHEV and D V SHIRKOV A new method in t he
theory of superconductivity Con sultant Bureau In c 1959)
2 FANO G and A TOMASINI NUOVO
Cimento
18 1247
(1960)
FANO G M
SAVOIA
and A TOMASΓ NΊ JNΓ UOVO Cimento 21 854 (1961)
3 E M E R Y V J and A M S E S SL E R Phys Rev 119 43 1960) 248 1960)
4 COOPER L N R L MILLS and A M SESSLER Phys Rev
114
1377 (1959)
5
KITAMTJRA
M
Progr Theor P hys
30 435
1963)
6 O D E H
F
IBM Journal Research Development USA)
8 187
1964)
7
D U N F O R D
N
and J T
SCH WARTZ Linear operators P ar t
I p 456 New York
Interscience Publishers Inc
P BILLARD G FANO
D e p a r t e m e n t
de
Physique Istituto
di
Fisica
Mathematique F aculte des Sciences delΓ Universita
Universite
dAix
Marseille Bologna I ta lia
Place Victor Hugo
F
13
Marseille
3
e
7232019 Fano-An Existence Proof 1968
httpslidepdfcomreaderfullfano-an-existence-proof-1968 26
G ap
Equ at ion 275
a n d m a k e t h e
following
h y p o t h e s e s o n t h e ke r n e l
K (k k )
K is a m e a s u r a b l e r e a l va l u e d b o u n d e d fu n c t io n o n R + x j β
+
le t
M gt
0 b e a b o u n d su c h t h a t
Kk k )
^
M
fo r
every
x +
( I )
[There
exists a compact interval =
k
ξ
1
^
k lt
f
2
C
R
+
( Π )
jfi lt 1 lt pound2 such
that
K(k k ) ^ 0 for (k k ) ζ l x J
[There
exist
three positive numbers a A ε ( 0 lt a lt A ε gt 0)
|such
that
th e
following
inequalities hold
ϊ) j dk
gt 1 +
ε
for
k ζ I
( I Π
a
) |g(t
gt
fe
) |983085 p= ^ ^ lt 4 g for
J V(^
2
983085 I )
2
+ A
2
~ ~
A
R + 983085 I
n ]
τ
( I Π
8
)
Γ |Z(Jfc ifc)l
1
= d fe
^ 1 for a l l
k ζ R +
j ) ( ^
2
983085 I )
2
+ ^ 4
2
C
( I V ) T h e r e
exists
a n L gt 0 su c h t h a t
f K(k
v
k)
983085 Z ( amp
2
k)983085
7
==ά ===rdk
l K V J
V 2gt I y
(
^
2
_
1 ) 2 +
^
2
fo r every (k
v
k
2
) ζ R + x JR +
F o r
t h e r e m a i n d e r of t h i s se c t io n w e will c o n s i d e r o n l y k e r n e l s v e r i 983085
fying c o n d it io n s ( I ) ( I V)
Definition 1 L e t ^ ( J R
+
) be t h e sp a c e of all c o n t in u o u s n u m e r ic a l
f u n c t i o n s o n R
+
w it h t h e t o p o lo g y of u n ifo r m c o n ve r ge n c e o n c o m p a c t s
$F R
+
) i s a F r e c h e t s p a c e
W e c o n s i d e r n o w t h e
following
s u b s e t o f
J f = ^Γ (li I
2 J
laquo
3
^ L ) = 1 ζ ^ ( B + )
real
valued
= sup|(jfc)| ^ 4inf (i) ^ α λ ( )= su p
I t i s
straightforward
t o
prove
th e
following
proposition
Proposition 1 J Γ is a convex compact
subset
of ^
r
(R+) and 0 $ X
Furthermore we have
Proposition 2
T he application T
X
983085 gt
amp (R
+
) defined for every
(T(f))(k)=
fκ kk)983085=
fk
]
983085 dyen
is a continuous mapping of J f ^ίo J^
7232019 Fano-An Existence Proof 1968
httpslidepdfcomreaderfullfano-an-existence-proof-1968 36
276
P
BILLARD
and G
F A N O
Proof By (I ) (T (f))
(k)
is defined for every
k ζ R
+
because
mdash for every
k ζ R
+
](amp
2
983085 I )
2
983085
and
Tf)
is real valued because of (I )
By (IV) and th e condition foo ^
A
we have
2
T( f ) (h) 983085 T(f) (k
2
) ltZ f K(k
v
V) 983085 K(k
2
k) y^j
^
L k
plusmn
983085 k
2
fo r (k
1
k
2
) ζR+ x
Therefore λ (ίΓ ()) ^ L which implies in particular
T(f)
t h e r m o r e
for every
k ζ R
+
we have by (IΠ
3
)
= j=
k
2
Fur983085
Consequently
||5
r r
( ) | |
0 O
^ ^4
F o r
amp ξ we have by (I I)
inf (amp) α
( I I I
2
) and the inequality
~ L gt
jy I jz(h h
f
tj ) lυ hi
mdash I
J fυ tίgt )
R+983085 I
y = J = = d k gt
α (l + β ) 983085 α β = α so inf
T(f) (k) a
Therefore T (
JΓ ) C Ct
I t still remains to prove the continuity of
T
As Jf
C lt^R
+
)
is a metrizable space in order to prove the continuity
of
T
on Jf it is sufficient to show that from
in
it follows that Γ (
n
) 983085 ^z^r ^( ) in ^
In order to see that this is the case lets
Ά x
an arbitrary number
η gt
0 and write
τ ( f
n
) (k ) 983085 Til)
(k)
)
r f i f
0
7232019 Fano-An Existence Proof 1968
httpslidepdfcomreaderfullfano-an-existence-proof-1968 46
G ap
Equation 277
If k
x
is chosen large enough so that
CO
Γ
l
2
983085
2
3 lts2
~
1
we have J
2
^ 9830854983085 independently of
k ζ R
+
and
n k
x
being fixed by this
c o n d i t i o n
there is uniform convergence of
t o zero on the interval [0 fej when n^oo This follows from the
inequality
bull |
Λ
V)
As Jj ^ M f |^
n
( i ) | ^^ J th^re exists an entire w
0
such that for n ^ w
0
o
J
x
^ 983085 | independently of Therefore w ^
0
=Φ [ Γ (
n
) 983085 ^( )||oo ^
η
which proves Proposition 2
Theorem Eq (1) admits at least one solution φ ζ Jf (Therefore in
particular
φ = 0 j
Proof The theorem follows immediately from Propositions 1 and 2
by applying the Schauder983085 Tychonoff theorem [7]
BemarJc
Condition IV holds if the following condition is verified
[There
exists
N gt 0 such that for every
fixed
h
r
ζ R
+
the function
( V )
K
k
Kvk) = Kh k) k
ζ Λ +) verifies
λ (K
r
)
^
N
This happens in particular if for every fixed k ζ R
+
the function
K
k
gt is continuous on R
+
difFerentiable on jR+ except at most for a de983085
numerable set of points of R
+
and the absolute value of this derivative
is majorized by N
I n general condition
I I I j
can be satisfied with a sufficiently small
a gt 0 and condition I I I
3
can be satisfied with a sufficiently large A gt 0
I n
order to produce a large clase of kernels
fulfilling
all the conditions
it is then sufficient to consider kernels which vanish sufficiently fast
outside of x (in order to
verify
condition I I I
2
) and which are suf983085
ficiently regular (in order to
verify
condition IV)
3
The Case of a
Positive
Kernel
If K(k k
f
) gt 0 for every (k
k
r
) ζR
+
x J+ one is tempted to put
= R
+
because the inequality I I I
2
is then automatically satisfied
7232019 Fano-An Existence Proof 1968
httpslidepdfcomreaderfullfano-an-existence-proof-1968 56
278 P B I L L A R D and G
However since in all reasonable physical cases lim
K (k k
f
)
= 0 it is not
β gtoo
possible in general to find in a such that the inequality 11^ written with
= R
+
is satisfied
I n order to avoid this difficulty we consider in the place of Jf the
following subset of JF (R
+
)
X
==
X
1
(a
A L) = ζ ^(R+) real valued gt 0
IIU = sup () lt A
f(k )
gt aK(k 1) (f) pound iλ
IceuroR+ )
nd we m ke the following hypotheses on the kernel
K(k k )
(Γ ) K is a measurable bounded function gt 0 on R
+
x R
+
let M
be a bound such that Kk k
f
) ^ M for every
(k k) ζR+ x β +
(II) There exists
a n α gt 0
such that the following inequality holds
r K V)KVl)
d yen
gt
ι
forall
k R+
K(h
9
1)
](A
2
983085 I )
2
+ aKψ I )
2
K(h
9
1)
](A
2
983085 I)
2
+ aKψ I)
2
( I I I ) There exists an L gt 0 such that
[Kk
1
k)983085K(k
2
k)983085=
A
983085dk
r
lt Lk
λ
983085 k
2
J
ι κ x
κ 2 n
V(k
2
983085 I)
2
+ A
2
~
for every (k
v
k
2
) ζ R
+
x R+
where A is a positive number
verifying
the inequalities
M f
l
=rdk
ltgt 1
AgtaK(ll)
I t
is straightforward to prove that propositions 1 and 2 as well as
the existence theorem hold equally well if we replace
Jf
by JΓ and we
take into account the new hypotheses (I )
( I I ) ( I I I )
on the kernel In
particular
if ζ Jf we have making use of the inequality
( I I )
gt f
Kkk
f
)~j983085
α Z (
^
1
gt
^ gt
aK(k)
~ R+ l^
2
983085
1
)
2
+
aK(V
I)
2
~
I
2
Example Let us consider the kernel
oo
k)=983085 ^j983085
dr e983085
ur
sπ ikr sinkr
o
k
2
[ α
2
+ (A + F )
2
] [α
2
+ (k 983085 k
7232019 Fano-An Existence Proof 1968
httpslidepdfcomreaderfullfano-an-existence-proof-1968 66
Gap Equation 279
This kernel arises naturally in physical situations (see ref 2) it
corresponds to an a tt ract ive two body po ten tial of the form
V(r) =
Ve~
ocr
i
r being the interpart icle distance I t is easy to verify tha t
K(kk) ^
2F
dK(kk)
dk
8 7 1
lt sup 1mdash983085
for
every (k k) ζ R+
x
R+
and therefore the kernel verifies the conditions (I) (III) (see the pre983085
ceding Remark) It is also
immediate
to see that the function
= mi
is continuous and strictly positive for
k gt
0
Therefore choosing
a
sufficiently small in order that
fD k )
j(amp
2
983085 I )
2
+
a
2
K(k
9
I )
2
also co ndition (I I ) is verified and the existence theorem applies
Acknowledgements
One of us
G
F)
would like
t o
thank P rofessor
D K A ST L E R
for
the
hospitality received
at the
University
of
Aix983085 Marseille
and
theD GRST
for financial support
References
1 BOGOLIUBOV N N V V TOLMACHEV and D V SHIRKOV A new method in t he
theory of superconductivity Con sultant Bureau In c 1959)
2 FANO G and A TOMASINI NUOVO
Cimento
18 1247
(1960)
FANO G M
SAVOIA
and A TOMASΓ NΊ JNΓ UOVO Cimento 21 854 (1961)
3 E M E R Y V J and A M S E S SL E R Phys Rev 119 43 1960) 248 1960)
4 COOPER L N R L MILLS and A M SESSLER Phys Rev
114
1377 (1959)
5
KITAMTJRA
M
Progr Theor P hys
30 435
1963)
6 O D E H
F
IBM Journal Research Development USA)
8 187
1964)
7
D U N F O R D
N
and J T
SCH WARTZ Linear operators P ar t
I p 456 New York
Interscience Publishers Inc
P BILLARD G FANO
D e p a r t e m e n t
de
Physique Istituto
di
Fisica
Mathematique F aculte des Sciences delΓ Universita
Universite
dAix
Marseille Bologna I ta lia
Place Victor Hugo
F
13
Marseille
3
e
7232019 Fano-An Existence Proof 1968
httpslidepdfcomreaderfullfano-an-existence-proof-1968 36
276
P
BILLARD
and G
F A N O
Proof By (I ) (T (f))
(k)
is defined for every
k ζ R
+
because
mdash for every
k ζ R
+
](amp
2
983085 I )
2
983085
and
Tf)
is real valued because of (I )
By (IV) and th e condition foo ^
A
we have
2
T( f ) (h) 983085 T(f) (k
2
) ltZ f K(k
v
V) 983085 K(k
2
k) y^j
^
L k
plusmn
983085 k
2
fo r (k
1
k
2
) ζR+ x
Therefore λ (ίΓ ()) ^ L which implies in particular
T(f)
t h e r m o r e
for every
k ζ R
+
we have by (IΠ
3
)
= j=
k
2
Fur983085
Consequently
||5
r r
( ) | |
0 O
^ ^4
F o r
amp ξ we have by (I I)
inf (amp) α
( I I I
2
) and the inequality
~ L gt
jy I jz(h h
f
tj ) lυ hi
mdash I
J fυ tίgt )
R+983085 I
y = J = = d k gt
α (l + β ) 983085 α β = α so inf
T(f) (k) a
Therefore T (
JΓ ) C Ct
I t still remains to prove the continuity of
T
As Jf
C lt^R
+
)
is a metrizable space in order to prove the continuity
of
T
on Jf it is sufficient to show that from
in
it follows that Γ (
n
) 983085 ^z^r ^( ) in ^
In order to see that this is the case lets
Ά x
an arbitrary number
η gt
0 and write
τ ( f
n
) (k ) 983085 Til)
(k)
)
r f i f
0
7232019 Fano-An Existence Proof 1968
httpslidepdfcomreaderfullfano-an-existence-proof-1968 46
G ap
Equation 277
If k
x
is chosen large enough so that
CO
Γ
l
2
983085
2
3 lts2
~
1
we have J
2
^ 9830854983085 independently of
k ζ R
+
and
n k
x
being fixed by this
c o n d i t i o n
there is uniform convergence of
t o zero on the interval [0 fej when n^oo This follows from the
inequality
bull |
Λ
V)
As Jj ^ M f |^
n
( i ) | ^^ J th^re exists an entire w
0
such that for n ^ w
0
o
J
x
^ 983085 | independently of Therefore w ^
0
=Φ [ Γ (
n
) 983085 ^( )||oo ^
η
which proves Proposition 2
Theorem Eq (1) admits at least one solution φ ζ Jf (Therefore in
particular
φ = 0 j
Proof The theorem follows immediately from Propositions 1 and 2
by applying the Schauder983085 Tychonoff theorem [7]
BemarJc
Condition IV holds if the following condition is verified
[There
exists
N gt 0 such that for every
fixed
h
r
ζ R
+
the function
( V )
K
k
Kvk) = Kh k) k
ζ Λ +) verifies
λ (K
r
)
^
N
This happens in particular if for every fixed k ζ R
+
the function
K
k
gt is continuous on R
+
difFerentiable on jR+ except at most for a de983085
numerable set of points of R
+
and the absolute value of this derivative
is majorized by N
I n general condition
I I I j
can be satisfied with a sufficiently small
a gt 0 and condition I I I
3
can be satisfied with a sufficiently large A gt 0
I n
order to produce a large clase of kernels
fulfilling
all the conditions
it is then sufficient to consider kernels which vanish sufficiently fast
outside of x (in order to
verify
condition I I I
2
) and which are suf983085
ficiently regular (in order to
verify
condition IV)
3
The Case of a
Positive
Kernel
If K(k k
f
) gt 0 for every (k
k
r
) ζR
+
x J+ one is tempted to put
= R
+
because the inequality I I I
2
is then automatically satisfied
7232019 Fano-An Existence Proof 1968
httpslidepdfcomreaderfullfano-an-existence-proof-1968 56
278 P B I L L A R D and G
However since in all reasonable physical cases lim
K (k k
f
)
= 0 it is not
β gtoo
possible in general to find in a such that the inequality 11^ written with
= R
+
is satisfied
I n order to avoid this difficulty we consider in the place of Jf the
following subset of JF (R
+
)
X
==
X
1
(a
A L) = ζ ^(R+) real valued gt 0
IIU = sup () lt A
f(k )
gt aK(k 1) (f) pound iλ
IceuroR+ )
nd we m ke the following hypotheses on the kernel
K(k k )
(Γ ) K is a measurable bounded function gt 0 on R
+
x R
+
let M
be a bound such that Kk k
f
) ^ M for every
(k k) ζR+ x β +
(II) There exists
a n α gt 0
such that the following inequality holds
r K V)KVl)
d yen
gt
ι
forall
k R+
K(h
9
1)
](A
2
983085 I )
2
+ aKψ I )
2
K(h
9
1)
](A
2
983085 I)
2
+ aKψ I)
2
( I I I ) There exists an L gt 0 such that
[Kk
1
k)983085K(k
2
k)983085=
A
983085dk
r
lt Lk
λ
983085 k
2
J
ι κ x
κ 2 n
V(k
2
983085 I)
2
+ A
2
~
for every (k
v
k
2
) ζ R
+
x R+
where A is a positive number
verifying
the inequalities
M f
l
=rdk
ltgt 1
AgtaK(ll)
I t
is straightforward to prove that propositions 1 and 2 as well as
the existence theorem hold equally well if we replace
Jf
by JΓ and we
take into account the new hypotheses (I )
( I I ) ( I I I )
on the kernel In
particular
if ζ Jf we have making use of the inequality
( I I )
gt f
Kkk
f
)~j983085
α Z (
^
1
gt
^ gt
aK(k)
~ R+ l^
2
983085
1
)
2
+
aK(V
I)
2
~
I
2
Example Let us consider the kernel
oo
k)=983085 ^j983085
dr e983085
ur
sπ ikr sinkr
o
k
2
[ α
2
+ (A + F )
2
] [α
2
+ (k 983085 k
7232019 Fano-An Existence Proof 1968
httpslidepdfcomreaderfullfano-an-existence-proof-1968 66
Gap Equation 279
This kernel arises naturally in physical situations (see ref 2) it
corresponds to an a tt ract ive two body po ten tial of the form
V(r) =
Ve~
ocr
i
r being the interpart icle distance I t is easy to verify tha t
K(kk) ^
2F
dK(kk)
dk
8 7 1
lt sup 1mdash983085
for
every (k k) ζ R+
x
R+
and therefore the kernel verifies the conditions (I) (III) (see the pre983085
ceding Remark) It is also
immediate
to see that the function
= mi
is continuous and strictly positive for
k gt
0
Therefore choosing
a
sufficiently small in order that
fD k )
j(amp
2
983085 I )
2
+
a
2
K(k
9
I )
2
also co ndition (I I ) is verified and the existence theorem applies
Acknowledgements
One of us
G
F)
would like
t o
thank P rofessor
D K A ST L E R
for
the
hospitality received
at the
University
of
Aix983085 Marseille
and
theD GRST
for financial support
References
1 BOGOLIUBOV N N V V TOLMACHEV and D V SHIRKOV A new method in t he
theory of superconductivity Con sultant Bureau In c 1959)
2 FANO G and A TOMASINI NUOVO
Cimento
18 1247
(1960)
FANO G M
SAVOIA
and A TOMASΓ NΊ JNΓ UOVO Cimento 21 854 (1961)
3 E M E R Y V J and A M S E S SL E R Phys Rev 119 43 1960) 248 1960)
4 COOPER L N R L MILLS and A M SESSLER Phys Rev
114
1377 (1959)
5
KITAMTJRA
M
Progr Theor P hys
30 435
1963)
6 O D E H
F
IBM Journal Research Development USA)
8 187
1964)
7
D U N F O R D
N
and J T
SCH WARTZ Linear operators P ar t
I p 456 New York
Interscience Publishers Inc
P BILLARD G FANO
D e p a r t e m e n t
de
Physique Istituto
di
Fisica
Mathematique F aculte des Sciences delΓ Universita
Universite
dAix
Marseille Bologna I ta lia
Place Victor Hugo
F
13
Marseille
3
e
7232019 Fano-An Existence Proof 1968
httpslidepdfcomreaderfullfano-an-existence-proof-1968 46
G ap
Equation 277
If k
x
is chosen large enough so that
CO
Γ
l
2
983085
2
3 lts2
~
1
we have J
2
^ 9830854983085 independently of
k ζ R
+
and
n k
x
being fixed by this
c o n d i t i o n
there is uniform convergence of
t o zero on the interval [0 fej when n^oo This follows from the
inequality
bull |
Λ
V)
As Jj ^ M f |^
n
( i ) | ^^ J th^re exists an entire w
0
such that for n ^ w
0
o
J
x
^ 983085 | independently of Therefore w ^
0
=Φ [ Γ (
n
) 983085 ^( )||oo ^
η
which proves Proposition 2
Theorem Eq (1) admits at least one solution φ ζ Jf (Therefore in
particular
φ = 0 j
Proof The theorem follows immediately from Propositions 1 and 2
by applying the Schauder983085 Tychonoff theorem [7]
BemarJc
Condition IV holds if the following condition is verified
[There
exists
N gt 0 such that for every
fixed
h
r
ζ R
+
the function
( V )
K
k
Kvk) = Kh k) k
ζ Λ +) verifies
λ (K
r
)
^
N
This happens in particular if for every fixed k ζ R
+
the function
K
k
gt is continuous on R
+
difFerentiable on jR+ except at most for a de983085
numerable set of points of R
+
and the absolute value of this derivative
is majorized by N
I n general condition
I I I j
can be satisfied with a sufficiently small
a gt 0 and condition I I I
3
can be satisfied with a sufficiently large A gt 0
I n
order to produce a large clase of kernels
fulfilling
all the conditions
it is then sufficient to consider kernels which vanish sufficiently fast
outside of x (in order to
verify
condition I I I
2
) and which are suf983085
ficiently regular (in order to
verify
condition IV)
3
The Case of a
Positive
Kernel
If K(k k
f
) gt 0 for every (k
k
r
) ζR
+
x J+ one is tempted to put
= R
+
because the inequality I I I
2
is then automatically satisfied
7232019 Fano-An Existence Proof 1968
httpslidepdfcomreaderfullfano-an-existence-proof-1968 56
278 P B I L L A R D and G
However since in all reasonable physical cases lim
K (k k
f
)
= 0 it is not
β gtoo
possible in general to find in a such that the inequality 11^ written with
= R
+
is satisfied
I n order to avoid this difficulty we consider in the place of Jf the
following subset of JF (R
+
)
X
==
X
1
(a
A L) = ζ ^(R+) real valued gt 0
IIU = sup () lt A
f(k )
gt aK(k 1) (f) pound iλ
IceuroR+ )
nd we m ke the following hypotheses on the kernel
K(k k )
(Γ ) K is a measurable bounded function gt 0 on R
+
x R
+
let M
be a bound such that Kk k
f
) ^ M for every
(k k) ζR+ x β +
(II) There exists
a n α gt 0
such that the following inequality holds
r K V)KVl)
d yen
gt
ι
forall
k R+
K(h
9
1)
](A
2
983085 I )
2
+ aKψ I )
2
K(h
9
1)
](A
2
983085 I)
2
+ aKψ I)
2
( I I I ) There exists an L gt 0 such that
[Kk
1
k)983085K(k
2
k)983085=
A
983085dk
r
lt Lk
λ
983085 k
2
J
ι κ x
κ 2 n
V(k
2
983085 I)
2
+ A
2
~
for every (k
v
k
2
) ζ R
+
x R+
where A is a positive number
verifying
the inequalities
M f
l
=rdk
ltgt 1
AgtaK(ll)
I t
is straightforward to prove that propositions 1 and 2 as well as
the existence theorem hold equally well if we replace
Jf
by JΓ and we
take into account the new hypotheses (I )
( I I ) ( I I I )
on the kernel In
particular
if ζ Jf we have making use of the inequality
( I I )
gt f
Kkk
f
)~j983085
α Z (
^
1
gt
^ gt
aK(k)
~ R+ l^
2
983085
1
)
2
+
aK(V
I)
2
~
I
2
Example Let us consider the kernel
oo
k)=983085 ^j983085
dr e983085
ur
sπ ikr sinkr
o
k
2
[ α
2
+ (A + F )
2
] [α
2
+ (k 983085 k
7232019 Fano-An Existence Proof 1968
httpslidepdfcomreaderfullfano-an-existence-proof-1968 66
Gap Equation 279
This kernel arises naturally in physical situations (see ref 2) it
corresponds to an a tt ract ive two body po ten tial of the form
V(r) =
Ve~
ocr
i
r being the interpart icle distance I t is easy to verify tha t
K(kk) ^
2F
dK(kk)
dk
8 7 1
lt sup 1mdash983085
for
every (k k) ζ R+
x
R+
and therefore the kernel verifies the conditions (I) (III) (see the pre983085
ceding Remark) It is also
immediate
to see that the function
= mi
is continuous and strictly positive for
k gt
0
Therefore choosing
a
sufficiently small in order that
fD k )
j(amp
2
983085 I )
2
+
a
2
K(k
9
I )
2
also co ndition (I I ) is verified and the existence theorem applies
Acknowledgements
One of us
G
F)
would like
t o
thank P rofessor
D K A ST L E R
for
the
hospitality received
at the
University
of
Aix983085 Marseille
and
theD GRST
for financial support
References
1 BOGOLIUBOV N N V V TOLMACHEV and D V SHIRKOV A new method in t he
theory of superconductivity Con sultant Bureau In c 1959)
2 FANO G and A TOMASINI NUOVO
Cimento
18 1247
(1960)
FANO G M
SAVOIA
and A TOMASΓ NΊ JNΓ UOVO Cimento 21 854 (1961)
3 E M E R Y V J and A M S E S SL E R Phys Rev 119 43 1960) 248 1960)
4 COOPER L N R L MILLS and A M SESSLER Phys Rev
114
1377 (1959)
5
KITAMTJRA
M
Progr Theor P hys
30 435
1963)
6 O D E H
F
IBM Journal Research Development USA)
8 187
1964)
7
D U N F O R D
N
and J T
SCH WARTZ Linear operators P ar t
I p 456 New York
Interscience Publishers Inc
P BILLARD G FANO
D e p a r t e m e n t
de
Physique Istituto
di
Fisica
Mathematique F aculte des Sciences delΓ Universita
Universite
dAix
Marseille Bologna I ta lia
Place Victor Hugo
F
13
Marseille
3
e
7232019 Fano-An Existence Proof 1968
httpslidepdfcomreaderfullfano-an-existence-proof-1968 56
278 P B I L L A R D and G
However since in all reasonable physical cases lim
K (k k
f
)
= 0 it is not
β gtoo
possible in general to find in a such that the inequality 11^ written with
= R
+
is satisfied
I n order to avoid this difficulty we consider in the place of Jf the
following subset of JF (R
+
)
X
==
X
1
(a
A L) = ζ ^(R+) real valued gt 0
IIU = sup () lt A
f(k )
gt aK(k 1) (f) pound iλ
IceuroR+ )
nd we m ke the following hypotheses on the kernel
K(k k )
(Γ ) K is a measurable bounded function gt 0 on R
+
x R
+
let M
be a bound such that Kk k
f
) ^ M for every
(k k) ζR+ x β +
(II) There exists
a n α gt 0
such that the following inequality holds
r K V)KVl)
d yen
gt
ι
forall
k R+
K(h
9
1)
](A
2
983085 I )
2
+ aKψ I )
2
K(h
9
1)
](A
2
983085 I)
2
+ aKψ I)
2
( I I I ) There exists an L gt 0 such that
[Kk
1
k)983085K(k
2
k)983085=
A
983085dk
r
lt Lk
λ
983085 k
2
J
ι κ x
κ 2 n
V(k
2
983085 I)
2
+ A
2
~
for every (k
v
k
2
) ζ R
+
x R+
where A is a positive number
verifying
the inequalities
M f
l
=rdk
ltgt 1
AgtaK(ll)
I t
is straightforward to prove that propositions 1 and 2 as well as
the existence theorem hold equally well if we replace
Jf
by JΓ and we
take into account the new hypotheses (I )
( I I ) ( I I I )
on the kernel In
particular
if ζ Jf we have making use of the inequality
( I I )
gt f
Kkk
f
)~j983085
α Z (
^
1
gt
^ gt
aK(k)
~ R+ l^
2
983085
1
)
2
+
aK(V
I)
2
~
I
2
Example Let us consider the kernel
oo
k)=983085 ^j983085
dr e983085
ur
sπ ikr sinkr
o
k
2
[ α
2
+ (A + F )
2
] [α
2
+ (k 983085 k
7232019 Fano-An Existence Proof 1968
httpslidepdfcomreaderfullfano-an-existence-proof-1968 66
Gap Equation 279
This kernel arises naturally in physical situations (see ref 2) it
corresponds to an a tt ract ive two body po ten tial of the form
V(r) =
Ve~
ocr
i
r being the interpart icle distance I t is easy to verify tha t
K(kk) ^
2F
dK(kk)
dk
8 7 1
lt sup 1mdash983085
for
every (k k) ζ R+
x
R+
and therefore the kernel verifies the conditions (I) (III) (see the pre983085
ceding Remark) It is also
immediate
to see that the function
= mi
is continuous and strictly positive for
k gt
0
Therefore choosing
a
sufficiently small in order that
fD k )
j(amp
2
983085 I )
2
+
a
2
K(k
9
I )
2
also co ndition (I I ) is verified and the existence theorem applies
Acknowledgements
One of us
G
F)
would like
t o
thank P rofessor
D K A ST L E R
for
the
hospitality received
at the
University
of
Aix983085 Marseille
and
theD GRST
for financial support
References
1 BOGOLIUBOV N N V V TOLMACHEV and D V SHIRKOV A new method in t he
theory of superconductivity Con sultant Bureau In c 1959)
2 FANO G and A TOMASINI NUOVO
Cimento
18 1247
(1960)
FANO G M
SAVOIA
and A TOMASΓ NΊ JNΓ UOVO Cimento 21 854 (1961)
3 E M E R Y V J and A M S E S SL E R Phys Rev 119 43 1960) 248 1960)
4 COOPER L N R L MILLS and A M SESSLER Phys Rev
114
1377 (1959)
5
KITAMTJRA
M
Progr Theor P hys
30 435
1963)
6 O D E H
F
IBM Journal Research Development USA)
8 187
1964)
7
D U N F O R D
N
and J T
SCH WARTZ Linear operators P ar t
I p 456 New York
Interscience Publishers Inc
P BILLARD G FANO
D e p a r t e m e n t
de
Physique Istituto
di
Fisica
Mathematique F aculte des Sciences delΓ Universita
Universite
dAix
Marseille Bologna I ta lia
Place Victor Hugo
F
13
Marseille
3
e
7232019 Fano-An Existence Proof 1968
httpslidepdfcomreaderfullfano-an-existence-proof-1968 66
Gap Equation 279
This kernel arises naturally in physical situations (see ref 2) it
corresponds to an a tt ract ive two body po ten tial of the form
V(r) =
Ve~
ocr
i
r being the interpart icle distance I t is easy to verify tha t
K(kk) ^
2F
dK(kk)
dk
8 7 1
lt sup 1mdash983085
for
every (k k) ζ R+
x
R+
and therefore the kernel verifies the conditions (I) (III) (see the pre983085
ceding Remark) It is also
immediate
to see that the function
= mi
is continuous and strictly positive for
k gt
0
Therefore choosing
a
sufficiently small in order that
fD k )
j(amp
2
983085 I )
2
+
a
2
K(k
9
I )
2
also co ndition (I I ) is verified and the existence theorem applies
Acknowledgements
One of us
G
F)
would like
t o
thank P rofessor
D K A ST L E R
for
the
hospitality received
at the
University
of
Aix983085 Marseille
and
theD GRST
for financial support
References
1 BOGOLIUBOV N N V V TOLMACHEV and D V SHIRKOV A new method in t he
theory of superconductivity Con sultant Bureau In c 1959)
2 FANO G and A TOMASINI NUOVO
Cimento
18 1247
(1960)
FANO G M
SAVOIA
and A TOMASΓ NΊ JNΓ UOVO Cimento 21 854 (1961)
3 E M E R Y V J and A M S E S SL E R Phys Rev 119 43 1960) 248 1960)
4 COOPER L N R L MILLS and A M SESSLER Phys Rev
114
1377 (1959)
5
KITAMTJRA
M
Progr Theor P hys
30 435
1963)
6 O D E H
F
IBM Journal Research Development USA)
8 187
1964)
7
D U N F O R D
N
and J T
SCH WARTZ Linear operators P ar t
I p 456 New York
Interscience Publishers Inc
P BILLARD G FANO
D e p a r t e m e n t
de
Physique Istituto
di
Fisica
Mathematique F aculte des Sciences delΓ Universita
Universite
dAix
Marseille Bologna I ta lia
Place Victor Hugo
F
13
Marseille
3
e