fano-an existence proof 1968

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



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^



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



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



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



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



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^



276

P

BILLARD

and G

F A N O


(k)


k ζ R

+

because

mdash for every

k ζ R

+

](amp

2

983085 I )

2

983085

and

Tf)



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


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


inf (amp) α

( I I I

2


~ 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


T

As Jf

C lt^R

+

)


of

T


in


n

) 983085 ^z^r ^( ) in ^


Ά x

an arbitrary number

η gt

0 and write

τ ( f

n

) (k ) 983085 Til)

(k)

)

r f i f

0



G ap

Equation 277

If k

x


CO

Γ

l

2

983085

2

3 lts2

~

1

we have J

2


k ζ R

+

and

n k

x

being fixed by this

c o n d i t i o n



inequality

bull |

Λ

V)

As Jj ^ M f |^

n


0

such that for n ^ w

0

o

J

x


0

=Φ [ Γ (

n

) 983085 ^( )||oo ^

η



particular

φ = 0 j



BemarJc


[There

exists


fixed

h

r

ζ R

+

the function

( V )

K

k

Kvk) = Kh k) k

ζ Λ +) verifies

λ (K

r

)

^

N


+

the function

K

k


+



+


is majorized by N


I I I j



3


I n


fulfilling

all the conditions



verify

condition I I I

2



verify

condition IV)

3

The Case of a

Positive

Kernel

If K(k k

f

) gt 0 for every (k

k

r

) ζR

+


= R

+


2






K (k k

f

)

= 0 it is not

β gtoo


= R

+

is satisfied



+

)

X

==

X

1

(a


IIU = sup () lt A

f(k )


IceuroR+ )


K(k k )


+

x R

+

let M


f

) ^ M for every

(k k) ζR+ x β +

(II) There exists

a n α gt 0


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


[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+


verifying

the inequalities

M f

l

=rdk

ltgt 1

AgtaK(ll)

I t



Jf

by JΓ and we


( I I ) ( I I I )

on the kernel In

particular


( 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


oo

k)=983085 ^j983085

dr e983085

ur

sπ ikr sinkr

o

k

2

[ α

2

+ (A + F )

2

] [α

2

+ (k 983085 k



Gap Equation 279



V(r) =

Ve~

ocr

i


K(kk) ^

2F

dK(kk)

dk

8 7 1

lt sup 1mdash983085

for

every (k k) ζ R+

x

R+



immediate


= mi


k gt

0

Therefore choosing

a


fD k )

j(amp

2

983085 I )

2

+

a

2

K(k

9

I )

2


Acknowledgements

One of us

G

F)

would like

t o

thank P rofessor

D K A ST L E R

for

the


at the

University

of

Aix983085 Marseille

and

theD GRST


References




Cimento

18 1247

(1960)

FANO G M

SAVOIA




114

1377 (1959)

5

KITAMTJRA

M

Progr Theor P hys

30 435

1963)

6 O D E H

F


8 187

1964)

7

D U N F O R D

N

and J T


I p 456 New York


P BILLARD G FANO


de

Physique Istituto

di

Fisica


Universite

dAix


Place Victor Hugo

F

13

Marseille

3

e



276

P

BILLARD

and G

F A N O


(k)


k ζ R

+

because

mdash for every

k ζ R

+

](amp

2

983085 I )

2

983085

and

Tf)



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


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


inf (amp) α

( I I I

2


~ 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


T

As Jf

C lt^R

+

)


of

T


in


n

) 983085 ^z^r ^( ) in ^


Ά x

an arbitrary number

η gt

0 and write

τ ( f

n

) (k ) 983085 Til)

(k)

)

r f i f

0



G ap

Equation 277

If k

x


CO

Γ

l

2

983085

2

3 lts2

~

1

we have J

2


k ζ R

+

and

n k

x

being fixed by this

c o n d i t i o n



inequality

bull |

Λ

V)

As Jj ^ M f |^

n


0

such that for n ^ w

0

o

J

x


0

=Φ [ Γ (

n

) 983085 ^( )||oo ^

η



particular

φ = 0 j



BemarJc


[There

exists


fixed

h

r

ζ R

+

the function

( V )

K

k

Kvk) = Kh k) k

ζ Λ +) verifies

λ (K

r

)

^

N


+

the function

K

k


+



+


is majorized by N


I I I j



3


I n


fulfilling

all the conditions



verify

condition I I I

2



verify

condition IV)

3

The Case of a

Positive

Kernel

If K(k k

f

) gt 0 for every (k

k

r

) ζR

+


= R

+


2






K (k k

f

)

= 0 it is not

β gtoo


= R

+

is satisfied



+

)

X

==

X

1

(a


IIU = sup () lt A

f(k )


IceuroR+ )


K(k k )


+

x R

+

let M


f

) ^ M for every

(k k) ζR+ x β +

(II) There exists

a n α gt 0


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


[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+


verifying

the inequalities

M f

l

=rdk

ltgt 1

AgtaK(ll)

I t



Jf

by JΓ and we


( I I ) ( I I I )

on the kernel In

particular


( 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


oo

k)=983085 ^j983085

dr e983085

ur

sπ ikr sinkr

o

k

2

[ α

2

+ (A + F )

2

] [α

2

+ (k 983085 k



Gap Equation 279



V(r) =

Ve~

ocr

i


K(kk) ^

2F

dK(kk)

dk

8 7 1

lt sup 1mdash983085

for

every (k k) ζ R+

x

R+



immediate


= mi


k gt

0

Therefore choosing

a


fD k )

j(amp

2

983085 I )

2

+

a

2

K(k

9

I )

2


Acknowledgements

One of us

G

F)

would like

t o

thank P rofessor

D K A ST L E R

for

the


at the

University

of

Aix983085 Marseille

and

theD GRST


References




Cimento

18 1247

(1960)

FANO G M

SAVOIA




114

1377 (1959)

5

KITAMTJRA

M

Progr Theor P hys

30 435

1963)

6 O D E H

F


8 187

1964)

7

D U N F O R D

N

and J T


I p 456 New York


P BILLARD G FANO


de

Physique Istituto

di

Fisica


Universite

dAix


Place Victor Hugo

F

13

Marseille

3

e



G ap

Equation 277

If k

x


CO

Γ

l

2

983085

2

3 lts2

~

1

we have J

2


k ζ R

+

and

n k

x

being fixed by this

c o n d i t i o n



inequality

bull |

Λ

V)

As Jj ^ M f |^

n


0

such that for n ^ w

0

o

J

x


0

=Φ [ Γ (

n

) 983085 ^( )||oo ^

η



particular

φ = 0 j



BemarJc


[There

exists


fixed

h

r

ζ R

+

the function

( V )

K

k

Kvk) = Kh k) k

ζ Λ +) verifies

λ (K

r

)

^

N


+

the function

K

k


+



+


is majorized by N


I I I j



3


I n


fulfilling

all the conditions



verify

condition I I I

2



verify

condition IV)

3

The Case of a

Positive

Kernel

If K(k k

f

) gt 0 for every (k

k

r

) ζR

+


= R

+


2






K (k k

f

)

= 0 it is not

β gtoo


= R

+

is satisfied



+

)

X

==

X

1

(a


IIU = sup () lt A

f(k )


IceuroR+ )


K(k k )


+

x R

+

let M


f

) ^ M for every

(k k) ζR+ x β +

(II) There exists

a n α gt 0


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


[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+


verifying

the inequalities

M f

l

=rdk

ltgt 1

AgtaK(ll)

I t



Jf

by JΓ and we


( I I ) ( I I I )

on the kernel In

particular


( 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


oo

k)=983085 ^j983085

dr e983085

ur

sπ ikr sinkr

o

k

2

[ α

2

+ (A + F )

2

] [α

2

+ (k 983085 k



Gap Equation 279



V(r) =

Ve~

ocr

i


K(kk) ^

2F

dK(kk)

dk

8 7 1

lt sup 1mdash983085

for

every (k k) ζ R+

x

R+



immediate


= mi


k gt

0

Therefore choosing

a


fD k )

j(amp

2

983085 I )

2

+

a

2

K(k

9

I )

2


Acknowledgements

One of us

G

F)

would like

t o

thank P rofessor

D K A ST L E R

for

the


at the

University

of

Aix983085 Marseille

and

theD GRST


References




Cimento

18 1247

(1960)

FANO G M

SAVOIA




114

1377 (1959)

5

KITAMTJRA

M

Progr Theor P hys

30 435

1963)

6 O D E H

F


8 187

1964)

7

D U N F O R D

N

and J T


I p 456 New York


P BILLARD G FANO


de

Physique Istituto

di

Fisica


Universite

dAix


Place Victor Hugo

F

13

Marseille

3

e





K (k k

f

)

= 0 it is not

β gtoo


= R

+

is satisfied



+

)

X

==

X

1

(a


IIU = sup () lt A

f(k )


IceuroR+ )


K(k k )


+

x R

+

let M


f

) ^ M for every

(k k) ζR+ x β +

(II) There exists

a n α gt 0


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


[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+


verifying

the inequalities

M f

l

=rdk

ltgt 1

AgtaK(ll)

I t



Jf

by JΓ and we


( I I ) ( I I I )

on the kernel In

particular


( 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


oo

k)=983085 ^j983085

dr e983085

ur

sπ ikr sinkr

o

k

2

[ α

2

+ (A + F )

2

] [α

2

+ (k 983085 k



Gap Equation 279



V(r) =

Ve~

ocr

i


K(kk) ^

2F

dK(kk)

dk

8 7 1

lt sup 1mdash983085

for

every (k k) ζ R+

x

R+



immediate


= mi


k gt

0

Therefore choosing

a


fD k )

j(amp

2

983085 I )

2

+

a

2

K(k

9

I )

2


Acknowledgements

One of us

G

F)

would like

t o

thank P rofessor

D K A ST L E R

for

the


at the

University

of

Aix983085 Marseille

and

theD GRST


References




Cimento

18 1247

(1960)

FANO G M

SAVOIA




114

1377 (1959)

5

KITAMTJRA

M

Progr Theor P hys

30 435

1963)

6 O D E H

F


8 187

1964)

7

D U N F O R D

N

and J T


I p 456 New York


P BILLARD G FANO


de

Physique Istituto

di

Fisica


Universite

dAix


Place Victor Hugo

F

13

Marseille

3

e



Gap Equation 279



V(r) =

Ve~

ocr

i


K(kk) ^

2F

dK(kk)

dk

8 7 1

lt sup 1mdash983085

for

every (k k) ζ R+

x

R+



immediate


= mi


k gt

0

Therefore choosing

a


fD k )

j(amp

2

983085 I )

2

+

a

2

K(k

9

I )

2


Acknowledgements

One of us

G

F)

would like

t o

thank P rofessor

D K A ST L E R

for

the


at the

University

of

Aix983085 Marseille

and

theD GRST


References




Cimento

18 1247

(1960)

FANO G M

SAVOIA




114

1377 (1959)

5

KITAMTJRA

M

Progr Theor P hys

30 435

1963)

6 O D E H

F


8 187

1964)

7

D U N F O R D

N

and J T


I p 456 New York


P BILLARD G FANO


de

Physique Istituto

di

Fisica


Universite

dAix


Place Victor Hugo

F

13

Marseille

3

e

fano-an existence proof 1968

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