SOME APPLICATIONS IN ATOMIC PHYSICS OF THE THEORY OF COUPLED DIFFERENTIAL EQUATIONS

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SOME APPLICATIONS IN ATOMIC PHYSICS OF THE THEORY OF COUPLED DIFFERENTIAL

EQUATIONS

C. Chuan, S. Bougouffa

To cite this version:

C. Chuan, S. Bougouffa. SOME APPLICATIONS IN ATOMIC PHYSICS OF THE THEORY OF

COUPLED DIFFERENTIAL EQUATIONS. Journal de Physique Colloques, 1988, 49 (C1), pp.C1-

251-C1-254. �10.1051/jphyscol:1988153�. �jpa-00227473�

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JOURNAL DE PHYSIQUE

Colloque C1, Supplement au n03, Tome 49, Mars 1988

SOME APPLICATIONS IN ATOMIC PHYSICS OF THE THEORY OF COUPLED DIFFERENTIAL EQUATIONS

C.X. CHUAN and S.BOUGOUFFA(~)

Institute of Physics, BP 260, Constantine, Algeria

La thbozie sur la shparation d'un systbrm d'hquations diffhzentielles couplhes est appliquhe A deux exemples c q o z t a n t chacun deux bquations. Nous prbsentons une discussion des rdsultats n d r i q u e s ainsi qu'une ghnhzalisation A un s y s t h avec un nombze quelconque d'hquations.

The theory on the separation of a system of coupled differential equations is applied to two e x w l e s each having two equations. N m z i c a l results are discussed and a generalisation to a system of any q d e z of equations will be outlined.

The system of coupled diffezentiel equations (CDE) to be considered here is :

The qeneral theory has already been presented in previous papezs

L7-j ,

[2] ,we continue to keep in the present one the, s m notations and

~onventions. Its iplemzntation as well as the test of its efficiency will be discussed belw.

The t w o equation case:Uhen n=2

,

with the use of the t z a n s f o m t i o n matrix T(a,& = T(a)T(dl) wheze a(& ) and dl(&, d ) are already defined in

-[22

we have :

,

unit m t r i x

, z =[$)

-,

ddl d dL& + 2 . . -

[?,dl] = - 2 dz dr

.

The theory provides a m a n to choose the dr

azbitrary p a z m t e z such that effect of the operatoz [l+d12 ]-'[?,dl] bec- negligible c q a z e d to the 2hs of the above equation so that convergence of the iterated solution is ensuzed. This choice is guided b y the follwinq properties of the function

dl

( M , S ) :

- dl(&,%) is always analytic even in the case whece the coupling function 8 1 2 ) have singulazities

.

-

7 f

is mnotonic and is a decrea~i,:~ function as is frequently the case in dltcmic 3hysics

,

then dl will also be a mnotonic function

.

- It m y be verified that :

(l'0n leave from INES Mecanique. Batna, Algeria

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988153

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JOURNAL DE PHYSIQUE

I n t h e cases undez c o n s i d e z a t i o n

,

i t i s p o s s i b l e t o d e t e r n i n e g z a p h i - c a l l y such t h a t e f f e c t s of t h e o p e z a t o z [ l + dl2]-'[?,dl] i n t h e 2 . h . s . of

(2) can be n e g l e c t e d a t f z i s t ozdez a p p z o x i m t i o n ccqoazed t o t h e 1 . h . s .

.

I n another wozds , t h e e q u a t i o n s can be c o n s i d e z e d as s e p a r a t e d and i t s s o l u t i o n c a n be used a s a t z i a l f u n c t i o n

"

i n ozdez t o i n i t i a t e t h e i n t e z a t i o n pzocess f o z h i g h e z ozdez a p p z o x i m t i o n i f m z e accuzacy i s needed

.

The m i x i n g p a z m t e z

X

i s t h e n d e f i n e d by

302 e x q o l e i t m y be checked t h a t :

- i f

qp/&' =x

i s independent o f

,

W / d z = 0 and t h e e q u a t i o n s can always be e x a c t l y sepazated as s t a t e d by t h e theozern on t h e s e p a z a t i o n

LI].

- i f B y y O t h e n 6 ( = 0 and a = I ,

A = /

we have t h e n

X

= 0 ( n o m i x i n g )

- i f h f = ( e x a c t zesonance) d,f-> V , a -4, A- 1

t h e n

2 -a

¹( m x i m m coup 1 i n g )

.

The c o u p l e d i n t e g r o - d i f f e r e n t i a l e q u a t i o n s : C o n s i d e z t h e f o l l a u i n g s y s t e m :

Wheze a n a l y t i c i t y and s y m t z y a r e ass.cumd f o z t h e k e z n e l s K . ^,.An

l d

e x t e n s i o n of t h e t h e o r e m on t h e s e p a z a t i o n of a s y s t e m of ChE i s p o s s i b l e heze and m y be s t a t e d as f o l l w e d : 131.

"702 any a n a l y t i c f o m of t h e f u n c t i o n s f i / , B i a n d y i , a c m l e t e

s e p a z a t i o n of t h e e q u a t i o n i s p o s s i b l e i f and o n l y i f B . . / ( f .-f

.I= ^K ,/i~.

^-

^{~ i )}

Y I L L I I1

and aze independent of z " .

The u s u a i pzoceduze t o s o l v e t h i s t y p e of e q u a t i o n s i s a g a i n i n t e z a t i o n based on an a p p r o p r i a t e c h o i c e of t h e " t z i a l f o n c t i o n " w h i c h m y be c o n s t z u c t e d w i t h t h e pzesent m t h o d . I n f a c t , we have f o u n d t h a t f o z t h e n = 2 case,

i g n o z i n g f i z s t t h e i n t e g z a l t e r n and s o l v i n g t h e

a,

we m y zecsuqr t h e M o i s w i c h t z i a l f u n c t i o n f o z t h e 3Ltck's m d e l .

N m r i c a l r e s u l t s : l n t h e g e n e r a l i s e d 3luck's m d e l [43,

.

^[5]^{we have}^:

So t h a t t h e r e s u l t i n g C2)€ c a n a l w a y s b e s e p a r a t e d a n d s o l v e d a n a l y t i c a l l y l e a d i n g t o t h e i h a L d s W c a & s s , s ~ c t i c y g Q ~ - ~ . We d i s p l a y i n 3 1 9 - 7 -

1

t h e z e s u l t s o b t a i n e d f o z k ' = 1 .

,

k l = 0.5

,

a , = l.',and 1 = 0

,

1

,

2,,

v e z s u s Log

c'.

dls c a n b e n o t i c e d , a g r e e m e n t w i t h B u c k ' s p z e u i o u s c a l c u l a - t i o n s i n q u i t e e x c e l l e n t . M o z e o v e z , i n g o i n g f u t h e z b e y o n d t h e r a n g e i n i t i a l l y i n v e s t i g a t e d b y t h i s a u t h o r , w e o b s e z u e c l e a z l y u n d u l a t i o n s o f

4En

w h i c h m a y b e e x p l a i n e d s i m p 1 y i n p z a s e n t h h e o z y . I n f a c t , t h e s o l u -

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tions o f the sepazated e q u a t i o n s a r e lineaz c o m b i n a t i o n o f t z i g o n o m e t z i c

( 1

= 0) oz spherical Bessel f u n c t i o n s

( 1 #

0) and m u s t s a t i s f y continuity zequizements at = a. 3 0 2

1

= 0 a n d w i t h C))ko / k l , w e f i n d that t h e a n a l y t i c e x p z e s s i o n o f Q'* is o f t h e f o z m :

1 ,

T h i s shows t h a t ~ ' ~ w h e n 3 0 C 7- o~ except at the

"

resonances

"

d e f i n e d by

6

⁼⁽²ⁿ⁺ n = O , 1 , 2

...

-

A t t h e s e zesonances,

Q i n

is independent o f C

.

T h e s e q u a l i t a t i v e c o n c -

0

lusions are c o n f i z m e d w i t h q u i t e g o o d accuzacy by e x a c t c a l c u l a t i o n s pzesented in 3ig.,7-.

I n t h e Lane, L i n schematic m o d e [ 6 ] w e have o n t h e othez hand : 812 ( z ) = A'/ t 2 being the co@pling stzenght

.

W e d i s p l a y in 3ig.-2- the inelastic c r o s s section uezsus 1 f o z vazious values o f the enezqy gap 5 k 2 = 0 . , .01

,

, 0 3 6

,

.5evc:luateded at f izst ozdez o f appzoximatlon.

@"(in units X a o ) 2

Huck (1=0)

...

I

~ i ~ ( i n units x a 2 )

0

7.

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C1-254 JOURNAL

DE

PHYSIQUE

Here it c a n also be seen that exact agreement w i t h the results inferred f z o m the zesonance distoztion method given by these authozs is obtained foz small values o f ~ k '

.

This method however become inadequate foz lazge energy gap

.

^{3 0 2}^example

^,

in our w o z k , the chazactezistic m a x i m u m o f

q i n

is not observed for

nk2

⁼ ^{.5 ev} .%Ldulations o f the cross

1

section

Q 1

w i t h increasing

A

aze also observed for f i x e d values o f 1

.

Second ozdez calculationsate actually in progcess.

g e n e r a l i s a t i o n : T h e case n > 2 zequize a unitary tzansformation matrix

In

w h i c h , in our case

,

will need at least n/2 unknown parametezs.

Exact separation is t h e r e f o r e expected for anly vezy special cases

.

Howeoez it is possible t o show that the strong coupling problem

,

in a f z i s t stage

,

can always be converted into a w e a k coupling one foz w h i c h the usual iterative methods become e f f i c i e n t

.

This appzoach ( C x C to be published ) w h i c h is an extension o f the one dezived f o z the two equations case , relies on the pzoperties o f the nxn t ~ a n * ~ ~ m a t i o n matrix

T

constzu-

".& f z o m the 2x2 matzix defined above. O n this new base

,

the inltial system is cast into a f z o m such that the coupling t e r m s , aze ordered in a sequence o f increasing stzenght and the problem zeduced to the c a s e o f n

'

2 coupling equations

.

The operation may be repeated again until sepaza- tion o f all equations i s reached

.

The original solution can be zecooered by the same sequence o f inverse transformation

.

sf

^erences

[ 1 1 C a o x c 7.7hys.dtMathgene 1069 (1981)

[2] Cao x c J.?hy~.

A

Math gene

15

2727 (1982)

, 1

⁶⁰⁹ ⁽¹⁹⁸⁴⁾

[3] Bougouffa S ,Cao x c Contributions to

STdtT?WjS

16 Boston (1986) C4] Bougouf f a S ThQse Univers .Constant ine (1 985)

[5] Huck

2.2.

7roc.?hys.Soc.

dt

369 (1957) [6] Lane

N.?. ,

fin

C.C.

3hys.2ev.

I33