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1/5

quant-

ph/9502019

19

Fe

b

1995

V a r i a t i o n a l p e r t u r b a t i o n e x p a n s i o n f o r s t r o n g - c o u p l i n g c o e c i e n t s o f t h e a n h a r m o n i c

o s c i l l a t o r

W . J a n k e

1 2

a n d H . K l e i n e r t

2

1

I n s t i t u t f u r P h y s i k , J o h a n n e s G u t e n b e r g - U n i v e r s i t a t M a i n z , S t a u d i n g e r W e g 7 , 5 5 0 9 9 M a i n z

2

I n s t i t u t f u r T h e o r e t i s c h e P h y s i k , F r e i e U n i v e r s i t a t B e r l i n , A r n i m a l l e e 1 4 , 1 4 1 9 5 B e r l i n

( F e b r u a r y 1 8 , 1 9 9 5 )

A s a n a p p l i c a t i o n o f a r e c e n t l y d e v e l o p e d v a r i a t i o n a l p e r -

t u r b a t i o n t h e o r y w e n d t h e r s t 2 2 t e r m s o f t h e c o n v e r g e n t

s t r o n g - c o u p l i n g s e r i e s e x p a n s i o n f o r t h e g r o u n d s t a t e e n e r g y

o f t h e q u a r t i c a n h a r m o n i c o s c i l l a t o r .

V a r i a t i o n a l p e r t u r b a t i o n t h e o r y y i e l d s u n i f o r m l y a n d e x -

p o n e n t i a l l y f a s t c o n v e r g i n g e x p a n s i o n s f o r a n y q u a n t u m

m e c h a n i c a l s y s t e m 1 { 3 ] . I n t h i s n o t e w e a d a p t t h i s

t h e o r y t h e p r o b l e m o f c a l c u l a t i n g s t r o n g - c o u p l i n g e x p a n -

s i o n s . A s a n e x a m p l e , w e c o n s i d e r t h e q u a n t u m m e c h a -

n i c a l a n h a r m o n i c o s c i l l a t o r a n d n d t h e e x p a n s i o n c o -

e c i e n t s w i t h a h i g h d e g r e e o f a c c u r a c y . T h e i n p u t i s

p r o v i d e d b y t h e e x a c t R a y l e i g h - S c h r o d i n g e r p e r t u r b a t i o n

c o e c i e n t s o f t h e g r o u n d - s t a t e e n e r g y u p t o o r d e r 2 5 1 ,

o b t a i n e d f r o m t h e r e c u r s i o n r e l a t i o n s o f B e n d e r a n d W u

4 ] . T h e r e s u l t i n g s t r o n g - c o u p l i n g c o e c i e n t s w i l l b e c a l -

c u l a t e d u p t o t h e 2 2 n d o r d e r w i t h a p r e c i s i o n o f a b o u t

2 0 d i g i t s .

T h e p o t e n t i a l o f t h e a n h a r m o n i c o s c i l l a t o r i s

V ( x ) =

!

2

2

x

2

+

g

4

x

4

( !

2

g > 0 ) ( 1 )

T h e R a y l e i g h - S c h r o d i n g e r p e r t u r b a t i o n t h e o r y y i e l d s a

p o w e r s e r i e s e x p a n s i o n

E ( g ) = !

1

X

= 0

e

B W

g = 4

!

3

( 2 )

w h e r e e

B W

a r e r a t i o n a l n u m b e r s

1

2

3

4

2 1

8

3 3 3

1 6

3 0 8 8 5

1 2 8

; : : : : ( 3 )

T h e s u p e r s c r i p t B W r e f e r s t o B e n d e r a n d W u . U s i n g t h e

s y m b o l i c a l g e b r a p r o g r a m M A P L E w e g e n e r a t e t h e r s t

2 5 1 t e r m s e x a c t l y .

T h e s e r i e s ( 2 ) c a n n o t b e u s e d f o r a n e v a l u a t i o n o f t h e

e n e r g y | i t h a s a z e r o r a d i u s o f c o n v e r g e n c e d u e t o t h e

f a c t o r i a l g r o w t h o f t h e c o e c i e n t s e

B W

. O n l y f o r s m a l l

c o u p l i n g s g

7/29/2019 9502019

2/5

F r o m t h e a p p r o x i m a t e e n e r g i e s ( 9 ) i t i s e a s y t o d e r i v e

s i m p l e f o r m u l a s f o r t h e c o e c i e n t s o f t h e s t r o n g - c o u p l i n g

e x p a n s i o n 3 ] . W e s i m p l y e x p a n d t h e f u n c t i o n w

N

( g !

2

)

i n p o w e r s o f ^ !

2

= ( g = !

3

)

2 = 3

g

2 = 3

a n d n d

W

N

= ( g = 4 )

1 = 3

"

0

+

1

g = 4

!

3

2 = 3

+

2

g = 4

!

3

4 = 3

+

#

( 1 2 )

w i t h t h e c o e c i e n t s

n

=

1

n

g

( 2 n 1 ) = 3

w

( n )

N

( g 0 ) ( 1 3 )

H e r e w

( n )

N

( g 0 ) d e n o t e s t h e n ' t h d e r i v a t i v e o f w

N

( g !

2

)

w i t h r e s p e c t t o ^ !

2

a t !

2

= 0 . T o c a l c u l a t e t h e s e d e r i v a -

t i v e s , w e n o t e t h a t ^ !

2

e n t e r s t h e r e e x p a n s i o n c o e c i e n t s

( 8 ) i n t h e f o r m

e =

X

j = 0

e

B W

j

( 1 3 j ) = 2

l j

( 4 ( ^ !

2

1 ) = g )

j

( 1 4 )

T h e q u a n t i t i e s w

( n )

N

( g 0 ) a r e t h e r e f o r e g i v e n b y

w

( n )

N

( g 0 ) = (

d

d !

2

)

n

e

^! = 0

=

X

j = 0

e

B W

j

( 1 3 j ) = 2

l j

( 1 )

n

n

n 1

Y

m = 0

( l j m )

!

( 4 = g )

j

( 1 5 )

T h e o p t i m a l v a l u e o f

N

h a s t h e N - d e p e n d e n c e ( s e e

R e f . 3 ] )

3

N

= g c N

1 + 6 8 5 = N

2 = 3

( 1 6 )

w h e r e t h e c o e c i e n t c i s t h e s o l u t i o n o f a s i m p l e t r a n -

s c e n d e n t a l e q u a -

t i o n ( c = 0 1 8 6 0 4 7 2 7 2 9 8 7 3 9 7 5 1 2 9 8 4 5 5 4 7 4 0 4 6 2 . e y e -

b a l l t t o e x t r e m a l v a l u e s o f . T h e c o r r e s p o n d i n g o r d e r -

d e p e n d e n t v a l u e s o f t h e d i m e n s i o n l e s s c o u p l i n g c o n s t a n t

g g =

3

N

a r e i n s e r t e d i n t o ( 1 3 ) a n d p r o d u c e t h e c o e -

c i e n t s s h o w n i n T a b l e I . O u r r e s u l t f o r

0

a g r e e s t o a l l

2 3 d i g i t s w i t h t h e m o s t a c c u r a t e v a l u e f o r

0

a v a i l a b l e i n

t h e l i t e r a t u r e 6 ] :

0

= 0 6 6 7 9 8 6 2 5 9 1 5 5 7 7 7 1 0 8 2 7 0 9 6 2 0 1 6 9 1 9 8 6 0

1 9 9 4 3 0 4 0 4 9 3 6 9 8 4 0 6 0 4 5 5 9 7 6 6 6 3 8 0 ( 1 7 )

F o r

1

a n d

2

, o u r r e s u l t s a r e c o n s i s t e n t w i t h b u t c o n -

s i d e r a b l y m o r e a c c u r a t e t h a n p r e v i o u s r e s u l t s i n R e f . 7 ]

a n d R e f s . 8 , 9 ]

1

= 0 1 4 3 6 6 8 8 0 0 ,

2

= 0 0 0 8 6 2 7 5 8 ] .

A s a f u r t h e r c h e c k w e h a v e e v a l u a t e d o u r s t r o n g -

c o u p l i n g s e r i e s a t g = 4 = 0 1 0 3 0 5 1 2 ( s e t t i n g ! = 1 )

a n d c o m p a r e d t h e n u m b e r s w i t h t h e v e r y p r e c i s e l o w e r

a n d u p p e r b o u n d s o f V i n e t t e a n d

C

i z e k 6 ] . T a b l e I I

s h o w s t h a t t h e e n e r g i e s a r e a c c u r a t e t o a b o u t 2 0 d i g i t s

f o r a l l c o u p l i n g s g = 4 1 . T h e a c c u r a c y i s l i m i t e d b y t h e

p r e c i s i o n o f t h e

n

N o t e t h a t o u r s t r o n g - c o u p l i n g e x p a n s i o n g i v e s g o o d

r e s u l t s d o w n t o v e r y s m a l l c o u p l i n g s g | e v e n a t g = 4 =

0 1 , t h e r e s u l t a g r e e i n g t o 7 d i g i t s w i t h t h e b o u n d s i n

6 ] . I f o n e w a n t s t o w o r k a t s u c h s m a l l c o u p l i n g s , o n e

c a n e a s i l y c a l c u l a t e h i g h e r c o e c i e n t s

n

t o i n c r e a s e t h e

a c c u r a c y .

I n F i g . 1 w e s h o w t h e a p p r o a c h o f

n

t o t h e a s y m p t o t i c

v a l u e g i v e n i n T a b l e I b y p l o t t i n g

N

(

n

)

N

n

( 1 8 )

o n a l o g a r i t h m i c s c a l e . T h e p e r i o d i c s t r u c t u r e i n t h e d a t a

i s c a u s e d b y a n o s c i l l a t o r y a p p r o a c h o f (

n

)

N

n

t o z e r o

( s e e F i g . 2 ) . T h e e n v e l o p e o f t h e s e o s c i l l a t i o n s f o l l o w s

t h e c u r v e

N

= e x p

0

1

N

1 = 3

( 1 9 )

F o r t h e l e a d i n g t e r m

0

o f t h e s t r o n g - c o u p l i n g e x p a n -

s i o n , t h e b e h a v i o r ( 1 6 ) s u g g e s t s a c o e c i e n t

1

9 7

3 ] . F r o m t h e e y e b a l l t s s h o w n i n F i g . 1 w e e x t r a c t

1

9 4 2 9 0 5 8 0 5 , a n d 7 1 8 f o r

n

w i t h n = 0 1 5 a n d

1 0 , r e s p e c t i v e l y . T h e s e e s t i m a t e s s h o u l d b e t a k e n w i t h

s o m e c a r e , h o w e v e r , s i n c e w e d o n o t k n o w h o w c l o s e w e

a r e t o t h e a s y m p t o t i c r e g i m e f o r N 6 5 2 5 1 , w h e r e

t h e n a t u r a l e x p a n s i o n p a r m e t e r i s 1 / N

1 = 3

0 2 5 0 1 6

T h e N - d e p e n d e n c e o f t h e c o e c i e n t

0

, f o r i n s t a n c e , i s

u s t a s w e l l t t e d b y t h e d a s h e d l i n e w i t h

1

= 9 7

W e c o n c l u d e b y m e n t i o n i n g t h a t v a r i a t i o n a l p e r t u r b a -

t i o n t h e o r y a l l o w s a l s o f o r a c a l c u l a t i o n o f t h e i m a g i -

n a r y p a r t s o f e n e r g i e s o n t h e l e f t - h a n d c u t i n t h e c o m -

p l e x c o u p l i n g c o n s t a n t p l a n e 1 0 ] . T h e r e s u l t s n o t o n l y

i m p r o v e t h e s e m i c l a s s i c a l v a l u e s 1 1 ] i n t h e t u n n e l i n g r e -

g i m e , b u t a l s o d e t e r m i n e s t h e i m a g i n a r y p a r t a t l a r g e

n e g a t i v e g 1 0 , 1 2 ] , w h e r e t h e d e c a y p r o c e e d s b y s l i d i n g

r a t h e r t h a n t u n n e l i n g .

W . J . t h a n k s t h e D e u t s c h e F o r s c h u n g s g e m e i n s c h a f t f o r

a H e i s e n b e r g f e l l o w s h i p .

T A B L E I . S t r o n g - c o u p l i n g p e r t u r b a t i o n c o e c i e n t s

n

n

n

0 0 . 6 6 7 9 8 6 2 5 9 1 5 5 7 7 7 1 0 8 2 7 0 9 6

1 0 . 1 4 3 6 6 8 7 8 3 3 8 0 8 6 4 9 1 0 0 2 0 3

2 0 . 0 0 8 6 2 7 5 6 5 6 8 0 8 0 2 2 7 9 1 2 8

3 0 . 0 0 0 8 1 8 2 0 8 9 0 5 7 5 6 3 4 9 5 4 3

4 0 . 0 0 0 0 8 2 4 2 9 2 1 7 1 3 0 0 7 7 2 2 1

5 0 . 0 0 0 0 0 8 0 6 9 4 9 4 2 3 5 0 4 0 9 6 6

6 0 . 0 0 0 0 0 0 7 2 7 9 7 7 0 0 5 9 4 5 7 7 5

7 0 . 0 0 0 0 0 0 0 5 6 1 4 5 9 9 7 2 2 2 3 5 4

8 0 . 0 0 0 0 0 0 0 0 2 9 4 9 5 6 2 7 3 2 7 1 2

9 0 . 0 0 0 0 0 0 0 0 0 0 6 4 2 1 5 3 3 1 9 5 4

1 0 0 . 0 0 0 0 0 0 0 0 0 0 4 8 2 1 4 2 6 3 7 8 7

1 1 0 . 0 0 0 0 0 0 0 0 0 0 0 8 9 4 0 3 1 9 8 6 7

1 2 0 . 0 0 0 0 0 0 0 0 0 0 0 1 2 0 5 6 3 7 2 1 5

1 3 0 . 0 0 0 0 0 0 0 0 0 0 0 0 1 3 0 3 4 7 6 5 0

1 4 0 . 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 7 6 0 0 8 9

2

7/29/2019 9502019

3/5

1 5 0 . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 4 5 8 9 0 1

1 6 0 . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 8 9 8 9 8

1 7 0 . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 9 1 9 6 0 0

1 8 0 . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 2 8 8 1 3

1 9 0 . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 2 9 6 2

2 0 0 . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 4 4 3 8

2 1 0 . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 2 3 0 5

2 2 0 . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 4

T h e s o l i d s t r a i g h t l i n e s a r e t h e b e s t e y e b a l l t s t o t h e e n v e -

l o p e o f t h e d a t a . T h e d a s h e d l i n e i n t h e

0

- d a t a h a s a s l o p e

e q u a l t o t h e t h e o r e t i c a l l y e x p e c t e d v a l u e 9 7 3 ] .

3

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4/5

F I G . 2 . O s c i l l a t o r y b e h a v i o r a r o u n d t h e e x p o n e n t i a l a p -

p r o a c h t o t h e l i m i t i n g v a l u e o f

0

T A B L E I I . G r o u n d - s t a t e e n e r g i e s f r o m s t r o n g - c o u p l i n g s e -

r i e s e x p a n s i o n . T h e l i n e s l a b e l e d \ l b " a n d \ u b " a r e t h e l o w e r

a n d u p p e r b o u n d s o f R e f . 6 ] .

g = 4 n

m a x

E

0

0 . 1 1 5 0 . 5 5 9 1 4 6 5 9 7 5 0 3 5 6 2 1 8 7 0

2 0 0 . 5 5 9 1 4 6 2 0 1 2 0 1 8 0 5 5 4 4 6

2 2 0 . 5 5 9 1 4 6 3 4 4 3 7 3 8 7 3 1 2 6 9

l b 0 . 5 5 9 1 4 6 3 2 7 1 8 3 5 1 9 5 7 6 3

u b 0 . 5 5 9 1 4 6 3 2 7 1 8 3 5 1 9 5 7 6 7

0 . 3 1 5 0 . 6 3 7 9 9 1 7 8 3 1 7 8 5 3 6 0 2 5 3

2 0 0 . 6 3 7 9 9 1 7 8 3 1 7 1 2 3 6 1 4 9 3

2 2 0 . 6 3 7 9 9 1 7 8 3 1 7 1 2 8 0 3 8 1 8

l b 0 . 6 3 7 9 9 1 7 8 3 1 7 1 2 7 8 5 2 8 3

u b 0 . 6 3 7 9 9 1 7 8 3 1 7 1 2 7 8 5 2 9 6

0 . 5 1 5 0 . 6 9 6 1 7 5 8 2 0 7 6 5 1 9 1 5 1 6 9

2 0 0 . 6 9 6 1 7 5 8 2 0 7 6 5 1 4 5 8 8 7 5

2 2 0 . 6 9 6 1 7 5 8 2 0 7 6 5 1 4 5 9 2 8 8

l p 0 . 6 9 6 1 7 5 8 2 0 7 6 5 1 4 5 9 2 5 1

u p 0 . 6 9 6 1 7 5 8 2 0 7 6 5 1 4 5 9 2 8 5

1 . 0 1 5 0 . 8 0 3 7 7 0 6 5 1 2 3 4 2 7 3 8 1 2 0 4 7 6

2 0 0 . 8 0 3 7 7 0 6 5 1 2 3 4 2 7 3 7 6 9 3 5 0 9

2 2 0 . 8 0 3 7 7 0 6 5 1 2 3 4 2 7 3 7 6 9 3 5 4 1

l b 0 . 8 0 3 7 7 0 6 5 1 2 3 4 2 7 3 7 5 6

u b 0 . 8 0 3 7 7 0 6 5 1 2 3 4 2 7 3 7 8 6

2 . 0 1 5 0 . 9 5 1 5 6 8 4 7 2 7 2 9 5 0 0 0 1 1 1 8 4 2 1 3 6 9

2 0 0 . 9 5 1 5 6 8 4 7 2 7 2 9 5 0 0 0 1 1 1 4 6 9 3 0 2 7

2 2 0 . 9 5 1 5 6 8 4 7 2 7 2 9 5 0 0 0 1 1 1 4 6 9 3 0 5 2

l b 0 . 9 5 1 5 6 8 4 7 2 7 2 9 4 9 9 9

u b 0 . 9 5 1 5 6 8 4 7 2 7 2 9 5 0 0 1

4

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1 ] H . K l e i n e r t , P h y s . L e t t . A 1 7 3 , 3 3 2 ( 1 9 9 3 ) ; J . J a e n i c k e

a n d H . K l e i n e r t , P h y s . L e t t . A 1 7 6 , 4 0 9 ( 1 9 9 3 ) ; H . K l e i -

n e r t a n d H . M e y e r , P h y s . L e t t . A 1 8 4 , 3 1 9 ( 1 9 9 4 ) .

T h e m e t h o d i s a s y s t e m a t i c e x p a n s i o n o f t h e r s t - o r d e r

a p p r o a c h o f R . P . F e y n m a n a n d H . K l e i n e r t , P h y s . R e v .

A 3 4 , 5 0 8 0 ( 1 9 8 6 ) a n d o f R . G i a c h e t t i a n d V . T o g n e t t i ,

P h y s . R e v . L e t t . 5 5 , 9 1 2 ( 1 9 8 5 ) ; I n t . J . M a g n . M a t e r . 5 4 -

5 7 , 8 6 1 ( 1 9 8 6 ) ; R . G i a c h e t t i , V . T o g n e t t i , a n d R . V a i a ,

P h y s . R e v . B 3 3 , 7 6 4 7 ( 1 9 8 6 ) .

R e l a t e d e x p a n s i o n s h a v e b e e n p r o p o s e d b y

R . S e z n e c a n d J . Z i n n - J u s t i n , J . M a t h . P h y s . 2 0 , 1 3 9 8

( 1 9 7 9 ) ; T . B a r n e s a n d G . I . G h a n d o u r , P h y s . R e v . D 2 2

9 2 4 ( 1 9 8 0 ) ; B . S . S h a v e r d y a n a n d A . G . U s h e r v e r i d z e ,

P h y s . L e t t . B 1 2 3 , 3 1 6 ( 1 9 8 3 ) ; P . M . S t e v e n s o n , P h y s .

R e v . D 3 0 , 1 7 1 2 ( 1 9 8 5 ) ; D 3 2 , 1 3 8 9 ( 1 9 8 5 ) ; P . M . S t e -

v e n s o n a n d R . T a r r a c h , P h y s . L e t t . B 1 7 6 , 4 3 6 ( 1 9 8 6 ) ;

A . O k o p i n s k a , P h y s . R e v . D 3 5 , 1 8 3 5 ( 1 9 8 7 ) ; D 3 6 , 2 4 1 5

( 1 9 8 7 ) ; W . N a m g u n g , P . M . S t e v e n s o n , a n d J . F . R e e d ,

Z . P h y s . C 4 5 , 4 7 ( 1 9 8 9 ) ; U . R i t s c h e l , P h y s . L e t t . B 2 2 7

4 4 ( 1 9 8 9 ) ; Z . P h y s . C 5 1 , 4 6 9 ( 1 9 9 1 ) ; M . H . T h o m a ,

Z . P h y s . C 4 4 , 3 4 3 ( 1 9 9 1 ) ; I . S t a n c u a n d P . M . S t e v e n -

s o n , P h y s . R e v . D 4 2 , 2 7 1 0 ( 1 9 9 1 ) ; R . T a r r a c h , P h y s .

L e t t . B 2 6 2 , 2 9 4 ( 1 9 9 1 ) ; H . H a u g e r u d a n d F . R a u n d a ,

P h y s . R e v . D 4 3 , 2 7 3 6 ( 1 9 9 1 ) ; A . N . S i s s a k i a n , I . L . S o -

l i v t o s v , a n d O . Y . S h e y c h e n k o , P h y s . L e t t . B 3 1 3 , 3 6 7

( 1 9 9 3 ) ; A . D u n c a n a n d H . F . J o n e s , P h y s . R e v . D 4 7

2 5 6 0 ( 1 9 9 3 ) .

2 ] H . K l e i n e r t , P f a d i n t e g r a l e i n Q u a n t e n m e c h a n i k , S t a t i s t i k

u n d P o l y m e r p h y s i k ( B . I . - W i s s e n s c h a f t s v e r l a g , M a n n -

h e i m , 1 9 9 3 ) .

3 ] H . K l e i n e r t , P a t h I n t e g r a l s i n Q u a n t u m M e c h a n i c s , S t a t i -

s t i c s a n d P o l y m e r P h y s i c s , 2 n d e d i t i o n ( W o r l d S c i e n t i c

P u b l i s h i n g C o . , S i n g a p o r e , 1 9 9 5 ) .

4 ] C . M . B e n d e r a n d T . T . W u , P h y s . R e v . 1 8 4 , 1 2 3 1 ( 1 9 6 9 ) ;

P h y s . R e v . D 7 , 1 6 2 0 ( 1 9 7 3 ) .

5 ] N o t e t h a t o u r a p p r o x i m a t i o n i s f a r m o r e a c c u r a t e t h a n

t h e o n e u s e d i n m a n y p a p e r s o n t h e a c c e l e r a t i o n o f c o n -

v e r g e n c e o f p e r t u r b a t i o n t h e o r y s u c h a s

Y a m a z a k i , J . P h y s . A 1 7 , 3 4 5 ( 1 9 8 4 ) . T h e a p p r o x i m a t i -

o n s i n t h e s e p a p e r s a r e r e p r o d u c e d i f w e u s e , f o r a l l N

=

1

a s a v a r i a t i o n a l f r e q u e n c y .

6 ] T h e m o s t a c c u r a t e e n e r g i e s h a v e b e e n c a l c u l a t e d b y

F . V i n e t t e a n d J .

C

i z e k , J . M a t h . P h y s . 3 2 , 3 3 9 2 ( 1 9 9 1 ) .

7 ] F . T H i o e a n d E . W . M o n t r o l l , J . M a t h . P h y s . 1 6 , 1 9 4 5

( 1 9 7 5 ) .

8 ] B . B o n n i e r , M . H o n t e b e y r i e , a n d E . H . T i c e m b a l , J .

M a t h . P h y s . 2 6 , 3 0 4 8 ( 1 9 8 5 ) .

9 ] M . A . S m o n d y r e v , J . O . I . N . T . p r e p r i n t , D u b n a ( 1 9 8 4 ) .

T h i r d t e r m , a s c i t e d b y B o n n i e r e t a l .

1 0 ] H . K l e i n e r t , P h y s . L e t t . B 3 0 0 , 2 6 1 ( 1 9 9 3 ) .

1 1 ] A . I . V a i n s h t e i n , D e c a y i n g S y s t e m s a n d t h e D i v e r g e n c e o f

P e r t u r b a t i o n S e r i e s , N o v o s i b i r s k R e p o r t ( 1 9 6 4 ) , i n R u s -

s i a n ( u n p u b l i s h e d ) .

J . S . L a n g e r , A n n . P h y s . 4 1 , 1 0 8 ( 1 9 6 7 ) .

1 2 ] R . K a r r l e i n a n d H . K l e i n e r t , P h y s . L e t t . A 1 8 7 , 1 3 3

( 1 9 9 4 ) .

5