D:\User\sbornik_38_pdf\24.DVI


Mathematical Problems of Computer Science 38, 59{60, 2012.

P auly M atr ix and T r ansfor mation Oper ator s for Dir ac

System

T. N . H a r u t yu n ya n , H . H . A z iz ya n

Yerevan State University, e-mail: hartigr@yahoo.co.uk
Armenian State Agrarian University, e-mail: hermineazizyan@mail.ru

L e t ¾1 =

Ã
0 i
¡i 0

!
; ¾2 =

Ã
1 0
0 ¡ 1

!
; ¾3 =

Ã
0 1
1 0

!
a r e we ll kn o wn P a u ly m a t r ix a n d

E =

Ã
1 0
0 1

!
. It is kn o wn t h a t t h e s o lu t io n y = '( x; ¸; ®) o f Ca u s c h y p r o b le m

f¾1
1

i

d

dx
+ ¾2p ( x ) + ¾3q ( x ) gy = ¸y; ¸ 2 C

y ( 0 ) =

Ã
sin®
¡cos®

!
;

c a n b e r e p r e s e n t e d in t h e fo r m

Ã
'0 ( x; ¸; ® ) =

Ã
sin( ¸x + ®)
¡cos( ¸x + ® )

!!

'( x; ¸; ® ) = '0 ( x; ¸; ®) +
Z x
0

K ( x; t ) '0 ( t; ¸; ® ) dt = ( E + K ) '0:

Op e r a t o r E + K is c a lle d t h e t r a n s fo r m a t io n o p e r a t o r . U n d e r d i®e r e n t c o n d it io n s o n s c a la r
fu n c t io n s p a n d q t h is o p e r a t o r a n d h is ke r n e l K ( x; t) wa s in ve s t ig a t e d in d i®e r e n t p a p e r s
( s e e [1 ]-[6 ]) .

T heor ema. L et p; q 2 L1loc ( 0 ; 1) : Then the kernel K ( x; t ) and the kernel H ( x; t ) of
inverse operator '0 ( x; )̧ = '( x; ¸ ) +

R x
0 H ( x; t ) '( t; ¸ ) dt can be represented in the form

K ( x; t ) = a¾1 + b¾2 + c¾3 + d ¢ E

H ( x; t) = ~a¾1 + ~b¾2 + ~c¾3 + ~d ¢ E;
where the functions (of two variables ( x; t ) )) a; b; c; d and ~a; ~b; ~c; ~d are represented by functions
p and q:

5 9



6 0 Pauly Matrix and Transformation Operators for Dirac System

R e fe r e n c e s

[1 ] Ga s ym o v M. G., L e vit a n B . M., D e t e r m in a t io n o f a d i®e r e n t ia l e qu a t io n b y t wo o f it s
s p e c t r a , U s p . Ma t . N a u k v.1 9 , N 2 , 1 9 6 4 , p p .3 -6 3

[2 ] Ma r c h e n ko V . A ., S t u r m -L io u ville o p e r a t o r s a n d t h e ir a p p lic a t io n s , N a u ko va D u m ka ,
K ie v, 1 9 7 7 .

[3 ] Me lik-A d a m ya n F. E ., On t h e c a n o n ic d i®e r e n t ia l o p e r a t o r s in H ilb e r t s p a c e , Iz ve s t .
A N A r m . S S R , Ma t h e m a t ic s , v.X II, N 1 , p p .1 0 -3 0 .

[4 ] L e vit a n B . M., S a r g s ya n I. S ., S t u r m -L io u ville a n d D ir a c o p e r a t o r s , N a u ka , Mo s c o w,
1 9 8 8 .

[5 ] H a r u t yu n ya n T. N ., Tr a n s fo r m a t io n s o p e r a t o r s fo r c a n o n ic D ir a c s ys t e m , D i®e r e n t ia ln ie
u r a vn e n iya , v.4 4 , N 8 , 2 0 0 8 , p p . 1 0 1 1 -1 0 2 1 .

[6 ] A lb e ve r io S ., H r in iv R ., Mikit u k Y a ., In ve r s e s p e c t r a l p r o b le m s fo r D ir a c o p e r a t o r s wit h
s u m m a b le p o t e n t ia ls , R u s . J. o f Ma t h . P h ys . v.1 2 , N 4 , 2 0 0 5 , p p .4 0 6 -4 2 3 .