POTENTIALS
ON
A
WITH
BACKGROUND
I.
M.
ZERO
OF
COEFFICIENT
OF
FINITE-ZONE
REFLECTION
POTENTIALS
Krichever
V e r y r e c e n t l y , the c l a s s of f i n i t e - z o n e p o t e n t i a l s u(x) f o r the S t u r m - L i o u v i l l e o p e r a t o r - ( d 2 / d x 2) +
u(x) h a s b e e n s t u d i e d f r o m v a r i o u s p o i n t s of v i e w (see [1, 2, 3]). In this p a p e r , we give a n a l g e b r a i c g e o m e t r i c a l c o n s t r u c t i o n of p o t e n t i a l s of w h i c h both the f i n i t e - z o n e mud t h e w e l l - k n o w n r a p i d l y d e c r e a s i n g p o t e n t i a l s with z e r o c o e f f i c i e n t of r e f l e c t i o n a r e p a r t i c u l a r c a s e s . In t h e g e n e r a l c a s e , t h e s e p o t e n t i a l s c o r r e s p o n d to p o t e n t i a l s without r e f l e c t i o n on a b a c k g r o u n d of f i n i t e - z o n e p o t e n t i a l s . The c o n s t r u c t i o n of a
s c a t t e r i n g t h e o r y f o r a s y m p t o t i c a l l y f i n i t e - z o n e p o t e n t i a l s w i l l be g i v e n i n a s u c c e e d i n g p a p e r .
It s h o u l d be n o t e d that, f o r n o n r e f l e c t i v e p o t e n t i a l s , the i d e a of the p r e s e n t c o n s t r u c t i o n c o i n c i d e s
with the i d e a of i n t e r p o l a t i o n [4], to which the a u t h o r was d i r e c t e d by A. B. Shabat and which s t i m u l a t e d
further investigation.
1. Let E be a r a t i o n a l f u n c t i o n with s i m p l e p o l e s o n a s m o o t h a l g e b r a i c c u r v e X. A c o m p l e x f u n c t i o n u(x), x E (a, b), h a s r e g u l a r a n a l y t i c p r o p e r t i e s if t h e r e e x i s t ~: Y ~ X, a t w o - s h e e t e d c o v e r i n g of X,
and a f u n c t i o n ~(x, P), P E Y, s u c h that: 1 °) e x c e p t at the p o l e s of E = ~*E, the f u n c t i o n is m e r o m o r p h i c ,
and its p o l e s do n o t d e p e n d on x; 2°) in a n e i g h b o r h o o d of the p o l e s of E, ~r (x, p) e~¢E(~-~-x0) is a r e g u l a r f u n c t i o n with v a l u e 1 a t t h e s e p o l e s ;
3°)
-- ~" (x, P) + u (z) • (x, p) = ~ (p) • (x, p).
(1)
B e f o r e f o r m u l a t i n g the f i r s t t h e o r e m , we i n t r o d u c e , f o r e v e r y e f f e c t i v e d i v i s o r D = ~ k s P , >/0, i . e . ,
ks -> 0, on Y the c o n c e p t of a n a d m i s s i b l e d i v i s o r . L e t T be the i n v o l u t i o n of Y which t r a n s p o s e s the s h e e t s ,
D + = T ' D , a n d -'2Doo be the d i v i s o r of p o l e s of E . We denote by ~- (~) the s u b s p a c e of f u n c t i o n s odd with
r e s p e c t to T * i n the l i n e a r s p a c e ~ (~) of the d i v i s o r ~ = D -~ D+--~- Doo . We r e c a l l t h a t the l i n e a r s p a c e of
a d i v i s o r is the s p a c e of r a t i o n a l f u n c t i o n s f o r w h i c h the s u m of the g i v e n d i v i s o r with t h e i r d i v i s o r s of
z e r o s and p o l e s is a n e f f e c t i v e d i v i s o r . We a d m i t a d i v i s o r d -> 0 f o r D if deg d = dim i_ (~) -- t, w h i l e
dim (~- (~) N ~ (~ -- d)) = 1.
T H E O R E M 1. A f u n c t i o n ~I,(x, P) w h i c h s a t i s f i e s c o n d i t i o n s 1° and 2 ° s a t i s f i e s Eq. (1) with s o m e p o t e n t i a l u(x) if a n d o n l y if t h e r e e x i s t s a d i v i s o r d = ~, l~×s, a d m i s s i b l e f o r its d i v i s o r D of p o l e s , s u c h t h a t
di
I~ = i,
dz i (~F(~F+)-~)
i = 0. . . . . Is -- i.
(2)
Here 9 + = T*~.
U n d e r the p r e m i s e s of the t h e o r e m , f o r e v e r y x the W r o n s k i a n F = ~Z'~+ -- ~ + ' ~ ~- (~). The a s s e r t i o n of the t h e o r e m is e q u i v a l e n t to the f a c t t h a t F does not depend on x. H e r e Eq. (2) holds at the z e r o s
of F. (We a g r e e to c h o o s e , b e t w e e n the p o s s i b i l i t i e s , a d i v i s o r d, a d m i s s i b l e f o r D, on the u p p e r s h e e t . )
C o n v e r s e l y , by the d e f i n i t i o n of d, it follows f r o m (2) t h a t F is c o n s t a n t .
D e f i n i t i o n . The d i v i s o r s D, d w i l l b e c a l l e d the s c a t t e r i n g d a t a f o r u(x).
T H E O R E M 2. F o r an a r b i t r a r y s e t of s c a t t e r i n g d a t a , the i n v e r s e p r o b l e m is s o l v a b l e if and o n l y if
E is a f u n c t i o n with one s i m p l e pole on a r a t i o n a l c u r v e .
M o s c o w State U n i v e r s i t y . T r a n s l a t e d f r o m F u n k t s i o n a l ' n y i A n a l i z i Ego P r i l o z h e n i y a , Vol. 9, No. 2,
pp. 77-78, A p r i l - J u n e , 1975. O r i g i n a l a r t i c l e s u b m i t t e d A u g u s t 23, 1974.
@19 75 Plenum Publishing Corporation, 22 7 West 1 7th Street, New York, N. Y. 10011. No part o f this publication may be reproduced,
stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming,
recording or otherwise, without written permission o f the publisher. A copy o f this article is available from the publisher for $15 00.
161
T h e p r o o f of T h e o r e m 2 f o l l o w s f r o m c o m p a r i s o n of t h e d i m e n s i o n of t h e s p a c e f o r m e d b y t h e f u n c t i o n s • (x, P), w h i c h s a t i s f y c o n d i t i o n s 1 ° a n d 2 ° and h a v e d i v i s o r of p o l e s D, f o r f i x e d x, and t h e n u m b e r
of E q s . (2), i . e . , d e g d.
On a h y p e r e l l i p t i c c u r v e of g e n u s g, f o r a d i v i s o r D of d e g r e e N, t h e r e e x i s t s an a d m i s s i b l e d i v i s o r
d if a n d o n l y if N -< g. H e r e d e g d = N - g. To f i n i t e - z o n e p o t e n t i a l s c o r r e s p o n d s the c o n d i t i o n d e g D = g.
T h e n d e g d = 0, and u(x) is u n i q u e l y d e t e r m i n e d b y the d i v i s o r D. To t h e c a s e of n o n r e f l e c t i v e p o t e n t i a l s
c o r r e s p o n d s a h y p e r e l l i p t i c c u r v e of g e n u s 0.
Remark.
It i s e a s y to o b t a i n t h e " f o r m u l a f o r t r a c e s " f o r u(x) g i v e n b y the d i v i s o r s D and d = ~×s
2g+1
on t h e h y p e r e l l i p t i c c u r v e
Fg(y2= H
(E--E~))
~1
2g~-1
N--g
N
u(x)= ~, El-}-2 Z E(us)--2 ~ 7k(x)"
H e r e t h e ~k(X) a r e t h e v a l u e s of E a t t h e z e r o s of O2(x, p ) .
C O R O L L A R Y . L e t -- oo = E, ~ ... ~ E~g+l<oo b e r e a l , t h e ~ s l i e in t h e i n t e r v a l s (E2n, E2n+l), and t h e
p o i n t s of D l i e s u c h t h a t one i s in e a c h of t h e i n t e r v a l s o b t a i n e d ; t h e n u(x) w i l l b e a s m o o t h r e a l f u n c t i o n
a s x -* • % e x p o n e n t i a l l y a p p r o a c h i n g f i n i t e - z o n e p o t e n t i a l s u~(x). T h e p o t e n t i a l u+(x) is g i v e n by t h e e f f e c t i v e d i v i s o r e q u i v a l e n t to D - d, and u_(x) by the d i v i s o r e q u i v a l e n t to D - d +.
T h u s , the d i v i s o r d d e t e r m i n e s a d i s p l a c e m e n t o n t h e s e t of f i n i t e - z o n e p o t e n t i a l s . F o r it to b e z e r o
(u+(x) = u_(x)), i t i s n e c e s s a r y t h a t d e g d >- g + 1. (Our a t t e n t i o n w a s d i r e c t e d b y V. B. M a t v e e v to the
p r e s e n c e of a d i s p l a c e m e n t in the c a s e of s o l i t o n p e r t u r b a t i o n s of s i n g l e - z o n e p o t e n t i a l s , the s t u d y of w h i c h
f r o m o t h e r p o i n t s of v i e w w a s u n d e r t a k e n in [5].)
2. In t h i s p a r a g r a p h , we g i v e e x p l i c i t f o r m u l a s f o r a k - s o l i t o n p o t e n t i a l on t h e b a c k g r o u n d of a n n z o n e p o t e n t i a l and a l s o an a n a l o g of t h e s u p e r p o s i t i o n l a w s f o r n o n r e f l e c t i v e p o t e n t i a l s [1].
L e t t h e p o t e n t i a l u(x) b e g i v e n b y the d i v i s o r s D = P1 + • • • + P n + k a n d d = )¢1 + • • • + ~ k on t h e h y p e r e l l i p t i c c u r v e F n . We d e n o t e by ui(x) t h e n - z o n e p o t e n t i a l s g i v e n by the d i v i s o r s P1 ÷
. ÷ Pn-~ ÷ P~.~,
0 ~ i ~ k; ; t h e ~ i ( x , P) a r e t h e i r c o r r e s p o n d i n g B l o c h f u n c t i o n s .
T H E O R E M 3.
Let
K(x)= u(x)dx. .Ki(x) =
Xo
u~(z)dx; t h e n g(x) = ~
Xo
a~(z)K¢(x), w h e r e
the ai(x) a r e t h e
i~O
s o l u t i o n s of the s y s t e m
<
0
=
i
(3)
/
T h e f u n c t i o n s ai(x) a r e r a t i o n a l f u n c t i o n s of t h e ~ i ( x , n s ) - ~ ; ( x , n s ) . T h e s e , in t u r n , c a n b e e x p r e s s e d r a t i o n a l l y in t e r m s of s i n g l e - s o t i t o n p o t e n t i a l s o n a b a c k g r o u n d of n - z o n e p o t e n t i a l s . Only t h e
a w k w a r d n e s s of t h e e x p r e s s i o n s o b t a i n e d f o r c e s us to confine o u r s e l v e s to the f o r m u l a t i o n in t h e t h e o r e m .
T H E O R E M 4. K(x) is a r a t i o n a l f u n c t i o n o f i n t e g r a l s of n - z o n e p o t e n t i a l s a n d of s i n g l e - s o l i t o n p o t e n t i a l s on a b a c k g r o u n d of n - z o n e p o t e n t i a l s .
To o b t a i n e f f e c t i v e f o r m u l a s in t h e c a s e of k - s o l i t o n p e r t u r b a t i o n s of s i n g l e - z o n e p o t e n t i a l s , i t i s
n e c e s s a r y to u s e , in a d d i t i o n t o T h e o r e m 3, t h e f a c t t h a t t h e B l o c h f u n c t i o n c o r r e s p o n d i n g to a s i n g l e - z o n e
potential given by a point z 0 is
xF (X, z) ~
Z (z - - ZO - - i ( x - - Xli))
"
H e r e , z (~) .... z (~ I ~, ~') a n d ~ (z) = ~ (~ I ~, z')
z (z -
are the Weierstrass
LITERATURE
Xo
2.
3.
162
ei~(z)(x-x,,).
zo)
or- a n d ~ - f u n c t t o n s .
CITED
S. P. Novikov, F u n k t s i o n a l ' . A n a l i z i E g o P r i l o z h e n . , 8, No. 3, 5 4 - 6 6 (1974).
B. A. D u b r o v i n , F u n k t s i o n a l ' . A n a l i z i Ego P r i l o z h e n . , 9, No. 1, 6 5 - 6 6 (1975).
A. R . I t - s a n d V . B. M a t v e e v , F u n k t s i o n a l ' . A n a l i z i E g o P r i l o z h e n . , 9, No. 1, 6 9 - 7 0 (1975).
4o
5.
A. B. Shabat, Dinamika Sploshnoi Sredy, No. 5; 130-145 !1970).
E. A. Kuznetsov and A° V. Mikhailov, Preprint No. 19, Institute of Automation and Electrometry of
the Siberian Branch, Academy of Sciences of the USSR, Novosibirsk (1974).
163