I L NUOV0
CIMENTO
VOL. XIV, N. 5
1o Dicembre 1959
Introduction to Complex Orbital Momenta.
T. REGGE
Ma,r-Planck-Institut f,~r P h y s i k and Astrophysik - Mi~nche~ (*)
(ricevuto il 18 Luglio 1959)
-In this paper the orbital momentum j, until now considered
as an integer discrete parameter in the radial SchrSdinger wave equations, is allowed to take complex values. The purpose of such an enlargement is not purely academic b u t opens new possibilities in discussing
the connection between potentials and scattering amplitudes. In particular it is shown t h a t under reasonable assumptions, fulfilled b y most
field theoretical potentials, the scattering amplitude at some fixed energy
determines the potential uniquely, when it exists. Moreover for special
classes of potentials V(x), which are analytically continuable into a
function V(z), z = x+,iy, regular and suitable bounded in x > 0, the scattering amplitude has the remarcable p r o p e r t y of being continuable for
a r b i t r a r y negative and large cosine of the scattering angle and therefore
for a r b i t r a r y large real and positive t r a n s m i t t e d momentum. The range
of validity of the dispersion relations is therefore much enlarged.
Summary.
1. - I n t h e f o l l o w i n g w e s h a l l c h o o s e d i m e n t i o n l e s s v a r i a b l e s , b y p u t t i n g
kr, w h e r e r is t h e d i s t a n c e f r o m t h e origin, k t h e w a v e n u m b e r (fixed).
W e c a n w r i t e t h e n S e h r 6 d i n g e r ' s e q u a t i o n as f o l l o w s :
x=
x2
~+~--U(x)%o=O.
H e r e 2 is a g e n e r a l i z e d c o m p l e x o r b i t a l m o m e n t u m ; w h e n 2 a s s u m e s posit i v e h a l f - i n t e g e r v a l u e s ( h e r e a f t e r r e f e r r e d t o as t h e p h y s i c a l v a l u e s ) we shall
write 2 = j +1.
E q . (1.1) is e v e n in 2.
F o r b r e v i t y w e s h a l l single o u t all q u a n t i t i e s eor-
(') Now at tile Palmer Physical L a b o r a t o r y - Princeton, N.J.
952
T. REOG]~
responding to U ( x ) ~ 0 with the index o. We shall also suppose
Ix 2U(x) I< M = const.
xlV(x) idx < Q < ~ ,
(1.2)
0
A l t h o u g h m u c h of t h e t h e o r y here developed could be r e t a i n e d u n d e r
w e a k e r condition, some interesting results would not hold or would require
too l e n g t h y proofs. W e list here a set of solutions of (1.1) with the a p p r o p r i a t e
b o u n d a r y conditions :
(1.3)
(2, x)
x -~ 0
,-- x ~÷~ .
Clearly :
F(~, 0, x) = (/(~, x) ;
(1.4)
~
2, -~, x
= S()t, x)
/~(2, 9, x) = cos ~q(2, x) + sin ~ S()t, x ) ,
F(~, 9, x) = F ( - - ~ , 9, x ) .
A slight generalized f o r m of a t h e o r e m of Poincar6 states t h a t F(~, 9, x)~
for fixed ~ a n d x, is an entire function of 2. I f U(x) = O(x-~+w), C > 0 (*)
small x, t h e n ~(~,x) is a n a l y t i c in R ( 2 ) > - - C
if C < I
and in R ( 2 ) > - - 1
if C > 1. Therefore ~(2, x) and ~ ( - - 2 , x) h a v e a c o m m o n domain of analyticity in the strip: IR().)I<C, ( < 1 ) .
F o r those values for which ~(2~ x) exists we m u s t h a v e a relation of the kind=
(1.5)
~(2, x) = 2~t [C()~)s(~t, x) - - S ( ~ ) C ( ) , x ) ] ,
C(2), S(2) depend on 2 only and are analytic in R ( 2 ) > C ( - - 1 ) with t h e
possible exception of 2 = 0, where there m a y be a simple pole. W e shall n o t
(') In the following we shall use, when there is no danger of confusion, the same
letter C for quite different constants in different formulas.
I N T R O D U C T I O N TO C O M P L E X O R B I T A L M O M E N T A
953
reproduce here the details of the proofs of all these statements which are mostly
by-products of more careful estimates contained in the Appendix.
We now observe t h a t the Wronskian of any two solutions of (1.1) is a
¢onstant. We have for instance ~rom the small x limit:
(1.6)
~(2, x ) ( ( - 2 ,
x) - ~ ( - 2 , x)~'(2, x) = - 22 = w[~(2, x), ~ ( - 2 , x)].
Similarly :
(1.7)
w[C(2, x), ~(2, x)] = 1 .
Introducing (1.5) into (1.6) and using (1.7) we find:
1
C(~t) S ( - - 2) - - 8 ( 2 ) G ( - - ~) = 2-~ "
(1.8)
This identity is of p a r a m o u n t importance in our theory. F r o m it and (1.5)
we can write C(2, x) and ~(2, x) in terms of qo(2, x) and ~(--2, x):
{
C(2, x) = C(2)~(--2, x) + C(--2)~(2, x),
(1.9)
~(x, 2) = 5'(2)~(--2, x) + ,_~(--2)~(2, x).
F r o m (1.9) if R(2) > 0 we deduce:
and similarly for 8(2).
C(2) = lira x x -½ C(2, x)
X-->0
Finally we list here the unperturbed functions:
~o° (2, x) = x½F(2 + 1)2~J~.(x) ,
C°(2,x)
(1.1o)
=
~
~
x½
1
2 cos (~2/2 i [J~(x) + J_~(X)],
S°(2, x ) - - ~ x½2 sin 1(z2/2) [J~ (x) -- J_~(x)],
co(2)
S ° (2)
= ~
27r
2~
2' sin ~ F(2)
2
'
'
2. - If 2 = j+½, ~0(2, x) is the only solution which satisfies the boundary
conditions required for the physical interpretation. The phase 6 is defined
954
T. R~GGE
then from its asymptotic behaviour at large distances:
~(X, x) ~ 2 ~ T ( ~ ) cos x - ~
(2.1)
+ ~(~) ,
We define (~(2) through (2.1) also when 2 is generally complex. Comparing(2.1) with (1.5) we find:
(2.2)
K(~)-
S(2)C(2)
ctg [ ~ - - ~ ( ~ ) ] .
The so defined (~(~) will be hereafter referred to as the (( interpolation )) of
the phase-shifts in the physical points. ~ o t all analytic functions which interpolate the phase shifts can be generated by a potential. Some necessary
conditions are:
a) K(~) is regular analytic in R ( ~ ) > 0 with the exception of simple
poles.
b) K(~) is an hermitian function, that is K(,~)=[K(2*)J*=K*(~).
(Cleurly this follows from C(2), S(~) being also hermitian), if 2 is
real then K(2) is also real.
a) and b) are by no means sufficient. To see it let us evaluate the integral
(see Appendix):
(2.3)
P(2)=
]q(~, x)l~ ~7 > 0
R(~) > 0.
0
(~otice that P(2) is not an analytic function of 4!).
The result can be stated as follows:
P(~)
R(
(2~)"]T2(2)Isinh[~II2)-2I((~)]>O'
~ complex K ( 2 ) > 0 .
The following inequalities can be therefore derived:
c)
d6(~)
d--~- ~ 2
d)
I(4)[~H2) -- 2I(8)] > 0
(2.5)
~ ~ 0 real,
1(4) ~ 0.
INTRODUCTION
TO C O M P L E X
ORBITAL MOMENTA
955
Consider now the function W(L) : K(2), L : 22. The hMf-plane R(2) ~ 0
is m a p p e d into the w h o l e / 5 plane cut along the negative real axis (cut plane).
a) ... d) can be t h e n t r a n s l a t e d into properties of W(L)
a) W(L) is regular analytic in the cut plane with the exception of simple
poles.
b) W(L)--W*(L). I n particular W is real if L is real ~ 0.
c) if L real ~ 0 t h e n
dW/dL ~ 0 ff the derivative is defined.
d) W(L) is ~ following function in the sense of ~VIGNER~t h a t is I(W)
and I(L) h a v e always the same sign, m o r e o v e r W is real only if L
is real and it can h a v e poles or zeros only on the real axis.
All these properties agree in characterizing W(L) as W i g n e r ' s function of
L (a), slightly generalized in the sense t h a t continuous distributions of singularities are a d m i t t e d on the cut while W i g n e r ' s function R(E) was merom o r p h y c . Cond. a)... d) restrict considerably the (..lass of interpolations b u t
t h e y are not y e t sufficient since some limitation has still to be imposed on
the growth of W or K for large 2. W e shall derive t h e m in the n e x t p a r t .
We only observe t h a t e) implies a limitation also on the physical phase shifts.
Indeed let us integrate (2.5) c) between 2----j÷½ and 2 : j ÷ ~ ;
we find:
(2.6)
Oj ~ - - ~ < 2 "
This is a weak b u t simple condition on the phuse shifts in order t h a t t h e y
m a y be produced b y a potential.
3. - If 2 is large we expect the p e r t u r b a t i o n created b y U(x) to decrease
so t h a t p e r t u r b e d functions will e v e n t u a l l y a p p r o a c h the u n p e r t u r b e d at c~.
Although this s t a t e m e n t is in general true there is a n u m b e r of cases in which
it is grossly false so t h a t g r e a t care has to be t a k e n in deriving results along
this line. F o r instance it is not generally true t h a t 5(2) Vanishes in the limit
of large 2. We shall see t h a t this holds for real 2 only, if no condition is impo sed on U(x), being valid along a n y r a y in ~/2 ~ arg 2 ~>-- ~/2 for a v e r y special
class of potentials. We shall not b o t h e r here with a detailed e x p l a n a t i o n of
how these results are derived. The general procedure is the following. We
t r a n s f o r m (1.1) into an equivalent Green's integral equation of the Volterra
t y p e in which the b o u n d a r y conditions as already included. Suitable u p p e r
bounds are t h e n placed u p o n the sum of the p e r t u r b a t i v e expansion using
T i t c h m a r s h ' s lemma. I n the A p p e n d i x we show the essentiM points in our
procedure. I t m u s t be n o t e d t h a t sometimes the lack of simple formulae for
(1) E. WI,;NER and J. NEUMANN: Annals
o] Math.. 59, 418 (1954).
956
T. REGGE
the Bessel functions of general complex order m a k e s it impossible to derive
s o m e limit directly. I t is t h e n m u c h b e t t e r to derive the result, usually a
condition on the growth of some analytic function in an angle, on the b o u n d a r y
of the domain a n d t h e n to e x t e n d it in the inside using the P h r a g m e n - L i n d e l o e f
t h e o r e m or its b y - p r o d u c t s . Using this technique it is possible to derive some
m o r e conditions on the interpolation which we state as follows:
(3.1)
7g
exp [ - - i~2] (exp [2i 8(2)] - - 1) --> 0,
- - ~ < arg 2 < 0 .
W e can write it into the equivalent f o r m
alim [ K ( 2 ) - - K ° ( 2 ) ] = 0 ,
arg 2 ve 0,
/
arg 2 = 0,
e):
I f U(x) us specialized t h e n we h a v e stronger results.
F o r instance if U(x) a d m i t s some b o u n d of the kind: (A > 0, C > 0, B > 1):
(3.2)
U(x) < C exp [ - - Ax ~] ,
x > x0 = const.
Then
(K(2) -- K°(2)) 1< -- (1 -- B)
21211n1~]+ 0(]21).
This b o u n d is essentially the one derived b y CARTER (z) for 2 = ~-k½. I f
B ----1, similar bounds of some use can be derived. A second w a y of restricting
U(x), of r e m a r k a b l e interest in connection w i t h dispersion relations, is prov i d e d b y the t h e o r e m :
(~ L e t U(x) a d m i t an a n a l y t i c continuation U(z), z = x+iy, regular in the
sector arg z < a - k e < ~/2, a n d such t h a t :
¢oei~:
f ]z][dz]]U(z) I< o o ,
o
is u n i f o r m l y b o u n d e d within the sector, and m o r e o v e r
U(z)= 0 (z-~+~°), C> O,
(2) D. S. CARTER; Thesis, Princeton; N. KRu~I: Phys. Rev. 107 1148 (1957). We
have had no opportunity of examining directly CARTER'S work and we know of his
result only throug h KHURI'S paper.
INTRODUCTION
TO
COMPLEX
ORBITAL
957
MOMENTA
- - ~ 0 , in the sector, t h e n :
IK(X) - - K°(A) I = 0 (exp [ - - 2~/(A)-I).
(3.3)
P o t e n t i a l s satisfying the conditions of this t h e o r e m will be n a m e d a-potentials
those which do not 0-potentials. ))
The proof of this t h e o r e m is based u p o n the W K B m e t h o d . I n d e e d if one
takes for g r a n t e d the first W K B a p p r o x i m a t i o n t h e n the a b o v e results already
follow.
(3.2) and (3.3) are in a w a y c o m p l e m e n t a r y conditions on if(x), indeed
the first implies an u p p e r b o u n d on 8(2) along arg~L----0 and nothing along
argX----± z/2, the second one works the opposite way. I n the derivation of
(3.3) it is essential to h a v e a < n/2, correspondingly it can be p r o v e d t h a t
a higher value of a cannot i m p r o v e the bound. Actually no potential attains:
iK(2)--Ko(~)[=O(exp[--(~@s):~X[),
(3.4)
s>0,
argZ=
~.
7g
This can be seen b y a p p l y i n g Carleman's t h e o r e m to b o t h sides of (3.4)
or using Carlson's t h e o r e m on the function F+(2) defined in A p p e n d i x IV. One
can include ~/2 in our discussion using the weaker s t a t e m e n t t h a t if some
potentials is a a-potential, where a is a n y angle < ~/2, then:
iK(2) - - K°(2) [ = O(exp [ - - (~r - - s ) I ( 2 ) ] ) ,
s>O
~' can be t a k e n small at will.
Finally we report some limit:
lim ,~o(~)
S(~) _ ~lim Co(~)
C(~) _ 1
~_~
(3.5)
arg ~ ~- 0,
R(it) > 0.
We shall give here a m e t h o d of construction of U(x) f r o m the interpolation K(~). We sketch here the essential points only w i t h o u t giving detailed proofs of our s t a t e m e n t s . We wish to point out also t h a t some more
research will be needed in order to cast the t h e o r y into a fully satisfactory
a n d rigorous form. W e believe, however, t h a t this is m o r e p r o p e r l y a m a t h e m a t i c i a n ' s t a s k and t h a t w h a t we are going to show here is already enough
for the physicist's needs.
The s t a r t i n g point is, as in Gel'fand and L e v i t a n ' s t h e o r y (3.5) an integral
4.
-
(~) I. IV[. GEL%'AND and LEVITAN: Amer. Math,. Soc. Trans., Sec. 2.1.250 (1955).
(~) R. JOST ~nd W. KOHN: Math. Phys. Medd., 27, (9) (1953).
{5) L. D. FADDEEV: Soviet Phys. Dokl. Transl., 3, 747 (1959),
62
- I1 N*~ovo C i m e t d u .
~o~
T.
REGGE
equation of the Volterra t y p e which relates ~(~, x) to ~0(~, x) t h r o u g h a kernel
which does n o t depend on ~:
(4.1)
x) =
x) + f K(x, y)90(2,
y) dy
Y~
.
o
One can justify (4.1) f r o m two points of view. The first is t h a t b y v i r t u e
of the a s y m p t o t i c expansion (see A p p e n d i x ) :
(4.2)
~(~, x) ----x~+½(1 ÷ D)
where
Qx 2
D < -cl;~l
-
we h a v e the Mellin representation:
(4.3)
~(~, x) --~ x ~÷~ ÷ f H (x,
y)y ~+'~7"'
. dy
0
a n d of course a similar for ~o(~, x). I t can be shown t h a t the latter can be
solved in x ~+½ a n d if this function is fed into (4.3) t h e n (4.1) follows. The
second w a y will be a p p a r e n t later. The y-~ factor is merely inserted for symmetry.
I n considering t h e functions ~(~, x) or ~o(2, x) the n a t u r a l p r o b l e m arises
w h e t h e r t h e y can be considered a complete a n d orthogonal set in the i n t e r v a l
0... oo. This p r o b l e m has been considered since long b y H. WEIL and several
other m a t h e m a t i c i a n s . The reader Will find a fully satisfactory overall view
on this subject in (% We shall r e p o r t here the results found with the help
of (+). T h e +(2, x) are eigenfunctions of the linear operator: (R(2) ~ 0!)
(4.4)
~x -- x~ ~
+ 1
- -
U(x)
,
subjected to the condition of being of class L 2 in 0 ... co if we a d o p t the n o r m :
co
(4.5)
(~, ~) =
(x)~*(x) x-~
0
( I t m u s t be clear t h a t u n d e r this n o r m all solutions of (1.1) are L ~ in a n y
i n t e r v a l H ... c~). Using the s t a n d a r d t e r m i n o l o g y we are in t h e so called
(6) E. C. TITCHMARS~: Eigen]unction Expansions, associated with second order
differential equations I, II. (Oxford, 1946).
INTRODUCTION
TO C O M P L E X
ORBITAL
959
MOMENTA
limit circle case. If no other condition is implied our set is clearly o v e r c o m p l e t e
a n d not orthogonal. We m u s t find a subset which satisfies these conditions.
This can be accomplished b y imposing a n o t h e r b o u n d a r y condition at co,
which c a n n o t be square integrability as in q u a n t u m mechanics, because it
is already satisfied. The new condition is t h a t all eigenfunctions should h a v e
the same a s y m p t o t i c phase # or, in other words, t h e y m u s t be multiples of
/7(4, tt, x). )Cow there are only two types of choices of 4 for this to h a p p e n :
A) R ( ~ ) - - 0 , we refer to it as the c o n t i n u u m
B) R ( ~ ) ~ 0, and F ( 4 ) ~ C(4) c o s # - F S ( ~ ) s i n / ~ = 0.
I n this case
/7(4,/~, x) =/7(-- ~)¢(4, x)
# is here real and 0 ~ t t ~ n ;
for each value of # we h a v e a different expansion. I t can be checked t h a t all functions in A, B) satisfy the orthogonality relations :
(/7(4, #, x), /7(4',/~, x))
= 4n 5 ( 4 ' - - 4'2) /7(4) /7(-- 4)4,
if R ( 4 ) = 0 ,
(4.6)
(F(2, #, x),/7(2n, #, x))
= 0
R(4')--~0,
F(4,.) = 0,
/7(4,) = 0,
1
24/7(4,. #
--
C~/2))/7'(4n, ,u)"
Moreover there are infinitely m a n y zeros of F(4), all real.
respondingly the expansion t h e o r e m :
(4.7)
~(x - y) = ~
We h a v e cor-
(d~ ~(4,/! F(4, #, y)
2i:~ J
x
y/7(4, #)
P
The p a t h P is shown in Fig. 1. This p a t h can be deformed into R(4) = 0,
and to loops which enclose the zeros of F ( 2 , # ) . F r o m R ( 4 ) = 0 we h a v e
the A terms, f r o m the loops the B
terms. Some r e m a r k s are needed for
(4.7); it is clearly a symbolical for
m u l a and s t a t e m e n t s of the kind:
P path
the Dirac function in (4.7) is indeD ~
p e n d e n t of #, and so n m s t be the se' ~ piano
cond term, are in general meaningless
physical point3
or false since ( ~ ( x - y) is a distribution in a space whose definition depends on ,u. H o w e v e r , under some
Fig. 1.
~~
960
REGG]~
m.
conditions, whose exact form will not be stated here, it is true t h a t
¢o
(4.s)
1(x) =
~g(~) x F ( ~ , #) '
p
0
in particular (4.8) holds ff ](x) decreases fast enough. In (4.1) we have always
x ~ y; we define K(x, y) also when y ~ x through the reversed equation:
y
(4.9)
q~o(4, y) : ~(~, x) --t K(x, y)q~(~, X ) Z
.
-2
dx.
/
0
Under these assumptions we find the following integral representation from
(4.9), (4.1) and (4.7):
(4.10)
K(x, y) =
2 [ /~o(2, #) ~(2, x) -- -F(2, #) ~0(~, y) .
P
F r o m (4.10) one can verify the differential equation:
(4.11)
~ I ; ( x , y) = QyK(x, y) ,
(4.1), (4.9) and (4.11) are consistent with (1.1) provided
x
(4.12)
K(x, x) =-~X ty U(y) dy ,
K(x, y) = O(x) ,
O(y),
x, y -~ O.
,2
0
Conversely (4.11), (4.12) can be used to define K(x, y) as the solution of
a certain differential partial equation with appropriate b o u n d a r y conditions.
The so defined K(x, y), if fed into (4.1) generates a function which satisfies
(1.1) with the appropriate boundary condition. This is a second way of
justifying (4.1). By rewriting (4.10) as follows ( # ~ 0):
(4.13)
1
+i¢o
K(x, y) = ~ j
[- ~ i
~
1
j V(~, x) v°(2, y) -4-
--ico
1
1
C0(2:-~°(2)
| ~ o j ~(2' x)~0(2, y)
loop8
where the (<loops >) integral encloses the zeros of C(~) and C°(k) only, avoiding
INTRODUCTION
TO
COMPLEX
ORBITAL
961
MOMENTA
those of S(;t), S°(2), and defining Q(x, y) through the same formula where
~(~, x) has been replaced everywhere b y ~0°(~, x) the integral equation follows:
x
(4.14)
K(x, y) = q(x, y) +
(x, z) q(x, y) ~ .
0
If K(x, y) is regarded as an u n k n o w n in (4.14) this equation is of the Fredholm type. I t can be shown t h a t the homogeneous c o u n t e r p a r t of (4.14) has
no non-triviM solutions and we assume therefore t h a t (4.14) can be Mways
solved under v e r y large conditions. F r o m Q(x~ y) we can deduce therefore
K(x~ y) and If(x, x) in particular. (4.12) yields then U(x). Moreover, Q(x, y)
involves ~°(2, x), which is an e l e m e n t a r y function, and essentiMly the ratios
C(--2)/C(2), N(--2)/S(2), having noticed from (1.8) t h a t
(7(-- ~,)
,s'(-,,t)
~_
c(~)
s(2)
2~ c(~)s(2)
The knowledge of these ratios
implies t h a t of U(x). We are left
with the task of connecting t h e m to
the interpolation K(2). To carry it
out we observe t h a t in virtue of
C(ia) = C(-- ia)* and S(ia) = S(-- is)*
these ratios are simply exponential
functions of arg U(ia) and arg S(ia).
At the same time from (1.8) we
know t h a t :
1
K ( ~ ) - - K ( - - i ) - - '-'~ C ( ~ ) C ( ~ ] )
)t p/ane
O path
"
We can calculate ln lC(~)I and
Fig. 2.
similarly ln]S(2) i from the interpolation. However, owing to (3.5) arg
C(~)/C°(~) and ln lC(~)/C°(~)I ff ~t = ia are in some sense conjugate functions
if one forgets a b o u t the zeros of C(1) and C°(1). We can still relate t h e m
b y taking the real and imaginary part of Cauchy's formula:
(4.15)
1 f~
~i
1
C(~)
is - ~ in ~ o ~ d~ =
1n CC(ia)
0~'
~ > 0 small.
(2 means here a p a t h which avoids the branch points of the integrand, as shown
in Fig. 2. The result expresses a r t C/Co as an Hilbert transform of ln lC/Col
962
T. REGGE
with an infinite series of additional terms which depend on the location of
the b r a n c h points, which in t u r n are perfectly known from K(X). This series
converges because the zeros of C(2) and those of C°(k) t e n d to be v e r y close
when ~ is large and to cancel each other. Similarly one proceeds with S(~).
We h a v e completed the chain from K(~) up to U(x). The only problem left
now is the actual construction of K(2) from the values t h a t it takes at
~ = j + ~ . This problem is still unsolved although we are well on the way to
do it. We shall discuss it in the n e x t part.
5. - We shall give here some heuristic arguments on the interpolation
problem. Our starting point is Carleman's theorem (5):
(5.1, ~ ( h ~
~--~2)sinO~--,n (kin
lntL('e'°,lc°s°d° +
~-~2)s i n O ' - -~12
B
f(1
÷
-
1
in i L(ia)
ida +
R
[L'(O)],
o
L()~) is in this theorem a general analytic function of ~; regular in R(2) ~ 0
with the exception of poles in k~ exp [iO~] and h a v i n g zeros at h~ exp [iOn].
Suppose now t h a t we have two interpolations of the same set of phase shifts:
K(2) and H(~). The difference K ( ~ ) - - H ( X ) must have zeros at least in
= j ~-½ and poles at most at the poles of K(X) and H(2). (We cannot exclude
t h a t K(X) and H(2) coincide elsewhere and t h a t either a new zero arises or
two poles coalesce). The poles are distributed with the density 1. Supposing
L(;~) = K ( ~ ) - - H ( 2 ) the contributions of zeros and poles in (5.17 are nearly
opposite and of the order of in H + e o n s t . Their algebrical sum will t e n d to
a constant limit. ~Vhat can be said a b o u t the contribution of the b o u n d a r y ?
F o r the sake of simplicity let us restrict ourselves to H ( ~ ) = K°(2) so t h a t
the problem now is to find a potential (in the following V°(x)) which produces
no scattering at a given energy. We have now some v e r y useful estimate of
the decrease of Z(2) on the boundary. Quite generally L(X) will decrease in
such a fashion as to make :Y(H) eventually negative and J(H) decreasing.
If, however, condition (3.2) is used we see t h a t :Y(H) can be made arbitrarily
large and negative if H is chosen sufficiently large, so t h a t unless K ( 2 ) = K ° ( 2 )
we face a contradiction in (5.1). Similarly, if V°(x) were a a-potential
J(H) could be made arbitrarily large and negative and (5.1) would be again
an absurdity. We see therefore t h a t V°(x) must be a a-potential which does.
not decrease faster t h a n any exponential. The usual field theoretical potentials
are therefore excluded. These results can be generalized and it can be shown
t h a t there is at most one a-potential which yields a given set of phase shifts
a t a given energy. I n some sense we find unieity in the inversion problem
963
I N T R O D U C T I O N TO C O M P L E X O R B I T A L M O M E N T A
u n d e r r a t h e r b r o a d conditions, t h a t is, analiticity r e q u i r e m e n t s in an arbitrarily small angle.
A specific e x a m p l e of V°(x) can be constructed as follows. We t a k e
Ho()~) = (KO(tt)--])/(KO(~t)-F1), H°(Jl) is again u W i g n e r ' s function of ~t2 and
it satisfies therefore all the same requirements of K°(~). I t has a l t e r n a t i v e l y
zeros and poles in the physie:~l points. I f H ( ~ ) - - ( K ( ~ I ) - - I ) / ( K ( J l ) ~ - I ) then
the ansatz H ( ~ ) = H ° ( Z ) ' [ I + ( C / ( 2 2 - 2~))], where C and 40 are constants, still
satisfies the correct properties and yields vanishing .phases since b o t h H(Jt)
~nd H°(~) t a k e the same values at the physical points, tto and C are not entirely a r b i t r a r y otherwise H(~) m a y be somewhere a non-increasing function.
Since it is enough here to show the existence of at least two different interpolations we shall be satisfied with one e x a m p l e only which is provided b y
t a k i n g H ° ( ~ ) < 0 and C small as readily checked. The general p r o b l e m is
still unsolved and we hope to tackle it in a f o r t h c o m i n g paper. I n analogy
to B a r g m a n ' s potentials which are solvable for all energies b u t for a single
partial wave it is possible to give potentials which are solvable for all values
of the angular m o m e n t u m at a fixed energy. Since these potentials are a r a t h e r
academic case we shall deal with t h e m in some future work.
6. - I n this p a r t we shall establish some results in the field of dispersion
relations. As well-known these relations are s t a t e m e n t s of analiticity of the
scattering a m p l i t u d e as function of the energy and the t r a n s m i t t e d m o m e n t u m .
Although here the energy is k e p t fixed it is still possible to derive for special
classes of potentials enough properties as to g u a r a n t e e for the exista~lce of
such relations. W e restrict ourselves to a-potentials ( a ¢ 0) and m o r e o v e r
we impose some additional condition on U(x) as follows:
(6.1)
[ U(x) i< C exp [-- ~x],
x > x~.
X
Correspondingly, the phase shift will decrease like exp [--£2] along the
real ~, where ~ ' = In (1 ~-(m2/2)) < ~ .
Under these conditions we h a v e the following a s y m p t o t i c estimate:
(6.2)
exp[2i~(2)]--l,~O(exp[i(z--2(~)~--~'~]),
7g
--~<arg~<0.
I n these rough e x t i m a t e s we h a v e neglected powers of ~ which are not
essential in our discussion. (6.2) is v e r y useful if the Legendre expansion of
the scattering a m p l i t u d e :
1
(6.3)
t(z)
co
](cos 0) = ,~ j~=0(2] ~- 1) (exp [2i~;] - - 1) P~(z),
z:
cos0,
964
T. REGGE
is transformed into the integral:
(6.4)
](z) =
d2 (exp [2i a(2)] -- ]) co~s~). P~-½ ( - z).
This artifice is due to WATSO~~
and it was used b y SOMMERFELD(7)
"
-"-:'~"-~ °
poles of e ~(~'
A p/ane
in some wave propagution problems.
5omrnecfe/d integco/
The integration p a t h is shown in
b
# etc-"
Fig. 3. I t encloses the zeros of cos ~2
and avoids the poles of the integrand in the upper quadrant. The
i m p o r t a n t fact a b o u t it is t h a t it
m a y converge outside the c u s t o m a r y
Fig. 3.
Legendre ellipse. To prove it, one
needs an u s y m p t o t i c expansion of Legendre functions of large order:
io
(6.5)
Pa_½(z)
1
V'2= sin0
exp [ g : i2(0 - - n)] .
•2
One m u s t choose the sign which yields the larger result. F r o m (6.5) and
(6.2) one sees t h a t it is possible to deform the p a t h in the lower q u a d r a n t into
2 = ia, a < 0, provided one has (0 -- 00 +i01)
(6.6)
01 < £
and
0o > ~ - - 2~.
If the first condition is dropped the lower purl of the deformed integral
still makes sense b u t the original expansion diverges. We have to deal now
with the upper p a r t where we did not apply immediately the same artifice
because of the poles in the integrand. F r o m (6.2) we see t h a t :
exp [2i ~(2)] -> 1
if
0 < - - arg 2 < 20,
where
tg 2o --
7c-- 2a
Correspondingly, the s~me limit holds in 0 < arg 2 < 20.
I t follows t h a t within this sector there is at most ~ finite n u m b e r of poles
(7) ~k. SOMMERFELD: Partielle Di]/erential-Gleichungen der Physik (Leipzig, 1947),
p. 285.
INTRODUCTION
TO
COMI)L:EX
ORBITAL
MOMENTA
965
of exp [2i($(~)]. The upper p a r t of the p a t h can be deformed along arg). = ~0,
and the poles in the sector yield a finite sum of residues which are amdytic
functions (Legendre functions) of z, in the z plane cut along z real ~ 1.
I f we now t r y to shift the upper p~th along )t= ia~ a ~ 0 we find that, a p a r t
fi'om the contribution of poles, (6.4) still makes sense provide~ 0o > 0 which
is included in (6.6}. The infinitely m a n y poles yield a converging series if
the values of the integral along an arc of arbitrarily large radius A indented
in ~ = iA and ~ = A exp [i~0], vanishes for A - > c~. This can be achieved if
the condition
(6.7)
0~ < ~
[4(7~ - - a) - - 00],
is satisfied. Now particularly interesting are the values 0 = ~ ~-iO 1 which
correspond to z real ~ 1 or in the dispersion relation language to r ~ 2k,
T being the t r a n s m i t t e d m o m e n t u m and k the wave number, (x=-kr). If a
(tan be t a k e n arbitrarily near to 7~/2 then 0~ m a y be arbitrarily large and correspondingly there is analyticity in z along the whole negative axis and, for
arbitrary large real values of T.
This result holds in particular for all those potentials which can be included
into the formul~
co
(6.s)
x U(x) = f e x p [ - - px] ](p~ d p ,
o;
Now from the previous literature (see (2.s)) one knows t h a t ](k, ~) is, for
v fixed real~ and an analytic function of k in the whole upper plane
I(k) > 0 (*) approaching ]~(v) {Born approximation) for large k, i~ v < 20:.
If the latter condition is not satisfied, a domain D arises where analyticity cannot be proved with the usual methods. This domain D = D(v)
however can be enclosed in a sufficiently large semicircle with center in
k = 0. Moreover~ ](k, ~) still approaches ]B(v) when k is large. We know t h a t
](k, v), if k is real, is analytic in a region which in terms of z = 1 - - O~/2k ~-)
is the sum of the interior of the ellipse of convergence and of the domain
defined in (6.7) and the second of {6.6). I n particular if k is kept fixed real
and # 0, ](k, ~) is a n analitic function of T for real a r b i t r a r y positive values
of ~ and in a suitable neighbourhood of them. The same result can be proved
with a limiting procedure when k = 0. I t is remarkable then t h a t analyti-
(s) S. GASIOROWICZand tI. P. NoY]~s: Nuovo Cimento, 10, 78 (1958).
(*) Bound states will not be considered here. However these can be easily included
into the theory.
9 66
T. R:EGGE
city follows in the whole I ( k ) > 0 plane. I n d e e d one can take a semicircle
enclosing D(~), where ~ < T : constant. Applying Cauehy's t h e o r e m to this
semicircle we get
[l(k, T)
(6.9)
](k, T) ~- 2 u i J
h--k
dh.
I f ~ ~ 2~ this f o r m u l a still defines a function, analytic in k and T~ the
l a t t e r a n a l y t i c i t y follows f r o m the fact t h a t all values on the b o u n d a r y are
a n a l y t i c functions of ~. (6.9) defines a function also within D which m u s t
be the analytic continuation of the original function because it coincides with
it outside D where is was defined. T a k e n a m e l y k outside D ( v ) b u t inside the
semicircle and split the semicircle into two closed loops, the first enclosing D
b u t not k, the l a t t e r k b u t not D(v). The contribution of the first is an analytic
function of v which vanishes if v ~ 2~ and therefore always the last just
yields the original function. The result follows. The a b o v e a r g u m e n t s are
s t a n d a r d ideas f r o m the t h e o r y of several complex variables. F o r special
potentials one gets still stronger results. I n d e e d suppose t h a t I V ( z ) [ ~ H / z 2
within a n d along t h e b o u n d a r y of l a r g z l ~ , ~ / 2 .
L e t 4 ° be a pole of
exp [2i ~(2)]. F r o m the discussion a b o v e we k n o w t h a t R(~ °) > 0, I ( 2 °) > 0 .
M o r e o v e r ~(2o, z)~-~ C exp [iz] if z --~ c~. Also
2
(6.10)
~0"(~°, z) ~- ~(~o, z)
T a k e now z : i y ;
(6.11)
y real.
Then ~ C e x p [ - - y ]
4 02
~--qD
1
4o z- 2- ~ ~(~o, z) - - U(z)~(~o, z) = o .
--
Y~
the dots now refer to y derivatives.
(6.12)
~* _ ~*
and
q ~ - U(iy)qJ - O ,
The conjugate reads
4o,~ _ 1 q~* _ U ( - - i y ) q J * : O .
y~
W e m u l t i p l y now (6.11) b y ~* and s u b t r a c t (6.12) multiplied b y ~.
resulting equation is t h e n i n t e g r a t e d between 0 ... c~. We obtain
co
co
(6.13)
0
0
I f I ( U ( z ) ) is replaced b y M / y 2 we obtain
(6.14)
R(~o) f(~o) < _M.
2
The
INTH, ODUCTION TO COMPLEX ORBITAL MOMENTA
967
This is a restriction on the position on poles: this result enables us to say
t h a t only a finite n u m b e r of poles h a v e R(~ °) > m + 1 where m q-½ is some positive constant. I n the Sommerfeld integral one can use R ( ~ ) = m q-½ as path,
the contribution of the e x t r a poles can be e v a l u a t e d and yields a finite sum
which diverges a t m o s t like some power of - - c o s 0 or v when those quuntities are large. The contribution of the R ( 2 ) = mq-½ p a t h can be best evaluated b y using (see App. V) a s y m p t o t i c expansions of Lcgendre functions:
co
I fF (A +F(A)
/(cos O) = 2V~
½) cos nA CdC{exp[2i ~(~)]--1}"
(6.15)
-co
-exp
[i~Ot+ ~(0.--~)]" (-- 2 cos 0) "~ ,
2 = m + i~ ÷ ½ + contr, e x t r a poles.
This contribution is therefore seen to be growing at m o s t like some power
of - - c o s 0 or v when these variables are large in modulus (the cut cos 0
real > 1 being excluded). The M a n d e l s t a m representation can then be derived.
The ~uthor wishes to t h a n k Dr. SYMA~Z~K for m a n y suggestions and
c o n s t a n t advice. This work was partially s u p p o r t e d b y the A m e r i c a n Air
Force.
APPE~'DIX
[
I n order to calculate P(~) we notice t h a t ~(~, x ) * = ~(~*, x) and t h a t :
(A.1)
1
cf(Lx)cf"(~*,x)-cf"(Lx)q~*(Lx) = ~ {~*~- ~1 [~(~, x)i~,
or if ). is real:
(A.2)
~(~, x) ~ ~"(~. x) - - ~"(~, x)
~
2~
~(), x) = ~-~ ~(~t, x) 2 .
I n t e g r a t i o n of these indentities between 0 ... oo yields the desired formulas.
The a s y m p t o t i c b e h a v i o u r of ~(2, x) for large x can be obtained t h r o u g h (15)
and (1.3).
968
T. REGGL
APPE~I)IX 1I
Most of the bounds used in this paper can be derived from suitable integral
equations of the Volterra type and from the following lemma (5)-
Lemma:
Let ] ( x ) ~ 0, g ( x ) ~ 0 and let ](x) be continuous g(x) integrable
in 0 ~ x ~ X.
Let ](x) < 0 + I](t) g(t) dt, 0 < x ~ X . Then ](x) < 0 exp ( Ig(t)dt}
d
O~x<X.
This l e m m a will be used also in the interval X ... co with obvious modifications. W e list here the integral equations which we h a v e used and t h e
functions which in our derivation correspond to ](x) and g(x).
1/ (U(y) --
(i)
Ix-1
p(2, x) ---- x ~+1 + ~
1) p(2, y ) [ y ~
~7 ] d y .
0
W e have if R ( 2 ) > 0: ( y < x)
y~+l
X).+I
x~
<
2
Y ~lx
l;
we take /(x)=tp()~,x)x ~-11 and g(x)= t U ( x ) - - l l Ixl(1/IAI):the result is essentiMly (4.2). A similar bound holds for p'(2, x)
co
here we replace the sines and cosines with i and take C = 1, ](x)= ]C(2, x) l~
g(x) = I((22--¼)/x ~) + U(x)[. W e obtain a result of the form:
I 0(2, x)
l<
C exp x
'
2 large.
Similarly we find a corresponding bound on 0'(2, x) and on S(2, x) and
S'(2, x). Introducing these bounds in the identity:
220(4) ---- 0(2, x)p(2, x)
--
0'(4, x)p(2, x)
and taking x = 2 we find
(A.3)
1o(2) 1< 12~0~1,
INTRODUCTION
TO
COMPLEX
ORBITAL
969
MOMENTA
where C is some constant. This bound, although v e r y rough, can be sharpened
with the help of P h r a g m e n - L i n d e l o e f ' s theorem(9):
co
(111)
c()., x) = C°()., x) + t U ( y ) C()., y)[V0()., x) S°()., y) - - C°()., y) S°()., x)] d y .
W e consider this equation for ). = ia (a real) only, since otherwise no reliable b o u n d on the u n p e r t u r b e d solutions is available. W e k n o w f r o m the
Poisson integral representation of Bessel functions t h a t :
C°(k, x),
S°()., x) < Ev'~-x ,
~ = ia~
where E ) 1 is a suitable constant.
The b o u n d holds u n i f o r m l y in x, 2 since E does n o t depend on t h e m . W e
co
t a k e C°()., x) = ] ( x ) E ~ x ,
]l~ve:
g(x) = 2xU(x) z E ~ a n d since fxIU(x)]=- Q < c~, we
0
C()., x) < E ~ / ~ exp [2~E2Q] = E' V ~ x ,
(AA)
The s~me results follows for S()°, x),
which is of the same type.
(IV) ~o(), x)=~oo(~, x) + ~
g(y)~0(~, y)[~o(,~, x)~oo(- ,~, y)_q~o(,~,_ x)q~o(,~,y)]dy.
0
As in ( I I I ) we suppose ). =-ia.
We h a v e the p r e l i m i n a r y inequality:
F eonst.
we t a k e t h e n of()., x ) = ](x)_FV/~~x and g ( x ) = 2xU(x)nF 2 and
derive u b o u n d of the kind:
(A.5)
~()., x) ~ F ' ~ / Z n x
as before we
F ' is a constant.
F r o m (A.5) and the formula:
co
(a.6)
co(k) +
Iv(x) co().,
x)
=
.
0
we can prove (3.5) along R ( ) . ) = 0, a--~ oo. I n d e e d if we split the interval
•of integration in 0 ... ~/~ and V/g... c~ we find the following b o u n d using (A.5):
co
lq
co
1/N
(9) R. P. BOAS: Entire t~unetions (New York, 1955).
970
T. REGG]~
This p a r t is o(X/2) and negligible [C°(h) is o(V/2-2X)]. I n 0 ... v/g we e x p a n d
Co(h, x) following (1.9) and use (4.2). i f 0 < x < %/~ from (4.2) we deduce:
large.
~(h, x) = x~+~(1 + 0(1))
Two terms arise from (A.6):
v~
¢°(h)fxU(x)(1 + o(1)),
0
and
V~
C°(-- ~)f xU(x)x~(1 ~- 0(1)) d x .
0
B o t h are negligible. The limit follows. We have shown here the simplest
obtainable bounds. A more refined estimate can be derived using the W K B
m e t h o d b u t we deliver it to the n e x t Appendix.
APPENDIX II.I
The W K B m e t h o d gives fairly simple estimates of the wave function when
is large b u t it is difficult to estimate the error when ~ and x are simultaneously
complex. F o r t u n a t e l y it is enough for our purposes to take 2----ia and x
complex. I n this case, provided a is large enough, there are no turning points
if U(x) is a a-potential. I n d e e d it is clear t h a t in this case U(z) admits the
Cauchy's r a p r e s e n t a t i o n :
coexp
[ia']
1 fz' g(z')
zU(z) = ~ J ~ z ' dz'
oa e x p [ - i l ~ ' ]
The integration p a t h is stretched along the two half-lines ÷ o o exp [-- ia']...0
and 0 . . . o o e x p [ i a ' ] .
Now within I a r g z l ~ < a we have l z - z ' l ~ z s i n s
and
clearly:
M
M
IzU(z) l< ]z]sin e
or
U(Z)<z2sin-~e ,
oaexp [ia]
were M = (1/~)ilz !l dzU
II (z)
Similarly
v(~'lzl< Z2~-n?
;u~.
M~ const.
INTRODUCTION
TO
COMPLEX
ORBITAL
97I
MOMENTA
I n order to avoid turning points the function a2+z~÷z~U(z) m u s t have
no zeros in the sector B ~arg z~< a. Suppose now t h u t there is Z such t h a t
a2 + Z2U(Z) = - - Z ~. W e h a v e Z2U(Z) < ( M / s i n e ) = M'. ~gow Z 2 = - - a 2 + p e i~,
where p < M ' and if a is t a k e n large enough
z~
~
ia +
pexp[i~]~
2ia
]
and
argz~4-~.
7g
Hence Z will e v e n t u a l l y fall out of the sector r which will be free of zeros.
I f we define s~(a, z ) = a ~ ÷ z 2 4 - z ~ U ( z ) one can p r o v e with similar a r g u m e n t s
t h a t there exists, for sufficiently large a, a c o n s t a n t C such t h a t :
(C ind. of a and z).
is(z) l > Ca
(A.7)
I t follows t h a t if we defines ~(z) t h r o u g h the equutions:
de
dz
A'(a, z)
,
z
lim ( ~ ( z ) - - a l n z ) = 0
z in X,
z--->a
~(z) is a single valued regular analytic function of z in Z.
~°(z) = a l n z - - a l n ( a + ~ / a
2 ÷ z 2) + ~ / a 2 + z
I n particular
2-a÷aln2a.
F r o m (A.7) one can p r o v e t h e n
(i.8)
, ~(z) - - ~°(z) [ < H ,
a
where H is i n d e p e n d e n t of z and a. All these inequalities require explicitly
i a l< n/z equality beeing excluded. L e t us define also w(~) = ~/s(z). ~(z), where
v,(z) is a n y solution of (1.1). (1.1) t r a n s f o r m s t h e n into:
d2w
d~ 2 -- w : - - J(z) w ,
(A.9)
where
P u t t i n g then X(z) = V~'(z)/az q~(2, z), we readily derive the integral e q u a t i o n :
~
(A.10)
X(z) = exp [i~(z)] [ s i n
J
[~(z) - - ~(t)] s (t)X(t)s(t) d t .
t
o
Take now X~(z) = exp [-- i~]X(z).
(A.11)
X~(z) satisfies:
X~ (z) = 1 - - f 1 - - exp [2i(~(z) - - ~(t))] J(t) X~ (t) s(t)
2i
R
"
972
T. REOGE
We choose a p a t h R of integration such t h a t along it I(~°(z))< I(tO(t)).
I t can be p r o v e d f r o m the f o r m u l a for t°(z) t h a t I(t°(t))increases along
a r g t ----eonst a n d decreases along I(v/aZ-k t *) ----eonst. Suppose I (tO(z)) = aa.
A suitable p a t h is t h e n R,: arg t----a a n d the line R~:
I(Vt~ + a~) = s ( V ~ + a~).
W i t h this position
lexp [2i(t°(z) - t°(t))] I< 1.
Consequently f r o m (A.8) we h a v e when a is large:
1 - - exp [2i(t(z)2i - - ~(t))] < K = c o n s t .
R has a corner T where R1 and R2 m e e t . W e h a v e T : O(a). W e need now
some simple b o u n d on J(t). I f 0 < t < T
one can use (A.7), if t > T
then
s(t) ~ C't is also needed. O m i t t i n g here the details we arrive at the conclusion t h a t :
f , J (t) s(t) Id--/= O(a-1) .
R
The a b o v e e s t i m a t e holds u n i f o r m l y for all z such t h a t I(t(z))= a. I t
lollows f r o m the l e m m a t h a t on this line Q one has X l ( z ) ~ 0(1) uniformly.
Also on Q we h a v e :
]q~(ia, z) I< CV~>> exp [ - - a a ] ,
where C is some constant, z> the largest between
Finally f r o m (A. 6) we derive:
z, a.
co
(A.12)
c(~)so(~)_ co(~)s(~) _
~2/v(~)vo(~, ~)v(~, ~)dz.
0
I n (A.12) the integration can be curried out along Q and the a b o v e found
inequality introduced. Dividing b o t h sides b y C(2)C°(k) t h e result follows.
Y o r e details on this technique can be f o u n d in (6).
APPENDIX
IV
W e suppose now t h a t A is large positive. Our starting point is eq. (IV)
(Appendix I I I ) . W e t a k e firstly x ~ 2. F r o m the general t h e o r y of Bessel
functions it is easy to p r o v e t h a t ~o(2, x ) > 0 and t h a t for sufficiently large
INTRODUCTION"
TO
COMPLEX
ORBITAL
4, q~,°(2,x)x -~, M > ½ is an increasing function.
MOMENT&
973
From the identity:
G(x, y, 4) = ~o(~, x)~0o(--4, y) --~0~(~, y ) ~ o ( - - 4 , x) =
)/q~ dz
22~°(Y, x)~°(2, Y o() -,:)..,,
-
oo~y,
y
replacing q0°(2, z) with the smaller quantity ~o°(2, y)(z/y) M we obtMn the bound:
y
~0(2, x)
I¢(~, x, y) l< ~ _ ½ ~o(~, y)
Putting then ~,(~, x) = ~(~,, x)~o(), x) we find the inequality:
x
]~(~, x ) ] < 1 + f ] y U ( y ) ] ~ ( ) " Y ) M 1 2~ d y .
o
From the lemma it follows then
~'(L x) ~ ~('()~, x) 0(1)
uniformly in 0 < x < 4. Ta.ke then the identity (A.6):
C()~)
(;°(2) + ~
U(x) C"(), x) ~°()~, x)O(1) dx ~0
co
÷/U(x)_ C°(), x)[ C().),'~(), x ) -
S()~)C(~, x)]dx /. j
In this fornmla C°()., x)cf°(,~, x ) ~ C"(2)/(~/'3 -- (z/~)") as it can be deduced
from the power series or from the asymptotic expnnsions of Bessel functions.
It can be moreover shown from the Schlaefli representation that:
(A.14)
(:.(). ,r)!]'! is,,().,
--~5~7--'
V~ :,.) < c = constant, x ~ .
From eq. (III) (Appendix II) it follows that the same kind of inequality
holds for C()~, ;r) and ~.~(), x). We are now in condition to write:
C(2)[_1 ,@ 0(1)] 4~ S(2) 0(1) =: C°().)[1 @ O(I)]
,rod simil,~rly
,%'()~)[1 ~ O(1)] + C ( ) . ) 0 ( 1 )
N°(),)[1 -- 0 ( 1 ) ] .
974
2. REGGE
I t follows:
c(1) = c0(1)[1 + o(1)] + s0(~) o(1)
et(.. ~
and
llm C(1)2 ÷ S(i)~ = 1 .
).--->coCo(l) 2 _~ k~o(1)2
P r o b a b l y our estimate here can be anaeliorated at the expense of a d d i t i o n a l
complication. I t would be also desirable to h a v e a simpler proof. We t u r n
our a t t e n t i o n now to (A.12). P u t t i n g C ( ~ ) ~ + S ( i ) 2 = T ( i ) 2 we can write it as
follows:
co
sin d(t) - - _
;t 2
(x)- To(~ )
T(Z)
0
If x>t
t h e n f r o m the corresponding bounds (A.14) we deduce:
l~o(i,x)]<2¢iT°(~).V~;
I~(i,x)I<2¢'lT(i)'V~,
C, C' being, as usual, two suitable constants, i t follows t h a t the c o n t r i b u t i o n
to sin 5 coming f r o m the interval 1 ... oo is of the order of
co
cC'fx]U(x) [dx = 0 ( 1 ) .
2
Actually a m o r e elaborate analysis shows t h a t this b o u n d is pessimistic
and one has actually 0(i-1). Between 0 . . . i one has
l~(~, x) T-l(x) I< H l cz°( ~, x) To-'(i) I .
This p a r t of the integral is certainly smaller t h a n :
co
1 2H f IU(x) l ¢0(~,
4,~~ ( 2 ) x)
~ dx.
o
Since I U ( x ) ! < Mx -2 f r o m (2.4) we get the u p p e r b o u n d :
H~M
42
I f U(x) is of the (3.2) t y p e t h e n the a b o v e technique yields easily the s t a t e d
result. I t is not clear to us if the a b o v e analysis is included in Carter's resul~
in the sense t h a t his b o u n d m a y hold regardless of I being not an half-integer.
A few words on the use of t h e P h r a g m e n - L i n d e l o e f t h e o r e m in e x t e n d i n g
these results to complex t are also needed.
Take
C(i) + iS(k)
T(t)
F+(1) = Co(1) + iSo(t) -- TO(k) exp [ - - i~(~t)].
INTRODUCTION
TO C O M P L E X
ORBITAL
975
MOMENTA
This function is easily seen to be regular in R ( 2 ) > 0. We h a v e also
F + ( 2 ) - ~ 1 Mong a r g 2 = 0 and arg 2 - - ( z / 2 ) . /'~+(2) is also b o u n d e d b y an
exponential function in the i n t e r m e d i a t e angles and is therefore b o u n d e d b y a
constant. F r o m Montel's t h e o r e m the limit holds in all
0 >arg 2 > -- 2 ~
F , ( ~ ) has also no zeros in this sector since if F-(2) is the adjoint function, t h e n :
F+(2)
exp[-
ia]
and also this function would vanish since (/~+, /7-) c a n n o t vanish simultaneously, for also C(2) and S(2) would vanish and also (2(2, x) because of (1.5).
But
' e x p [_-- 2i d(2)] 1 = exp ~2I[d(2)]~ > exp [~I(2)] > 0
and the above claim is clearly impossible, Along arg 2 - - n / 2
one has
]F+(2) i - - O ( e x p [ ~ ( 2 ) - - 2 a ( 2 ) ] ) . If one applies Carleman's t h e o r e m then unless
- - u / 2 , one has always zeros of /~+ in this sector. The properties of F-(2)
are of course the same of F÷(2) provided one changes arg 2 into - - a r g L
All extensions to complex 2 can be carried out with the same technique.
APPENDIX
V
W e shall sketch here briefly the low energy limit. The potential will b e
supposed to decrease exponentially. I n this case a simple generalization of
k n o w n a r g u m e n t s yields the limit:
(A.15)
sin d(2) ~-- k2~/(2),
k~->0
where v](2) is finite.
This limit follows f r o m (A.12) and f r o m the similar i n d e n t i t y :
vo
(A.16)
¢(2) ,q0(_ 2) - - Co(_ 2) AS'().)
~
@ ~
J
o
~(r)r ~+½q~(2,
r) d r -- ~2~
"
0
U(r) is here the usual p o t e n t i a l and r the distance ( x = k r ) .
= U(x). ~0(2, r) is t h a t solution of:
0
)2
q~"
0 __
1
r2
which behaves like r~.+½ for small r.
--
0
Also
=
976
~. REGGE
(A.12) can be correspondingly written as (k small):
¢o
C(2) S°(2)--S(2)C°(2) -
~-~ f Uo(r)r~+½cp(~,r)dr.k2~= fl(R)k ~".
o
I t follows t h a t if R ( ~ ) ~ 0:
3o~(~)~(2) _/~(~)
~(~.) '
~(~) _ ~(~).
~o(~.)
Therefore ~t(~) is a m e r o m o r p h y c function of 2 in R (~) ~ 0. I f 2 is large
u(x) -:: 1 and fl(~) becomes in general large. A good a p p r o x i m a t i o n to ~(2, r)
is t h e n simply r ~+½. Also one can show t h a t if U(r) is a a potential t h e n
T0(~)2~().) = 0(exp [ - - 2a12]]) for 2 lurge imaginary. Take now S o m m e r f e l d ' s
integral (6.4). I n the low energy limit
3 2
--Z--
if 3 is k e p t constant.
T2
2k 2
1 ~ 2k~
B u t if - - z is large positive then:
F(~) T e'~-Ik 1-2~
P~_+(--z) ~
v / ~ F ( 2 + ~)
A factor k -1 was o m i t t e d for simplicity in (6.4). H a v i n g care of this factor
a n d others, Sommerfeld's representation takes the limiting f o r m :
ico
(A.17)
](O, 31 :
• dZ ~ )
\~] F(a) F(X + ½) cos zX "
-ico
The discussion of this integral in no point is essentially different f r o m the
general case already t r e a t e d in Section 6, and it is actually simpler because
it involves the theory of e l e m e n t a r y Mellin transforms.
I n (A.17) the p a t h avoids the zeros of ~. The resulting analiticity domain
is the sector J.argvl~ a plus the in'terms of the circle 131< ~.
RIASSUNT0
Nel presente lavoro viene definita l'interpolazione degli sfasamenti nello scattering
da potenziale per valori generalmente complessi del momento orbitale. Tale definizione
si presta particolarmente a discutere le proprieth analitiche (tra cui la rappresentazione
di ~fandelstam) dell'ampiezza di scattering giovandosi all'uopo di un metodo dovuto
a Watson sueccessivamente perfezionato da Sommerfeld.