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Analyticity of the jost functions for the coulomb potential in the complex angular momentum plane
Nuclear Physics 53 (1964) 519--528; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
ANALYTICITY OF THE JOST FUNCTIONS FOR THE COULOMB POTENTIAL IN THE COMPLEX ANGULAR MOMENTUM PLANE C. A. L 6 P E Z and I. SAAVEDRA Institute de Flsica y Matemdticas, Universidad de Chile, Santiago, Chile t Received 10 October 1963 Abstract: The Jest functions for the Coulomb potential and complex angular momentum are con-
structed and their analytic properties discussed in detail.
1. Introduction The Jest functions f(k), for fixed angular momentum (l = 0), were first introduced 1) as a way of separating the poles of the scattering matrix corresponding to bound states from the redundant poles 2), which have normally been assumed to have no much physical significance (in this connection see, however, sect. 3). The generalization of these functions to arbitrary angular momentum has been discussed by Newton a). In what follows we shah always be referring to Jest functions for arbitrary angular momentum. The purpose of this note is to study the analytic structure of these functions in the complex angular momentum plane in a case for which they have not been explicitly constructed previously, namely the case of the Coulomb potential. It is important to remark that this is the only physically interesting case where this can be done. Jest functions can also be given explicitly for cut-off potentials, but these are unlikely to have much physical significance; the same can probably be said about potentials like 1/x 2, for instance, for which Jest functions can also be constructed. Most of the properties of these functions which have been discussed recently t,, refer to the case of the Yukawa potential. It is interesting therefore to compare these results, some of which have been conjectured to be true in general, with the corresponding ones for the case we consider. Finally, the underlying motivation of this work should be pointed out. We think the Jest functions may have a deeper physical meaning than the one the historical background may suggest - the separation of poles of the S-matrix. They may have information not contained in the scattering matrix, which is defined as a ratio between Jest functions, and therefore they may well be a more adequate tool in scattering theory. The possibility of characterizing the interaction by the appearance of t Present address: Departarnento de Fisica, Institute de Ciencias, Universidad de Chile, Santiago, Chile. tt See, e.g., Bottino et al. 4), and Newton 6). 519
520
¢. A. L(~PEZ AND I . SAAVEDRA
"dynamical" poles (as opposed to "kinematical" ones), as discussed in sect. 3, can perhaps be interpreted as a hint pointing along this line of thought. In sect. 2 we construct explicitly the Jost functions for the Coulomb potential and arbitrary energy and angular momentum, Their analytic structure is discussed in detail in the remaining sections. 2. Jost Functions for the Coulomb Potential
We write the radial SchrSdinger equation as d2R + 2 d R + [k2+ t/ dr 2 rd--r r
l(/+1)7 V J R =0,
(1)
where
k 2 = 21.tE]h2 and t/ ----2/ze2/h 2, # being the reduced mass of the system. The function ~p(/, k, the differential equation r
r2
r) - rR(l, k, r), satisfies
cp = 0,
(2)
which has a regular solution, defined by the boundary condition ~p(l, k, r) oc rz÷l,
(3)
r--~O
and given by
q~(l,k,r) = rl+le'k'F I / + 1 - 2k'ir/2l+2;-2ikr],
(4)
F being the confluent hypergeometric function. Besides this regular solution one can obtain two linearly independent solutions of eq. (2), irregular at the origin, namely,
~(l,k,r)= rt+le'krW2 [ / + 1 - - 2 k ' i t / 2 1 + 2 ; - 2 i k r ] ,
(5) ,(/, --k, r) = rt+le'k'wl [ l + l - - 2 k ' i r / 2 / + 2 ; - 2 i k r
1.
The functions W1 and T412are familiar in scattering theory t and defined by
W~(a' b; Z) - (b-1)l (-Z)-" fc (1Wz(a'b;Z)-(b-1)!(+Z)'-'eZfc(l*-z)2ni * See, e.g., ref. 6).
t\a-b
e't-"dt,
#OSTFUNCTIONS
521
the contour C being taken from - oo, along the real t-axis, round the origin, and back to - o0. Introducing now the functions
G(l, + k, r) - -T-2ik~(l, + k, r),
(6)
one finally arrives at the equation tp(l, k, r) --- 1 [G(l, -k, r)-G(l, k, r)],
(7)
which we shall use to define the Jost functions. Let us first establish the asymptotic behaviour of ~a(l, k, r). The asymptotic behaviour of the functions G(I, +_k, r) is
G(l, k, r) ,,v
r(21+2)
e_~.me~ze_,tu+t.m),os 2~.~,
(8)
e- ~"/4keitk'+ tn/2~) log2k,J.
(9)
"~°° F ( l + l - ~kk)(2ik)' G(I, - k, r) ..~
F(2I + 2)
" ~ r ( l + l + ~k)(2ik )' Using eqs. (8) and (9), and introducing the phase shift 5(1, k) in the usual way, we finally obtain
tp(l, k, r) ~, e½i~le-i~t,, k) 1 e- ~/,k r(21+ 2) ,~ oo k F(l + 1 + i~l/2k) x(2ik)-' sin
[k r - ½ ~ l +
2--k
oo)
As is well known, the logarithmic phase appearing in eq. (10) is an unavoidable consequence of the infinite range of the Coulomb potential. Before proceeding, we shall introduce at this point the Regge notation ~. = l+½. Eq. (2) is then written as ;~2 ¼-] ,',+ r r 2 -] rp = 0, (2') and the solution regular at the origin is 9(2, k, r), satisfying q~(2, k, r) oc r ~+½.
(3')
r'-*0
As eq. (2') is a function of 2 2, (#(-- ,~, k, r) is another independent solution for 2 # 0. In fact, one shows that the Wronskian W[go(2, k, r), cp(-)., k, r)] = - 2 2 ,
(11)
a result known for the case of Yukawa type potentials (the reason for obtaining the
522
C.A. LOPEZ AND L SAAVEDRA
same result in both eases being that for both potentials the regular solutions have the same behaviour at the origin). Let us now introduce the functions f(;t, + k , -
r)
r(2+½T-ittl2k)
=
r(2,t+l)
(T-Eik)a-½G(2, +_k, r),
(12)
which clearly are solutions of eq. (2'). Using eqs. (8) and (9) one finds W[f(2, k, r),f(2, - k , r)] =
2ike -~/2k,
(13)
which proves that for k ~ 0 the solutions f(2, k, r) and f(2, - k , r) are linearly independent. We can then define the Jost functions f ( 2 , + k ) by the expressions f(2, k) = f(2, - k ) =
e+~/Z~W[f(2, k, r), ¢p(;t, k, r)],
(14)
e+~/2kWEf(2, --k, r), q~(2, k, r)].
(15)
The evaluation of the Wronskian is straightforward and leads finally to f(2, ± k ) =
/'(2,~+1)
+½± i /2k)
(+2ik)_(~_½) '
(16)
where - k is defined + as - k = ke - l ' . These are the Jost functions for the Coulomb potential. The S-matrix is then introduced as
S(2, k) = f(~' k)e""-~) - r(2 +½-i~I/2k) f(;t, ke -'z) F ( 2 + ½ + i~l/2k)'
(17)
a well-known result. Finally, the regular solution tp(;t, k, r) can be written in the usual form: ~0(2, k,
r)
If(2, k)f(2,
- k , r)-f(;¢, - k)f(2, k, r)].
(18)
3. Poles and Zeros o f the Jnst Functions
From eq. (17) we see that the S-matrix poles correspond to zeros o f f ( 2 , -k), which are in turn poles of F(2+½-irl/2k): these are the Regge poles. The zeros of S(2, k), on the other hand, correspond to zeros off(;t, + k) and are given by the poles of F(2 + ½+ itl/2k). The Jost functionsf(;t, ± k ) exhibit simple poles at the points ;t = - ½ ( n + l ) ,
n = 0, 1,2 . . . . .
(19)
which arise from the factor F(2;t+ 1) in the numerator of eq. (16), and, as they are k-independent, divide out and do not appear in the expression for S(2, k). t Following Bottino et aL, ref. 4). The cut appearing for complex 2 is kinematical, in the sense that it remains when the interaction is switched off (i.e. for ~ = 0).
JOST FUNCTIONS
523
When the potential is removed (t/ = 0), the Coulomb Jost function reduces to the known "free" Jost function 4), namely, t fo(2, k) = 24 V 2~r ( 2 + 1)k-(~-½'e -½'''~-~r',
(20)
which has simple poles only at the points 2 = -n-I,
n = 0, 1,2 . . . .
(21)
We shall call these poles, which do not depend on the interaction, "kinematical" poles. They are of course included among the Coulomb Jost functions poles given by eq. (19). The remaining poles in this expression, that is to say, those appearing when the interaction is switched on (tl # 0), we shall call "dynamical" poles. The Jost functions poles have thus been classified into two groups: a) kinematical, located at the points 2 = - 1 , - 2 , - 3 . . . . , or, equivalently, l = -½, - ~ , - ½ , . . ; and b) dynamical, located at the points 2 = -½, -½, - ~ , . . . ,
or/=
-1, -2, -3,....
It is important to remark that, in the 2-plane, the poles of the Jost functions are determined solely by the behaviour of the potential near the origin. They are known to exist, for instance, for the Yukawa potential case 7), which has the same behaviour as the Coulomb potential when r ~ 0. We next find that the position of the poles of the Jost functions correspond exactly to those points where the S-matrix can take the indeterminate form 0/0 (ref. s)). These points are the simultaneous solutions of the equations f(2, k) = O,
f ( 2 , - k ) = O,
and one readily verifies that they are given by 2 =
-½(n+l),
n = O, 1 , 2 . . . .
,
as in eq. (19). Furthermore, we see that these are the only points in the complex 2plane where a pole and a zero of the scattering matrix can coincide. This last result can also be obtained directly from eq. (18). If, for some k,f(A, k) andf(2, - k ) vanish simultaneously, then, to keep ~p(A, k, r) @ 0, as it must be, the functions f(2, _+k) must also have a pole at that point. It is interesting to draw an analogy between this situation, where the Jost functions are functions of both energy and angular momentum, for a potential with constant parameters (i.e. r / = constant), with a situation where the angular momentum is kept fixed (i.e. l = constant) and the Jost functions are considered as functions of the t One obtains eq. (20) from eq. (16) by using the duplication formula for the gamma-function.
524
C. A. L6PEZ AND I. SAAVEDRA
energy and the parameters of the potential. Consider the ease of the exponential potential V(r) = - Voe-'/° (22) for fixed l -- 0. Then the Jost function is
f ( k ) = e-+"~l°~(°2V°)F(1+2aik)J+2~k(2aVo~).
(23)
In the k-plane this function has simple poles on the imaginary axis, which are also poles of the S-matrix: these are the redundant poles 2). The explicit form of the S-matrix is
S(k) -- S2''(2a Vo*)r(l + 2ink) e- 2u,k ,o, (.2vo), J _ 2,o,(2a Vo~)r(1 - 2iak)
(24)
and there are redundant poles at points satisfying the relation
2iak = - n ,
n = 1, 2, 3 , . . . .
(25)
Now, as
J_.(~) = (-)"j.(~), J_.(¢¢) = 0 implies J.(~) = 0 and (eq. (24)) we find again the situation in which a pole and a zero of the scattering matrix coincide, and thus cancel each other. The meeting points are the solutions of the equation
J _ .( 2a Vo~) = O, considered as a function of the potential strength Vo. In these cases the redundant poles acquire a very important physical meaning, namely, that of preserving the bound states, which are also given by the equation J_.(2aV~o) = 0. At a meeting point of a zero and a pole, only the redundant pole survives, and becomes itself the bound state. For the case of cut-off potentials, there are no redundant poles and the only meeting point of zeros and poles of the S-matrix is the origin *. Finally, it is interesting to recall that, in the k-plane, the poles of the Jost functions are determinded solely by the tail of the potential. We can compare this situation with what happens in the 2-plane, where the singularities of the Jost functions are determined solely by the behaviour of the potential near the origin. These are remarkable properties, which suggest to us that the Jost functions f(2, k) may well be more fundamental to the description of a physical system than the scattering matrix.
4. The Regge Trajectories The "Regge trajectories" are the loci in the ;t-plane of the poles of S(2, k) when k varies on the positive real and imaginary axis. Analogously, we shall speak of the "zero trajectories", referring to the loci of the zeros of S(2, k). +This has been shown in a general way by Humblet x,) and explicitly for a square well by Nussenzveig 9).
JOST FUNCTIONS
525
The Regge trajectories (in the/-plane) are given by the equation 1 = - n + i_~, 2k
(26)
where n = 1, 2, 3 , . . . , and have been described by Singh lo) and Fenster 11). When k is real and varies in the range (0, + oo) the trajectories are straight lines perpendicular to the real axis at the negative integers, the poles being all at the same distance of the real axis. When k ~ oo all poles arrive simultaneously at the negative integers, which correspond to the dynamical poles of the Jost functions. At the point k -- oo we expect the scattering matrix to satisfy S(2, oo) = 1,
(27)
as setting k = oo is equivalent to putting r / = 0 in eq. (17), which corresponds to the fact that at infinite energy the particle does not "see" the potential. This is immediately proved to be so by using the generalized unitarity condition S*(,~*, k*)S(,t, Ic) = 1,
(28)
which for real k says that to every S-matrix pole at the point l there corresponds a zero at the point l*. Therefore, the zero trajectories are symmetrical (with respect to the real axis) to the Regge trajectories, that is to say, for k = oo a zero and a pole arrive simultaneously at the negative integers, cancelling each other and thus giving rise to eq. (27). When k varies on the positive imaginary axis the Regge poles move along the real /-axis towards the origin, maintaining a constant distance (actually equal to 1) between two successive ones; the zeros, on the other hand, travel in the opposite direction, but with exactly the same speed. Therefore, poles and zeros meet at points which correspond precisely to the position of the kinematical poles of the Jost functions. A special situation arises for the energy value irl/2k = ½, when all poles, except the first one (i.e. the one corresponding to n = 1) are "killed" by a zero at the points l = - 3, - { . . . . . The first pole, however, does not meet any zero, and, consequently, the scattering matrix has a single pole at I = - ½ (i.e. 2 = 0). This result is of course also immediately obtained from the difference equation satisfied by the gamma-function: F(z+ 1) = zF(z). However, the existence of this single pole does not appear to have physical significance, its contribution to the scattering amplitude being finite on account of the vanishing of the factor (2l+ 1) appearing in the pole expansion of the amplitude t 5. The Dynamical Jost Function Bottino et al. 4) and, particularly, De Alfaro et al. 13), have considered in considerable detail the properties of the Jost functions for arbitrary energy and angular mot We refer to the expansion given by Mandelstam 12) (eq. 3.8).
526
c.A.
L 6 P E Z A N D i . SAAVEDRA
mentum, led by the general viewpoint we have also adopted in this paper. The potential studied in their work is given by a superposition of Yukawa potentials. In ref. ~a), instead of working directly with the Jost functions, a new function was introduced: F(2, k) ----f(2, k) fo0[, k)'
(29)
wherefo(2, k)is the free Jost function given by eq. (20). Then the only poles of the function F(2, k) thus defined are the dynamical poles of the corresponding Jost function, and one can therefore refer to it as the "dynamical" Jost function. As the properties of F(2, k) listed by De Alfaro et aL refer explicitly to the Yukawa potential (as already mentioned), it is::of interest to see if these properties are also valid for the Coulomb case. This we do in this section. Insertion ofeqs. (16) and (20) into eq. (29) yields k) =
F(2 +½+ i~i/2k)
.
(30)
From this expression the following properties of F(2, k) are immediately obtained t: (a)
limF(~, k) = I,
(31)
k--* oO
(b) F*(2, k) = F(2*, e-"k*),
(32)
which hold for the Yukawa dynamical Jost function as well. It has also been proved that this function satisfies the identity la) etXXF(2,e-iXk)F(-2, k ) - e - ~ F ( 2 , k)F(-2, e-~Zk) = 2i sin ~A.
(33)
We now prove that F(2, k) satisfies a very similar relation in the Coulomb case. Using the important property of the gamma-function =
,
sin ~z one shows e'ZXF(2, e - i ' k ) F ( - 2 , k) = eI'~ sin ~(2+½-itl/2k) sin n(2 + 3) ' e-iXXF(2, k)F(--2, e-~k) = e -i~ sin it(2+½+ irl/2k) sin n(2 + 3) ' whence one finally obtains
e~'~F(A, e-l~k)F(-2, k ) - e - l ~ F ( 2 , k)F(-2, e-i'k) = 2ie ~/2~ sin hA.
(34)
T In ref. (13) it is also s h o w n t h a t lim F()., k) = 1 2--* oO
for a certain range o f values o f ;t a n d k. T h i s relation does n o t hold in o u r case, as c a n be seen f r o m t h e expression F(z)/r(z+a)
~
z -~.
z . . ~ oo
JOST FUNCTIONS
~2,.~
Identities (33) and (34) differ only in the irrelevant factor e"~/2t, arising from the asymptotic behaviour of the functions f(2, + k , r) for the Coulomb case. Finally, let us remark that an identity similar to eq. (33) has also been discussed by Newton 5), the difference between his expression and eq. (33) arising from the different definitions used for the Jost functions (different boundary conditions for ~0(I, k, r) when r ---, 0). Using Newton's notation our eq. (34) is written as
FN(2, k)F~v(-- 2, -- k) - Ft¢(2, - k)F~¢( - 2, k) = - /k e-*~/2k sin 2n2,
(35)
this expression differing from the one given by Newton by the factor e -z~/2k o n the right-hand side. Newton's identity is thus valid only for Yukawa-type potentials, but the conclusions he arrives at for the Coulomb case are nevertheless correct, due to the essential identity of the equations satisfied by F(2, k) in both these cases. Appendix In this appendix we give the Jost function for a superposition of a Coulomb potential and a "hard core" term. The potential we consider is
V(r) =
A rz
--
B -
-
r
(A.I)
and, as is well-known, the only modification of the problem amounts to a redefinition of the "centrifugal barrier" term. Writing ~2 - - ~2#A -,
(A.2)
the Jost function is readily shown to be f(3., k) =
F ( 2 x / ~ + ~ 2 + 1) (2ik) -('/x'+g'-t), F(x/22 + ~2 + ~ + itl/2k )
(A.3)
that is to say, the presence of the A/r 2 t e r m introduces a cut on the imaginary axis of the 2-plane, running from + i~ to - i~, i. e., depending only on the strength of the hard core. If the sign of the 1/r 2 term is reversed (attractive potential), then the same cut is found, this time running along the real 2 axis.
References l) 2) 3) 4) 5)
R. Jost, H¢lv. Phys. Acta 7.0 0947) 256 S. T. Ma, Phys. Rev. 71 0947) 195 R. G. Newton, J. Math. Phys. 1 0960) 319 A. Bottino, A. M. Longoni and T. Rcgg¢, Nuovo Cim. 23 0962) 954 R. G. Newton, J. Math. Phys. 3 (1962) 867
528 6) 7) 8) 9) 10) 11) 12) 13) 14)
c. A. L6PEZ AND I. SAAVEDRA N. F. Mott and H. S. W. Massey, The theory of atomic collisions (Oxford University Press, 1949) Hung Cheng, Phys. Rev. 127 (1962) 647 A. O. Barut and F. Calogero, Phys. Rev. 128 (1962) 1383 H. M. Nussenzveig, Nuclear Physics 3 (1955) 499 V. Singh, Phys. Key. 127 (1962) 632 S. Fenster, Nuclear Physics 38 (1962) 638 S. Mandelstam, Ann. Phys. 19 (1962) 254 V. De Alfaro, T. Regge and C. Rossetti, Nuovo Cim. 26 (1962) 1029 J. Humblet, Mere. in-8 ° Soe. Roy. So. Libge 12, no. 4 (1952)
ANALYTICITY OF THE JOST FUNCTIONS FOR THE COULOMB POTENTIAL IN THE COMPLEX ANGULAR MOMENTUM PLANE C. A. L 6 P E Z and I. SAAVEDRA Institute de Flsica y Matemdticas, Universidad de Chile, Santiago, Chile t Received 10 October 1963 Abstract: The Jest functions for the Coulomb potential and complex angular momentum are con-
structed and their analytic properties discussed in detail.
1. Introduction The Jest functions f(k), for fixed angular momentum (l = 0), were first introduced 1) as a way of separating the poles of the scattering matrix corresponding to bound states from the redundant poles 2), which have normally been assumed to have no much physical significance (in this connection see, however, sect. 3). The generalization of these functions to arbitrary angular momentum has been discussed by Newton a). In what follows we shah always be referring to Jest functions for arbitrary angular momentum. The purpose of this note is to study the analytic structure of these functions in the complex angular momentum plane in a case for which they have not been explicitly constructed previously, namely the case of the Coulomb potential. It is important to remark that this is the only physically interesting case where this can be done. Jest functions can also be given explicitly for cut-off potentials, but these are unlikely to have much physical significance; the same can probably be said about potentials like 1/x 2, for instance, for which Jest functions can also be constructed. Most of the properties of these functions which have been discussed recently t,, refer to the case of the Yukawa potential. It is interesting therefore to compare these results, some of which have been conjectured to be true in general, with the corresponding ones for the case we consider. Finally, the underlying motivation of this work should be pointed out. We think the Jest functions may have a deeper physical meaning than the one the historical background may suggest - the separation of poles of the S-matrix. They may have information not contained in the scattering matrix, which is defined as a ratio between Jest functions, and therefore they may well be a more adequate tool in scattering theory. The possibility of characterizing the interaction by the appearance of t Present address: Departarnento de Fisica, Institute de Ciencias, Universidad de Chile, Santiago, Chile. tt See, e.g., Bottino et al. 4), and Newton 6). 519
520
¢. A. L(~PEZ AND I . SAAVEDRA
"dynamical" poles (as opposed to "kinematical" ones), as discussed in sect. 3, can perhaps be interpreted as a hint pointing along this line of thought. In sect. 2 we construct explicitly the Jost functions for the Coulomb potential and arbitrary energy and angular momentum, Their analytic structure is discussed in detail in the remaining sections. 2. Jost Functions for the Coulomb Potential
We write the radial SchrSdinger equation as d2R + 2 d R + [k2+ t/ dr 2 rd--r r
l(/+1)7 V J R =0,
(1)
where
k 2 = 21.tE]h2 and t/ ----2/ze2/h 2, # being the reduced mass of the system. The function ~p(/, k, the differential equation r
r2
r) - rR(l, k, r), satisfies
cp = 0,
(2)
which has a regular solution, defined by the boundary condition ~p(l, k, r) oc rz÷l,
(3)
r--~O
and given by
q~(l,k,r) = rl+le'k'F I / + 1 - 2k'ir/2l+2;-2ikr],
(4)
F being the confluent hypergeometric function. Besides this regular solution one can obtain two linearly independent solutions of eq. (2), irregular at the origin, namely,
~(l,k,r)= rt+le'krW2 [ / + 1 - - 2 k ' i t / 2 1 + 2 ; - 2 i k r ] ,
(5) ,(/, --k, r) = rt+le'k'wl [ l + l - - 2 k ' i r / 2 / + 2 ; - 2 i k r
1.
The functions W1 and T412are familiar in scattering theory t and defined by
W~(a' b; Z) - (b-1)l (-Z)-" fc (1Wz(a'b;Z)-(b-1)!(+Z)'-'eZfc(l*-z)2ni * See, e.g., ref. 6).
t\a-b
e't-"dt,
#OSTFUNCTIONS
521
the contour C being taken from - oo, along the real t-axis, round the origin, and back to - o0. Introducing now the functions
G(l, + k, r) - -T-2ik~(l, + k, r),
(6)
one finally arrives at the equation tp(l, k, r) --- 1 [G(l, -k, r)-G(l, k, r)],
(7)
which we shall use to define the Jost functions. Let us first establish the asymptotic behaviour of ~a(l, k, r). The asymptotic behaviour of the functions G(I, +_k, r) is
G(l, k, r) ,,v
r(21+2)
e_~.me~ze_,tu+t.m),os 2~.~,
(8)
e- ~"/4keitk'+ tn/2~) log2k,J.
(9)
"~°° F ( l + l - ~kk)(2ik)' G(I, - k, r) ..~
F(2I + 2)
" ~ r ( l + l + ~k)(2ik )' Using eqs. (8) and (9), and introducing the phase shift 5(1, k) in the usual way, we finally obtain
tp(l, k, r) ~, e½i~le-i~t,, k) 1 e- ~/,k r(21+ 2) ,~ oo k F(l + 1 + i~l/2k) x(2ik)-' sin
[k r - ½ ~ l +
2--k
oo)
As is well known, the logarithmic phase appearing in eq. (10) is an unavoidable consequence of the infinite range of the Coulomb potential. Before proceeding, we shall introduce at this point the Regge notation ~. = l+½. Eq. (2) is then written as ;~2 ¼-] ,',+ r r 2 -] rp = 0, (2') and the solution regular at the origin is 9(2, k, r), satisfying q~(2, k, r) oc r ~+½.
(3')
r'-*0
As eq. (2') is a function of 2 2, (#(-- ,~, k, r) is another independent solution for 2 # 0. In fact, one shows that the Wronskian W[go(2, k, r), cp(-)., k, r)] = - 2 2 ,
(11)
a result known for the case of Yukawa type potentials (the reason for obtaining the
522
C.A. LOPEZ AND L SAAVEDRA
same result in both eases being that for both potentials the regular solutions have the same behaviour at the origin). Let us now introduce the functions f(;t, + k , -
r)
r(2+½T-ittl2k)
=
r(2,t+l)
(T-Eik)a-½G(2, +_k, r),
(12)
which clearly are solutions of eq. (2'). Using eqs. (8) and (9) one finds W[f(2, k, r),f(2, - k , r)] =
2ike -~/2k,
(13)
which proves that for k ~ 0 the solutions f(2, k, r) and f(2, - k , r) are linearly independent. We can then define the Jost functions f ( 2 , + k ) by the expressions f(2, k) = f(2, - k ) =
e+~/Z~W[f(2, k, r), ¢p(;t, k, r)],
(14)
e+~/2kWEf(2, --k, r), q~(2, k, r)].
(15)
The evaluation of the Wronskian is straightforward and leads finally to f(2, ± k ) =
/'(2,~+1)
+½± i /2k)
(+2ik)_(~_½) '
(16)
where - k is defined + as - k = ke - l ' . These are the Jost functions for the Coulomb potential. The S-matrix is then introduced as
S(2, k) = f(~' k)e""-~) - r(2 +½-i~I/2k) f(;t, ke -'z) F ( 2 + ½ + i~l/2k)'
(17)
a well-known result. Finally, the regular solution tp(;t, k, r) can be written in the usual form: ~0(2, k,
r)
If(2, k)f(2,
- k , r)-f(;¢, - k)f(2, k, r)].
(18)
3. Poles and Zeros o f the Jnst Functions
From eq. (17) we see that the S-matrix poles correspond to zeros o f f ( 2 , -k), which are in turn poles of F(2+½-irl/2k): these are the Regge poles. The zeros of S(2, k), on the other hand, correspond to zeros off(;t, + k) and are given by the poles of F(2 + ½+ itl/2k). The Jost functionsf(;t, ± k ) exhibit simple poles at the points ;t = - ½ ( n + l ) ,
n = 0, 1,2 . . . . .
(19)
which arise from the factor F(2;t+ 1) in the numerator of eq. (16), and, as they are k-independent, divide out and do not appear in the expression for S(2, k). t Following Bottino et aL, ref. 4). The cut appearing for complex 2 is kinematical, in the sense that it remains when the interaction is switched off (i.e. for ~ = 0).
JOST FUNCTIONS
523
When the potential is removed (t/ = 0), the Coulomb Jost function reduces to the known "free" Jost function 4), namely, t fo(2, k) = 24 V 2~r ( 2 + 1)k-(~-½'e -½'''~-~r',
(20)
which has simple poles only at the points 2 = -n-I,
n = 0, 1,2 . . . .
(21)
We shall call these poles, which do not depend on the interaction, "kinematical" poles. They are of course included among the Coulomb Jost functions poles given by eq. (19). The remaining poles in this expression, that is to say, those appearing when the interaction is switched on (tl # 0), we shall call "dynamical" poles. The Jost functions poles have thus been classified into two groups: a) kinematical, located at the points 2 = - 1 , - 2 , - 3 . . . . , or, equivalently, l = -½, - ~ , - ½ , . . ; and b) dynamical, located at the points 2 = -½, -½, - ~ , . . . ,
or/=
-1, -2, -3,....
It is important to remark that, in the 2-plane, the poles of the Jost functions are determined solely by the behaviour of the potential near the origin. They are known to exist, for instance, for the Yukawa potential case 7), which has the same behaviour as the Coulomb potential when r ~ 0. We next find that the position of the poles of the Jost functions correspond exactly to those points where the S-matrix can take the indeterminate form 0/0 (ref. s)). These points are the simultaneous solutions of the equations f(2, k) = O,
f ( 2 , - k ) = O,
and one readily verifies that they are given by 2 =
-½(n+l),
n = O, 1 , 2 . . . .
,
as in eq. (19). Furthermore, we see that these are the only points in the complex 2plane where a pole and a zero of the scattering matrix can coincide. This last result can also be obtained directly from eq. (18). If, for some k,f(A, k) andf(2, - k ) vanish simultaneously, then, to keep ~p(A, k, r) @ 0, as it must be, the functions f(2, _+k) must also have a pole at that point. It is interesting to draw an analogy between this situation, where the Jost functions are functions of both energy and angular momentum, for a potential with constant parameters (i.e. r / = constant), with a situation where the angular momentum is kept fixed (i.e. l = constant) and the Jost functions are considered as functions of the t One obtains eq. (20) from eq. (16) by using the duplication formula for the gamma-function.
524
C. A. L6PEZ AND I. SAAVEDRA
energy and the parameters of the potential. Consider the ease of the exponential potential V(r) = - Voe-'/° (22) for fixed l -- 0. Then the Jost function is
f ( k ) = e-+"~l°~(°2V°)F(1+2aik)J+2~k(2aVo~).
(23)
In the k-plane this function has simple poles on the imaginary axis, which are also poles of the S-matrix: these are the redundant poles 2). The explicit form of the S-matrix is
S(k) -- S2''(2a Vo*)r(l + 2ink) e- 2u,k ,o, (.2vo), J _ 2,o,(2a Vo~)r(1 - 2iak)
(24)
and there are redundant poles at points satisfying the relation
2iak = - n ,
n = 1, 2, 3 , . . . .
(25)
Now, as
J_.(~) = (-)"j.(~), J_.(¢¢) = 0 implies J.(~) = 0 and (eq. (24)) we find again the situation in which a pole and a zero of the scattering matrix coincide, and thus cancel each other. The meeting points are the solutions of the equation
J _ .( 2a Vo~) = O, considered as a function of the potential strength Vo. In these cases the redundant poles acquire a very important physical meaning, namely, that of preserving the bound states, which are also given by the equation J_.(2aV~o) = 0. At a meeting point of a zero and a pole, only the redundant pole survives, and becomes itself the bound state. For the case of cut-off potentials, there are no redundant poles and the only meeting point of zeros and poles of the S-matrix is the origin *. Finally, it is interesting to recall that, in the k-plane, the poles of the Jost functions are determinded solely by the tail of the potential. We can compare this situation with what happens in the 2-plane, where the singularities of the Jost functions are determined solely by the behaviour of the potential near the origin. These are remarkable properties, which suggest to us that the Jost functions f(2, k) may well be more fundamental to the description of a physical system than the scattering matrix.
4. The Regge Trajectories The "Regge trajectories" are the loci in the ;t-plane of the poles of S(2, k) when k varies on the positive real and imaginary axis. Analogously, we shall speak of the "zero trajectories", referring to the loci of the zeros of S(2, k). +This has been shown in a general way by Humblet x,) and explicitly for a square well by Nussenzveig 9).
JOST FUNCTIONS
525
The Regge trajectories (in the/-plane) are given by the equation 1 = - n + i_~, 2k
(26)
where n = 1, 2, 3 , . . . , and have been described by Singh lo) and Fenster 11). When k is real and varies in the range (0, + oo) the trajectories are straight lines perpendicular to the real axis at the negative integers, the poles being all at the same distance of the real axis. When k ~ oo all poles arrive simultaneously at the negative integers, which correspond to the dynamical poles of the Jost functions. At the point k -- oo we expect the scattering matrix to satisfy S(2, oo) = 1,
(27)
as setting k = oo is equivalent to putting r / = 0 in eq. (17), which corresponds to the fact that at infinite energy the particle does not "see" the potential. This is immediately proved to be so by using the generalized unitarity condition S*(,~*, k*)S(,t, Ic) = 1,
(28)
which for real k says that to every S-matrix pole at the point l there corresponds a zero at the point l*. Therefore, the zero trajectories are symmetrical (with respect to the real axis) to the Regge trajectories, that is to say, for k = oo a zero and a pole arrive simultaneously at the negative integers, cancelling each other and thus giving rise to eq. (27). When k varies on the positive imaginary axis the Regge poles move along the real /-axis towards the origin, maintaining a constant distance (actually equal to 1) between two successive ones; the zeros, on the other hand, travel in the opposite direction, but with exactly the same speed. Therefore, poles and zeros meet at points which correspond precisely to the position of the kinematical poles of the Jost functions. A special situation arises for the energy value irl/2k = ½, when all poles, except the first one (i.e. the one corresponding to n = 1) are "killed" by a zero at the points l = - 3, - { . . . . . The first pole, however, does not meet any zero, and, consequently, the scattering matrix has a single pole at I = - ½ (i.e. 2 = 0). This result is of course also immediately obtained from the difference equation satisfied by the gamma-function: F(z+ 1) = zF(z). However, the existence of this single pole does not appear to have physical significance, its contribution to the scattering amplitude being finite on account of the vanishing of the factor (2l+ 1) appearing in the pole expansion of the amplitude t 5. The Dynamical Jost Function Bottino et al. 4) and, particularly, De Alfaro et al. 13), have considered in considerable detail the properties of the Jost functions for arbitrary energy and angular mot We refer to the expansion given by Mandelstam 12) (eq. 3.8).
526
c.A.
L 6 P E Z A N D i . SAAVEDRA
mentum, led by the general viewpoint we have also adopted in this paper. The potential studied in their work is given by a superposition of Yukawa potentials. In ref. ~a), instead of working directly with the Jost functions, a new function was introduced: F(2, k) ----f(2, k) fo0[, k)'
(29)
wherefo(2, k)is the free Jost function given by eq. (20). Then the only poles of the function F(2, k) thus defined are the dynamical poles of the corresponding Jost function, and one can therefore refer to it as the "dynamical" Jost function. As the properties of F(2, k) listed by De Alfaro et aL refer explicitly to the Yukawa potential (as already mentioned), it is::of interest to see if these properties are also valid for the Coulomb case. This we do in this section. Insertion ofeqs. (16) and (20) into eq. (29) yields k) =
F(2 +½+ i~i/2k)
.
(30)
From this expression the following properties of F(2, k) are immediately obtained t: (a)
limF(~, k) = I,
(31)
k--* oO
(b) F*(2, k) = F(2*, e-"k*),
(32)
which hold for the Yukawa dynamical Jost function as well. It has also been proved that this function satisfies the identity la) etXXF(2,e-iXk)F(-2, k ) - e - ~ F ( 2 , k)F(-2, e-~Zk) = 2i sin ~A.
(33)
We now prove that F(2, k) satisfies a very similar relation in the Coulomb case. Using the important property of the gamma-function =
,
sin ~z one shows e'ZXF(2, e - i ' k ) F ( - 2 , k) = eI'~ sin ~(2+½-itl/2k) sin n(2 + 3) ' e-iXXF(2, k)F(--2, e-~k) = e -i~ sin it(2+½+ irl/2k) sin n(2 + 3) ' whence one finally obtains
e~'~F(A, e-l~k)F(-2, k ) - e - l ~ F ( 2 , k)F(-2, e-i'k) = 2ie ~/2~ sin hA.
(34)
T In ref. (13) it is also s h o w n t h a t lim F()., k) = 1 2--* oO
for a certain range o f values o f ;t a n d k. T h i s relation does n o t hold in o u r case, as c a n be seen f r o m t h e expression F(z)/r(z+a)
~
z -~.
z . . ~ oo
JOST FUNCTIONS
~2,.~
Identities (33) and (34) differ only in the irrelevant factor e"~/2t, arising from the asymptotic behaviour of the functions f(2, + k , r) for the Coulomb case. Finally, let us remark that an identity similar to eq. (33) has also been discussed by Newton 5), the difference between his expression and eq. (33) arising from the different definitions used for the Jost functions (different boundary conditions for ~0(I, k, r) when r ---, 0). Using Newton's notation our eq. (34) is written as
FN(2, k)F~v(-- 2, -- k) - Ft¢(2, - k)F~¢( - 2, k) = - /k e-*~/2k sin 2n2,
(35)
this expression differing from the one given by Newton by the factor e -z~/2k o n the right-hand side. Newton's identity is thus valid only for Yukawa-type potentials, but the conclusions he arrives at for the Coulomb case are nevertheless correct, due to the essential identity of the equations satisfied by F(2, k) in both these cases. Appendix In this appendix we give the Jost function for a superposition of a Coulomb potential and a "hard core" term. The potential we consider is
V(r) =
A rz
--
B -
-
r
(A.I)
and, as is well-known, the only modification of the problem amounts to a redefinition of the "centrifugal barrier" term. Writing ~2 - - ~2#A -,
(A.2)
the Jost function is readily shown to be f(3., k) =
F ( 2 x / ~ + ~ 2 + 1) (2ik) -('/x'+g'-t), F(x/22 + ~2 + ~ + itl/2k )
(A.3)
that is to say, the presence of the A/r 2 t e r m introduces a cut on the imaginary axis of the 2-plane, running from + i~ to - i~, i. e., depending only on the strength of the hard core. If the sign of the 1/r 2 term is reversed (attractive potential), then the same cut is found, this time running along the real 2 axis.
References l) 2) 3) 4) 5)
R. Jost, H¢lv. Phys. Acta 7.0 0947) 256 S. T. Ma, Phys. Rev. 71 0947) 195 R. G. Newton, J. Math. Phys. 1 0960) 319 A. Bottino, A. M. Longoni and T. Rcgg¢, Nuovo Cim. 23 0962) 954 R. G. Newton, J. Math. Phys. 3 (1962) 867
528 6) 7) 8) 9) 10) 11) 12) 13) 14)
c. A. L6PEZ AND I. SAAVEDRA N. F. Mott and H. S. W. Massey, The theory of atomic collisions (Oxford University Press, 1949) Hung Cheng, Phys. Rev. 127 (1962) 647 A. O. Barut and F. Calogero, Phys. Rev. 128 (1962) 1383 H. M. Nussenzveig, Nuclear Physics 3 (1955) 499 V. Singh, Phys. Key. 127 (1962) 632 S. Fenster, Nuclear Physics 38 (1962) 638 S. Mandelstam, Ann. Phys. 19 (1962) 254 V. De Alfaro, T. Regge and C. Rossetti, Nuovo Cim. 26 (1962) 1029 J. Humblet, Mere. in-8 ° Soe. Roy. So. Libge 12, no. 4 (1952)