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A simple reduction procedure for the three- and N-body scattering equations
Volume 69B, number 1
PHYSICS LETTERS
18 July 1977
A SIMPLE REDUCTION PROCEDURE FOR T H E THREE- AND N-BODY SCATTERING EQUATIONS
B.R. KARLSSON
CERN, Geneva,Switzerland Received 1 March 1977 A generalized form of the two-body Kowalski-Noyes method is shown to provide a both simple and powerful umtary reduction of the three- and N-body scattering equations. Employing generahzed half-off-shell functions that satisfy off-shell but real and non-singular integral equations, the reduction directly leads to on-shell integral equations for the scattering amplitudes. Physically, it is a sunple example of how the scattering problem can be split into an internal and an external part.
It was first shown by Kowalski [1 ] and Noyes [1 ] that the off-shell two-body t matrix can be written as a sum of a separable term and a real remainder, and that these terms can be obtained from real, non-singular integral equations. Noyes also pointed out that this representation might be useful in the study (and solution) of the threebody Faddeev equations. This idea has subsequently been explmted by several authors [2], but a straightforward three- (and N) body generalization of the Kowalski-Noyes (KN) method itself has up to now not been carried out. It is, therefore, the aim of this note to show how this can be done, and to point out some useful features of such an approach. First, let us briefly recall the two-body results. From the Dppmann-Schwinger equation for the half-off-shell t matrix (e.g. for s waves; ~'2 = k2/2p)
q2 dq
trio, k; ~2) = v(p,k) - f v(p, q)
fz
0
t(q,k;~2 )
one subtracts the same equation, with p = k and multiphed by v(p,
f(p,k) -°(p'k)
v(k, k)
/[
v(p, q) -
v(p'k)v(k'q) 1 v(k, k)
(1)
i0
q2dq
~2 2 ~ 2 - L
k)/o(k, k), 10
to get
f(q,k),
(2)
m terms of the real half-off-sheU function f(p, k) = tO), k; ~2)/t(k, k; ~'2). In contrast to (1), this is a real, non-singular integral equation. Takmg p = k in (1) one furthermore finds that f is related to the on-shell t through
t(k, k; ~2) = o(k, k)
/[1 + f0 v(k, q)
Eqs. (2) and (3) are frequently used instead of (1) for practical evaluation of t(k, k, ~'2). For the special case of a separable potential, (2) reduces to f(p, k) = v(p, k)/v(k, k), and the only non-trivial part of the two-body problem is the evaluation of the integral in (3). It can also be shown that t(k, p'; ~2) = t(k, k; ~2)f(p', k) with the same half-off-shell function as above, and that the completely off-shell t matrix has the representation t(p, p', ~2)
=f(p, k)t(k, k; ~ 2 ) f ( p ' ,
k) + r(p, p'; ~'2).
(4)
In (4), r is a real function that satisfies the integral eq. (2) with the drxving term replaced by v(p, p') 13
Volume 69B, number,
PHYSICS LETTERS
18 July 1977
- v(p, k)v(k, p')/v(k, k). The function r vamshes on- and half-off-shell, and is identically zero for a separable potential. In the three-body case the analogue of eq. (1) IS the half-off-shell Faddeev equation [for simplicity of presentation we will only consider s waves J l = X = 0, where J = ! + k, i.e., only s wave Interactions. As initial state we choose three free particles m channel 1 with relative momenta q0 (for a pair) and p0 (for the third particle relative to the pair). In the half-off-shell scattering amplitude T~(q~, p~;qO, pO;E ) the initial state is on-the-energy-shell ~,0= + ~0 2 = E ] . Suppressing initial state labels m T~ we have (6~7 "= 1 "- 8¢.r) =
oo
- p°),l(ql, ql°;
Pl
+f '
0 '
Z )q2 dq
q°, (s)
,2 d
,
,2 d
P__LP_~q_v
,
q__u
,
,
- 3'~ ~ 0f to(q~, q; ~'t~)q 2 dq DO~t(q, p~, q~, py) q"r2 + ~'q2 _ E - IO T~r(qy' p~) r
where 7 2 = E - , ~ , and where Dr>r is a standard channel recoupling coefficient. Note that D~(q~,p~,q~,p'~)is diagonal in the over-all energye i.e., proportional to 6(~'~ +,ot~ - q'r ~'2 - Pv ~'2 )' so that the t matrices in both the driving terms are half-off-shell. In analogy with the two-body case we introduce the representation
o~
T~(q~, Pt3) =
x(q~, P~; Px) Px dPx T (Kx, Px),
(6)
o for a half-off-shell scattering amplitude component in terms of the on-shell amplitude components (K is always understood to be related to p through the three-body on-shell condition ~'2 = E - if2). Taking qt3 = Kt~ in (6), multiplying byfo(q~, K~) and subtractmg the result from (6) we obtain an equation for the half-off-shell function F ~ , which is analogous to (2)
F~x(qg ' P~" p~) = 8~ x 1 8(p(~ - p'~)f(q~, K~)
(7) oo
- ~
"2d " "2-
',
,, ,, q'r q'r Pv ctP'r F ~ '. . . . . f r~(q~, q, ~'2)q2 dq Di3y(q, p~;qy, py) ~7~2-+~-2---£--7,, yxtq~l, P~;Px)"
~r~ 0
q'r
P'r - t~ - 1u
We have here assumed that for 72 < 0 the function t(q, q'; ~'2) has a representatmn similar to (4) m terms of real f u n c t m n s f , [ and ~,f(~, g) = 1, such that the bound state pole singularity of t(q, q', ~2) is completely contained in the separable term (recall that for 72 < 0 the imaginary part of t comes from this pole). Evidently, there is consaderable freedom in the choice of this representation. One simple alternative ~s to take f(q, g) and i(g, ~;22) as the analytic continuatmn o f f ( q , ~) and t(~, ~; ~'2) from real to imaginary ~. This can in general only be done up to some point ~ = il~c{ between the lowest energy bound state and the beginning of the potential cut Or, t(g, ~;~2) and r are no longer real on the cut). For I m g > I~cl one can simply taker(q, ~) = 0. For other alternatives we refer to the literature [2, 6, 7]. Returning to eq. (7) we note that the 8 function in D&r allows us to write the propagator in (7) as
( / ~ +~'2 __ ~7-- 10) - t = ( ~ 2 -- ~'~ --10)--1. Recalling that rt~ in (7) vanishes when q = ~ (half-on-shell), we conclude that eq. (7) is a real, non-singular, two variable integral equation for FCx. In particular, F~x is itself a real function. In order to obtain the on-shell TI~ from F~.~, 1.e., the analogue of eq. (3), we simply take qt~ = g~ in (5) and use (6) to get 14
Volume 69B, number 1
PHYSICS LETTERS
18 July 1977
oe
T#(Kf'P#)=T~o("#'Pfl)-
f [Kfx(P#'Px)+R#x(P#'Px)] PX dPx T I.Kx, px),
(8)
0 where we have split the kernel into an on-shell piece K#x [corresponding to the on-shell part of Fflx, 6#x(1/p~)6(p# - pfl)], and an off-shell piece Rfx. The three-body analogue of (3) is therefore not an algebraic expression but a one-variable integral equation for the on-shell scattering amplitude m terms of the half-off-shell function Fflx. In the particularly simple case of a separable two-body interaction, the function r in (4) and (7) is absent, and Fflx is trivially given by the driving term in (7). Therefore, the solution of (8) is in this case the only non-trivaal part of the three-body problem. Tins observation can be given an interesting Interpretation. the three-body scattering amplitudes that are obtained using the separable approximation are essentially unitarlzed [via eq. (8)] Born approximations to eq. (7) #. In such a perspective the well-known, but not equally well understood, success of the separable approximation IS less astonishing. One can now proceed and derive a representation analogous to (4) for the fully-off-shell three-body scattering amplitude. Tins should not be done directly for T# but for the more symmetric Faddeev components Mfl~ [which satisfy the equation Mil s = t#6fl~ - tflG 0 ZT¢# M. m and the relation T fl = ~ce~/#u]. Generalizing the two-body manipulations that lead to (4) one finds t
c~
' ' E )= ~ /Ffl ,(Pfl'qf;P"f)P'r . . . . . 2 d P~ Mflc~(qf'Pf;qc~'Pa; "},,X "
(9) ....... ' . . . . . . px X M ~.h(K.r,pT;~h,Px;E)
2 d PX " ' F hc~tPx;qc~'Pc,)+Mfl~(qfl'Po'qc~'Pc~ + ....... r " ' ' ; E),
where Fo is the real half-off-shell function, F + stands for its right-hand side counterpart, and where M . ,,w'/. ,. ,,, . ha "rxt~'r' P'r; Kx ' PX, E ) is on-shell. The remainder function M~a satisfies an equation similar to (7) but with the driving term replaced by
5 #x rf(q o, q'~; "~2) (1/p2)5 (po - p'~). It is, therefore, a real function that vanishes on- and half-off-shell, and (9) is indeed a sum of a "separable" part a and a real remainder. It should now be clear for anyone familmr with the four- and N-body Faddeev-Yakubovski formahsm [4, 5] that the above results directly generalize to the N-body case if only the N body equations are written down for properly chosen operators. In view of the large number of operators available, it is a slightly non-trivial result that the appropriate choice is the Yakubovski generalization of the three-body Mflc~ operator (in the four-body case, this is the Y#~. of ref. [5]); It satisfies the equations - 5 o r M o#a r°" # a -- 5arMa#a - ~ K;,Go6-6-fi YP: P,7
_ ~ P,'Y
Y°P~C^'K r "#3'- ~v'-~"
For this choice the N body versions of (5)--(9) can more or less be written down by inspection. It should be mentioned that there are some anomalies connected with the simplest version of the two-body KN method [6] [e.g., divergence o f f ( p , k) when t(k, k; ~'2) = 0]. Such anomahes might need some attention also in the more general case considered here, but they are not expected to limit the applicability of the method more than they have done in the two-body case. The reduction of the Faddeev equations worked out here is a special (and particularly simple) case of a more * ThB point of vmw is simdar to that of Amado [3] who argues that the three-body equations in the separable approxlmahon mainly serve to provide the scattering amplitudes with appropriate analytlclty properties.
15
Volume 69B, number 1
PHYSICS LETTERS
18 July 1977
general reduction scheme [7] based on an analysis of the physically intuitive multiple scattering structure of the Faddeev equations. Thas fact provides some insight into the physical content of the equations obtained here (e.g., they represent a separation of the scattering problem into an internal and an external part, see ref. [7]) and is taken as an excuse for leaving out such aspects of the reduction in this note. On the other hand, the present work extends the analysis of ref. [7] by providing an explicit example of how the function R~x m eq. (8) is constructed. Thas point was not discussed m detail in the more general context of ref. [7]. The splitting of the three-body problem into two parts is characteristic of the reduction procedure for the Faddeev equations developed m this note: the solution of a real, non-singular, two-variable mtegral equation [eq. (7)], and the subsequent solution of an on-shell, one-variable integral equation [eq. (8)]. The last equation is similar in structure to the three-body equations in the separable approximation and mainly serves to unitarize the solution t~ the ftrst equation. In particular, eq. (8) can be combined with any approximate (but real) solution to the more complicated eq. (7) to generate unitary scattering amplitudes (see ref. [7] ), Let us conclude wlth the following remark. The natural separation of the unitarizatlon into a comparatively simple equation is clearly a very powerful feature of the reduction presented here, but on the other hand, any reasonable unitary reduction scheme ought to have this feature. It might, therefore, be considered an equally important feature that the equationthat here carries the details o f the dynamics, i.e., eq. (7), is completely free from propagator singularities, and therefore has a much simpler structure than the original Faddeev equations (it might be recalled that in the three-body K matrix formahsm [8] which is a competing unitary reduction scheme, the corresponding equation contains awkward principle part type singularities). In fact, it is this feature that has made the original Kowalski-Noyes method a standard tool for solving the two-body scattering equations.
References [1 ] H.P. Noyes, Phys. Rev. Lett. 15 (1965) 538, K.L. Kowalski, Phys. Rev. Lett. 15 (1965) 798,908E. [21 D.D. Brayshaw, Phys. Rev.C13 (1976) 1024; M.G. Fuda, Phys. Rev. C3 (1971) 485, K.L. Kowalskl, Phys. Rev. C8 (1973) 1973, H.P. Noyes, Phys. Rev. D5 (1972) 1547; E.O. Air, P. Grassberger and W. Sandhas, Nucl. Phys. B2 (1967) 167. [3] R.D. Amado, Phys. Rev. Lett. 33 (1974) 333. [4] O.A. Yakubovskl, Yad. Fiz. 5 (1967) 1312 [Soy. J. Nucl. Phys. 5 (1967) 937]. [5] B.R. Karlsson and E.M. Zelger, Phys. Rev. D9 (1974) 1761. [6] T.A. Osborn, Nucl. Phys. A138 (1968) 305; D.D. Brayshaw, Phys. Rev. 182 (1969) 1658. [7] B.R. Karlsson, to be pubhshed. [8] R.T. Cahdl, Nucl. Phys. A194 (1972) 599; K.L. Kowalskl, Phys. Rev. D5 (1972) 395; T. Sasakawa, Nucl. Phys. A203 (1973) 496; B.R. Karlsson, ITP Goteborg report 76-12 (1976).
16
PHYSICS LETTERS
18 July 1977
A SIMPLE REDUCTION PROCEDURE FOR T H E THREE- AND N-BODY SCATTERING EQUATIONS
B.R. KARLSSON
CERN, Geneva,Switzerland Received 1 March 1977 A generalized form of the two-body Kowalski-Noyes method is shown to provide a both simple and powerful umtary reduction of the three- and N-body scattering equations. Employing generahzed half-off-shell functions that satisfy off-shell but real and non-singular integral equations, the reduction directly leads to on-shell integral equations for the scattering amplitudes. Physically, it is a sunple example of how the scattering problem can be split into an internal and an external part.
It was first shown by Kowalski [1 ] and Noyes [1 ] that the off-shell two-body t matrix can be written as a sum of a separable term and a real remainder, and that these terms can be obtained from real, non-singular integral equations. Noyes also pointed out that this representation might be useful in the study (and solution) of the threebody Faddeev equations. This idea has subsequently been explmted by several authors [2], but a straightforward three- (and N) body generalization of the Kowalski-Noyes (KN) method itself has up to now not been carried out. It is, therefore, the aim of this note to show how this can be done, and to point out some useful features of such an approach. First, let us briefly recall the two-body results. From the Dppmann-Schwinger equation for the half-off-shell t matrix (e.g. for s waves; ~'2 = k2/2p)
q2 dq
trio, k; ~2) = v(p,k) - f v(p, q)
fz
0
t(q,k;~2 )
one subtracts the same equation, with p = k and multiphed by v(p,
f(p,k) -°(p'k)
v(k, k)
/[
v(p, q) -
v(p'k)v(k'q) 1 v(k, k)
(1)
i0
q2dq
~2 2 ~ 2 - L
k)/o(k, k), 10
to get
f(q,k),
(2)
m terms of the real half-off-sheU function f(p, k) = tO), k; ~2)/t(k, k; ~'2). In contrast to (1), this is a real, non-singular integral equation. Takmg p = k in (1) one furthermore finds that f is related to the on-shell t through
t(k, k; ~2) = o(k, k)
/[1 + f0 v(k, q)
Eqs. (2) and (3) are frequently used instead of (1) for practical evaluation of t(k, k, ~'2). For the special case of a separable potential, (2) reduces to f(p, k) = v(p, k)/v(k, k), and the only non-trivial part of the two-body problem is the evaluation of the integral in (3). It can also be shown that t(k, p'; ~2) = t(k, k; ~2)f(p', k) with the same half-off-shell function as above, and that the completely off-shell t matrix has the representation t(p, p', ~2)
=f(p, k)t(k, k; ~ 2 ) f ( p ' ,
k) + r(p, p'; ~'2).
(4)
In (4), r is a real function that satisfies the integral eq. (2) with the drxving term replaced by v(p, p') 13
Volume 69B, number,
PHYSICS LETTERS
18 July 1977
- v(p, k)v(k, p')/v(k, k). The function r vamshes on- and half-off-shell, and is identically zero for a separable potential. In the three-body case the analogue of eq. (1) IS the half-off-shell Faddeev equation [for simplicity of presentation we will only consider s waves J l = X = 0, where J = ! + k, i.e., only s wave Interactions. As initial state we choose three free particles m channel 1 with relative momenta q0 (for a pair) and p0 (for the third particle relative to the pair). In the half-off-shell scattering amplitude T~(q~, p~;qO, pO;E ) the initial state is on-the-energy-shell ~,0= + ~0 2 = E ] . Suppressing initial state labels m T~ we have (6~7 "= 1 "- 8¢.r) =
oo
- p°),l(ql, ql°;
Pl
+f '
0 '
Z )q2 dq
q°, (s)
,2 d
,
,2 d
P__LP_~q_v
,
q__u
,
,
- 3'~ ~ 0f to(q~, q; ~'t~)q 2 dq DO~t(q, p~, q~, py) q"r2 + ~'q2 _ E - IO T~r(qy' p~) r
where 7 2 = E - , ~ , and where Dr>r is a standard channel recoupling coefficient. Note that D~(q~,p~,q~,p'~)is diagonal in the over-all energye i.e., proportional to 6(~'~ +,ot~ - q'r ~'2 - Pv ~'2 )' so that the t matrices in both the driving terms are half-off-shell. In analogy with the two-body case we introduce the representation
o~
T~(q~, Pt3) =
x(q~, P~; Px) Px dPx T (Kx, Px),
(6)
o for a half-off-shell scattering amplitude component in terms of the on-shell amplitude components (K is always understood to be related to p through the three-body on-shell condition ~'2 = E - if2). Taking qt3 = Kt~ in (6), multiplying byfo(q~, K~) and subtractmg the result from (6) we obtain an equation for the half-off-shell function F ~ , which is analogous to (2)
F~x(qg ' P~" p~) = 8~ x 1 8(p(~ - p'~)f(q~, K~)
(7) oo
- ~
"2d " "2-
',
,, ,, q'r q'r Pv ctP'r F ~ '. . . . . f r~(q~, q, ~'2)q2 dq Di3y(q, p~;qy, py) ~7~2-+~-2---£--7,, yxtq~l, P~;Px)"
~r~ 0
q'r
P'r - t~ - 1u
We have here assumed that for 72 < 0 the function t(q, q'; ~'2) has a representatmn similar to (4) m terms of real f u n c t m n s f , [ and ~,f(~, g) = 1, such that the bound state pole singularity of t(q, q', ~2) is completely contained in the separable term (recall that for 72 < 0 the imaginary part of t comes from this pole). Evidently, there is consaderable freedom in the choice of this representation. One simple alternative ~s to take f(q, g) and i(g, ~;22) as the analytic continuatmn o f f ( q , ~) and t(~, ~; ~'2) from real to imaginary ~. This can in general only be done up to some point ~ = il~c{ between the lowest energy bound state and the beginning of the potential cut Or, t(g, ~;~2) and r are no longer real on the cut). For I m g > I~cl one can simply taker(q, ~) = 0. For other alternatives we refer to the literature [2, 6, 7]. Returning to eq. (7) we note that the 8 function in D&r allows us to write the propagator in (7) as
( / ~ +~'2 __ ~7-- 10) - t = ( ~ 2 -- ~'~ --10)--1. Recalling that rt~ in (7) vanishes when q = ~ (half-on-shell), we conclude that eq. (7) is a real, non-singular, two variable integral equation for FCx. In particular, F~x is itself a real function. In order to obtain the on-shell TI~ from F~.~, 1.e., the analogue of eq. (3), we simply take qt~ = g~ in (5) and use (6) to get 14
Volume 69B, number 1
PHYSICS LETTERS
18 July 1977
oe
T#(Kf'P#)=T~o("#'Pfl)-
f [Kfx(P#'Px)+R#x(P#'Px)] PX dPx T I.Kx, px),
(8)
0 where we have split the kernel into an on-shell piece K#x [corresponding to the on-shell part of Fflx, 6#x(1/p~)6(p# - pfl)], and an off-shell piece Rfx. The three-body analogue of (3) is therefore not an algebraic expression but a one-variable integral equation for the on-shell scattering amplitude m terms of the half-off-shell function Fflx. In the particularly simple case of a separable two-body interaction, the function r in (4) and (7) is absent, and Fflx is trivially given by the driving term in (7). Therefore, the solution of (8) is in this case the only non-trivaal part of the three-body problem. Tins observation can be given an interesting Interpretation. the three-body scattering amplitudes that are obtained using the separable approximation are essentially unitarlzed [via eq. (8)] Born approximations to eq. (7) #. In such a perspective the well-known, but not equally well understood, success of the separable approximation IS less astonishing. One can now proceed and derive a representation analogous to (4) for the fully-off-shell three-body scattering amplitude. Tins should not be done directly for T# but for the more symmetric Faddeev components Mfl~ [which satisfy the equation Mil s = t#6fl~ - tflG 0 ZT¢# M. m and the relation T fl = ~ce~/#u]. Generalizing the two-body manipulations that lead to (4) one finds t
c~
' ' E )= ~ /Ffl ,(Pfl'qf;P"f)P'r . . . . . 2 d P~ Mflc~(qf'Pf;qc~'Pa; "},,X "
(9) ....... ' . . . . . . px X M ~.h(K.r,pT;~h,Px;E)
2 d PX " ' F hc~tPx;qc~'Pc,)+Mfl~(qfl'Po'qc~'Pc~ + ....... r " ' ' ; E),
where Fo is the real half-off-shell function, F + stands for its right-hand side counterpart, and where M . ,,w'/. ,. ,,, . ha "rxt~'r' P'r; Kx ' PX, E ) is on-shell. The remainder function M~a satisfies an equation similar to (7) but with the driving term replaced by
5 #x rf(q o, q'~; "~2) (1/p2)5 (po - p'~). It is, therefore, a real function that vanishes on- and half-off-shell, and (9) is indeed a sum of a "separable" part a and a real remainder. It should now be clear for anyone familmr with the four- and N-body Faddeev-Yakubovski formahsm [4, 5] that the above results directly generalize to the N-body case if only the N body equations are written down for properly chosen operators. In view of the large number of operators available, it is a slightly non-trivial result that the appropriate choice is the Yakubovski generalization of the three-body Mflc~ operator (in the four-body case, this is the Y#~. of ref. [5]); It satisfies the equations - 5 o r M o#a r°" # a -- 5arMa#a - ~ K;,Go6-6-fi YP: P,7
_ ~ P,'Y
Y°P~C^'K r "#3'- ~v'-~"
For this choice the N body versions of (5)--(9) can more or less be written down by inspection. It should be mentioned that there are some anomalies connected with the simplest version of the two-body KN method [6] [e.g., divergence o f f ( p , k) when t(k, k; ~'2) = 0]. Such anomahes might need some attention also in the more general case considered here, but they are not expected to limit the applicability of the method more than they have done in the two-body case. The reduction of the Faddeev equations worked out here is a special (and particularly simple) case of a more * ThB point of vmw is simdar to that of Amado [3] who argues that the three-body equations in the separable approxlmahon mainly serve to provide the scattering amplitudes with appropriate analytlclty properties.
15
Volume 69B, number 1
PHYSICS LETTERS
18 July 1977
general reduction scheme [7] based on an analysis of the physically intuitive multiple scattering structure of the Faddeev equations. Thas fact provides some insight into the physical content of the equations obtained here (e.g., they represent a separation of the scattering problem into an internal and an external part, see ref. [7]) and is taken as an excuse for leaving out such aspects of the reduction in this note. On the other hand, the present work extends the analysis of ref. [7] by providing an explicit example of how the function R~x m eq. (8) is constructed. Thas point was not discussed m detail in the more general context of ref. [7]. The splitting of the three-body problem into two parts is characteristic of the reduction procedure for the Faddeev equations developed m this note: the solution of a real, non-singular, two-variable mtegral equation [eq. (7)], and the subsequent solution of an on-shell, one-variable integral equation [eq. (8)]. The last equation is similar in structure to the three-body equations in the separable approximation and mainly serves to unitarize the solution t~ the ftrst equation. In particular, eq. (8) can be combined with any approximate (but real) solution to the more complicated eq. (7) to generate unitary scattering amplitudes (see ref. [7] ), Let us conclude wlth the following remark. The natural separation of the unitarizatlon into a comparatively simple equation is clearly a very powerful feature of the reduction presented here, but on the other hand, any reasonable unitary reduction scheme ought to have this feature. It might, therefore, be considered an equally important feature that the equationthat here carries the details o f the dynamics, i.e., eq. (7), is completely free from propagator singularities, and therefore has a much simpler structure than the original Faddeev equations (it might be recalled that in the three-body K matrix formahsm [8] which is a competing unitary reduction scheme, the corresponding equation contains awkward principle part type singularities). In fact, it is this feature that has made the original Kowalski-Noyes method a standard tool for solving the two-body scattering equations.
References [1 ] H.P. Noyes, Phys. Rev. Lett. 15 (1965) 538, K.L. Kowalski, Phys. Rev. Lett. 15 (1965) 798,908E. [21 D.D. Brayshaw, Phys. Rev.C13 (1976) 1024; M.G. Fuda, Phys. Rev. C3 (1971) 485, K.L. Kowalskl, Phys. Rev. C8 (1973) 1973, H.P. Noyes, Phys. Rev. D5 (1972) 1547; E.O. Air, P. Grassberger and W. Sandhas, Nucl. Phys. B2 (1967) 167. [3] R.D. Amado, Phys. Rev. Lett. 33 (1974) 333. [4] O.A. Yakubovskl, Yad. Fiz. 5 (1967) 1312 [Soy. J. Nucl. Phys. 5 (1967) 937]. [5] B.R. Karlsson and E.M. Zelger, Phys. Rev. D9 (1974) 1761. [6] T.A. Osborn, Nucl. Phys. A138 (1968) 305; D.D. Brayshaw, Phys. Rev. 182 (1969) 1658. [7] B.R. Karlsson, to be pubhshed. [8] R.T. Cahdl, Nucl. Phys. A194 (1972) 599; K.L. Kowalskl, Phys. Rev. D5 (1972) 395; T. Sasakawa, Nucl. Phys. A203 (1973) 496; B.R. Karlsson, ITP Goteborg report 76-12 (1976).
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