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The Bateman method for multichannel scattering theory (*)

Department of Physics, Purdue University - West Lafayette, IN 47907-1396, USA

(ricevuto il 7 Gennaio 1997; approvato il 25 Marzo 1997)

Summary. — Accuracy and convergence of the Bateman method are investigated for

calculating the transition amplitude in multichannel scattering theory. This approximation method is applied to the calculation of elastic amplitude. The calculated results are remarkably accurate compared with those of exactly solvable multichannel model.

PACS 03.65.Nk – Nonrelativistic scattering theory.

1. – Introduction

During the past two decades, there has been rapid development of the Schwinger variational method for studying scattering process, bound, and resonance state [1-5]. In this paper, we present a Bateman method [6] which is the special case of the Schwinger variational method [7-9], to calculate transition amplitudes in a multichannel scattering theory.

In sect. 2, we briefly describe the Bateman method. In sect. 3, we describe an exactly solvable model for multichannel scattering problem which will be used to access the accuracy of the Bateman method. The Bateman approximation method for multi-channel scattering problem is described in sect. 4. Application of the Bateman method is described and comparisons of results are presented in sect. 5. Summary and conclusions are given in sect. 6.

2. – Bateman method

We consider the formal identity for the potential V,

V 4VV21_{V 4}

!

i , jVNib aiNV

21

Njb a jNV , (1)

(*) The authors of this paper have agreed to not receive the proofs for correction.

(**) Permanent address: Department of Physics, Cheju National University, Cheju 690-756, Republic of Korea.

where Nib and Njb are, in general, different complete sets. If we truncate the summation over the complete sets in eq. (1), we obtain the separable approximation

V(M) 4

!

where dij4 axiNVNhjb. Equation (2) represents an interpolation process since

V(M)

Nhib 4VNhib and axiNV(M)4 axiNV. One can show [9] that the well-known

Bubnov-Galerkin, Hilbert-Schmidt and Bateman methods are special cases of the expression eq. (2) for appropriate choice of the functions Nhb and Nxb. The separable approximation equation (2) is closely related to Weinstein’s method in the formulation of Bazley and Fox [10].

In the case of multichannel theory, the total Hamiltonian can be written as H 4

H01 V, where anNH0Nmb is a diagonal matrix and anNVNmb is a non-diagonal matrix,

and Nnb and Nmb are channel (basis) functions. The Bateman expression for anNVNmb can be written as [6] anNVNmb 4

!

where, dij4 aaiNVNajb and n , m 40, 1, 2, R. The main problem is how to choose

points a0, a1, a2. . . , aM, when M is not large. When anNVNmb is analytical function of

n and m, we propose to use a condition, ¯NfN

2

¯ai

4 0, to fix a0, a1, a2, . . . aM ( f is the

amplitude).

3. – Exactly solvable model for multichannel scattering problem

The Schrödinger equation for a multichannel system can be written as 2 ˇ2 2 m˜ 2_C a(r) 1

!

where m is the reduced mass and eathe excitation energy in a-th channel. The solution

of eq. (4) can be expressed in terms of the integral Ca(r) 4fa(r) da02 m 2 pˇ2



!

ˇ

k

(r KQ), eq. (5) can be written as

Ca(r) KeikaQ rda01 f0 Ka

eikar

r ,

(6)

where f_{0 Ka}is the scattering amplitude given by

f_{0 Ka4 2} m

2 pˇ2

!

We can rewrite eq. (5) as NCab 4Nfab da01 1 E 2ea2 H01 ie

!

where gab are the channel-coupling strengths, ua(r) 4 arNuab and ub(r8) 4 ar8 Nubb are

scalar functions of the relative (projectile-target) coordinate, and Cais given by

Ca4

!

gabaubNCbb ,

(11)

which satisfies the following relation:

Ca4

!

!

into eq. (12), we can obtain for fa(E) the following equation:

!

b

[dab2 gabGb] Q fb(E) 4ga0.

(15)

Using eqs. (7), (9), (11) and (14), we can express the scattering amplitude f_{0 Ka}as

f_{0 Ka4 2} m

2 pˇ2afaNuab au0Nf0b fa(E) . (16)

If we define transition amplitude T_{0 Ka}, by renormalizing the scattering amplitude

f_{0 Ka}as

T_{0 Ka4 2}2 pˇ

2

m f0 Ka,

(17)

the transition amplitude can be written as

T_{0 Ka4 af}aNuab au0Nf0b fa(E) ,

where afaNuab is given by

afaNuab 4



1

( 2 p)3



(19)

Using the following relation:

fa(k) 4



(20)

eq. (19) can be written as

afaNuab 4ua(ka) 4 akaNuab .

(21)

Therefore, the transition amplitude, eq. (18), can be expressed as

T_{0 Ka4 ak}aNuab au0Nk0b Q fa(E) ,

(22)

where fa(E) can be calculated from eq. (13) and eq. (15). The coefficient Gbrequired for

calculating fa(E) is given by

Gb4 1 ( 2 p)3



For the function ub(k), we will take the Yamaguchi form factor [11]

ub(k) 4 1 k2_{1 g}b2 , (24) which leads to Gb4 2 m 4 pˇ2 1 gb(gb2 ikb)2 . (25)

For the coupling strength, we take the expression given by

gab4 2d exp

k

g

h

l

where the parameter a measures the extent to which the strength of the coupling is distributed among distant channels. By inserting eq. (26) into eq. (15), eq. (15) becomes

!

y

k

z

and

fAb(E) 42

fb(E)

d .

(29)

In eq. (28), g and D are parameters given by D 4 ˇ k2 mgb, g 4

o

In this study, we will treat gb as constant, as Breitschaft et al. [12].

4. – Bateman approximation for multichannel scattering problem

As an approximated method to solve fa(E) in eqs. (27) and (29), we use the Bateman

approximation to gAab, gAabK gAab(M)4

!

where M EN21 (N is the number of channel), dij4 gAlilj. And l04 0 and l1, l2, R , lM

are parameters to be determined. Inserting eq. (31) into eq. (27), we obtain the following relation: fAa(M)(E) 4 gAa02

!

!

!

which satisfies the following relation:

!

k

!

l

!

In the case of M=0, we can rewrite

S04 B A , (35) where A 4 gA001

!

!

We can calculate fAa( 0 )by inserting eq. (35) into eq. (32). The transition amplitude T0 Ka( 0 )

can be calculated by inserting fa( 0 )(E)

(

)

Si, i.e. S0and S1are found by solving eq. (34), therefore, fAa( 1 )(E) are obtained by using

eq. (32). By inserting fa( 1 )(E)

(

)

T_{0 Ka}( 1 ) can be calculated.

5. – Applications and comparisons

As in the preceding sections, we have presented Bateman approximation method to calculate the transition amplitude in multichannel model. Using this approximation, we have obtained the simple form for fa(E) related to the transition amplitude. We have

applied the Bateman approximation method to the calculation of elastic transition amplitude (T_{0 K0}). For the comparison, we have also calculated the corresponding transition amplitudes by using both exactly solvable multichannel model and Born approximation. Four parameters (D , g , E and a) are needed in order to calculate transition amplitude for exactly solvable multichannel model (T_{0 K0}( ex ) _{) and for the M=0}

case of Bateman approximation (T_{0 K0}( 0 ) _{). In the case of calculation for transition}

amplitude by using Bateman approximation (T_{0 K0}( 1 ) _{) for the M=1 case, l}

1values should

be added to the above four parameters and this parameter is determined by the

TABLE I. – Calculated results of real and imaginary parts of exactly solvable model (T_{0 K0}( ex )_), Bateman approximation (T_{0 K0}(M) _{for M 40 and 1) and Born approximation (T0 K0}( Born )_{) for elastic} transition amplitude. The number of channel is 10.

E a/l1 Re (T_{0 K0}( ex )_{) / Re (T} 0 K0( 0 ) ) / Re (T0 K0( 1 ) ) / T0 K0( Born ) Im (T_{0 K0}( ex )_{) Im (T} 0 K0( 0 ) ) Im (T0 K0( 1 ) ) 20.0 20.0 23 .64392 3 1023 _{23 .28912 3 10}23 _{23 .64388 3 10}23 _{29 .73092 3 10}23 6 28 .90100 3 1025 _{29 .84224 3 10}25 _{28 .90120 3 10}25 20.0 35.0 23 .23887 3 1023 _{23 .09725 3 10}23 _{23 .23887 3 10}23 _{29 .73092 3 10}23 6 29 .15909 3 1025 _{29 .55015 3 10}25 _{29 .15909 3 10}25 20.0 50.0 23 .12126 3 1023 _{23 .04820 3 10}23 _{23 .12126 3 10}23 _{29 .73092 3 10}23 6 29 .26564 3 1025 29 .46992 3 1025 29 .26565 3 1025 60.0 20.0 26 .73015 3 1024 26 .58269 3 1024 26 .73013 3 1024 21 .08193 3 1023 6 26 .23148 3 1026 26 .67876 3 1026 26 .23152 3 1026 60.0 35.0 26 .41088 3 1024 26 .35223 3 1024 26 .41088 3 1024 21 .08193 3 1023 6 26 .62051 3 1026 26 .79687 3 1026 26 .62052 3 1026 60.0 50.0 26 .32169 3 1024 26 .29149 3 1024 26 .32169 3 1024 21 .08193 3 1023 6 26 .73312 3 1026 26 .82376 3 1026 26 .73313 3 1026 100.0 20.0 22 .83969 3 1024 22 .81241 3 1024 22 .83969 3 1024 23 .89548 3 1024 6 21 .49004 3 1026 21 .56718 3 1026 21 .49005 3 1026 100.0 35.0 22 .75283 3 1024 22 .74192 3 1024 22 .75283 3 1024 23 .89548 3 1024 6 21 .59695 3 1026 21 .62753 3 1026 21 .59695 3 1026 100.0 50.0 22 .72873 3 1024 22 .72309 3 1024 22 .72873 3 1024 23 .89548 3 1024 6 21 .62704 3 1026 21 .64278 3 1026 21 .62704 3 1026

TABLE II. – Calculated results of real and imaginary parts of exactly solvable model (T_{0 K0}( ex )_), Bateman approximation (T_{0 K0}(M) _{for M 40 and 1) and Born approximation (T0 K0}( Born )_{) for elastic} transition amplitude. The number of channel is 50.

E a/l1 Re (T0 K0( ex ))/ Re (T0 K0( 0 ) )/ Re (T0 K0( 1 ) )/ T0 K0( Born ) Im (T_{0 K0}( ex )_{) Im (T} 0 K0( 0 ) ) Im (T0 K0( 1 ) ) 20.0 20.0 23 .44832 3 1023 _{22 .44840 3 10}23 _{23 .42753 3 10}23 _{29 .73092 3 10}23 5 27 .92161 3 1025 _{28 .36509 3 10}25 _{28 .09301 3 10}25 20.0 35.0 22 .58618 3 1023 _{21 .55274 3 10}23 _{22 .56136 3 10}23 _{29 .73092 3 10}23 4 26 .60354 3 1025 _{26 .06174 3 10}25 _{26 .77477 3 10}25 20.0 50.0 22 .13334 3 1023 _{21 .16738 3 10}23 _{22 .12978 3 10}23 _{29 .73092 3 10}23 8 25 .82355 3 1025 _{24 .83510 3 10}25 _{25 .87656 3 10}25 60.0 20.0 26 .22221 3 1024 _{25 .50678 3 10}24 _{26 .21971 3 10}24 _{21 .08193 3 10}23 9 25 .74412 3 1026 _{27 .02731 3 10}26 _{25 .74637 3 10}26 60.0 35.0 25 .04252 3 1024 _{24 .05255 3 10}24 _{25 .04197 3 10}24 _{21 .08193 3 10}23 14 25 .63032 3 1026 _{26 .62088 3 10}26 _{25 .63370 3 10}26 60.0 50.0 24 .33010 3 1024 _{23 .29302 3 10}24 _{24 .32960 3 10}24 _{21 .08193 3 10}23 21 25 .23299 3 1026 26 .00652 3 1026 25 .23442 3 1026 100.0 20.0 22 .63854 3 1024 22 .47202 3 1024 22 .63819 3 1024 23 .89548 3 1024 11 21 .48346 3 1026 21 .81366 3 1026 21 .48336 3 1026 100.0 35.0 22 .22419 3 1024 21 .95505 3 1024 22 .22391 3 1024 23 .89548 3 1024 18 21 .59015 3 1026 21 .96114 3 1026 21 .59070 3 1026 100.0 50.0 21 .95048 3 1024 21 .65456 3 1024 21 .95014 3 1024 23 .89548 3 1024 24 21 .56966 3 1026 21 .92096 3 1026 21 .56985 3 1026 following condition: ¯NT0 K0( 1 ) N2 ¯l1 4 0 , (38) where T_{0 K0}( 1 ) 4 Re (T0 K0( 1 ) ) 1i Im (T0 K0( 1 ) ) . (39)

For a case of E 420.0, a450.0 and channel number 490, the extremum point does not exist for l1D 0. Thus, we choose l1value for which the derivative of NT0 K0( 1 ) N2is minimum

for l1D 0. The parameters used in this calculations are taken from Breitschaft et

al. [12]: D 40.1, g42.094 and equally spaced spectrum given by ea4 0.1a. In this

schematic calculation, we choose units such that c 4ˇ4m41. As examples of applica-tions, three kinds of collision energies E (20.0, 60.0 and 100.0) are taken together with three different a values which are contained in coupling strength gAabin eq. (26).

The calculations of the elastic transition amplitude (T_{0 K0}) for three numbers of channel of 10, 50 and 90, and three a values of 20, 35 and 50, respectively are performed for three different energies. The calculated results are shown in tables I-III. For each channel, each table displays also calculated results of exactly solvable multichannel model (T_{0 K0}( ex ) _{), the Bateman approximation (T}

0 K0(M) for M= 0 and M=1) and the Born

approximation (T_{0 K0}( Born )_{). In each table, we can find that our Bateman approximation for}

TABLE III. – Calculated results of real and imaginary parts of exactly solvable model (T_{0 K0}( ex )_), Bateman approximation (T_{0 K0}(M) _{for M 40 and 1) and Born approximation (T0 K0}( Born )_{) for elastic} transition amplitude. The number of channel is 90.

E a/l1 Re (T0 K0( ex ))/ Re (T0 K0( 0 ) )/ Re (T0 K0( 1 ) )/ T0 K0( Born ) Im (T_{0 K0}(ex)_{) Im (T} 0 K0( 0 ) ) Im (T0 K0( 1 ) ) 20.0 20.0 23 .44825 3 1023 _{22 .44840 3 10}23 _{23 .42750 3 10}23 _{29 .73092 3 10}23 5 27 .92179 3 1025 _{28 .36509 3 10}25 _{28 .09316 3 10}25 20.0 35.0 22 .58266 3 1023 _{21 .54510 3 10}23 _{22 .54031 3 10}23 _{29 .73092 3 10}23 3 26 .61188 3 1025 _{26 .04220 3 10}25 _{26 .80287 3 10}25 20.0 50.0 22 .10501 3 1023 _{21 .10608 3 10}23 _{22 .04474 3 10}23 _{29 .73092 3 10}23 1 25 .72373 3 1025 _{24 .65069 3 10}25 _{25 .98817 3 10}25 60.0 20.0 26 .22214 3 1024 25 .50678 3 1024 26 .21968 3 1024 21 .08193 3 1023 9 25 .74448 3 1026 27 .02731 3 1026 25 .74651 3 1026 60.0 35.0 25 .03457 3 1024 24 .04009 3 1024 25 .02559 3 1024 21 .08193 3 1023 12 25 .62551 3 1026 26 .61387 3 1026 25 .66008 3 1026 60.0 50.0 24 .29882 3 1024 23 .17680 3 1024 24 .28420 3 1024 21 .08193 3 1023 14 25 .30098 3 1026 25 .89488 3 1026 25 .37675 3 1026 100.0 20.0 22 .63854 3 1024 22 .47202 3 1024 22 .63818 3 1024 23 .89548 3 1024 11 21 .48350 3 1026 21 .81366 3 1026 21 .48340 3 1026 100.0 35.0 22 .22179 3 1024 21 .95040 3 1024 22 .22086 3 1024 23 .89548 3 1024 16 21 .59439 3 1026 21 .96137 3 1026 21 .59836 3 1026 100.0 50.0 21 .94466 3 1024 21 .60738 3 1024 21 .94340 3 1024 23 .89548 3 1024 20 21 .58728 3 1026 21 .90746 3 1026 21 .59573 3 1026

multichannel model. Furthermore, the results of M 41 case Bateman approximation agree remarkably well with results of exactly solvable multichannel model, showing substantial improvements with respect to the results of Bateman approximation for the

M 40 case. As expected, the calculated transition amplitudes in multichannel model

become much closer to those of Born approximation as the energy increases.

It is seen that the real parts of transition amplitude are much larger than imaginary parts, thus, these play an important role in the absolute value of transition amplitude. For a fixed number of channels and values of a , both the real and imaginary parts of the elastic transition amplitude decrease as the energy increases. Our calculated results with fixed values of energy and a depend on the number of channels used. For a fixed number of channels, the calculated results for the transition amplitude depend also on values of a and E.

6. – Summary and conclusions

In this paper, we have presented the Bateman approximation method to calculate the transition amplitude in multichannel scattering theory. This method produces accurate results compared with those of exactly solvable multichannel model. Particularly, the agreements between the results of M=1 case Bateman approximation and results of exactly solvable multichannel model are remarkably good. Thus, we conclude that our Bateman approximation method for the M 41 case

(

l1 is determined from eq. (38)

)

amplitude in multichannel scattering model. * * *

One of the authors (YJK) acknowledges a partial support provided by a faculty exchange agreement between Purdue University and Cheju National University.

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