Sequential projections of rearrangement channels

ANNALS

OF PHYSICS:

67, 389-405

Sequential

(1971)

Projections

of Rearrangement YUKAP

Department

of Physics,

University

Channels

HAHN

of Connecticut,

Stows,

Connecticut

06268

Received March 9, 1971

Proposed here is a stepwise procedure of constructing the totally closed channel Hamiltonian for scattering systems with rearrangement processes, in which the nonorthogonal rearrangement channel components are projected out in sequence. The connection of this method to that followed in the generalized variational bounds formulation is clarified, and several possible applications are indicated. A simpler yet rigorous procedure for the evaluation of resonance energies is discussed with the partially projected operators.

I. INTRODUCTION

AND SUMMARY

The difficulties of analyzing the many-particle scattering systemsare well known, and various theoretical approaches [ 1,2] which are mathematically consistent have been studied. However, it is not clear whether or not still simpler and more practical formulations are possible. For example, the hierarchy of equations constructed for the many-particle systems by Faddeev [3] and Weinberg [4] is not very easy to apply in practice and has so far produced only meager theoretical results. More tractable approaches seemto be those based on the unified reaction theory of Feshbach [5], which allows one to circumvent, at least formally, large parts of the hierarchy of scattering equations in terms of the channel projection operators, and have already yielded many significant theoretical calculations [8]. In applying the Feshbach theory in its original form to reactions involving particle rearrangements, however, one encounters a severe difficulty in constructing the open-channel projection operator P. This problem arises, as is well known, essentially from the nonorthogonality among the rearrangement channels. In a recent paper [6], this difficulty has been removed, and a detailed application of the formalism [7] (the generalized variational bounds-GVB) has been carried out to the positron-hydrogen pickup collision [8]. Since the explicit form of the open-channel operator P is assumed unknown, many important properties of the GVB could be proven only formally or implicitly stated. Likewise, in applications, many of the manipulations had to be handled with care in order to maintain certain orthogonality properties. 389

390

HAHN

We present here an alternative formulation of essentially the same theory from a more intuitive point of view, thus hopefully providing a better insight into the structure of the GVB and also improving the calculational procedures. The sequential projection procedure proposed here treats the many rearrangement channels one at a time, gradually building up the orthogonal operators channel by channel. The mathematical operation employed for this purpose is essentially the following: If we take a real symmetric operator A with the spectrum {a,} given by AX,

=

a,X,

(l-1)

)

some discrete and others continuous, then, the new operator A’ which is orthogonal to the X, state can be constructed as A’ = A - AX, > a;’ < &*A, with the property

(1.2)

that A’X,>

= <&*A’

= 0.

(1.3)

The well-known examples are A = H = energy Hamiltonian, A = QHQ = quasistate Hamiltonian describing the resonance states, and A = V = interaction potential as applied in the quasi-particle method [9], with slightly modified definition of X, and a;’ as obtained from the strength eigenvalue problem. The above procedure obviously cannot be applied directly to the present problem, since we have several open channels, each with full continuum spectrum, and all of which are to be projected out. As will be shown below (Section II), the necessary modification of (1.2) involves combined use of the single channel projection operators Pi and the Green’s functions PiGpPi = -(P,M,,P,)-l associated with Pi . Writing MO = H - E, the new operator M,’ defined by MO’ = Mi = M, - Mo(-G;Jl)

M,,

can be shown to be orthogonal to the Pi operator. Obviously, (1.4) can be generalized stepwise to include many rearrangement channels, whose mutual nonorthogonality has always been an obstacle in formulating a consistent theory. The operators derived in Sections II and III are all of the above form. Section II contains a detailed exposition of the procedure of sequential projections of rearrangement channels. As the channels are projected out one by one from MO , the resulting operators in the intermediate stages are orthogonal to all the open channels projected out up to that point, irrespective of whether those projected channels are mutually ‘commutative’. When all the open channels are projected out, the resulting operator M acts, by definition, only on the closedchannel space. The form of M would look highly asymmetric in its dependence on

391

SEQUENTIAL PROJECTIONS

the various open channel indices, since we have constructed it in a rather arbitrary sequence. However, we show in Section III that M is completely symmetric in all the open-channel labels, and is identical to the operator used in the GVB formulation [6, 71. Several possible applications of the new formalism are discussed in Section IV mainly to point out the calculational procedures involved. Throughout the paper, we regard the exchange channels with identical particles as a special case of the rearrangement channels, and limit our discussions only to channels involving two clusters. Possibility of treating the breakup channels in the present formalism is briefly mentioned at the end of Section II.

II. THE METHOD

OF SEQUENTIAL

PROJECTIONS

To simplify the discussion, we consider the two-channel process 1 + (3 + 2),-(1

channel 1

+ 310+ 2,

(2.1)

channel 2

where all three particles are assumed distinguishable and have finite masses. Generalizations to systems involving more particles and more open channels are straightforward, while the case with identical particles simplifies considerably the resulting hierarchy of equations. Furthermore, we restrict our study to channels with two clusters, and only a brief consideration will be given on the breakup problems at the end. After eliminating the kinetic energy of the total center of mass, we write the Hamiltonian H in the form H = Hi + Vi,

i=

1,2,

(2.2)

with Hi = hi(rjJ + ti(Ri), vi = vij + vi, ) i#j, where rja is the relative coordinate of the pair j and 3, and Ri is the coordinate from the center of mass of the (j3) pair to the i-th particle. The channels for the process (2.1) are specified by the projection operators p. 2 E Q(r.

33 )

><

qp’(*,: 0

)

33 9

where {#~~‘} are the sets generated by hi , as h% .#(i)(r n

33

) = &i)#(i)(r. 12 n

33’

)

(2.3)

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HAHN

The complementary

operators to Pi are then given by Qi = 6(rj, - ri3) - Pi E 1 - Pi .

(2.4)

The rearrangement channels such as those in (2.1) are in general not orthogonal, and it is well known that P, and Pz do overlap and

[PI 2P21# 0.

(2.5)

The commutativity of the rearrangement channel operators [PI, P,] = 0 is a much weaker relation than the orthogonality property PIP2 = P,P, = 0, which also does not hold. [For cases in which rn3 > m, , m2 , [PI , P,] = 0 holds but PIP2 # 0, and we can construct the open-channel operator P which projects both open channels such that Q = 1 - P is the closed channel operator.] We note, however, that, in the limits Ri -+ co, we have PIP, = P,P, = 0

[Q, >Q,l = 0,

and

(2.6)

which are essentially the asymptotic orthogonality [2] of the channels 1 and 2. There have been various attempts [5] to derive at least formally the open-channel projection operator P such that it has the following properties; first of all, when acted on the wavefunction Y, the operator P gives PY = PlY,

+ P,Yz )

p2 = p = pi,

e-7)

QP = PQ = 0,

(2.8)

where Pip = Pi , but PY f P,Y + P,Y, and with

QY=Y-PY,

does not contain any of the two open channels 1 and 2, i.e., PiQ = 0.

(2.9)

Such operators P and Q are difficult to obtain in practice because of the property (2.5), but they can be written down formally and their existence was sufficient to derive the GVB [7]. The approach to be discussed below does not have to assume the properties of P, although we will show its connection to the GVB in Section III. Since we will also show later that the particular sequence of channels chosen in the stepwise projections does not affect the final operator to be derived, we simply start with the channel 1 and work our way up. (i)

First, we consider the channel 1 by writing y = Ply

+ Q,Y

(2.10)

SEQUENTIAL

393

PROJECTIONS

which is simply a restatement of (2.4), and obtain the usual Feshbach equations P,M,P,!P

= -PIM,,QIY

(2.11a) (2.11b)

- PIN,, ,

Q&foQlY = -Q&foP,Y - QlNo , where we set, for convenience of notations, M,=H-E,

and the original scattering equation is written in the form Moyf(o) = -. No ,

(2.12)

with Y(O) = Y and No = 0. In (2.1 l), P,Y contains all the asymptotic contribution of the channel 1, while Q,Y describes the rest of the channels, both open and closed. In particular, the Q, space contains the channel 2 (and more if other channels are also open), so that special care is needed in approximately inverting the operator QIMoQl . On the other hand, PIYco) contains usually an unknown function of one variable and (2.11 a) can readily be solved once the right side of the equation is given. Thus, we formally solve for PIY(0) as PIYw

= P,Yc

+ G~MoQIY(o)

(2.13)

+ G,P’N, ,

where PlMOPlY2

= 0,

P,M,P,G,p’P,

= -P,

PIYopl 3 YC, ,

P,GcP,

and we apply the standing wave boundary conditions quantities real. Thus, for example, we set P,Yc

= @)(r2,)

(2.14)

= G,P’,

throughout

to make all (2.15)

z&R,)

with R&R&+0 R&

Substituting

+

I

a, sin k,R, - &)

i

2

= 0 + t$” cos (t&R, - $)]

CT1.

(2.16)

(2.13) into (2.1 lb), we have

QJMo + MoG,p’MoIQ,Yco) = -Q,M,P,Y~

-

QIMoG~No

- QINo ,

(2.17)

or, simply, with obvious notations, Q&f,Q1y’O’

= -QlNl,

(2.18)

394

HAHN

where Nl = M,,P,Yc

+ M,GcN,,

+ N,,

and Ml = M,, + M,GcM,

.

In (2.18), QIM,,Ql is still a many-particle operator and generally as difficult to invert as M,, , except the fact that now the channel 1 component is not present. The main underlying idea of the procedure under consideration is not to invert such operators until all the open channels are projected out. Now, from the dennitions (2.14) for !Yci and Gli, it is clear that we have the orthogonality properties

QlMl = MlQl = MI 3 P,M,

= M,P, = 0

(2.19)

and also

Q,N, = NIT

PINI = 0.

(2.20)

Thus, we can rewrite (2.18) in the form without the operator Q1 as M,Y’l’

= -Nl

(2.21)

,

where we added the superscript in Y to keep track of the intermediate stages involved, but otherwise we can write Y (I) = Y without loss of generality; the solution of (2.21) has to be in the Q, space because of (2.19) and (2.20). We should compare (2.21) with the original Eq. (2.12). The orthogonality property of Y(l) to P, is completely incorporated in the operator Ml , so that Y(l) needs no longer to be othogonalized. Therefore, we can essentially start all over again with (2.21), as we did with (2.12), keeping in mind that we do not have the channel 1 this time. [Consequently, the new Y in (2.21) should have the decaying boundary condition in the asymptotic region of the channel 1.1 (ii)

Next, we consider the channel 2, by writing Y’l’(&‘)

= P2yI’l’ + QzY’l’,

(3.22)

and obtain P,MIP,Y’l’ = -P,MIQ,Y’l’ Q2MlQ,Y’1’ = -Q2MlP,Y’1’

- P2Nl , - Q,Nl .

(2.23a) (2.23b)

[For a two-channel problem, QzMIQz is already a well-behaved operator in the closed-channel space. This will be elaborated on later in Section IV.] The solution of (2.23a) is P,Y”’

= P,YF

+ PzGf=PzMIQzYC1) + PzG;PaP2Nl,

(2.24)

SEQUENTIAL

395

PROJECTIONS

where P&f,P,Yp

= 0,

P2MlP2GpP2

PzYF

= -P,,

3 ?q,

(2.25)

PzG;P2Pz = G;PB.

Again, there is no ambiguity in the solution (2.24) and PzY(l) can be calculated once QzY1) is given, perhaps in some approximate forms. Substituting (2.24) into (2.23b), we obtain Q,[M,

+ MIGlp”MJ

&Y(l)

= -Q,M,P,Yp

-

QzMIG~N,

- Q,N, ,

(2.26)

or, simply write (2.26) in the form

QzK,QP’ where we explicitly

= -Q2N2, ,

(2.27)

have

M,, = Ml + M,GpM, = MO + M,G?M, N,, = Mly~

+ (M,, + M,,G,P’M,) Gp(M,

+ M,GpN,

+ M,G&lJ,)

(2.28)

+ Nl .

In exactly the same way as before, we find, using (2.25)

MzlQz = Q&f,, = M,, , P,M,,

(2.29)

= M,,PB = 0,

and

Q&J = N,, ,

P,N,,

= 0.

(2.30)

Thus, we can drop the Q, operator in (2.27) and write M21y’2’

= -N 21 >

(2.31)

where we set arbitrarily QzYu) = Y (2), but we recall that it can also be equal to Y, insofar as its use in (2.31) is concerned. (2.31) should now be compared with (2.21) and (2.12). Again, the orthogonality property of Y(l) to P2 is transferred from the wave function to the operator M,, and the function N,, . The advantage of this is that we do not have to keep track of the fact that Yc2) is originally from Yc2) = Q2Yc1) = Q2QIY(0). Rather, Yc2) in (2.31) can be of any arbitrary functional form satisfying the proper boundary conditions. The interesting property of n/r,,

396

HAHN

and N,, is that, although the noncommutativity from the property (2.19) for Ml ,

Q&k = MzlQl = Ma >

(2.5) is still present, we have,

Q&z1 = Nzl,

(2.29’)

and P,M,,

= M,,P,

= 0,

(2.30’)

PJV,, = 0.

Combining (2.29), (2.29’) and (2.30), (2.30’), it is clear that M,, and N,, are both orthogonal simultaneously to both P, and Pz . Therefore, we have finally constructed the operator which is entirely in the closed-channel space Q. (iii) We can easily generalize the above procedure to cases with JV open channels. When all the open channels are projected out, we have M

4v,Jv-1,....2,1

ye-m = -N

dv,#v-1....,2.1

l

(2.32)

where both M and N are orthogonal to all the Pi channels with i = 1,2,..., JV, and where all the intermediate M are essentially of the same structure as (1.4). This already suggests that, although the forms of M,, and N,, of (2.28) are highly asymmetric in the channel labels 1 and 2, they may actually be symmetric. This is indeed the case, as will be shown in Section III. (iv) We can now solve the hierarchy of equations step-by-step backward, starting from (2.31), followed by (2.24) and (2.13). Thus, for the process (2.1), RIYc2’ --, 0

(2.33)

as Ri --+ CO,and we have ylc2) E Q2@)

= GzNzl

,

(2.34)

where G,O;;= -(M2J-‘,

and thus p 2 ycl’ = p 21y’s + GP2M GQ”N 21' 1 1 21

(2.35)

Finally, then, we have the complete solution in terms of (2.34) and (2.35), as Y=PIY+Q,Y

(2.36)

= P,Y + P2Y(‘) + Q2Yc1),

where PIY = P,Yc

+ G,P’M,,(P,Y”’ + Q,Y(‘)).

(2.37)

SEQUENTIAL PROJECTIONS

397

The remarkable feature of the solution Y in the form given by (2.36) is that it is obtained by inverting the well-deBnedclosed-channel operator M,, , which is the only many-particle operator to be dealt with; all the rest of the quantities in (2.36) are of the potential scattering type and thus readily calculable, with no theoretical ambiguity present throughout the program. Thus, an approximation to (M&-l will give uniquely the resulting approximate Y. To summarize the result obtained so far, we have step-by-step projected out the open-channel components from M,, such that the final A4 is completely in the closed Q-space, without any of the orthogonality requirement explicitly imposed on the wave function. Obviously, we can carry out similar analysis by writing the wave function Y as Y=

PIY+Q,Y

= f’,Y + P,QlY

(2.38a) + Q,Q,Y.

(2.38b)

This is what we would have had from (2.10) and (2.22) if Q, was not incorporated into the operator M1 . Using (2.38b), we can again derive a set of coupled equations of the form [lo]

(QIQ&&Q~QJ

Q,y = -QIQ&W’IY

- Q,Qd4,PP1’,

(2.39)

and similarly for P,Y and PzY(l) = P,Q,Y. However, since [Q, , Q,] # 0 except in the asymptotic regions, the formalism cannot be quite symmetric under the exchange 1 t) 2, although obviously the operator Q,Q,M,,Q,Q, is again in the closed channels. For a more symmetric procedure along those lines with (2.38), we refer to the papers of Ref. [lo]. For casesin which only the breakup channel is open in addition to the two channels of (2.1), we can effectively apply the present formulation. Since A4,, of (2.31) is already orthogonal to P, and P, , we can solve the breakup problem assuming that only that channel is open. One has to be careful in applying the operator M,, to this problem, however, since any separation of M,, into two parts for the purpose of perturbative treatment would immediately destroy the orthogonality property. Therefore, the solution would necessarily involve approximations directly on the wave function Y(z) of (2.31). For example, we have Yc2) = CD- M-lM@ - M-lN for any reasonable function Qi, where M is no longer a hermitian operator and acts only on functions to its right. M-l involves the full three-particle scattering problem.

III. SYMMETRIC

FORMS FOR M,, AND N,,

The expression for the closed channel operator M,, given in Section II appears at first sight to be asymmetric under the exchange of channel labels, since we have 595/6712-4

398

HAHN

derived it by first projecting out the PI component from M,, = H - E, followed by the Pz projection (especially in view of the property [PI , P,] # 0). However, we have also seen that M,, is orthogonal to P, and P, simultaneously. We show in this section that M,, is symmetric and equal to the operator M derived earlier [6,7] in connection with the GVB. For this purpose, we briefly describe the formalism developed previously [7]. To begin with, we assume the existence [5] of the open-channel projection operator P with the following properties: PY = PlYI + P2Y2 ) p2 =

p =

Q E

p+,

1 -

p =

Qt

=

Q2,

(3.1)

PQ = 0,

and PiQ = QPi = 0,

(3.2)

PiP = PP( = Pi )

but QiQ = Q = QQi . Due to (2.5), such operators are not easily derivable explicitly, and we do not need them, but use them here merely to exhibit certain orthogonality properties. With Y = PY + QY, we have the coupled equations PiMoP,Yg = -P,MoPjYj

- PiMoQYp

6 f j>,

(3.3)

and Qh&,QY = -QM,P,Y/,

- QMoP2Y2 .

(3.4)

Thus, PiYi in (3.1) are defined by (3.3) with the appropriate boundary conditions. We further define the functions PYp and PGPP by PiMoPiYip = -P~M~PjYjpy

G f 3,

PiMoPGiPPi = -Pi ,

(3.5)

or, simply PM,PYP = 0

(3.5’)

PMoPGPP = -P,

(3.5”)

and where PYP E YP s PIYIP + P,Y2P, PGPP = GP = GIPPl f

GzPP, ,

GiPPi = PIG$Pi f P,GiiP,.

(3.6)

SEQUENTIAL

Explicitly,

399

PROJECTIONS

GzPP, , for example, satisfies the coupled equations

with the simplified

&WW;&

+ P,MoP,Gi’zP,

= --pz ,

P,M,,P,G;P,P,

f P,M,P,G;2P,

= 0,

notation G; = P,G;P, .

Formally

(3.7)

(3.8)

solving (3.7), we then have G,9, = G,P’M,,Gc2 ,

f’,M, + ~oG,P’MoIP,Gi’J’, = -Pz ,

(3.9) (3.10)

where G> = -(P,MzJ1.

The left side of (3.10) contains the operator which is simply Ml of (2.18), and thus Gi2 = Gp

(3.11)

and also Gr2 = G,P1M,,G,P2, GzPP, = Gp + G,P’M,,G,P2.

(3.12) (3.13)

Finally, then, we have for the total open-channel Green’s function GP given by GP = Gp + GpM,Gp

+ Gp + GpM,,Gc.

(3.14)

(Note that GIPP, # Gp). Similarly, we have the solution of (3.5) for Yp in the form (i # j) (3.14’) Above, we have written out in detail the forms of equations that Yp and GP satisfy, in order to show that these quantities can be obtained without the explicit use of the operator P. They can be obtained explicitly numerically without much difficulties, since the equations to be solved are always of the form of potential scatterings. Now, solving (3.3) formally using (3.5’) as PY = Yp + GPM,,QY

(3.15)

400

HAHN

and substituting it back to (3.4), we obtain (3.16)

Q[M, + MoGpMol QY = -Q&P’, or, simply QM’Z’QY

=

(3.17)

-QN’2’.

From (3.1), (3.2), and (3.5), we can show [6,7] that W2) and JV2) in (3.17) satisfy the following crucial orthogonality properties; PM=MP=O,

QM=MQ=M,

P,M = MPi = 0,

(3.18a)

PiN = 0.

(3.18b)

and QN=

N,

PN=O,

and

Thus, the Q-operator in (3.17) can ail be dropped, and Y in (3.17) now satisfies MWy’2,

=

-N(2),

(3.19)

where we have added the superscript in Y simply to denote the fact that the solution of (3.19) cannot contain the components of the channels 1 and 2. More generally, for a problem with JV open channels, we can readily construct the J-channel Green’s function GP and corresponding wave function YP, and write J@+‘-‘yy’“y’ = -N(x)

(3.20)

,

where Mcx’ NcN’

= M, -t- M,,GpM, 3 M,YP ,

= n/r, + S,

or simply MY

(3.21)

= -N.

Of course, it is not always necessary to project out from M, precisely the Jlr open channels; some of the closed channels may be included in the P-space and projected out in addition to all the JV open channels, in which case the resulting M is still in the closed-channel space. However, (3.20) is the simplest case which gives bounds on scattering parameters [7]. Now, we can return to M,, and N,, of Section II. From (2.28) and (2.25), M,, may be written explicitly as M,, = M,, + M,(Gp + GpM,Gp

+ GpM,,Gp

+ G,P’

+ GcM,,GpM,G,P’)

M, .

(3.23)

SEQUENTIAL

401

PROJECTIONS

Using the definitions for the various Green’s functions, (2.14) and (2.25), we can write (3.24) After expanding Gp on the right side of (3.24) and resumming the series, we also get Gp = Gp + G~M,G~M,G,Pe, (3.25) and similarly Gp = Gc + G,P’h&G,P2M,,G,P’.

(3.25’)

Thus, the first two terms in the square bracket of (3.23) become simply GZP, the third and the last terms combine to give G2 = G,P,, and the fourth term can be identified as Gfz . Combining these terms and comparing with (3.14), we finally have Al,, = AP’ = A4 = Ml, , (3.26) which is symmetric in the channel labels 1 and 2. The function N,, of (2.28) can also be written explicitly as N,, = kflY~

+ M,G~N,

= A4,(!f’~ + GpNl

+ NL + G,‘lM,Y~

(3.27) + G,‘lM,,G~N,

- P,YP1).

Noting from (3.14’) that tl’p + GPN, = P,Y/,’

(3.28)

and Plypl + G~M,(Y;‘z

we immediately

+ G,P2N,)= Ply”

+ G,P’MoY2P = PIYIP,

have N,, = MoPYp = N@’ = N = N,, ,

(3.29)

which is again completely symmetric under the exchange 1 f-) 2. This completes the proof that the earlier formulation outlined in (3.19) is identical to that given in Section II, although they are obtained by two different procedures. While the result (3.19) is derived by the ‘one-step’ projection using GP and Yp, (2.31) is constructed by the ‘step-by-step’ projections, thus indicating how the rearrangement channels are being taken out of M,, one-by-one. Since the resulting closed-channel operator is completely symmetric, the particular sequences followed are immaterial.

402

HAHN

Now that we have identified with N, it is a simple matter [7] of M and N. Defining as before of the reactance matrix elements

the operators M,, with M and the function N,, to write down the scattering parameters in terms the parameter h which is essentially a linear sum Kii , we have (3.30)

h = hP - W,, > KtNd,

where Xp is the value of X in the ‘coupled-static approximation’ with the QY term in (3.3) being set equal to zero. For details of the definitions of h and the general procedure for solving various coupled equations, we refer to the recent paper by Dirks and Hahn [8]. Incidentally, we can also write (3.30) in another form h - hP = (N,, , Y) = -(Y,

A&Y)),

(3.31)

which should also be useful for various purposes.

IV. THE SPECTRUMOF M,, AND APPROXIMATIONS Equation (2.31) is extremely useful in analyzing the rearrangement processes, since the operator M,, = M behaves in many ways like an operator for the bound state problem. However, A4 is still a complicated many-particle operator with the nonlocal interaction S I- MoGPM,, so that we have not only the task of constructing A4 and N, but also of devising some useful approximation procedures to solve (2.31). The most crucial point to remember in applying (2.31) is that both M and N have to be retained in their original forms and any modifications on them would immediately destroy the orthogonality properties (2.29), (2.29’), (2.30), and (2.30’). Only those approximations which adjust the wave functions are allowed, and, for this purpose, we first discuss the structure of the spectrum of M. From (2.29) and (2.29’), we have shown that A4 is an operator in the Q-space, so that its continuum part of the spectrum starts at the lowest excitation threshold E U* Thus, in the energy region E f E Li , we expect to have only the discrete spectra, most of them located above E for low enough E. If we write M&(E)

= EnYE) &a(E),

(xl 9-a

= hm ,

(4.1)

and E,O(E)

= EaM(E)

+ E,

(4.2)

then, for E =CEsQ(E) for all n, we obviously have M(E,

0) > 0.

(4.3)

SEQUENTIAL

403

PROJECTIONS

[The normalization of X, , given by (4.1), is not known in general because the explicit form for Q is not available. However, the function N of (3.29) may be used in the form (X, , N)(N, N)-l(N, X,), which does not affect the position EnM = 0.1 In (4.3), we have explicitly included the dependence of M on the energy E and the normalization parameter 6’ of the scattering wave functions which assume approximately the forms (in the limit of small coupling between PI and PJ P,y/ --f ai#f)(rti)[sin(k&

+ fl) + tan&

- 0) cos(k&

+ e)] ??JRi

(4.4)

as Ri + co, where ai specifies the initial conditions and & are the eigenphase shifts. Thus, for some particular choices of 0, we may have violated (4.3), thus producing the spurious states [8]. Consequently, h, P, X, and EnM, E,Q are all dependent on 8. In addition to the e-dependence, A4 and EnM are also functions of E due to the presence of the shift operator S in M. Therefore, EnM and EnQ form trajectories in E; these ‘shadow’ trajectories do not show their drastic effects on the cross section except when EmM(E)=

0,

i.e.,

EnQ(E) = E,

(4.5)

which are the conditions for resonances. We expect M-l and h in (3.30) to blow up at these energies and one of the eigenphase shifts assumes the value si = e + (2m + 1) n/2.

(i) The nearest-resonance approximation follows if we take one or more states of M for which EnM are small. Thus, from (3.30), we have

where The scattering amplitudes, which always satisfy the M-channel unitarity, can be obtained [8] from the K-matrix calculated by (4.6). Obviously, (4.6) is limited in accuracy, except when E - EaQ are very small, and perhaps more suited for an approximate parametrization of h. (ii) A much more powerful approach than (4.6) is to variationally estimate the operator M;’ using the Hylleraas-Undheim theorem, as (4.7)

where GL , Wd&)

= 6% - E) b .

404

HAHN

(4.7) follows from (4.3) and the Schwartz inequality, and X,, are the trial functions with the decaying boundary conditions asymptotically (i.e., square integrable). From (3.30) and (4.7), we immediately have (4.8) where This is the generalized variational bounds formulation, which was derived [7] and applied [8] earlier. As 01~increases, (4.8) is capable of yielding the correct scattering parameter h, including all the continuum and discrete spectrum of MS1 . We note that X,, need not be preorthogonalized to the P-space, since, in E,, and yat, M and N eliminate the P-components in X,, . The only complication in practice with (4.8) is the explicit appearance of YP and GP. As we have shown, Yp satisfies the coupled equations (3.5), while GP satisfies (3.7) which is much harder to obtain. We can avoid the direct GP evaluation by solving the auxiliary equations PM,PY,,

= -PM&,

for each X,, . Then, the matrix elements involving

Kt >~21-%3t)= G&t 3Mat

(4.9)

M,, become

+ r,tl).

(4.10)

This procedure greatly simplifies the actual calculations of the GVB. However, as a consequence of the alternate formulation presented in Section II, we propose a still simpler procedure which avoids both GP and Yp: (iii) We go back to (2.18) and consider the operator Ml which is orthogonal to PI . It is a much simpler operator than Mzl since we only have to solve for the simple Green’s function G2 of a one-channel problem. Similarly, N1 can be obtained easily by solving (2.14). Now, instead of incorporating the entire orthogonality property of Q into operators, as we did with M,, , we try to use the trial functions which are orthogonal to P, , i.e., we consider the operator Q2M1Qz in (2.23b). In fact, QzMIQz and its counterpart QIMzQl are also in the Q-space and work as well as M,, insofar as the bound properties are concerned. Thus, for low enough E, we have

QdGQl > 0, Q&GQ, > 0. [As remarked

earlier, the normalization

(4.11)

of the functions X,, should always be

405

SEQUENTIAL PROJECTIONS

defined in terms of the appropriate weight function N&N, now diagonalize a matrix with elements WJwm =
, Ni)-l(Ni

.] We can (4.12)

and obtain the trajectories Ez(E)

= E,att2’ - E,

which will generally be different from the ones obtained with M,, or QZMIQZ . Of course, En0 of (4.2) give the correct resonance solutions and E$“’ and E$” are only partially shifted. However, these are presumably better than the ones obtained with QHQ even if the operator Q is available. In the present case, we do not have Q, and are thus forced to use the more complicated but better operators such as M,, and the ones in (4.11). We can certainly go one step back and incorporate the entire property of Q in the wave function rather than splitting it as we did with QZMIQZ . Thus, following (2.39), we can diagonalize the matrix with the elements [lo] (4.14) However, the construction of Q,Q,Xn, is rather involved and, for purely practical reasons, we prefer the operators in (4.11) for the resonance energy estimates. Applications of (4.11) and (4.12) are in progress, especially in connection with the search for possible resonance states below the first excitation threshold of the positron-hydrogen scattering system [l I].

REFERENCES 1. R. G. NEWTON,

“Scattering Theory of Waves and Particles,” McGraw-Hill, New York, 1966. AND K. M. WATSON, “Collision Theory,” Chapt. 4 and Appendix C, John Wiley and Sons, New York 1964. L. D. FADDEEV, Z/z. Eksperim. Teor. Fiz. 39 (1960), 1459; English translation, JETP 12 (1961), 1014. S. WEINBERG, Phys. Rev. 133 (1964), B232. H. FFBHBACH, Ann. Phys. 19 (1962), 287. Y. HAHN, Phys. Left. B 30 (1969), 595. Y. HAHN, Phys. Rev. C 1 (1970), 12. J. F. Dmxs AND Y. HAHN, Phys. Rev. A 2 (1970), 1861; A 3 (1971), 310. We refer to these papers for many of the earlier references on the work of the variational bounds formulations and their numerous applications. S. WEINBERG, Phys. Rev. 131 (1963), 440. Y. HAHN, Phys. Rev. 169 (1968), 794; 159 (1967), 91; and 142 (1965), 603. Y. IIAHN, Phys. Rev. A, to appear.

2. M. 3. 4. 5. 6. 7.

8.

9.

10. 11.

L. GOLDBERGER