Families of strictly equivalent potentials for the elastic wavefunction

Nuclear Physics A530 (1991) 303-330 North-Holland

A

I S

S

I LY EQUIVALENT POTENTIALS ELASTIC AV I N C. MAHAUX and R. SARTOR

Institut de Physique BS, Université de Liège, B-4000 Liège l, Belgium

Received 25 February 1991 Abstract : It is shown that there exists a manifold infinite family of single-particle potential operators which all yield the exact elastic component of the many-body scattering wavefunction . These operators may have different analytical properties in the complex energy plane. Correspondingly, the real and imaginary parts of these potentials are connected by dispersion relations in which the dispersion integrals may run over different energy domains. Among these potentials, the mass operator is the most closely related to the phenomenological mean field, mainly because it also yields information on the bound states ofthe (A -1)-nucleon system. Moreover, the mass operator fulfills a simple dispersion relation similar to that used in recent practical applications; the other operators fulfill more complicated dispersion relations. Nuclear matter is discussed in detail . In that case, explicit expressions are derived in the framework of second-order perturbation theory .

1. ntr uction In his pioneering theory of nuclear reactions, Feshbach showed that the elastic component of the scattering wavefunction fulfills a one-body wave equation with a nonlocal energy-dependent potential operator, dubbed the "generalized opticalmodel potential" 1,2 ). This potential is hermitian for energies smaller than the lowest inelastic threshold 8t. A dispersion relation connects its real part at the energy E to a discrete sum over pole terms and to an integral which involves its imaginary part at all energies E'> E, At about the same time, it was shown that the elastic component of the scattering wavefunction also fulfills a one-body wave equation in which the potential is the "mass operator" of many-body theory 3-5). The latter takes into account the identity 6-13) . of nucleons, whose effects are not easy to incorporate in Feshbach's theory As the generalized optical-model potential of Feshbach's theory, the mass operator has an imaginary part for positive energies larger than the lowest inelastic threshold E, ; in addition, however, it also has an imaginary part for negative energies smaller than some value Eh <0. A dispersion relation enables one to express its real part at the energy E in terms of discrete pole contributions and of an integral which involves Eh 14,15) . The part of the dispersion its imaginary part at all energies E'> Et and E' < integral which runs over negative energies E' < Eh reflects the fact that the mass operator has a "left-hand cut" in the complex energy plane. In contrast, the generalized optical-model potential only has a "right-hand cut". The generalized 0375-9474/91/$03 .50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)

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optical-model potential of Feshbach's theory and the mass operator of many-body theory thus have different analytical properties in the complex energy plane and, consequently, differ from one another. e main purpose of the present work is fourfold. Firstly, v=e show that actually there exist an infinite number of one-body operators which all yield the exact elastic component of the wavefunction. Secondly, we study their analytical properties and the dispersion relations that they fulfill. Thirdly, we derive their explicit expressions in the case of nuclear matter, up to second-order perturbation theory. Finally, we discuss their relationship with the phenomenological mean field. ur presentation is as follows. In sect. 2, we write the time-ordered one-body Green function as the sum of a "particle Green function" and of a "hole Green function" and discuss some of their properties . n sect. 3, we review in detail the proof that the mass operator yields the exact elastic component of the scattering wavefunction in order to exhibit that this proof leaves room for introducing arbitrary parameters. e thereby show that an infinite number of single-particle operators exist which all yield the exact elastic component of the scattering wave function . show that these "wavefunction-equivalent" operators have different analytical properties d, relate y, fulfill different dispersion relations. In sect. 4, we illustrate the formal developments of sect. 3 by considering the case of nuclear matter. In particular, we derive explicit expressions for the equivalent operators in the framework of second-order perturbation theory ; we discuss in detail one of these operators because it is very akin to Feshbac 's generalized optical-model potential. Sect. 5 contains a discussion and a summary. cle

hole Green functions

2.1. DEFINITIONS

e set =1 and assume that the hamiltonian is time-independent. Let 0A~ denote the ground-state wavefunction of the A-nucleon system. We write the timeordered one-body Green function in the form G(

'; t - 0 = Gp( k 1 ; t - t') + Gh(, W; t -- t')

(2.1)

where the "particle Green function" Gp and the "hole Green function" Gh are defined as follows Gp (,

; t - t i ) = -io(t -

I

t')(VI oA),ak(t)ak ,(t') VfoA))

Gh( , I; t - tI) = i®(t, - ty

OA)jak'(t ')ak(t)f VfOA))

.

(2 .2a) 2.2b

ere, ®(r) is the step function : ®(T) =1 for T > 0 and ®(T) = 0 for T <0. The quantities ak(t) and ak(t) are annihilation and creation operators in the Heisenberg representation : ak e" . Wt ae-Wt (2.2c) ak( t ) = e k ak(t)=e~x~

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2.2 . SPECTRAL REPRESENTATIONS

Introducing Fourier transforms by means of e.g.

f.

Gp (k, k'; t - t') = (21r)-' one gets (77 -> +0) Gp(lk, k' ; w) _ (

0)Jak

Gh(k, k' ; w) _ (

0p(A)1 ak'

dw e -"(`") Gp(k, k'; w),

w - [H

~~A)

-1 (A)] . ak'l WôA) ) +tyl W0 1

(,~)

w _ C ~o

akl _ H] t1Î

(2.3)

(2Aa)

A) 0 )

2.4b

where the ground-state energy of the A-nucleon system. We introduce the eigenstates of the hamiltonians for (A + 1) and (A -1) nucleons. In the case of bound states, we write H(Ati) qf(Atl) = W(Atl) Vf(A&1) x A A ,

(2 .5a)

where A is a numbering index. In the case of scattering states, we write H(Atl)wc'(A$1)

c'(Atl)

2.5b

where the upper index c' specifies the boundary condition. This index is necessary because, for a. given energy 9, eq. (2 .5b) has as many linearly independent solutions as there exist different modes of decay of the (A :± 1)-nucleon system at that energy. In other words, the number of linearly independent scattering eigenstates with energy W is equal to the number of open channels. We shall adopt the same notation as in ref. 16) namely: the upper index c' means that there exists an incoming wave only in channel c' (and outgoing waves in all the open channels). The states are normalized in the following way f ~At1)l (V

AAtl)) =

'A' ,

(

(Atl)1,,~ çAtl)) _

s , g( C C

W _ V) .

(2 .Sc)

Spectral representations of the particle and hole Green functions are obtained by inserting complete sets of eigenstates in eqs. (2 .4a) and (2 .4b) : _

(Wolakl

+ Gh(k, k'; w) =Y a

~A+l

))(

_

c'(A+I)

d W'

( +1)lak,l A

ôA))

('Polakl9f9(A

A+1)lak,l +1) )(IW~

0A))

0

f-'_«A+1)

2.6a

âA-1) )C AA-1) lakl~OA)) _ (A-1) w_C (A) 0 ~a -ii

OA) lak'l

/ ~ÔA) f-'1_c'(A-1)

ç A_

1) %(W~(A-1) lakiV r r w( A

Iak'1IY~

0A) )

2.6b

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ere, 3-c ( At, ) denotes the minimum energy of the channel c'(A :E 1). The infinitesimals --t+q in the denominators specify how the integration contour should be drawn ref. 17 ) . when performing the Fourier transform (23), see e.g. a

2.3. ANALYTICAL PROPERTIES AN

re

OVERLAP FUNCTIONS

tion (21a) shows that the particle Green function has isolated poles on the xis (coming from the lower-half plane) at the "single-particle type energies" (A+1) (A+1) _ W(A) (23a) A

A

0

Likewise, t e hole green function has poles on the real w-axis (coming from the half- lane) at the "single-particle type energies" (23b)

{A-1D - Qp {AD _ {A-1, 1 - Wn Wà

e expression "single-particle type energies" refers to the fact that, in the indepen-particle model, they are identical to the energies associated with the bound ell-model orbits . They = 51 negative. The Fermi energy is defined to lie half-way tween the single-particle type energies associated with the ground states of the + I)- and (A - l)-nucleon systems: ir

C~All) (A-1) EF " : 2L "0 -E 0

(23c)

I-

t a single-particle type pole, the residue of the particle or of the hole Green function is the pmduct of so-called "overlap functions" defined as follows: {A+1)) jp {AD I ak J s r..(A+ 1)) (kjX(A-1))=(jp(A-1)j 11p(A) . (2.8a) A

A

A

ak

0

e quantitty jV (,A ")) plays an important role in the description ofdirect one-nucleon s ping reactions and the quantity JV(A-1)) in the description of direct one-nucleon A icku or knockout The norm y(Aj:1) = ( X A

A(A=1:1)jX(AA_-!:1))

(21b)

of the overlap function is the "spectroscopic factor" of the bound state W(A=A One can likewise associate overlap functions with scattering eigenstates of the (A ± I)-nucleon systems. We shall henceforth reserve the label "c" for the elastic channel, in which a nucleon is impinging on the ground state of the A-nucleon system. By definition, the corresponding overlap function (klXc,) = ( 1P 0(A )jak j jpc(A+1)) (25) 9

is the momentum representation of the "elastic part of the scattering wavefunction". It is that quantity which can be calculated from the generalized optical-model potential in Feshbach's theory. By analogy with eqs. (2.7a), (2.7b), it is convenient to label the scattering states of the (A+ I)-nucleon system by the positive "singleparticle type energy" E = W - W0(A) (110)

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and those of the (A -1)-nucleon system by the negative "single-particle type energy" E =

W

ôAj -

.

(2.11)

We shall use the same notation for quantities expressed either as functions of 9 or of E ; for instance, we shall indifferently write V É or X' . The contribution of the elastic scattering channel to the second term on the right-hand side of eq. (2 .6a) can then be written in the form

f "" O

dEr

(kjV E')(XE'I w iq - E'+

(2.12)

This shows that the particle Green function has a "right-hand" branch cut which runs below the real w-axis from 0 to oo. Likewise, the hole Green function has a "left-hand" branch cut which runs above the real w-axis, from -®o up to the maximum value of [ w(Aj - J..(A _jj; the latter maximum is determined by the threshold energy for particle emission from the (A -1)-nucleon system. 2.4. MASS OPERATOR

The mass operator (or "self-energy") N' is defined by the following relation .N'= (G o ) - ' - G-1 ,

(2.l3)

where Go is the time-ordered Green function associated with the independentparticle model in which the nucleons only feel an external single-particle potential U. The latter is introduced in order to localize the A-nucleon system 6 '' 8 ); here, we require that it contains A bound single-particle states with energies smaller than the Fermi energy EF . T.he definition (2.13) yields the "Dyson equation" G(k W; t - t') = G°(k, W; xX(k,,

t - t') +

k2 ; t2

1111

d 3 kl d 3 k2 di, dt2 G°(k, k, ;

- tj)G(k29 V; t l -

0-

t-

t2)

(2.l4)

3. Equivalent single-particle potentials In the present section, we construct families of single-particle operators which all yield the same elastic wavefunction XE when introduced as potential operators in a Schr6dinger equation. It has been proved that the mass operator fulfills this property 6) . In the literature, some steps of the proof are usually omitted. Here, we analyze it in detail in order to spot steps where arbitrary quantities can be introduced, thereby leading to families of equivalent operators. Our discussion will also encompass bound states of the (At 1)-nucleon systems.

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3.1.

SS OPERATOR AS SINGLE-PARTICLE FIELD

e follow the outline given by Koltun and Eisenberg '9). The scattering state at time t is related to the state at time t = 0 by I T(t)) = e" I W(0))

.

e the momentum of the incident nucleon and sk. = (k')'/2its kinetic energy. itial state is given by -(ffi~ = UM lim e -4it , (31) f_-00

I -

K

x

e_-Co

U

9

which yiel s I

- .(O))=

e

lim e' e-"

(AD

e"-00

. a tk.jVp(A)) o

(33)

entum representation of the elastic part of the scattering wavefunction at t is eaual to 0 (A) 1ak e iEo(A) t

y using e s. (3.1) and (3.3), one finds successively lim

k

e-00 =i

lim e -

(3.4a)

WW) (A)

0

k "I G.(

=i lim e -'r-l'

r'-0c)

(3-4b) (3 .4c) (3.4d)

e last eau ity derives from the fact that lim e.-00 Gh(

W; t - 0 = 0

(3.5)

because of the step factor 0(t'-- t) on the right-hand side of eq. (2.2b). ne wants to derive a Lippmann-Schwinger equation for (kl%k,(t)). By multiplying the Dyson equation (114) by exp (-Qt'), taking the limit t-> -oo and using eq. (3 .4d), one finds JA t» = (kle2,(t» +

d3 k1 kl d-3 k2 dt, dt2 GO(k, kl ; t - t2)

XX(ki , k2 ; t2 - tj)(

(31a)

where 1v0k ,(0)=i lim e_'60' GO(k, W; t - t') e"-00

(31b)

C. Mahaux and R. Sartor / Strictly equivalent elastic potentials

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is the momentum representation of the scattering wavefunction associated with the external potential U. Let s exhibit its time-dependence. One has G°(k, W; t - t') = Gp(k, '; t - t') + GO(k, '; t - t) ,

(3.7a)

. - EF) e-`ea (`- " ) ua()u.*('), Go( k '; t - t') = -t®(t - t') 1: ®(e

(3 .7b)

G°( k' ; t - t') = i®(t'- t) Y, ®(EF - ea) e -'e°(`-") u«()ul(') .

(3.7c)

a

Here, ua (k) = (k I ua) and ea are the eigenfunctions and eigenvalues of the sin e particle hamiltonian ho = -® 2/2m + U. (3.8a) n eq. (4.7b), the sum over a implicitly includes an integration over scattering states, for which ea = 6k':-- 6k' . Eq. (3.6b) yields . -EF) e-`e-' ua (k)ul(k') lim e'(ee- Ek . W . (3.8b) (kjVk'(t))=Y, ®(e Since the limit vanishes unless ea = Ek. , one has e-kWt . (kjVo'(t)) = (kluk')

(3.8c)

The Fourier transform of eq. (3.6a) then yields (klX&'(ek')) = (kluk)+ X

ff

d3kl d3 k2 G°( ki ; Ek)

X(kl, k2 ; E0(k2IXk'(Ek')) "

(3.9

This is the Lippmann-Schwinger equation for the scattering of a nucleon by the potential .N'+ U provided that the quantity G° can be replaced by the usual Green function g° fur the sca«ring of one nucleon by the potential U, namely by g°(k, kl ; Ek') =Ea

ua(k)ua*c(ki) . Ek'_ea +i7l

(3.10a)

This is in fact the case. Indeed, since Ek' > 0 and EF < 0, one can replace -iq by +i,q in the last term inside the square brackets of the Lehmann representation x

®(e,,, - EF) + 0(EF- ea) Ek' - e. + 11'% Ek' - ea ~71

.

(3 .10b)

This implies that (3 for Ek, > 0 . G°( k kl ; COW) = g°( k l ; Ek')

.10c)

'The proof that the mass operator yields the elastic part ofthe scattering wavefunction is thereby completed.

C

310 3 .2 .

FA

ahaux and

. Sartor / Strictly equivalent

elastic potentials

ILIES OF ELASTIC WAVEFUNCTION-EQUIVALENT OPERATORS

e tree main steps of the preceding proof are the following. e (i) elastic wavefunction is related by eq. (3.4d) to the time-ordered one-body reen wavefunction . (ii) e potential scattering wavefunction is related by eq. (3 .6b) to the timeor ere reen function associated with the independent-particle hamiltonian. (iii) or > , the time-ordered Green function °(co) is identical to the Green function "(w) of the one-body problem. Steps (i) an (ii) use the property that the hole Green function plays no role in the limit t'-> - , see e.g. the equivalence between eqs. (3 .4c) and (3 .4d). In the statements (i) and (ii) above, one could therefore have replaced "the time-ordered reen function" by "'the particle Green action plus any amount of the hole Green f cti '. n the present section, we exploit this freedom in order to construct families o single-particle operators which all yield the exact elastic wavefunction. et s introduce the quantities t - t'; a) = ,,( ®; t - t®)+ aGh(k, '; t - t') , (3 - t ; b) = O( °; t - t') + bGh( '; t - t , 3

.11a) .11b

where and b are arbitrary. Below, we take a and b to be independent of ' and t - t' i order to be able to derive explicit results. As argued above, eqs. (3 .4d) and (3.6) remain valid if one replaces G by G(a) and Go by G°(b) on their right-hand si es. e now define a '"modified mass operator" X(a, b, c) by means of the following yson-t e equation - t'; a) = Go(

'; t - t®; b) +

d3k, d3k2 dt, dt2 Go(k, k, ; t - t2)

XX(,, 2 ; t2 - t, ; a, b, c) G(k2 , k'; t, - t'; c) , (3

.12a)

where a, , and c are arbit ._°ary. y closely following the proof of sect. 3.1, one readily finds that the elastic-scattering wavefunction is an eigenstate of the singlearticle hamiltonia C h (E ; a, b, c) = ho +X(E ; a, b, c)

(3.12b)

or all values of a, and c. Note that the first factor in the integrand of the modified yson equation (3.12a) is the same as in the usual Dyson equation (2 .13b) . This is necessary in order to carry out the step (iii) of the proof. Henceforth, we shall call X= X(E; 1, 1, 1) the "mass operator proper" and the quantities X(E ; a, b, c) modified mass operators" when a, b or c 01 . e now derive an equation for the modified mass operators. Let us adopt the representation { ua } in which GP and Go are diagonal. e Fourier transform of the modified Dyson equation then reads 66

G«ß(w ; a) = S~ßG«a(w ; b)+Y, Gâa(w),Y«,(w ; a, b, c)Gyß(w ; c) , (3 Y

.13a)

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which gives

â G,,,,ß(w ; a)[G-'(w ; c)]ß, = G°f.(w ; b)[G-'(w; c)]~,.+

a, b, c) (3.13b)

provided that G(w; c) has an inverse. This is our sole assumption . t appears quite unrestrictive; for instance, it holds true in the case of nuclear matter studied in sect. 4. Since [GO,,,,,,, (w)]-' = w - e~, eq. (3.13b) yields the following expression for the modified mass operator a, b, c)

e jj j: G«ß(w; a)[G- '(w; c)]p, - G'.(w; b)[G-'(w; c)],, ß (3.14)

We consider two illustrative examples. (1) a=cO0 and b=1. Then,

a,1, a) = (w - ej & y - [G -'(w; a)],, ; this quantity has the same analytical properties as the mass operator proper: it has a left-hand cut as well as a right-hand cut. (2) a = c = 0 and b=1 . Then, X,(w ; 0, 1, 0) = (w - e )gy -[GP (3.15b) ; this quantity has the same analytical properties as the particle Green functien : it has a right-hand cut but no left-hand cut. It will be studied in detail in sect. 4.3, in the case of nuclear matter. 3.3. BOUND STATES

The mass operator proper enables one to calculate the energies and overlap functions of the gound state of the (A+ 1)- and (A -1)-nucleon systems in addition to the elastic-scattering wavefunction "). We now discuss to what extent the modified mass operators introduced in sect. 3.2 also have this property . We first derive the one-body wave equation fulfilled by G(w ; a). The modified Dyson equation (3.13a) yields [GO(w)] -'G(w ; a) -X(w ; a, b, c)G(w ; c) = [GO(w ]-'G'(w ; b) .

In the {ua} representation, the right-hand side of the relation reads

s,ß [®(e,,, -EF)+b®(EF-ea)]=[D(b)]aß,

(3.16a) (3.16b)

which defines the matrix D(b); the latter reduces to the unit matrix if b =1 . Eq. (3 .16x) can be written in the form (3 .16c) (w - ho) G(w ; a) -X(w ; a, b, c) G(w ; c) = D(b) , where ho is the single-particle hamiltonian of eq. (3 .8a) .

C.

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e enote y  the normalized overlap functions. Using eqs. (2.6a), (2.6b), ( .8a) an ( .8b), one finds that (w ; ) _

(a)

I

A>t

AI

w-EA tiri

+F(w; a)

(3.17a)

the vicinity of a bon single-particle energy  (of the type E A( A+I) or E(A- ')). (w ; ) is regular in e vicinity of A and we introduced the ere, otation i A-- (All) A , (3.17b) E( AA = if EA -') . e

g

{w ; a) about w =

(w; )= F( E, ; )+ (w - A) ci

F(' ) +O(w - EA)2 T

(3.17c)

e s. (3 .17x) and (3.17 ) in eq. (3.16c), one obtains w-EA 4

'c)-(,- o , ( (c) .,

(3 .18a)

( EA ;

, c)I A)(AI+O(w - EA) .

(3.18b)

e right-hand side of eq. (3.18a) is regular at w = E,,, the numerator of the e must vanish. This yields (E(A+l) A+l) ) = E (A-rl) A+l) fl+ . ; a, b, C)jjirA (3.19) ) Iil ue of a, an c or a, c ixs 0, one has C ; ( À i(A (EÂA-1) o+ , b, C)jjX((1A -')) - E AA-1) f A-1) ) . (3 .20) a (3.19) sows that one can calculate the energies and the normalized overlap functions of the bound states of the (A+ 1)-nucleon system from the modified mass operators, for any value of a, b and c. Eq. (3.20) shows that this also holds true for e bound states of the (A -1)-nucleon system provided that the parameters a and c both differ from zero. e to to the spectroscopic factors, i .e. to the norm of the overlap functions. Eq. (3 .18b) yields d YAW YA(a) 1YA(a) l~A~ dEA

(EA, a, b, c) 1 XA)

+(jAI( A - ho)F( EA ; a) - X(EA ; a, b, c)F ( EA ; c) 1 jA) (311) =(jAO D(b) J A) .

C. Mahaux and R Sartor / Strictly equivalent elastic potentials

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In the case a = c, the operator .N(w ; a, b, a) is symmetric in coordinate space and the following equation ca be derived from eq. (3 .16a): (,jAj(ho+X(E ; a, b, a) - E } =0 .

(3.22a)

Hence, the content of the curly brackets on the left-hand side of eq. (3.21) vanishes when a = c; one has then ~x(a)

=

(xajD(b)jX,) 1-(x~l(d/dEx)~(E~ ; a, b, a)Ixa>

(3.22b)

In the case b =1, Iß(1) is the unit matrix and `pa(a)= 1-(XAI

d X(Ex ; a,1,a)IXa ) dE

(3.22c)

The latter expression is formally identical to the one which had previously been derived in the case of the mass operator proper, see e.g. refs. 2' M). It is related 22) to the property that the operator X(E; a, l, a) is symmetric in coordinate representation. Eq. (3 .22c) is valid for any value of a; however, one should keep in mind that, in the case a = 0, eq. (3.22c) only applies to the bound states of the (A + 1)-nucleon system and not to those of the (A -1)-nucleon system. The derivations given in sect. 3.2 and 3.3 extend earlier works which dealt with the mass operator proper. They suffer from the same weakness as these earlier proofs, namely from the fact that recoil effects are not taken into account . This problem was emphasized by Bel1 6). It has not yet been satisfacto rily eradicated 2311. Above, it is made apparent for instance by the fact that the single-particle wave equation (3 .19) or (3 .20) for the overlap functions involves the nucleon mass rather than the reduced mass as one would intuitively expect . This is a consequence of the fact that we tied the A-nucleon ground state to a fixed point in space. One could try to introduce an operator a~(r) which creates a nucleon at the location r relative to the centre-of-mass of the (A+ 1)-nucleon system 24) but this type of formalism has not yet been worked out. There exist "conserving approximations" to the Green function and to the mass operator in which these two quantities are related in such a way that conservation of the total momentum is ensured, see e.g. refs. 25-28 ) . To the best of our knowledge, these approximation schemes have not yet been used to derive a one-body Schrôdinger equation for the elastic part of the scattering wavefunction or for the overlap functions of the bound states of the (A :t 1) nucleon systems. Nuclear matter The simple case of nuclear matter is instructive because it enables one to explicitly construct the modified mass operators and, thereby, to exhibit differences and analogies with the mass operator proper. It is also useful in order to establish contact

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with eshbach's generalized optical-model potential. The main simplification which occ rs in nuclear matter is that, because of translational and rotational invariance, all uantities are diagonal in k and, besides w, only depend upon k = Jkl. Henceforth, we no longer use bold face symbols for vectors. Below, 1k) is a plane wave-state with momentum k and lk i) a product of two plane-wave states, whose antisymrized product is called k, )t . Momenta smaller than the Fermi momentum kF re denoted by h l and h 2 , and momenta larger than kF by p, and p2 ; k can be either rger or smaller than kF - We write the hamiltonian in the form H = h® + (v - U), with ho = -(2 m)W'V2 + U( k') where U(k) is an "auxiliary" potential that we choose to be a continuous function of k Here, we use the same notation for a sum of 0 ratorn as for one of these operators. The eigenvalues of ho are e(k) = k2 12m + U(k) . (11) ASS

PERATIOR PROPER

1. L "itions and properties. We only recall some results and properties, see e.g. ref. ') . The time-ordered one-body Green function is related to the mass operator

proper by

52a) (k; w) = w - "k) - X-Yk; Q .

In the complex w-plane, G(k; Q and Mk; w) have a left-hand cut which runs above the real axis for w smaller than the Fermi energy EF and a right-hand cut which runs below the real axis for o) > EF . For (o real, we write X(k ; w)

ne has

Y(k ; Q + iW(k ; w)

for w > EF

Y(k ; w) - i7r(k ; w)

for (o < EF

(4.2b)

(k ; w) --- 0. The following dispersion relation holds n k ; Q = n k ; coo) + 71'

1 .

dio' W(k ; w)

1 lolf-tu

1 W - Wo

1,

(4.2c)

where the "subtraction energy" cvo is arbitrary and wherits the integral is a principal value. 4.1.2. Second-order perturbation theory. In second-ord..-r perturbation theory, the quantity .N'( k; w) is the sum of the diagrams shown in fig. i . Their algebraic expressions are the following Xu (k; w) = - U(k) , X,(k; w) = 1 (k, hl 1 vIk, hl ),, û» = -12 X2 ( _ ) (

E

k ; w) = -12 E

J(k, hilVIP29 pl ) tl2 w + e(h l ) - e(pl ) - e(P2) + i7j 5 J (k,

pi lvlh2, h J)J12

w + e(pj - e(h l ) -- e(h2) - i 7l *

(13a) (13b)

(130 (?3d)

C. Mahaux and R Sartor / Strictly equivalent elastic potentials

315

k ; co

NI

k ;w

k ;w

Fig . 1 . Diagrams which represent the first and second-order contributions to the crass operator. The cross corresponds to a "U-insertion" . Horizontal dashed lines represent the nucleon-nucleon interaction v. The solid lines with an upward pointing arrow are associated with particle states (p; > kF) and those with downward pointing arrows with hole states (h; < kF).

Here and below, the summations run over all the numbered momenta. The contribution Xu compensates the external potential U(k) contained in h o . The first order term X, is the Hartree-Fock contribution; it is real and independent of w. The diagram X2(,) involves a summation over two-particle-on-e-hole intermediate states ; in the complex (o-plane, it has a right-hand cut which runs below the real axis. The diagram X2(_) involves a summation over two-hole-one-particle intermediate states; it has a left-hand cut which runs above the real axis. For k _ 3 kF [ref. 3°)] these two cuts meet at w = eF = e(kF), i.e. at the first-order approximatioäi to the Fermi energy. As in eq. (4.2b), we write lim X2(t) (k ; w) = V2(t)(k ; w) t iIV2(t) (k ; w) ,

;;-+0

(4.4a)

one has IV2(t)(k; (o) ,0. The following dispersion relations hold V2(+)(k ; to) = ir-'

°° dw,

IV2(+)(k ; w' ) W' -w

eF

f eF _1

'

V2(-)(k ; w') .

(4.4b)

Setting 'V2(k ; w) = 'V2(+)(k ; w ) + V2( - )(k; w) W2(k ; w) = IV2(+)(k ; w) + 'W2(_)(k ; w),

(4 .4c)

316

C,

. Sartor / Strictly equivalent elastic potentials

ahaux and

e as W

2(k ;

w) = ar

-

r ac J _CO

dw

2(k; W)

(4.4d)

'-W

e henceforth adopt the artree- oc choice (k) = .N',(k) for the auxiliary tential. Most quantities of physical interest are given by "on-shell" values i.e., in sewn order, y values obtained by setting w = e(k) in the relevant expressions. et s for instance consider the energy momentum relation k

_

2M

+ XI (k) + ~r2 (k ;

).

(4.5)

n first order, it reduces to (4.6) elation ( .) defines the function e(k) of eq. (4.1); it can also be used to define a function k(e). p to second order, the potential energy of a nucleon with momentum k is given by 'V(k) =X,(k)+

,(k, e(k)) .

4.7a

ourier transform over k, is yields a energy-independent nonlocal potential operator r"(I - 'I). Correspondingly, up to second order the potential energy of a cleon with energy is given by (4 (E) = ,(k(E))+ 2(k(E) ; E) .

.7b)

is is the "local equivalent" of the nonlocal potential Y(k). Thus, remember that the k-dependence of X(k ; w) corresponds to a nonlocality in the spatial coordinates an its w-dependence to a "true" energy dependence i.e. to a nonlocality in time. ecause of correlations, the momentum distribution of the ground state differs from that of a Fermi gas. In second order, the occupation probability of a plane-wave state with momentum k is given by n 2 ( < kF) =1 + =I

-2

n2 (k > kF) = =2

a 'V2(+)(k ;

109W

w)

lto=e(k)

I(k, h,IVIP2, P I )J 2 , [e(k) + e(h l ) - e(P l ) - e(P2))2 a V2( - ) ( k; w) aw

to= e(k) 12

I(k, Pi IVIh2, hl), [e(k) + e(Pj - e(hj - e(h2)12

e slope of n 2 (k) is infinite at k = k, :E 0.

(4.8a)

.

(4.8b)

C. Mahaux and R. Sartor / Strictly equivalent elastic potentials

317

In nuclear matter, the "quasiparticle peak" is the analog of the peak of the distribution of spectroscopic factors in the finite case. The "quasiparticle strength" measures the amount of strength contained in a lorentzian approximation to the quasiparticle peak. In second order, it can be approximated by Z2(k) = 1 +

a

w

Ci

V2(k; w)

m=e(k)

(4.9)

.

For k close to kF , the width of the quasiparticle peak approaches zero and Z2(k) plays the same role as the spectroscopic factor of a single-particle excitation . 4.2. MODIFIED MASS OPERATORS

In the case of nuclear matter, eq. (3.14) becomes X(k ; w ; a, b, c) = [w - e(k)][G(k ; w ; a)G- '(k; w ; c) - G ®(k; w; b)G - '(k; w; c)] . (4.10)

The algebraic calculation of the right-hand side is rather tedious; it is described in the appendix. It yields the following expressions, in which the upper index (2) recalls that they are valid up to second order: .y(2)(k> kF ; w ; a, b, c)=X,(k)- -U(k) +X2(+)(k; w)+ir2(_)(k; e(k)) J(h2 , h,l vl k, P j) t l 2 +i[w e(k)~ [e(k) +e(P, )-e(hj-e(h2)]2 jJ, x L aw+e(P, )-e(h,)-e(h2)-i l? ./V(2)(k < kF ; w ; a, b, c) = a - b [w - e(k)] c + bc [XI(k) - U(k) +X2(+)(k ; e(k))+X2(_)(k ; w),

12 j(P2, P , I vIk, hl), -![w e(k) ] E +[e(k) +e(h,)-e(P i )-e(P2)]2 x

c-a+b w-e(k) __bl c2 w+e(hl)-e(P,)-e(P2)+cj

(4.l2)

Incidentally, the quantities a, b and c could be allowed to depend upon k; this corresponds to the remark made below eq. (3 .llb) in the finite case. It can be checked that, in the case a = b = c =1, one retrieves the expressions given in sect. 4.1 for the mass operator proper, as one should.

C

318

ahaux and R. Sartor / Strictly equivalent elastic potentials

e quantity ( `(k ; w ; , b, c) is discontinuous at k = kF. n the domain k > kF, the ass operator proper and the modified mass operator have the same on-shell . Furthermore, the quasiparticle strength is values (obtained by setting w = e()') ive y (k> kF) =1 +

a K(k; w;

b, c)

lco=e(k)

(4.l3)

for any value of the parameters a, , c .0 the domain k < kF, the modified ass operator 2)(k < kF ; w ; a, b, c) depends u on all tree parameters a, b and c It does not yield the same energy-momentum relation ( .5) as the mass operator proper, except in the case a = c for which the relation (k) +

2 (k < F; (o; a, b, a)

(4.14a)

is ivalent to eq. (4.5), up to second order. This is in keeping with eq. (3.19) in the fi ite case. Note, however, that ,j ( -''(k < kF ; e(k); b, a) vE X,(k)+ 2(+)(k; e(k))+X2(-)(k; e(k)) . (4.14b) 4.3. PARTICLE

ASS OPERATOR

Definition as analytical properties. e have seen insect. 3 that the particle part o1 the green function fully determines the elastic component of the scattering wavefunction as well as the energies, spectroscopic factors and overlap functions of the bound states of the (A+ 1)-nucleon system. If one is not interested in the ( -)-nucleon system, it may thus appear natural to introduce a single-particle I otential operator that only yields the particle part of the Green function . This is a n to the approach adopted in Feshbach's generalized optical-model theory; this is why we now discuss in detail the properties of the "particle mass operator" defined as follows: .3.1.

Xp(k; w) = X(k ; w ; 0, 1, 0) .

(4.16)

qs. (3 .12a) and (4.10) yield p(k; w) = GO(k; w) + GO(k; w)Xp(k;

w)Gp(k ; w)

(4.17a)

which amounts to Xp(k ; w) = &, - e(k) - Gp1(k; w) ,

(4.17b)

Gp(k ; w) = ( w - e(k) - Xp(k; w)} -1 .

(4.17c)

C. Mahaux and R Sartor / Strictly equivalent elastic potentials

319

The particle-mass operator has the same analytical properties as Gp(k; w) in the complex w-plane. It has o left-hand cut but has a right-hand cut which runs below the real axis, from the Fermi energy to infinity. In order to derive dispersion relations from this property, one must know the asymptotic behaviour of .N'p(k; w) for large w. We recall that G(k; w) = Gh (k; w) + Gp (k; a) , (4

.18)

where Gh and Gp are the hole and particle parts ofthe time-ordered Green function. It is known that G(k ; w) ---11w for large w, see e.g. ref. 21). In a similar way, one can show that Gp (k; w) -~-

1- n(k) , w

(4.19a)

where n (k) is the probability that the plane-wave state with momentum k be occupied in the correlated ground state of nuclear matter. Eqs. (4.17c) and (4.19a) imply the following asymptotic behaviour for the particle mass operator Xp(k; w) --- - n(k) w . 1-n(k)

(4.19b)

Xp (k; w) = 'Vp(k; w) + t7Vp (k; w) .

(4.190

For w, real, we set The following dispersion relation holds:

n(k) 'Vp(k ; w) = Vp(k; wo) - 1- n(k) + v-1

IEF

(w - wo)

dw' 9Wp(k; w')

,

1

w -w

-

w,

1 - wo

(4.19d)

This provides the following expression for the momentum distribution: 1-n(k)

aw

fECF

w, - (f)

I'

(4.19e)

The second term on the right-hand side of eq. (4.19d) did not appear in the dispersion relation originally derived from Feshbach's theory of nuclear reactions 1'31-33) ; it was thus not included in the corresponding applications 34,35) . Its absence in these early works refie--ts the fact that they did not fully include erects stemming from target correlations and/or from the identity between the scattered nucleon and the nucleons contained in the target. Its appearance is the price that one has to pay for dealing with a particle mass operator that has no left-hand cut. However, it would be incorrect to identify the second term on the right-hand side of eq. (4.19d) with the contribution of the left-hand cut in eq. (4 .2c). This can be shown as follows.

C Afahaux and R. Sartor / Strictly equivalent elastic potentials

320

since Q( k; w) is analytic in the vicinity of the real positive axis, G( k; W) and p( k; w) have the same discontinuity across the right-hand cut. This discontinuity yields the "particle part of the spectral function", namely :

i

SP(k ; (o > EF)= - [G(k ; w + iq) - G(k; w - i77)] 21r I

= - [ p( k; w + i7j) zV

p( k; w - i7j)] .

(4.20a)

is gives the followin identity which holds for all to > EF

VV

i(k; w) _ k VP (k+ [Wp(k; (0)]2 2m (120b) 9

U

owever, 'Y,,(k; 0 and (k ; w) or k; V and Qk ; w) an different. Or instance, eqs. (4.17b) and (4.20a) give, for w :-~p(k; w) - Qk ; w) = Im G(k; w)IlGp(k; or) JG(k; (4.21a)

or large positive

w eqs. (4.19b) and (4.20b) yield

(k ; w) - [1- n(k)]'7Vp(k ; w) .

(121b)

qs. (4.19d) and (4.21b) imply that, for w > EF,, the derivatives of gyp( k; w) and of

'P (k; w) with respect to k are both infinite at k = kFt 0- They also imply that 'P(k ; w) and R'p(k; w) are discontinuous at k = kF . They indicate that the nonlocality of the particle mass operator is more complicated than that of the mass operator proper. They severely hinder any practical application of the dispersion relation (4.19d). Moreover, the second derivative of gyp( k; w) with respect to w is infinite at w=EF +O . None of these complications exist in the case of the mass operator proper. 4.3.2 Second-order approximations. In the domain k < kF,, the quantity xP(2) is given by eq. (4.17b), with G(2 P ~(k < kF ; = 12 cE

l(P2,,pllvlk, k2>, 12 [w + e(h,) - e(y) - e(P2) + in][e(k) + e(hj ) - e(p,) - e(P2)]2 *

(4 .22a)

For large w, it has the asymptotic behaviour X(2) (k < kF ; (0 ) - P

n2 (k) n2(k) to 1-

(122b)

where n2(k
C. Mahaux and R. Sartor / Strictly equivalent elastic potentials

321

has only little physical interest. In particular, the quantity Xp)(k < k F ; e(k)) cannot be interpreted as the potential energy of a nucleon with momentum k Indeed, X()(k ; e(k)) can be ascribed that meaning only for e(k) > eF, which precludes the domain k < kF for any acceptable choice of U(k). We thus focus on the domain k > kF, in which eq. (4.11) yields X(2)(k > kF ; w) = XI(k) - U(k) +X2(+)(k ; w) + 'V2(-)(k ;

e(k)) + [w-e(k)]

ai

aw 'V2(-)(k; w) lco=e(k)

(4.23)

In second order, the quasiparticle strength Z2 (k) can be written as in eq. (4.9), namely Z2 (k > kF) = 1 + a aw

Vp)(k ;

w)

lcj=e(k)

(4.24)

It is instructive to compare eq. (4.23) with the second-order approximation to the mass operator proper, namely with (4 X(2)(k; w) = X,(k) - U(k) +X2(+)(k; w)+ .V'z(-)(k; w) .

.25)

Eqs. (4.23) and (4.25) show that, for k > kF and up to second order, the real part of the particle mass operator is obtained from that of the mass operator proper by expanding the correlation graph V2(-)(k; w) about w = e(k) and by retaining only the first two terms of this expansion. Note that eq. (4.25) is valid for all values of k, while eq. (4.23) only holds for k > kF . This reflects the general feature that the mass operator proper is a continuous function of k, in contrast to the particle mass operator which has a discontinuity at k = kF . In keeping with the general results derived in the previous section, Ar(p2)(k > kF ; w) only has a right-hand cut. It fulfills the following subtracted dispersion relation Vp2)(k>

kF ; w) = Vp) (k; wo) -(w-wo)n2(k) f Fdw' e

w - w w - wo

where n2(k) is the second-order approximation to the momentum distribution, see eq. (4.8b). In the framework of second-order perturbation theory, no subtraction need be performed and one has

cVp)(k> kF ; w) + U(k) =A(2)(k)-wn2(k)+ir-'

°°

dw' cW P(2),(k ; w') w -w eF

(4.27a)

where A(2)(k) = XI(k) + e(k)n2(k) +V2( -)( k ; e(k)) .

(4 .27b)

322

C %haux and R. Sartor / Strictly equivalent elastic potentials

he quantities A(2Q and W(k -I w) both have an infinite derivative with respect P P t k at k=kF+O- It appears doubtful that they can be approximated by simple ex cessions . As noted previously, these complications persist at all orders of pert rbation theory. They are in striking contrast with the simple nature ofthe dispersion i7clation fulfilled by the mass operator proper which reads, in second order, r2

jk ; A + U(k) = S(k) +

dw'

Mk; W)

(128)

e available calculations of the Hemcee-Fock contribution Xl (k) show that its de ndence upon k can be approximated by simple expressions. Moreover, eq. 28) holds for k < kF as well as for k > kF - Because of these main reasons, the dis rsion relation fulfilled by the mass operator proper lends itself to practical ap lications '5), in contrast to the one fulfilled by the particle mass operator. In evious applications 35) of the latter, one had in effect neglected both n2(k) = 0 I-Ak ; e(k)); this was unjustified, especially for k close to kF4.4. GENERALIZED OPTICAL-MODEL POTENTIAL

In the dispersion relation (4.2c), the domain of integration includes energies to' < eF which correspond to excited states ofthe (A - I)-nucleon system. This feature is in striking contrast with the dispersion relation which appears in the nonntisymmetrized version of Feshbach's theory of the generalized optical-model otential '). In second order, it originates from the contribution V2( -)(k; w) of fig. 1. or k > k F, this contribution has the following physical interpretation. Because of ground-state correlations, two nucleons with momenta h,, h 2 < kF can be excited above the Fermi sea, in states with momenta pl , p2 > kF . Because of the Pauli principle, this hinders the possibility of adding a nucleon with momentum k > kF on top of the correlated ground state. Hence, the contribution 'V2( -)(k > kF ; 0 ) escribes an "interference with a ground-state correlation's 36) . Below, we sketch that a similar contribution appears in the antisymmetrized version of Feschbach's theory 12,13,37). e limit ourselves to the real part of the generalized optical-model potential since the on-shell value X2( - )(k ; e(k)) is real in the domain k > kF which is of interest here. Let l A denote the unit operator in the Hilbert space spanned by the antisymmetrized eigenstates IA of the hamiltonian of the A-nucleon systeriti : 1A = E JAXAI (129a) e unit o erator 1A,1 can be written in the form

1 E a t lAa,, . (129b) 1A+l = A+ 1 ,, ' where the index a refers to the single-particle eigenstates of the model hamiltonian ho . We denote the scattering eigenstate of H for the (A+ I)-nucleon system by

C. Nahaux and R. Sartor / Strictly equivalent elastic potentials

323

I ~) = I ~` A+' ~) and introduce the one-body state vector IXV)) = ß (A lap I V)1 .8) .

(4 .30x)

The elastic single-particle component of the scattering wavefunction is lX(0) IX~) = ). In the identity (0I[aa, H]I V) = (E - ~0A))
we set E = V - Wâ) (see eq. (2 .10)) and readily find (ce lholxw)+

p,à

IV

.) ,

( A+l) - '(OIJaaßIA)(AlaOIW)=E(Ola~l9) .

(4.30b)

(4 .30c)

with J,, = [aa , v] . Eq. (4.30c) forms an infinite set of coupled equations for the quantities (A lap IV). In order to derive a generalized optical-model potential, one should "eliminate" the quantities associated with A 0 0 [ref. '3 )]. For our present limited purpose, it is sufficient to only retain the contribution of A = 0 in the sum on the left-hand side of eq. (4.30c). The latter is then approximated by (aIho+Xop, - U(k)IX') = E(OlajT) ,

(4.31a)

(a IXoptlß) = (OIJaaßl0) .

(4-31b)

(kIXaptlk`) = Skk-Xopt(k)

(4.32a)

.N'Opt(k) = J1 (k) - J2(k) ,

(4.32b)

with In order to obtain an explicit result, let us consider nuclear matter. T31en, with

`>,(k) _ (OlakvaZIO) ,

J2(k) = (Olvakakl0) -

(4.32c)

By using Wick's theorem for expanding J, (k) and J2(k) one finds that their leading terms are given by J1(k) = O(k-kF)[N,(k)+2X2(_)(k ; e(k))] ,

(4.33a)

J2(k) = O(k - kF).N'2(_)(k; e(k)) + 0(kF- k) .N'2( +)(k; e(k)) .

(4 .33b)

Eqs . (4.32b)-(4.33b) thus yield the following low-order contribution to the generalized optical-model potential, in the domain k > kF which is the one of physical interest : (4 ,N'op,(k > kF) = X,(k)+X2(_)(k > kF; e(k» .

.34)

The second term on the right-hand side of this relation is the on-shell value of the sought I'auli correlation graph. The polarization contribution X2(+)(k; (0) would appear if one would retain terms with A 5x` 0 on the left-hand side of eq. (4.30c)

324

C %ham and R. Sartor / Strictl~ equivalent elastic potentials

[ref. 01 . owever, the perturbation expansion of the generalized optical-model otential rapidly becomes unmanageable because it is difficult to truncate the sum over A in a way which would be consistent with the required order of perturbatici theory. Here, our purpose was limited to show that a contribution similar to the correlation graph X,( - ) occurs in the generalized optical-model potential when the auli orinciWe is taken into account. isCUU~Zn 0-

nu

In sect. we wrote the time-ordered Green function as the sum of a particle part . an a hole part Gh . The spectral representation of Gp involves the eigenstates of the ( + I)-nucleon hamiltonian, while that of the Gh involves the eigenstates of the ( ®1)- nucleon hamiltonian. In particular, these spectral representations invols.-e the energies of the bound eigenstates of the (A :k I)-nucleon systems and the projections of these eigenstates on the ground state of the A-nucleon system. These re actions are called the "overlap functions"; they play an important role in tile escription of stripping, pickup and knockout reactions . The spectral representation of G P also involves, in particular, the overlap function X',Ifi associated with the scattering of one nucleon with energy E from the A-nucleon ground state . This overlap function is called the "elastic part of the scattering wavefunction" ; it plays a basic role in the theory of the optical model . eshbach had shown that XE is an eigenstate of a single-particle hamiltonian with a potential operator that he called the "generalized optical-model potential" '°2 ) .

This potential also yields the energies of the bound states of the (A+ I)-nucleon system, whose spectroscopic factors can be expressed as expectation values of its 31) . energy derivative Originally, these properties were derived in the case when the projectile differs from the particles contained in the A-nucleon target. The formalism was later extended to the case of identical particles ' 3 ) but explicit expressions were not given for the overlap functions and spectroscopic factors. acts stemming from the identity of particles are more conveniently handled by using a second quantization formalism . In that framework it was shown that the expression "generalized optical-model potential" can be replaced by "mass operator" everywhere in the previous paragraph. Indeed, the mass operator also yields the elastic part of the scattering wavefunction as well as the energies and overlap functions of the bound states of the (A + I)-nucleon system, whose spectroscopic factors can be calculated by taking expectation values of its energy derivative. In addition, it possesses the following two properties . Firstly, the effects of antisymmetrization are now included . Secondly, the mass operator also yields the energies of the bound states of the (A -1)-nucleon system as well as the corresponding overlap functions and spectroscopic factors. The existence of the latter additional property indicates that the mass operator differs from the generalized optical-model potential. Formally, this is exhibited by

C. Mahaux and R. Sartor / Strictly equivalent elastic potentials

325

the fact that the generalized optical-model potential only has a right-hand cut in the complex energy plane, while the mass operator has a left-hand cut in addition to a right-hand cut. As a consequence, the dispersion relations which connect the real to the imaginary parts are different for these two operators. This is of practical interest, because dispersion relations have recently been used to extrapolate the phenomenological optical-model potential towards negative energy in such a way that it smoothly joins the shell-model potential. This implies that these works deal with the mass operator instead of the generalized optical-model potential since the latter contains no information on the bound states of the (A -1)-nucleon system or, consequently, on the shell-model potential. For instance, the mass operator enables one to describe not only the elastic component of the n + 2a$Pb scattering wavefunction but also the energies, overlap functions and spectroscopic factors of the bound states of 207 Pb as well as of 2o9Pb . In contrast, the generalized optical-model potential yields no information on the bound states of 207PbIn the process of investigating the relationship and difference between the mass operator and the generalized optical-model potential, we realized that an infinite number of operators exist which all yield the elastic component of the scattering wavefunction as well as the energies, overlap functions and spectroscopic factors ofthe (A + 1)-nucleon system. These "modified mass operators" contain three largely arbitrary operators that we denoted by a, b and c. In particular, these can be arbitrary numbers, which _s the case that we considered for definiteness . In eq. (3.13a), the modified operators are defined in a single-particle representation generated by a single-particle hamiltonian ho = _V212m + U, where U is an external potential. e latter has to be introduced in order to localize the target. In effect, the present as well as previous works imply a non-recoil approximation i.e. assume that the target is fixed. It appears that there does not yet exist a fully satisfactory microscopic treatment of centre-of-mass motion in the antisymmetrized theory of the elastic component of the wavefunction 23). In sect. 3.2, we showed that the modified mass operators X(E; a, b, c) all yield the exact elastic-scattering wavefunction, regardless of the values of a, b and c. The usual mass operator, that we called the "mass operator proper", is obtained by setting a = b = c =1 . If a, b and c are all different from zero, the modified mass operator has a left-hand cut in the complex E -plane, like the mass operator proper. If a = c = 0, the modified operator X(E ; 0, 1, 0) has no left-hand cut but only enables one to calculate the particle part of the Green function. As a consequence, this "particle mass operator" can be used to compute the elastic part of the scattering wavefunction as well as energies, overlap functions and spectroscopic factors of the bound states of the (A+ 1)-nucleon system; however, it yields no information on the eigenstates of the (A -1)-nucleon system. These properties are identical to those of Feshbach's generalized optical-model potential. In order to gain a better insight, we studied the simple case of nuclear matter in sect. 4. e derived explicit expressions for the second-order approximation

a aax and

. Sartor / Strictly equivalent elastic potentials

c) to the modified ass operators . In that approximation to nuclear tter, "ttie ( + 1)-nucleon system" essentially corresponds to momenta k > kF toe e ; ies > eF, were F is the e i momentum and eF = e(k F) the first ation to the e i energy. In that domain, the second-order approxier e ass operator and to the mass operator roper have the tio s to the "o e -s 11 values", i.e. are equal if one sets = e(k) = k2 /2 + U(k). e shell va s of their first derivatives with respect to are also equal. sect. we investigated in detail the properties of the modified mass operator t t is tained by setting = c = 0 and b =1 ; as indicated above, we called it the icle ass operator" because it enables one to compute only the particle part e ss operator. t is thus very in to Fes bach"s generalized optical-model tential e, however, that it includes the effects o antis metrization . In sect. .1, e st ie e general properties o this particle mass operator in nuclear , tter. e showed that it only has a right-hand cut. Accordingly, it fulfills a is rst relation in which the dispersion integral runs from the e i energy to +a". is is to e contrasts with the fact that the mass operator proper fulfills a rsio relation in which e integral so runs over negative energies because it as a left-' an cut in addition to a right-hand cut. e price that has to be paid for the isa e ance of e left-hand cut is that the dispersion relation fulfilled by t e ice ass operator involves a "background contribution" which is a linear f ction o the energy w d is a complicated function of the momentum; as a c sequence, this background to cannot e easily parametrized, which severely i its the practical usefulness of this dispersion relation . Moreover, the real part of t isle ass operator is discontinuous at k = k F and has no simple physical g for k < k F . These general features have been illustrated by means of explicit expressions derived in the framework of second-order perturbation theory. In sect. we sketched how the on-shell value of the Pauli correlation graph occurs in the a tis et ize version of Feshbac 's theory of the generalized optical-model potential. 1 the operators considered in the present work are energy-dependent . Therefore, they differ from the nonlocal energy-independent operator introduced by Kuo et 38,39)' e latter operator also yields the elastic component of the scattering wavefunction, as well as the overlap functions associated with the bound states of the ( + )-nucleon system 22) . However, it yields no information on the (A -1)nucleon system. t shares that limitation with the particle mass operator and with es ach's generalized optical-model potential. In these potentials, all or part of the energy dependence ofthe ass operator proper has been eliminated by a suitable modification ofthe nonlocality. As a consequence, the nonlocality of these potentials is quite complicated. Moreover, they are not easily amenable to perturbation expansions in powers of the strength of the nucleon-nucleon interaction. These remarks y no sans imply that Feshbac 's generalized optical-model potential is void of interest. Indeed, this potential has the advantage of involving the actual a

C. Mahaux and R Sartor / Strictly equivalent elastic potentials

32 7

excited states of the A-nucleon system. It thus suggests plausible approximation schemes for taking into account collective excitations of the target or for evaluating the optical-model potential energies at energies where the effects of antisymmetrization either become small or can be approximated by simple means. The existence of many operators which all yield the same elastic-scattering wavefunction raises the need of discussing which one among them is most closely related to the empirical mean field. The mass operator proper is a strong candidate because it provides a smooth transition between the positive energies associated with elastic scattering from the A-nucleon ground state on the one hand, and the negative energies associated with the bound states of the (A :E 1)-nucleon systems on the other hand . In other words, it provides a smooth transition from the optical-model potential to the shell-model potential as the energy decreases from positive to negative values. This is of practical importance in view of the current interest in determining the shell-model potential and the energies and spectroscopic factors ofbound states of the (A :f: 1)-nucleon systems by extrapolating phenomenological optical-model potentials towards negative energies'). These extrapolations make essential use of the dispersion relation fulfilled by the mass operator proper. Appendix We outline the derivation of expressions (4.11) and (4.12) for the modified mass operators. The contributions to the time-ordered Green function G(k; w) up to second-order perturbation theory are represented by the Feynman diagrams of fig. 2. We emphasize that the Feynman diagrams are not time-ordered ; in particular, the self energy insertion in the diagram G2 contains both .lire(+)(k ; w) and X2( -)( k; w) .

Go

GU

Gb

f G bU

G Ub

Gbb

GUU

G2

Fig. 2. Feynman diagrams which represent the contributions to the time-ordered Green function up to second order. The solid lines correspond to the unperturbed Green function G°(k; w). The crosses represent U-insertions and the horizontal dashed lines the nucleon-nucleon interaction.

C

32

ahaux and R. Sartor / Strictly equivalent elastic potentials

e expression of the particle part of the Green function can be obtained by the relation:

si

(k; w) =

t

21r

this yields w- e(k) +in

r _0 .

dw'

G(k,

w)

(A.1)

w - w' + in .

[ - e(k)+in ]`

[w -

I(P2, P11vI

e(k) +in]

h,),1 2

w-e(k)+i ]2[w+e(hl)-e(P,)-e(P2)+in] hllvi Pl),1 2 w- e(k) +in ][e(k)+e(P1) - e(hl)-e(h2)]2 [e(k)

p(

<

F;

1(h2, h1Ivj PI)J 2 +e(P,)-e(hl)-e(h2)][w-e(k)+in]2,

(A.2)

i

h 1 ),12 e(Pl)+e(P2)-e(hl)-e( )]2[w+e(hl)-e(P1)-e(P2)+irl] I( P2, p 1v1k

w) = ;

(A.3)

Similarly the hole part of the

reen function can be calculated from the expression :

(k; w) =-

2 -ff

_~

dw'

(A.4)

w-w'-in

is yields h(k

> k F ; w) [e(hl)+e(h2)-e(Pl)

Gh(k < kF ; w) =

-

1 e(k) -

-21 +2 +, ~ 2

in

+

2 I(h2, h l IVIk, Pl)J

e(hl) -e(h2) --in1'

e(k)]2[w+e(pl) /Y, (k)

+ [XI(k) - U(k)]3 U(k)2 [w - e(k) - in] [w - e(k) - in] I(P2,

pi 1vIk, hl>, 12

[e ( k) + e(hl) - e(P,) - e(P2) ] 2 [w - e(k) I(P2, P11vlk, h,),

12

in]

[e(k) +e(hl)-e(P1)-e(P2) ] [e" -e(k)-i~l)]2

[w

-

I(h2,hllvlk,Pl),12 e(k) - in] 2[w + e(pl) - e(hl) - e(h2) - in] *

(A.6)

C. Mahaux and R. Sartor / Strictly equivalent elastic potentials

329

The second-order approximation to G(k; w ; c) is readily obtained from the above expressions. Eq. (4.10) involves the second-order approximation to the inverse of G(k; w ; c), which is given by: G-'(k > kF ; (0 ; c) =w - e(k) - .ff,(k) + U(k) -i Y_ +[w - e(k) ] i

- 1 2: X

E

2

I(P2, pi 1v1k, h 1 ),1 . w+e(h,) - e(PJ-e(P2)+gn

J (h l , h21VIk, PI), 12

[e(k) +e(P,)-e(h,)-e(h2)]2

I(h2, h i IvIk'PI), 12 _2C[w_e(k)l2 e(k) +e(p,)-e(h,)-e(h2)

PI), 12 I(h2, h,I vI k, [w+e(P')-e(h,)-e(h2) -igj[e(h , )+e(h2)-e(P')-e(k)j2 (A.7)

G- '(k
c

Jw - e(k) - X,(k)+ U(k)+[w - e(k)fl

-z -i

2

I(P2, PlIvIk, h,),12 [e(k)+e(h,)-e(P,)- e( P2)J2

2:

I(P2, P l IvIk, h,), 1 e(k) +e(h,)-e(p,)-e(P2)

E

I(h2, hil vik, PI )1I2 _1 [w _ e(k)J2 w+e(p,)-e(h,)-e(h2)-irl 2c

1 I(P2, P l I vI k, h ),12 ] [e(P , )+e(P2)-e(k)-e(h :) 2[w+e(h,)-e(P i )-e(P2)+i7?j

.

(A.8)

Eqs. (4.11) and (4.12) are obtained by inserting the expressions of G(k ; w ; a) and G-'(k; w ; c) in eq. (4.10) and by dropping terms of order higher than second. Note that eq. (A .8) only holds for c 0- 0. Hence, this equation cannot be used to calculate Xp(k < kF ; w) as stated in the main text, when c = 0, one has to use (A.3) in conjunction with eq. (4 .17b) . efere ces 1) 2) 3) 4) 5) 6)

H. Feshbach, Ann. of Phys. 5 (1958) 357 H. Feshbach, Ann. Rev. Nuc. Sci. 8 (1958) 49 A. Klein and R. Prange, Phys. Rev. °112 (1958) 994 J.S. Bell and E.J. Squires, Phys. Rev. Lett. 3 (1959) 96 M. Namiki, Progr. Theor. Phys . 23 (1960) 629 J.S. Bell, in Lectures on the many-body problem, ed . E.R. Caianiello (Academic Press, New York, 1962) p. 91 7) H. Feshbach, Ann. of Phys. 19 (1962) 287

330 9) 10) 11) 12) 13) 14) 15) 16) 17) IS)

20) 21) 22) il) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37) 38) 39)

C

Kaw and R. Sartor / ,Strictly equivalent elastic potentials

L. Fetter and K.M. Watson, Adv. Theor. Phys. 1 (1965) 115 erman, in Lectures in Theoretical Physics, ed . by P.D. Kunz, D.A. Lind and W.E . Brittin cdon and Breach, New York, 1966) p. 565 Nguyen Van Giai, J. Sawicki and N . Vinh Mau, Phys. Rev . 141 (1966) 913 F. pillars, in Fundamentals in nuclear theory, ed. by A. De-Shalit and C. Villi (IAEA,Vienna, 1967) p. 269 .A. Friedman and K Feshbach, Racah memorial volume (North-Holland, Amsterdam, 1968) Friedman, Ann . of Phys . 45 (1967) , 265 Luttinger, Phys. Rev. 121 (19,60) 942 J. C. ahaux and R. Sartor, Adv . Nucl. Phys. 20 (1991) 1, and references therein C. ahaux and H.A. Weidenmüller, Shell-model approach to nuclear reactions (North-Holland, msterdao IM) C.A. Enixelbrecht an H.A. Weidenmüller, Nucl. Phys. A184 (1972) 385 J.W. Nagele and H. Orland, quantum many-particle systems (Addison-Wesley, Redwood City, 1988) ch. 5 .S. and J.M. Eisenberg, quantum mechanic of many degrees of freedom (Wiley, New York-, 1988) ch. I I G. Wegmann, Phys. Lett. B29 (1969) 218 J.P. Blaizot and R'plea, quantum theory of finite systems (MIT Press, Cambridge, Mass. USA, 1",1 86) ch. 14 B. Buck and R. Lipperheide, Phys . Left. B123 (1983) 1 E.F. Redish and F. Villars, Ann, of Phys. 56 (1970) 355, and references therein J. Thouless, Nuel. Phys. 75 (Iffl) 128 G. Bayrn and LP. Kadanoff, Phys. Rev. 124 (1961) 287 G. Baym, Phys. Rev. 127 (1962) 1391 1 Letourneux and R. Padjen, Nucl. Phys. A193 (1972) 257 C.M . Shakin and M.S. Weiss, Phys. Rev. C15 (1977) 1911 J.P. Jeukenne, X L*une and C. Mahaux, Phys. Reports C25 (1976) 83 R. Sartor, Nucl. Phys. A239 (1977) 329 Lipperheide, Nucl. Phys. 89 (1966) 97 M. Bertero and G. Passstore, Nuovo Cim . 2A (1971) 579 M. Bertero and G. Passstore, Z. Naturforsch . 28A (1973) 519 L Ahmad and W. raider, J. Phys. G2 (1976) L157 G. Passstore, in Nuclear optical model potential, ed. by S. Boffi and G. Passstore (Lecture Notes in Physics, vol. 55, Springer, Berlin, 1976) p. 177, and references therein G.F. Bartsch and T.T.S. Kuo, Nucl. Phys . A112 (1968) 204 C.M. Shakin and R.M. Thaler, Phys. Rev . C7 (1973) 494 T.T.S. Kuo, F. Osterfeld and S.Y. Lee, Phys. Rev. Left. 45 (1980) 786 S.Y. Lee, F . Osterfeld, K. Tam and T.T.S. Kuo, Phys. Rev. C24 (1981) 329

a