Scattered wave and Lippmann-Schwinger equation

Volume 83A, number 1

PHYSICS LETTERS

4 May 1981

SCATTERED WAVE AND LIPPMANN-SCHWINGER EQUATION Suprokash MUKHERJEE Saha Institute of Nuclear Physics, Calcutta, India

Received 10

March 1981

It is shown that a scattered wave is “everywhere outgoing” and the Lippmann—Schwinger equation has unique solutions.

In this paper we report contradictions we note when we examine Gerjuoy’s [1] analysis of the asymptotic nature of scattering states and the uniqueness of the solution of the Lippmann—Schwinger (LS) equation, in the light of the following lemma on wronskians involving the Green’s function of a many-body system: Lemma. For any arbitrary state vector In), the wronskians 9 (Ga(z), I ii)), 9(G(z), I ri)) etc. vanish identically for all points z in the complex plane, except (probably) for those belonging to the spectrum of Ha, H etc., where a wronskian is defined as 9(Ga(z), In)) = Ga(Z)IYOIfl)_ Ga(Z)i~0Ifl),

(1)

with H,H0 and V as the total hamiltonian, the kinetic and the potential energy operators, respectively; H = H0 + V=Ha + Va; V Va + Va;Ha H0 + Va; Ga(z)=(z _Ha)’, G(z)=(z—H)’;Ha isthehamiltonianin the a-channelwith eigenfunction pa(Ec~);Ha I Pa(Ea)) = Ea I Pa(Ea)). The direction of the arrow onH0 indicates the state or operator in that direction on which H0 operates first. Proof. When I ~) is an eigenvector ofHa with eigenvalue m the lemma may be proved simply as follows:

HaIfl)=mlfl),

Ga(z)Ifl)=”~,

(2)

(Z_iia)Ga(Z)=

1

(3)

9(Ga(z),

Ga(Z)(Z_lia),

In)) = Ga(Z)i~aIfl)

—

Ga(4Pain>

(4)

=Ga(z)m117)+ [1—zGa(z)]ifl>

(5)

=z~m in)+(i_zm)in>0.

(6)

Eqs. (4) and (6) use the equation for the Green’s function, eqs. (3) and (2), respectively. For an arbitrary vector In) the proof along similar lines is more lengthy. As an alternative, one notes that 9 (Ga(Z), I ri)) is an eigenfunction of Ha with eigenvalue z: (z i~a)9(Ga(Z), In>) = 0, (7) —

which may be verified directly by using the defmition in eqs. (1) and (3). Since the spectrum ofHa in a scattering problem consists of points on the real axis only [2], eq. (7) can have only trivial solutions when z does not belong to the spectrum of Ha. We, therefore, have 0 031—9163/81 /0000—0000/$ 02.50 © North-Holland Publishing Company

1

Volume 83A, number 1

PHYSICS LETTERS

4 May 1981

9(G

0(Z), Iii))=0

(8)

identically, for all complex z and also for those points on the real axis which do not belong to the spectrum of H0. For z belonging to the spectrum Of H0, G0(z) or the wronskian above is not defined. The proof follows similar lines for the vanishing of~(G(z),hi)) with H, G(z) replacing H0 and G0(z), respectively, in eqs. (1)—(8), the restriction on z now being with respect to the spectrum of H. This lemma seriously affects Gerjuoy’s conclusions [11 about the outgoing wave character of the solution = + ic1, e~> 0) the following equations [1]:

I ~,0(z~)) of the LS equation, which is given by (z1 kli0(z1))=iiG(z1)Ip0(E0)) =

(9)

Ip0(E0))+ Ix0(z1))

(10)

=Ip0(E0))+G0(zi)V0k110(zi))+2(G0(zi), 1x0(z1)))

(11)

I~0(E0))+G(z1)V0I~0(E0))+ 9(G(z1), 1x0(ziP), (z1 —Ji)k~0(z1))=ie1kc0(E0)).

(12)

=

(13)

The scattered parts of eqs. (11) and (12) correspond to eqs. (3.3b) and (3.3a) of Gerjuoy [1] and his equations on the real axis are 4~IE I~~’~E0)) Ip0~E0))+G~E0)V0 I~1i~’0))+9(G0 Ix~0~’0))) 0), (14) =

Ip0(E0))+G~(E0)V0Ip0(E0))+9(G~(E0), Ix~(E0))).

(15)

Gerjuoy expected that (see his section III)

IiP~(E0P e1—*0 lim I~i~(zi)),

(16)

which amounted to accepting the principle of analytic continuation of the wavefunction in the complex energy plane from the (upper rim of) the positive real axis which has at least a square root cut and forbids putting 6 = 0 for continuum energies. Ix,~(E0)),G~(E0)and G~(E0)are the limiting values of 1x0(zi)), G0(z1) and G(z1) for 1 0, respectively. To avoid zero results, Gerjuoy prescribed that the dummy variables in a wronskian like 9(G0(Z1), 1x0(zi))) should not be evaluated over the entire infinite volume, but within a large sphere of radius p, giving p-subscripted wronskian 9(G0(z1), I x0(z1 )))~, and then the limit e~-÷ 0 is performed first with p held constant, followed by p oo and this would give eqs. (14) and (15) from eqs. (11) and (12) as perhaps eq. (16) suggested. But this implied, for example, -+

—~

J 9(G~(E0),iX~(E0)))= lim00L~I° [lim 9(G0(zi),1x0(zi)))pl.

(17)

p~

Gerjuoy argued that the other alternative p (with e~held constant) followed by 61 0 would give zero for the wronskians of eqs. (11) and (12), thus giving the LS equation with its usual solution from them and not eqs. (14) and (15) on the real axis. But the above prescription of Gerjuoy does not actually save the situation or it can readily be verified that —~°°

-~

9(G 0(Z1),

Ix0(z1)))~is also an eigenfunction ofH0 as in eq. (7) and like”

9(G0(Z1), IX0(Zi)))p

=

0

.

(8) we get (18)

identically, with similar restrictions on z1, and another equation with G0(Z1) replaced by G(z1). From eqs. (17) and (18) we therefore have 9(G:(E0), 2

Ix~(E’0)))= 0,

(19)

Volume 83A, number 1

PHYSICS LETTERS

4 May 1981

9(G~(Ea),iX~(Ea)))= 0,

(20)

giving the IS equation and its solution from eqs. (14) and (15). Thus contrary to Gerjuoy’s claim we note that 1X(E’a)) is automatically “outgoing everywhere” and one does not have to impose eq. (20) separately on eq. (15) to obtain the usual solution of the IS equation. It should, however, be pointed out that Gerjuoy’s derivation of his equations, eqs. (14) and (15) above, on the real axis is done while “sitting on the real axis”. Thus G~(Ea)and IX’~(E0)>are first obtained from 61 0 per—~

formed on G0(z1) and IXa(Zi)) etc. and then put into eq. (1) to form 9(G~(E0),Ix~(E0)>),i.e. products of limits are evaluated in eq. (1) instead of the limit of the product of operators, as suggested by eq. (17). We, therefore, should write 9(G~iE0),Ix(E0)))= lim 9(Ga(Z2),IXa(Zi))),

(21)

Ej

instead of eq. (17), with z1 Using the identity 9(G0(z2), in))

=

=

E0

+

ie1, z2

=

Ea + ie2, and the limits e~-~ 0 and ~2 -+0 are taken independently.

[1+ (z~ z2)G0(z2)] 9(G0(z1), In>)

(22)

—

for any vector in), the two-pointwronskian 9(Ga(Zi), IXa(Z2))) can be evaluated in terms of the one-point wronskian 9(G0(z1), 1x0(zi))). Ifby any prescription we have a non-zero value for lim 9(Ga(Zi),IXa(Zi)>) e1

—o

Is),

then for e~ 0 and In) = IXa(Zi)> put into eq. (22) we get zero for the right side of eq. (22) when we take I~) = I~Pa(Ea))and I~) otherwise, in the limit ~2 0 when Lippmann’s identity [3] is used. We thus have either 9(G~(Ea),Ix~(E0)))= 0 directly or that eq. (21) reduces to eq. (17) which leads to eqs. (19) and (20). Thus both the definitions eqs. (17) and (21), give eqs. (19) and (20) due to our lemma or eqs. (8) or (18), upholding the conclusions that the scattered wave I X(Ea)) for any real continuum energy is automatically “everywhere outgoing”. Eqs. (19) and (20) also change Gerjuoy’s proof of the non-uniqueness of I i~I~(Ea)> for real energy. To accommodate the use of the two-point wronskian of eq.(21) which is closer to that used in eqs. (14) or (15), we rewrite eqs. (11) and (12) as —~

-~

I ~I1a(2i)) ipa(Ea)>+Ga(Z2)Vahi,lIa(Zi)) +9(Ga(Z2), IXa(Zi))) +(z2

z1)G0(z2)jx0(z1))

(23)

—I~a(Ea))+G(z2)VaIca(Ea)>+9(G(z2),Ixa(zi)))+(z2_zi)G(z2)Ixa(zi)>.

(24)

—

The right side of eq. (24) is identical to that of eq. (12), as can be checked by using eq. (22). The e~ 0, 62 0 limits performed in eqs. (23) and (24) give Gerjuoy’s equations on the real axis, eqs. (14) and (15), plus extra terms arising from the last terms of eqs. (23) and (24). They can also be obtained directly, while “sitting on the real axis”, by using the correct equatiqn for the Green’s function on the real axis. Eq. (3) in the complex plane leads to &Y~a(’~’a +ie1)= 1 _lEiGa(Ea +ie1)

(Ea

etc., so that the correct equation for the Green’s function on (the upper rim of) the real axis is (Ea

—

&)~(~‘a)

=

G~(Ea)(Ea

lim ~6iGa(Ea+ii)=~a(E’aL j

—

~a)

=

1

—

~~Ea),

(25)

(26)

—0

3

Volume 83A, number 1

PHYSICS LETTERS

4 May 1981

The limiting value operator .‘~(E0)gives non-zero results when operating on a function space with Dirac delta functions [4]and their effects are missing in Gerjuoy’s calculation on the real axis as he seemed to have used eq. (25) without the ~(E0) term. Writing I ~,1i) for the right side of eq. (24) and eliminating I p0(E0)> from it by using eq. (11) we get I~i)=Il~i~(zi))+ [G(z2)V0h,1i0(zi))_G0(zi)V0Ii1i0(zi))_G0(z2)V0G0(zi)V0IiP0(zi))J +9(G(z2), Ix0(zi)))

—

IA)— G(z2)V0IA)+(z2

—

z1)G(z2)1x0(z1)),

(27)

where we use 9(G0(z1), Ix0(z1)))

IA),

(28)

9(G(z2), Ix0(z1 ))) = ~(G(z2), G0(z1 ) V0I 9(G(z2), IA)) = IA)

+ G(z2) V0IA)

—

(z2

~,1i~(zi))) +

—

9(G(z2), IA))

z1)G(z2)IA)

,

(29)

,

(30)

9(G(z2),G0(z1)V0k1i0(z1)))= [G0(z1)V0I~i0(z1))— G(z2)V0k1i0(zi))+G(z2)V0G0(zi)V0k110(zi))] (31)

z2)G(z2)G0(z1)V0It,1i0(z1)).

From eqs. (24)—(3l), we get

iiP)I,110(zi))

(32)

exactly, for all e~and ~2• Thus we get that the solution I ~i0(z1))of eq. (11) is unique for all z1 and necessarily identical to that (I ,li)) given by the right side of eq. (24) which is the same as that of eq. (12). Eq. (3.4a) of Gerjuoy is given by the first two terms of eq. (27) in the limit Cl o~ 0. It may be noted that the square bracket term of eq. (27) is no longer required to be zero as in Gerjuoy’s case, but instead it is exactly cancelled by the wronskians and other terms following it in eq. (27) and these extra terms are missing in Gerjuoy’s eq. (3.4a) which forbade him to have the uniqueness eq. (32) for complex and real energies. If we now use our lemma to remove all the wronskians and then pass to the limits e~÷0, ~2 ~ 0 in eq. (27) we regain Gerjuoy’s eq. (3.4a) plus an extra term arising from the last term in eq. (27) which, of course, may not survive due to Lippmann’s identity. But irrespective of its survival, this term along with the square bracket term is zero, by virtue of our lemma used on eq. (31), for all 61,62 and hence so for 0, ~2 ~ independently. Thus we finally get uniqueness, eq. (32), for “real” energies as well. Further, in the absence of wronskians in the above equations, we conclude that for all energies, real and complex, the Chew—Goldberger solution [eq. (12) without the wronskian term] is the unique solution to the Lippmann—Schwinger equation [eq. (11) without the wronskian] and that this result will remain valid when rearrangement channels are open, for their presence does not alter eqs. (l)—(32) above. This result is similar to the one reached independently elsewhere [4] where rearrangement channels were treated directly. -+

—~

References [1] [2] [3] [4]

4

E. Gerjuoy,Phys. Rev. 109 (1958) 1806. R.G. Newton, Scattering theory of waves etc. (McGraw-Hill, New York, 1966). W. Sandhas, Few-body dynamics, eds. A.N. Mitra et a!. (North-Holland, Amsterdam, 1976). S. Mukherjee, Phys. Lett. 81A (1981) 207.