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Bound states embedded in the continuum and the formal theory of scattering
ANNALS
OF PHYSICS:
Bound
22,
States
123-132
(1963)
Embedded in the Continuum Formal Theory of Scattering LUCIANO
Istituto
di Fisica
dell’lJniversita’,
and
the
FONDA
Parma, Italy. Istituto Gruppo di Pavma, Italy
Nazionale
di Fisica
Nucleare,
The formal theory of scattering is developed when bound states embedded in the continuum are present. It is shown that it is still possible to define the scattering matrix due to the vanishing of the matrix element connecting the state vector describing the generic bound state embedded in the continuum with the unperturbed state pertaining to the same energy. The scattering matrix does not exhibit a polar singularity, or could even be not singular, at the energies of these bound states. I. INTRODUCTION
The theory of scattering in its formal developments (1-12)’ has not fully taken into consideration the possibility of having discrete eigenvalues of the total Hamiltonian falling onto the continuum region. It is the purpose of this paper to analyze this case in detail with particular regard to the definition of the scattering matrix. Needless to say that the states belonging to these eigenvalues of the total Hamiltonian H, describe “particles” which are stable, i.e., of infinite lifetime, as long as the time development of the system is given by H. A very characteristic feature of these states is represented by their spatial behavior: their eigenfunctions do not fall exponentially at infinity, but they oscillate with amplitudes decreasing as rPz (16). In Section II use is made of the Fock free particle representation to obtain the asymptotic spatial behavior of the Green’s function associated with the generalized equation for the unperturbed states. The result is then used to exploit the asymptotic behavior of the above mentioned eigenstates of H. It is shown that the matrix element connecting one of these states with the unperturbed state pertaining to the same energy is zero. This allows the definition of the scattering matrix, as it is shown in Section III, together with the realization that the scattering matrix does not exhibit a polar singularity at the energies of these bound states or could even be there not singular at all. Due to the vanishing of the aforementioned matrix element, no resonances 1 For a review refs. 14 and 16.
the reader
is referred
to the recent 123
book
by S. S. Schweber
(IS).
See also
124
FONDA
do appear in the scattering states of H near the energy of these bound states. They ought to be considered as limiting cases of very sharp resonances. II.
ASYMPTOTIC
BEHAVIOR
Suppose that the Hamiltonian H has a continuous plus a discrete spectrum and that some of its discrete eigenvalues fall onto the continuum region. We will say that these eigenvalues represent bound states embedded in the continuum. If the continuous spectrum of H is supposed to go from E = 0 to E = + M, we have for these states:
(Ee - WV% = 0, with
$e normalizable
0 < E, < +‘=
(2.1)
by definition: (A,
1clJ = 1
(2.2)
H will be split into two parts H = Ho + Hr where Ho is the “unperturbed” Hamiltonian and HI is the “interaction” Hamiltonian. For a generic physical problem Ho has a continuous spectrum starting from a certain value Eo and going up to + * . If this is the case, we say that HI induces an over-all energy shift of amplitude E. so that the continuous spectrum of H starts from E = 0. This happens, for example, in the theory of quantized fields and quantities of the kind of Eo are referred to as self-energies. It is customary then to renormalize the zero point energy of Ho so that its continuous spectrum coincides with the continuous spectrum of H. In what follows we will suppose that this operation has been already performed. In addition to the continuous spectrum, Ho may have a discrete spectrum with eigenvalues that could also fall onto the continuum region. The possibility of an accidental coincidence of some discrete eigenvalues of Ho with some discrete eigenvalues of H will however not be discussed here. The integral equation for $e reads therefore as follows: 1c/e =
w(Ee)
+
E
e
;
H
HIA 0
(2.3)
where the state p(E,) is an eigenstate of Ho belonging to the continuous eigenvalue E, , i.e., an unperturbed not normalizable state. We have still to specify how the contour of integration understood in the second term of Eq. (2.3) has to avoid the point E, . We write, therefore, quite in general:
$0 = w(E,) + [ME, - Ho) + G”‘l&$e
(2.4)
BOUND
STA4TES
AND
125
SCA!B!ERING
where by GCp) we mean: Gcp)
Qd and Q(E’) values of Ho , values of Ho, principal part follows :
E
P Ee - Ho = 5: E, f
being projection operators for the discrete and continuous eigenrespectively. The sum over d extends over all the discrete eigenthose embedded in the continuum included. P means that the of the integral has to be taken. The &function term in (2.4) is as
6(E, - Ho) = lm Q(E’)a(E, Substitution
of (2.5)
&U-L) = I dEJ)(dEJ
- E’) dE’ =
into (2.4)
$e = d&J with $i”
Ed
shows
I
(2.5)
that:
b + P((P(&),
H&)1
+ d”
(2.6)
given by $y’
= GcP)H&
(2.7)
We shall now show, by analyzing the asymptotic behavior of (2.7) combined with the normalization requirement (2.2)) that both a! = 0 and (p( EJ , HI+,) = 0 and consequently that #B = #y’. In order to see this we shall use the Fock free particle representation (1, 13). In what follows the elimination of the center-of-mass motion from Eq. (2.7) will be understood. P will denote the total linear momentum. The projection of (2.7) onto single particle states is normalizable and therefore does not yield any condition. In order to evaluate the projection of Eq. (2.7) on a two, three, . . . , N-particle state, consideration has to be given to the many particle structure of the Green’s function Gtp). We write G (P) = 3&J@+’ + G’-‘1 with G’+’ and G’-’ defined by 1 G’*” = lim .+o+ E, f ic - Ho We split the “unperturbed”
Hamiltonian
(2.8)
Ho as follows:
Ho=K+V where K is the kinetic energy operator. K and V are diagonal in the Fock representation. We then obtain for G’+’ and G’-’ the following equation: G(*) = I’&*) ( 1 + V@*‘)
(2.9)
Gk+’ and Gi-’ are defined by (2.8) with K in place of HO . We will now consider
126
FONDA
the element of Gjcf’ and Gg’ in the Fock representation corresponding to the N-particle state and we will analyze its asymptotic behavior in the coordinates of the various particles. Using coordinate representation, we have for this element :’
Gi*‘(rl,
rz , . . . , rN 1rll, rzl, . a . rN’)
=J
dp~ dpz - . . dpNg’*‘(pl
, PZ , . . . , PN)
.{rl,rz,...,r,lpl,pz,...,p,)(2*)36(pl+p+
es* +PN-PP) - (pl
where the propagator
(2.10)
, p2
, ’ ’ ’ , pN
1 rip
ri,
’ ’ ’
,rN’)
g(*) is given by:
&*‘hPZ,***,PN)
(2.11)
= (Ehiie-~p12+m12-Z/p22+m22--..-~pN2+mN2)-1
and the limit E -+ O+ is understood. mi is the mass of the ith particle in the considered N-particle state. Writing xi = ri - rl, and integrating over pN , we get: Gi*‘(r
1r’)
= (2~)~~~~~) exp (iP.xN)
*g’*)(pl
, p2,
* * * PN)
exP
(~PI’XIN
J dpl dpz * * * dp,-1 +
&?*XZN
-k
*.
.
(2.12)
+
~PN-I’XN-I,N),
where xiN = Xi - xN . We have indicated briefly by r and r’ the arguments Gi*‘. In Eq. (2.12) pN is a function of the other momenta: PN
=
pN(p1,
p2,
* * ’ , PN-1
Integration over the various infinity, gives : lim
Gj*‘(r
,
p)
=
p
-
PI
-
p2
-
*, . . -
of
(2.13)
PN-1
angles in the limit as xIN , xZN, . . . , xN--I,N tend to
1r’)
‘IN** x2N-m
...
JN-l,N+-
= (21r)~(~-~)il-~
exp
(CP.xN)
&‘~dPl~~p&t
**. ~*PN-&N-1 (2.14)
e
{PlzlN
_
e--iPlzlN
PN-1,
pN(*tpl
$W2N
_
, f
p2
e--iP2QN
, . ’ ’ , f eiPN--IxN--I.N
PN--1
7 ‘)> _
. . .
+O(&,, XlN
z We use natural
x2N
units
h = c = 1.
XN-1,N
e-iPN-l=‘N--l,N
1
BOUND
STATES
AND
127
SCATTERING
where the sign of the various pa in pN of g(*) is fixed in each factor of the square bracket by the corresponding sign of the exponential exp ( fipiriN). The direction of pi is given by: pi = piXiM/x,a . We shall now consider Eq. (2.14) for the Green’s function GA+‘. The integrancf in (2.14) as a function of pi has two poles, coming from the propagator g(+), one, in the first and one in the third quadrant of the complex pi-plane. Let us now take into consideration the generic term in the square bracket containing the negative exponential exp ( -ipaiN). It is clear that we can add to the contour from zero to + m in pi , the contour from --im to zero because on this new contour the considered exponential is exp ( --ip~~~) E exp (- 1pi 1 xiN) tending to zero as xiN goes to infinity. By the same token we can now close the so obtained contour with a limit quarter circle in the fourth quadrant and by the use of the Cauchy theorem we see that in the considered limits the contribution of all the terms with negative exponentials is zero. We have therefore: lim Gk+‘(r X2N-w. . ..
/ r’)
= (27r)2(1--N)?--N exp (iP.xN)
x1N-m XN-I.N-=
. exp
(~&WIN
-k @&QN f XlN’x2N
. . . -k
~PN-#N-1,~)
+
ok-3
’ ’ . XN-1,N
where pN in g’+’ is now given by Eq. (2.13) with the directions of the various pi’s as there specified. It is now convenient to integrate Eq. (2.15) with respect to pN--1. Adding the contour from +im to zero and a limit quarter circle in the first quadrant of the complex pN-,-plane (an argument similar to that given above shows that both contributions vanish in the limit xN-l,N ---f m ) by application of the Cauchy theorem we get: lim
GS’(r
1r’)
= (2*)3-2Ni2-N
exp (Z’P.xN)
pl dp~
s
XlN-‘W xsN+-= xN-.N,N-‘m
. /- p,dpz
...
1 -’
s
(2.16)
pN-~=COld. PN-liPN-l=COnat.
. eXp
(~P@IN
-k
ip2Xm
-k
GN.%N
where
pN-] is determined
E = dm
+
'\/pz2
. . .
-6
~PN-I-TN-I
by the equation +
m22 +
N) L-
+
0(x:;)
. . . XN-1,~
dI
+
... P
-
[g’+‘]-1
+ pl
dpZ,-1 -
p2
= 0, i.e.: +
-
’ ’ .
mL
(2.17) -
pN-1
I2 +
mN2
128
FONDA
In Eq. (2.16) the integrations extend from zero to + 00 ; it is clear however that only a certain physical range of variability for pl , p2 , . * . , pN.-2 will be allowed for fixed E. In fact from the conservation of energy Eq. (2.17) we see that in certain regions p&l will necessarily go over to i 1pN-l 1 contributing an exponentially decreasing faCtOr to the integral in the limit as xN-1-N goes to infinity. The limit on r for Gk” is immediately obtained from Eq. (2.16) : lim
Gk+’ (r ] r’ ) = (21) 1’2(3--N)i2-N exp (iP . rN )
TIN+= T2N-m
.. .
TN-l.N-‘=’
. exp
(‘@I
TIN
f
ifI2 r1N
- CX~~W*(P, 8”
QN * r2N
s, v; r’>
+
. . * +
’ * * TN-l, +
~PN-IAN--13
N)
N
O(rZ)
where riN = ri - rN, pi = piriN/riN , and the derivative of [g(+)]-’ is evaluated as specified in (2.16). (~~(p, s, u) is an eigenstate of K belonging to the continuous eigenvalue E, in which the particles 1, 2, . . . , N obtain momenta spin s with projection v on the x-axis. xSVis the real spin Pl , P2 , ’ * . > PN , total wave function. The complex conjugation of Eq. (2.18) combined with the exchange of P with -P furnishes the expression for GK(-). Use now of (2.9) together with (2.7) yields the asymptotic behavior of the eigenfunction associated with the bound state embedded in the continuum: lim
$iP’(r)
= (2rr)1’2(3-N)j2-N
exp (iPax,)
C s xsr 8”
TlN+= r2N*m
.. .
tN-LN-O”
.
exp
(ipl
?“lN
+
ip2
r2N
rlNr2N’ +
( _
)N-2
exp
+
’ ’ ’ +
( --ipl
rlN
-
ip2
r1N
* ((p’+‘( - p, s, v), H&) where the “remainder”
ipN-I
’ ’ ’ ‘TN-1)
1
r2N
TN-1,
N)
(P
(p, s, Y),H&)
N r2N
-
’ ’ ’ -
’ ’ ’ rAL-1)
ipN-l
rN-1,
N
+ remainder
goes to zero in each riN faster than (rJ1.
(2*1g)
BOUND
STATES
AND
129
SCATTERING
C-1 and (J+) are eigenstates of Ho belonging to the continuous eigenvalue E, cp satisfying an incoming and an outgoing wave boundary condition, respectively.” It is clear from (2.19) that no cancellation is now possible asymptotically between #i” and the (Y, @terms of (2.6). Consequently, in order that $e be normalizable, they must vanish separately in the limits considered above. The integral at the right hand side of (2.19) therefore vanishes identically. This implies both that ?, 5 (&,
s, ~1, HI&)
(2.20)
= 0
and that 01= 0. In (2.20) cp may satisfy either an outgoing or an incoming wave boundary condition. The directions of the various pi’s in (2.20) are arbitrary, as long as they make up a total linear momentum P, their moduli are in the physical range given by the limits of integration together with the conservation of energy relation (2.17) .6 III.
DEFINITION
OF
THE
SCATTERING
We can now evaluate the scalar products of H satisfying the equations: #*) (El = d*‘(E) i.e., “scattering 3&
states”
+,“rj
between
E *;
through
time
, ye ) =
(3.1)
combined
with
(2.7)
reversal:
operator. The transformation the eigenvalue zero for
WL.(s,
, eigenstates
_ H HIP’*‘(E)
lY@(p, s, V) = (-)“‘v$A+‘(-p, reversal simplicity
tiL,and J/(*)(E)
for the energy E. Use of (3.1)
and p(+) are related
with8 the time if we take for have
MATRIX
the
s, -v)
properties of $e under 8 are analogous; total linear momentum P, we then
(-)8~+v4(s.
, -7%)
where sE and Y< are the spin and its z-component of the considered bound state embedded in the continuum. It is then seen that the two matrix elements appearing at the right hand side of (2.19) are simply related: (‘d-‘(p,
4 If Hr
=
s, v),
HI
+
HI!b.(S,
HZ ,
, d)
use of (2.20)
=
(-)e+u+a,+ue(q(+)(-P,
s, -71,
together
with
(2.7)
(x(Ed,
Hz&)
= 0
shows
that
Hr!be(se
, -v,))*
also
where x(K) is eigenstate of HO + HI belonging to the continuous eigenvalue E, . 5 Note that Eq. (2.20) implies also that the Hamiltonian Ho behaves as an Hermitian operator when considered in scalar products such as (q, H&.). B In ref. 16 the vanishing of the matrix element (2.20) and the existence of these bound states embedded in the continuum have been established by the use of a many-channel theory (see Eq. (2.29) and Section V of that paper).
130
FONDA
gives :
which, because of (2.20), is always zero. Consequently the scattering states and the bound states embedded in the continuum are orthogonal vectors and the completeness relation for the eigenstates of H reads therefore as follows:
where the sum over b extends over the bound states (energy less than zero) and the sum over e over the bound states embedded in the continuum. We now want to see whether the presence of bound states embedded in the continuum influences the definition of the scattering matrix. For this purpose let us consider the following limits: lim U(0, t)& t-r+m
(3.4)
iXOirJiH(t-t,)e-iH,t U(lo , 1) = e e
(3.5)
where
is the operator which gives the time development of the state vectors in the interaction representation, and #pi is a general wave packet which at the end of the calculations will be made to tend to ~a, eigenstate of Ho belonging to the continuous eigenvalue Ei . We shall expand 9; in series of eigenstates of Ho :
with self-explanatory symbols for the integration variables. The second term at the right hand side of (3.6) will, however, give no contribution in the limit & ---f pi, so we can disregard it completely in the following developments.’ Substitution of (3.6) in (3.4) and use of the completeness relation (3.3) gives:
+
7 (yDd , ~6) = 0 also for states qd whose eigenvalue obtained from an equation similar to (3.2) with HI + v.
(3.7)
is embedded the substitutions
in the continuum as it is $, + pd , p -+ pR and
BOUND
STATES
AND
131
SCATTERING
The first term at the right hand side of (3.7) may be evaluated t ---) ‘f 03 by exhibiting the singularity inherent in ($i*‘, (o,) :’
in the limits
(3.8) and the result, in the limit ~$i-+ cpi , is #I*’ . As for the second term, since (I& , C,Q) = (Eb - EJ’( I/& , HIpa) and Et, < 0, the integrand does not exhibit any singuIarity in the range of integration over E, . Therefore use of the Riemann-Lebesgue lemma shows that this term vanishes in the limits t --+ F ~4. We come now to the discussion of the third term at the right hand side of Eq. (3.7). By projecting (2.7) onto pa and taking the complex conjugate we get.
Substituting (3.9) into (3.7) and integrating for the third term the following expression:
over the energy E, , we obtain
where the plus sign holds in the limit t * + 00 and the minus sign when t --+ - x . We have indicated by /3 the variables which together with the energy form a complete set of commuting observables. Use now of (2.20) tells us that Eq. (3.10) is identically equal to zero. It is seen therefore that the definition of the scattering matrix is by no means affected by the presence in the theory of bound states embedded in the continuum. Also in this case one gets the known result :
from which the scattering matrix can be obtained in the usual way. We can also show that the T-matrix, and consequently the scattering matrix, does not exhibit a polar singularity at E = E, . Infact, letustake for the T-matrix its expression in terms of the complete Green’s function (E + ic - H)-‘: TG
=
Making
(91,
HI&+‘)
=
(cpf , H19i)
use of the completeness Tfi
=
lim
C
c+o+ e where the “remainder” * The
procedure
sketched
+
c-of
relation
(9f
HI E + i’, _ H Hm
lim
) HIGH)
(me ) H’9”
+ remainder
E + ir - E,
is well
known,
(3.12)
(3.3) we get:
contains the contribution here
>
(3.13)
from the bound states and from
see refs.
7 and
13.
132
FONDA
the scattering states of H. Use of (2.20) shows now that the contribution from the bound states embedded in the continuum does not produce a polar singularity in the T-matrix. Moreover, if the scalar products (P,,~, $B) are finite also for E = E, , as we would expect on physical grounds, then the term in the T-matrix corresponding to the discrete eigenvalue E, of H is zero at E = E, , showing that the T-matrix is perfectly regular at the position of the bound states embedded in the continuum. There will appear no resonance-like peak in the cross sections at E = E, for scattering experiments whose description is given in terms of the Hamiltonian H. With respect to this Hamiltonian $e describes in fact a nonradioactive state, its stability being assured by the vanishing of the matrix element (2.20). For what ’ concerns the production of these states (it is clear that they cannot be created in the above mentioned type of experiments),one must resort to different kinds of scattering systems.g RECEIVED:
September 12, 1962 REFERENCES
1. C. M$LLER, Kgl. Danske Videnskab. Selskab, Mat.-Fys. Medd. 2. B. A. LIPPMANN AND J. SCHWINGER, Phys. Rev. 79, 469 (1950). 8. H. S. SNYDER, Phys. Rev. 83, 1154 (1951). 4. S. T. MA, Phys. Rev. 87, 652 (1952). 6. M. GELL-MANN AND M. L. GOLDBERGER. Phys. Rev. 91, 398 6. H. EKSTEIN, Phys. Rev. 94, 1063 (1954); ibid. 101, 880 (1956). 7. M. N. HACK, Phys. Rev. 96, 196 (1954). 8. S. SUNAEAWA, Progr. Theoret. Phys. (Kyoto) 14, 175 (1955). 9. H. E. MOSES, Nuovo Cinzento 1, 103 (1955). 10. T. IMAMURA, Progr. Theoret. Phys. (Kyoto) 18, 51 (1957). 11. J. M. JAUCH, Helv. Phys. Acta 31, 127,661 (1958). 12. J. M. JAUCH AND I. I. ZINNES, Nuovo Cimento 11,553 (1959). 18. S. S. SCHWEBER, “An Introduction to Relativistic Quantum
23, No.
1 (1945).
(1953).
Field Theory,” Chap. 11. Illinois, 1961. “Theory of Photons and Electrons,” Chap. 7. Addison14. J. M. JAUCH AND F. ROHRLICH, Wesley, Cambridge, Massachusetts, 1955. 16. W. BRENIQ AND R. HAAG, Fortschr. Physik 7, 183 (1959). 16. L. FONDA AND R. G. NEWTON, Ann. Phys. (N. Y.) 10, 490 (1960). Row-Peterson,
Evanston,
9 For example, a bound state embedded in the continuum seems to appear in the negative helium ion. The level 4P6,2 of this system lies in fact in the continuum and it is not liable to auto-ionization (E. Holeien and J. Midtdal, Proc. Phys. Sot. A, 68,815 (1955)). An analogous state seems to be present in the helium atom. I would like to express my appreciation to Prof. E. P. Wigner for having informed me on these states and for correspondence.
OF PHYSICS:
Bound
22,
States
123-132
(1963)
Embedded in the Continuum Formal Theory of Scattering LUCIANO
Istituto
di Fisica
dell’lJniversita’,
and
the
FONDA
Parma, Italy. Istituto Gruppo di Pavma, Italy
Nazionale
di Fisica
Nucleare,
The formal theory of scattering is developed when bound states embedded in the continuum are present. It is shown that it is still possible to define the scattering matrix due to the vanishing of the matrix element connecting the state vector describing the generic bound state embedded in the continuum with the unperturbed state pertaining to the same energy. The scattering matrix does not exhibit a polar singularity, or could even be not singular, at the energies of these bound states. I. INTRODUCTION
The theory of scattering in its formal developments (1-12)’ has not fully taken into consideration the possibility of having discrete eigenvalues of the total Hamiltonian falling onto the continuum region. It is the purpose of this paper to analyze this case in detail with particular regard to the definition of the scattering matrix. Needless to say that the states belonging to these eigenvalues of the total Hamiltonian H, describe “particles” which are stable, i.e., of infinite lifetime, as long as the time development of the system is given by H. A very characteristic feature of these states is represented by their spatial behavior: their eigenfunctions do not fall exponentially at infinity, but they oscillate with amplitudes decreasing as rPz (16). In Section II use is made of the Fock free particle representation to obtain the asymptotic spatial behavior of the Green’s function associated with the generalized equation for the unperturbed states. The result is then used to exploit the asymptotic behavior of the above mentioned eigenstates of H. It is shown that the matrix element connecting one of these states with the unperturbed state pertaining to the same energy is zero. This allows the definition of the scattering matrix, as it is shown in Section III, together with the realization that the scattering matrix does not exhibit a polar singularity at the energies of these bound states or could even be there not singular at all. Due to the vanishing of the aforementioned matrix element, no resonances 1 For a review refs. 14 and 16.
the reader
is referred
to the recent 123
book
by S. S. Schweber
(IS).
See also
124
FONDA
do appear in the scattering states of H near the energy of these bound states. They ought to be considered as limiting cases of very sharp resonances. II.
ASYMPTOTIC
BEHAVIOR
Suppose that the Hamiltonian H has a continuous plus a discrete spectrum and that some of its discrete eigenvalues fall onto the continuum region. We will say that these eigenvalues represent bound states embedded in the continuum. If the continuous spectrum of H is supposed to go from E = 0 to E = + M, we have for these states:
(Ee - WV% = 0, with
$e normalizable
0 < E, < +‘=
(2.1)
by definition: (A,
1clJ = 1
(2.2)
H will be split into two parts H = Ho + Hr where Ho is the “unperturbed” Hamiltonian and HI is the “interaction” Hamiltonian. For a generic physical problem Ho has a continuous spectrum starting from a certain value Eo and going up to + * . If this is the case, we say that HI induces an over-all energy shift of amplitude E. so that the continuous spectrum of H starts from E = 0. This happens, for example, in the theory of quantized fields and quantities of the kind of Eo are referred to as self-energies. It is customary then to renormalize the zero point energy of Ho so that its continuous spectrum coincides with the continuous spectrum of H. In what follows we will suppose that this operation has been already performed. In addition to the continuous spectrum, Ho may have a discrete spectrum with eigenvalues that could also fall onto the continuum region. The possibility of an accidental coincidence of some discrete eigenvalues of Ho with some discrete eigenvalues of H will however not be discussed here. The integral equation for $e reads therefore as follows: 1c/e =
w(Ee)
+
E
e
;
H
HIA 0
(2.3)
where the state p(E,) is an eigenstate of Ho belonging to the continuous eigenvalue E, , i.e., an unperturbed not normalizable state. We have still to specify how the contour of integration understood in the second term of Eq. (2.3) has to avoid the point E, . We write, therefore, quite in general:
$0 = w(E,) + [ME, - Ho) + G”‘l&$e
(2.4)
BOUND
STA4TES
AND
125
SCA!B!ERING
where by GCp) we mean: Gcp)
Qd and Q(E’) values of Ho , values of Ho, principal part follows :
E
P Ee - Ho = 5: E, f
being projection operators for the discrete and continuous eigenrespectively. The sum over d extends over all the discrete eigenthose embedded in the continuum included. P means that the of the integral has to be taken. The &function term in (2.4) is as
6(E, - Ho) = lm Q(E’)a(E, Substitution
of (2.5)
&U-L) = I dEJ)(dEJ
- E’) dE’ =
into (2.4)
$e = d&J with $i”
Ed
shows
I
(2.5)
that:
b + P((P(&),
H&)1
+ d”
(2.6)
given by $y’
= GcP)H&
(2.7)
We shall now show, by analyzing the asymptotic behavior of (2.7) combined with the normalization requirement (2.2)) that both a! = 0 and (p( EJ , HI+,) = 0 and consequently that #B = #y’. In order to see this we shall use the Fock free particle representation (1, 13). In what follows the elimination of the center-of-mass motion from Eq. (2.7) will be understood. P will denote the total linear momentum. The projection of (2.7) onto single particle states is normalizable and therefore does not yield any condition. In order to evaluate the projection of Eq. (2.7) on a two, three, . . . , N-particle state, consideration has to be given to the many particle structure of the Green’s function Gtp). We write G (P) = 3&J@+’ + G’-‘1 with G’+’ and G’-’ defined by 1 G’*” = lim .+o+ E, f ic - Ho We split the “unperturbed”
Hamiltonian
(2.8)
Ho as follows:
Ho=K+V where K is the kinetic energy operator. K and V are diagonal in the Fock representation. We then obtain for G’+’ and G’-’ the following equation: G(*) = I’&*) ( 1 + V@*‘)
(2.9)
Gk+’ and Gi-’ are defined by (2.8) with K in place of HO . We will now consider
126
FONDA
the element of Gjcf’ and Gg’ in the Fock representation corresponding to the N-particle state and we will analyze its asymptotic behavior in the coordinates of the various particles. Using coordinate representation, we have for this element :’
Gi*‘(rl,
rz , . . . , rN 1rll, rzl, . a . rN’)
=J
dp~ dpz - . . dpNg’*‘(pl
, PZ , . . . , PN)
.{rl,rz,...,r,lpl,pz,...,p,)(2*)36(pl+p+
es* +PN-PP) - (pl
where the propagator
(2.10)
, p2
, ’ ’ ’ , pN
1 rip
ri,
’ ’ ’
,rN’)
g(*) is given by:
&*‘hPZ,***,PN)
(2.11)
= (Ehiie-~p12+m12-Z/p22+m22--..-~pN2+mN2)-1
and the limit E -+ O+ is understood. mi is the mass of the ith particle in the considered N-particle state. Writing xi = ri - rl, and integrating over pN , we get: Gi*‘(r
1r’)
= (2~)~~~~~) exp (iP.xN)
*g’*)(pl
, p2,
* * * PN)
exP
(~PI’XIN
J dpl dpz * * * dp,-1 +
&?*XZN
-k
*.
.
(2.12)
+
~PN-I’XN-I,N),
where xiN = Xi - xN . We have indicated briefly by r and r’ the arguments Gi*‘. In Eq. (2.12) pN is a function of the other momenta: PN
=
pN(p1,
p2,
* * ’ , PN-1
Integration over the various infinity, gives : lim
Gj*‘(r
,
p)
=
p
-
PI
-
p2
-
*, . . -
of
(2.13)
PN-1
angles in the limit as xIN , xZN, . . . , xN--I,N tend to
1r’)
‘IN** x2N-m
...
JN-l,N+-
= (21r)~(~-~)il-~
exp
(CP.xN)
&‘~dPl~~p&t
**. ~*PN-&N-1 (2.14)
e
{PlzlN
_
e--iPlzlN
PN-1,
pN(*tpl
$W2N
_
, f
p2
e--iP2QN
, . ’ ’ , f eiPN--IxN--I.N
PN--1
7 ‘)> _
. . .
+O(&,, XlN
z We use natural
x2N
units
h = c = 1.
XN-1,N
e-iPN-l=‘N--l,N
1
BOUND
STATES
AND
127
SCATTERING
where the sign of the various pa in pN of g(*) is fixed in each factor of the square bracket by the corresponding sign of the exponential exp ( fipiriN). The direction of pi is given by: pi = piXiM/x,a . We shall now consider Eq. (2.14) for the Green’s function GA+‘. The integrancf in (2.14) as a function of pi has two poles, coming from the propagator g(+), one, in the first and one in the third quadrant of the complex pi-plane. Let us now take into consideration the generic term in the square bracket containing the negative exponential exp ( -ipaiN). It is clear that we can add to the contour from zero to + m in pi , the contour from --im to zero because on this new contour the considered exponential is exp ( --ip~~~) E exp (- 1pi 1 xiN) tending to zero as xiN goes to infinity. By the same token we can now close the so obtained contour with a limit quarter circle in the fourth quadrant and by the use of the Cauchy theorem we see that in the considered limits the contribution of all the terms with negative exponentials is zero. We have therefore: lim Gk+‘(r X2N-w. . ..
/ r’)
= (27r)2(1--N)?--N exp (iP.xN)
x1N-m XN-I.N-=
. exp
(~&WIN
-k @&QN f XlN’x2N
. . . -k
~PN-#N-1,~)
+
ok-3
’ ’ . XN-1,N
where pN in g’+’ is now given by Eq. (2.13) with the directions of the various pi’s as there specified. It is now convenient to integrate Eq. (2.15) with respect to pN--1. Adding the contour from +im to zero and a limit quarter circle in the first quadrant of the complex pN-,-plane (an argument similar to that given above shows that both contributions vanish in the limit xN-l,N ---f m ) by application of the Cauchy theorem we get: lim
GS’(r
1r’)
= (2*)3-2Ni2-N
exp (Z’P.xN)
pl dp~
s
XlN-‘W xsN+-= xN-.N,N-‘m
. /- p,dpz
...
1 -’
s
(2.16)
pN-~=COld. PN-liPN-l=COnat.
. eXp
(~P@IN
-k
ip2Xm
-k
GN.%N
where
pN-] is determined
E = dm
+
'\/pz2
. . .
-6
~PN-I-TN-I
by the equation +
m22 +
N) L-
+
0(x:;)
. . . XN-1,~
dI
+
... P
-
[g’+‘]-1
+ pl
dpZ,-1 -
p2
= 0, i.e.: +
-
’ ’ .
mL
(2.17) -
pN-1
I2 +
mN2
128
FONDA
In Eq. (2.16) the integrations extend from zero to + 00 ; it is clear however that only a certain physical range of variability for pl , p2 , . * . , pN.-2 will be allowed for fixed E. In fact from the conservation of energy Eq. (2.17) we see that in certain regions p&l will necessarily go over to i 1pN-l 1 contributing an exponentially decreasing faCtOr to the integral in the limit as xN-1-N goes to infinity. The limit on r for Gk” is immediately obtained from Eq. (2.16) : lim
Gk+’ (r ] r’ ) = (21) 1’2(3--N)i2-N exp (iP . rN )
TIN+= T2N-m
.. .
TN-l.N-‘=’
. exp
(‘@I
TIN
f
ifI2 r1N
- CX~~W*(P, 8”
QN * r2N
s, v; r’>
+
. . * +
’ * * TN-l, +
~PN-IAN--13
N)
N
O(rZ)
where riN = ri - rN, pi = piriN/riN , and the derivative of [g(+)]-’ is evaluated as specified in (2.16). (~~(p, s, u) is an eigenstate of K belonging to the continuous eigenvalue E, in which the particles 1, 2, . . . , N obtain momenta spin s with projection v on the x-axis. xSVis the real spin Pl , P2 , ’ * . > PN , total wave function. The complex conjugation of Eq. (2.18) combined with the exchange of P with -P furnishes the expression for GK(-). Use now of (2.9) together with (2.7) yields the asymptotic behavior of the eigenfunction associated with the bound state embedded in the continuum: lim
$iP’(r)
= (2rr)1’2(3-N)j2-N
exp (iPax,)
C s xsr 8”
TlN+= r2N*m
.. .
tN-LN-O”
.
exp
(ipl
?“lN
+
ip2
r2N
rlNr2N’ +
( _
)N-2
exp
+
’ ’ ’ +
( --ipl
rlN
-
ip2
r1N
* ((p’+‘( - p, s, v), H&) where the “remainder”
ipN-I
’ ’ ’ ‘TN-1)
1
r2N
TN-1,
N)
(P
(p, s, Y),H&)
N r2N
-
’ ’ ’ -
’ ’ ’ rAL-1)
ipN-l
rN-1,
N
+ remainder
goes to zero in each riN faster than (rJ1.
(2*1g)
BOUND
STATES
AND
129
SCATTERING
C-1 and (J+) are eigenstates of Ho belonging to the continuous eigenvalue E, cp satisfying an incoming and an outgoing wave boundary condition, respectively.” It is clear from (2.19) that no cancellation is now possible asymptotically between #i” and the (Y, @terms of (2.6). Consequently, in order that $e be normalizable, they must vanish separately in the limits considered above. The integral at the right hand side of (2.19) therefore vanishes identically. This implies both that ?, 5 (&,
s, ~1, HI&)
(2.20)
= 0
and that 01= 0. In (2.20) cp may satisfy either an outgoing or an incoming wave boundary condition. The directions of the various pi’s in (2.20) are arbitrary, as long as they make up a total linear momentum P, their moduli are in the physical range given by the limits of integration together with the conservation of energy relation (2.17) .6 III.
DEFINITION
OF
THE
SCATTERING
We can now evaluate the scalar products of H satisfying the equations: #*) (El = d*‘(E) i.e., “scattering 3&
states”
+,“rj
between
E *;
through
time
, ye ) =
(3.1)
combined
with
(2.7)
reversal:
operator. The transformation the eigenvalue zero for
WL.(s,
, eigenstates
_ H HIP’*‘(E)
lY@(p, s, V) = (-)“‘v$A+‘(-p, reversal simplicity
tiL,and J/(*)(E)
for the energy E. Use of (3.1)
and p(+) are related
with8 the time if we take for have
MATRIX
the
s, -v)
properties of $e under 8 are analogous; total linear momentum P, we then
(-)8~+v4(s.
, -7%)
where sE and Y< are the spin and its z-component of the considered bound state embedded in the continuum. It is then seen that the two matrix elements appearing at the right hand side of (2.19) are simply related: (‘d-‘(p,
4 If Hr
=
s, v),
HI
+
HI!b.(S,
HZ ,
, d)
use of (2.20)
=
(-)e+u+a,+ue(q(+)(-P,
s, -71,
together
with
(2.7)
(x(Ed,
Hz&)
= 0
shows
that
Hr!be(se
, -v,))*
also
where x(K) is eigenstate of HO + HI belonging to the continuous eigenvalue E, . 5 Note that Eq. (2.20) implies also that the Hamiltonian Ho behaves as an Hermitian operator when considered in scalar products such as (q, H&.). B In ref. 16 the vanishing of the matrix element (2.20) and the existence of these bound states embedded in the continuum have been established by the use of a many-channel theory (see Eq. (2.29) and Section V of that paper).
130
FONDA
gives :
which, because of (2.20), is always zero. Consequently the scattering states and the bound states embedded in the continuum are orthogonal vectors and the completeness relation for the eigenstates of H reads therefore as follows:
where the sum over b extends over the bound states (energy less than zero) and the sum over e over the bound states embedded in the continuum. We now want to see whether the presence of bound states embedded in the continuum influences the definition of the scattering matrix. For this purpose let us consider the following limits: lim U(0, t)& t-r+m
(3.4)
iXOirJiH(t-t,)e-iH,t U(lo , 1) = e e
(3.5)
where
is the operator which gives the time development of the state vectors in the interaction representation, and #pi is a general wave packet which at the end of the calculations will be made to tend to ~a, eigenstate of Ho belonging to the continuous eigenvalue Ei . We shall expand 9; in series of eigenstates of Ho :
with self-explanatory symbols for the integration variables. The second term at the right hand side of (3.6) will, however, give no contribution in the limit & ---f pi, so we can disregard it completely in the following developments.’ Substitution of (3.6) in (3.4) and use of the completeness relation (3.3) gives:
+
7 (yDd , ~6) = 0 also for states qd whose eigenvalue obtained from an equation similar to (3.2) with HI + v.
(3.7)
is embedded the substitutions
in the continuum as it is $, + pd , p -+ pR and
BOUND
STATES
AND
131
SCATTERING
The first term at the right hand side of (3.7) may be evaluated t ---) ‘f 03 by exhibiting the singularity inherent in ($i*‘, (o,) :’
in the limits
(3.8) and the result, in the limit ~$i-+ cpi , is #I*’ . As for the second term, since (I& , C,Q) = (Eb - EJ’( I/& , HIpa) and Et, < 0, the integrand does not exhibit any singuIarity in the range of integration over E, . Therefore use of the Riemann-Lebesgue lemma shows that this term vanishes in the limits t --+ F ~4. We come now to the discussion of the third term at the right hand side of Eq. (3.7). By projecting (2.7) onto pa and taking the complex conjugate we get.
Substituting (3.9) into (3.7) and integrating for the third term the following expression:
over the energy E, , we obtain
where the plus sign holds in the limit t * + 00 and the minus sign when t --+ - x . We have indicated by /3 the variables which together with the energy form a complete set of commuting observables. Use now of (2.20) tells us that Eq. (3.10) is identically equal to zero. It is seen therefore that the definition of the scattering matrix is by no means affected by the presence in the theory of bound states embedded in the continuum. Also in this case one gets the known result :
from which the scattering matrix can be obtained in the usual way. We can also show that the T-matrix, and consequently the scattering matrix, does not exhibit a polar singularity at E = E, . Infact, letustake for the T-matrix its expression in terms of the complete Green’s function (E + ic - H)-‘: TG
=
Making
(91,
HI&+‘)
=
(cpf , H19i)
use of the completeness Tfi
=
lim
C
c+o+ e where the “remainder” * The
procedure
sketched
+
c-of
relation
(9f
HI E + i’, _ H Hm
lim
) HIGH)
(me ) H’9”
+ remainder
E + ir - E,
is well
known,
(3.12)
(3.3) we get:
contains the contribution here
>
(3.13)
from the bound states and from
see refs.
7 and
13.
132
FONDA
the scattering states of H. Use of (2.20) shows now that the contribution from the bound states embedded in the continuum does not produce a polar singularity in the T-matrix. Moreover, if the scalar products (P,,~, $B) are finite also for E = E, , as we would expect on physical grounds, then the term in the T-matrix corresponding to the discrete eigenvalue E, of H is zero at E = E, , showing that the T-matrix is perfectly regular at the position of the bound states embedded in the continuum. There will appear no resonance-like peak in the cross sections at E = E, for scattering experiments whose description is given in terms of the Hamiltonian H. With respect to this Hamiltonian $e describes in fact a nonradioactive state, its stability being assured by the vanishing of the matrix element (2.20). For what ’ concerns the production of these states (it is clear that they cannot be created in the above mentioned type of experiments),one must resort to different kinds of scattering systems.g RECEIVED:
September 12, 1962 REFERENCES
1. C. M$LLER, Kgl. Danske Videnskab. Selskab, Mat.-Fys. Medd. 2. B. A. LIPPMANN AND J. SCHWINGER, Phys. Rev. 79, 469 (1950). 8. H. S. SNYDER, Phys. Rev. 83, 1154 (1951). 4. S. T. MA, Phys. Rev. 87, 652 (1952). 6. M. GELL-MANN AND M. L. GOLDBERGER. Phys. Rev. 91, 398 6. H. EKSTEIN, Phys. Rev. 94, 1063 (1954); ibid. 101, 880 (1956). 7. M. N. HACK, Phys. Rev. 96, 196 (1954). 8. S. SUNAEAWA, Progr. Theoret. Phys. (Kyoto) 14, 175 (1955). 9. H. E. MOSES, Nuovo Cinzento 1, 103 (1955). 10. T. IMAMURA, Progr. Theoret. Phys. (Kyoto) 18, 51 (1957). 11. J. M. JAUCH, Helv. Phys. Acta 31, 127,661 (1958). 12. J. M. JAUCH AND I. I. ZINNES, Nuovo Cimento 11,553 (1959). 18. S. S. SCHWEBER, “An Introduction to Relativistic Quantum
23, No.
1 (1945).
(1953).
Field Theory,” Chap. 11. Illinois, 1961. “Theory of Photons and Electrons,” Chap. 7. Addison14. J. M. JAUCH AND F. ROHRLICH, Wesley, Cambridge, Massachusetts, 1955. 16. W. BRENIQ AND R. HAAG, Fortschr. Physik 7, 183 (1959). 16. L. FONDA AND R. G. NEWTON, Ann. Phys. (N. Y.) 10, 490 (1960). Row-Peterson,
Evanston,
9 For example, a bound state embedded in the continuum seems to appear in the negative helium ion. The level 4P6,2 of this system lies in fact in the continuum and it is not liable to auto-ionization (E. Holeien and J. Midtdal, Proc. Phys. Sot. A, 68,815 (1955)). An analogous state seems to be present in the helium atom. I would like to express my appreciation to Prof. E. P. Wigner for having informed me on these states and for correspondence.