- Email: [email protected]
Properties of the bound states embedded in the continuum
ANNALS
OF
Properties
PHYSICS:
26, 246-246
of the
(1964)
Bound
States
Embedded
LUCIANO Istituto
di Fisica
dell’Universitic, Gruppo
in
the
Continuum
FONDA
Parma; Istituto Nazionale di Parma, Italy
di Fisica
Nucleare,
AND GIAN Istituto
di Fisica
CARLO
GHIRARDI
dell’Universitd,
Trieste,
Italy
The energy dependence of the matrix element connecting the state vector describing the generic hound state embedded in the continuum with the unperturbed states is investigated in detail. The energy behavior is such that the scattering matrix turns out to be not infinite at the energy of the bound state embedded in the continuum. The asymptotic spatial behavior of the state vectors relative to these kinds of eigenvalues is analyzed.There results that the two-particle projected part of these vectors determines in most cases the asymptotic behavior of such states. I. INTRODUCTION
As it has been recently shown (1) , when discrete eigenvalues happen to be embedded in the continuum part of the spectrum of the Hamilton operator, the matrix element connecting the corresponding normalizable eigenstates with the unperturbed state pertaining to the same energy is zero. This circumstance allows the definition of the scattering matrix, together with the realization that this matrix does not exhibit a polar singularity at the energies of these bound states. The main purpose of this paper is to investigate the behavior of the aforementioned matrix element when the energy E of the unperturbed state is made to approach E, , the energy of the bound state embedded in the continuum. As will be shown in Section II, this matrix element behaves like (E, - JT)~“+‘, with c positive. This result makes the scattering matrix not infinite at E = E, . In Section III consideration has been given to the asymptotic spatial behavior of the wave functions of the bound states embedded in the continuum. It is found that the two-particle projected portion of these states behaves as r-312--r (t > 0) times an oscillatory term of wave number pe , relative momentum for the two particles of center of mass total energy E, . If E < 35, this oscillatory 240
BOUND
STATES
ON
THE
241
CONTINUUM
term makes the wave function infinite in the momentum representation just at the point p, . On the contrary, if the energetic threshold of t’he considered twoparticle state is higher t’hen E, , the asymptotic spatial behavior is exponentially decreasing. The three- or more particle projected portions of these bound states have a different asymptotic behavior in the relative coordinates of the various particles. In fact they vanish at least as Y2 if their energetic thresholds are lower or equal to E, , and they vanish exponentially if their energetic thresholds occur at energies higher than E, . II.
EXERGk-
DEPEKDEKCE
We will assume that some of the discrete eigenvalues of the total Hamiltonian, H, fall onto its continuous spectrum, which is supposed to stretch from the mass, ~2, of the lightest particle to E = + 00. For the eigenstates belonging to these discrete eigenvalues of H we have
The total Hamiltonian is split into two parts, Ho and HI , which are the free and the interaction Hamiltonian, respectively.’ In ref. (1) , by making use of the Fock free-particle coordinate representation, consideration has been given to the spatial behavior of the states #/, ; from the normalization condition it has been established both that (~a(&),
H&!
(2.2)
= 0
and that +
.
(3.3)
is the eigenstate of Ho belonging to the eigenvalue E, and @‘(E,)
G@“(E,)
is given
(2.4)
= 6 E 1 H , e 0
where CPmeans that the Cauchy principal value has to be taken for the integral over the continuous spectrum of Ho understood at the right-hand side of (2.4 ) . In order to study the behavior of the element (p(E), HItie) for E --+ I!‘, , we make use of the Fock free-particle momentum representation. In what follows we shall restrict ourselves for simplicity to spinless particles. The introduction of spin and other degrees of freedom would make our notations very cumbersome, and it is anyway straightforward. 1 The continuous spectra of Ho and H are suitably h = c = 1 will be used.
chosen
to be the same.
Natural
units
242
FONDA
AND
GHIRARDI
We expand $, as follows: A = al PI + NT2 /- d3pl d3p, . . . d$i,-I U&PI , pz , . . - , PM)
-
.(pN(Pi
, P2,
. * * , PN-11,
N free particles where (ON(PI , pz , . . * , pN--I) is an eigenstate of Ho containing with linear relative momenta pl , p2 , . - - , pN-1, constructed in the usual way.3 The (ON’s satisfy h&l
>P2,
. . ’ PN-11,
+%U(pi,
pi,
. . ’ , &f-l) =
&fNs(pl
> -
&)@2
-
pz’>
-‘*
and (ol is normalized. The elimination of the center of mass motion and the zero eigenvalue for the total linear momentum is assumed Evaluating the norm of the state vector #, and requiring that normalized, one obtains the following condition on the meEicients sion (2.5) :
6(pN-1
-
&-1)
is understood, for simplicity. this state be of the expan-
* In Eq. (2.5) a sum over all possible kinds of one-particle (many-particle) states is understood. An accidental coincidence of the energies of some of these free one-particle states with some of the discrete eigenvslues of H will not be considered here. a One of the possible waps of constructing this system of relative momenta is to start with a set of relative coordinates chosen as follows : = the distance between any two chosen particles to be called 1 and 2, rl = the distance between particle 3 and the center of mass of the system (1 + 2), r2 r,%J-1 = the distance between particle N and the center of mass of the system ... + N - l), (1+2+ = the distance of the center of mass of the total system (1 + 2 + .f. + N) from the rN origin of the chosen frame of reference. The generating function 8’2 for the canonical transformation connecting the Cartesian coordinates ri’ and the new coordinates ri (i = 1, 2, ... , N) is given by Fz = 5 i=l
pk(pk’) are the momenta canonically is simply constructed from the recipe then
5 Ah; ri’.pk
.
k=l
conjugate to the coordinates given above; its determinant
rk(rk’). The is equal to +l.
matrix A We have
N
rk = f!?‘f = c Akdri’; pi’ = a; = gl Aki I aPk i=l The connection of the system)
between the relative and the old momenta
pk.
momenta (note that pN is the total linear momentum pk’ (k = 1, 2, . . . , N) is therefore given by pi = $l(A-l)aipk’
BOUND
STATES
ON
d3p2 ... s&I1 r?p,-1)
THE
UN(P1,
243
CONTINUUM
p2,
. . . ,
PN-1)
I2 5
0.6)
1.
By using Eq. (2.3) one obtains for the unr’s: dPl,PZ,
*.*
, PN-1)
=
((PN(PI
, P2,
*. * , PN-d,
=
(cpnr(pl
, PZ , . . . , ~~-11,
ik) HI
A)
E, - E
(2.7)
>
where E is the relativistic energy for the system of N free particles with relative momenta p1 , p2 , . . * , pN-l , and total linear momentum equal to zero. We have dropped the principal value symbol 6 since Eq. (2.2) holds. Substitution of Eq. (2.7) into (2.6) yields
s
d3pl d3p2
1 (ndp1,
. . . d3p,-l
~2,
. . . , ~~-11,
HI
tie)
(E, - E)3
I2 <
= I.
If we make the substitution pl , p2 , . . . pN-i + E’, (Y~, (~2 , . . . , a3N-4 , where the CX’Sare the eigenvalues of a set of variables which together with the energy operator H form a complete set of commuting observables, the above inequality transforms into the following: &3N-4 ’
<
L;+m2+...+mN
(j-j c
“E,,2
j
da1
da2
J
Ep’
’ ’ ’
P2 al(Y2
* ’ ’ PN--1 "'
(Y3N-4
>
(2.8)
. I (pN(E, a), HI $4 I2 I 1. J( Pl;LZ’PN;&-4) is the Jacobian of the transformation from the pi’s to the E, a's variables and is a function which never vanishes. ~11 , mz , . . . , )nN are the masses of the particles in the considered N-particle state. As the integral in E must converge, the integral in the ar’smust behave like (E:, - E)‘+’ with E > 0, if E, 2 ml + ~1~+ 3. . -I- ?)lN. Due to the fact that the integrand in Eq. (2.8) has a definite sign, it follows: 1 w+r From Eq. (2.7) we have also that
f&&E, a) z'ye (E, - E)-I'"+'.
(2.10)
Obviously, Eqs. (2.9) and (2.10) hold only for those states (PN(E,a) which are present at the energy E, , i.e., for those satisfying ml + na2 -f + . . -/- mN $ E, . It follows from Eqs. (2.9) and (3.13) of ref. (1) that the scattering amplitude is not infinite at E = h', . III.
SPATIAL
BEHAVIOR
For the study of the spatial behavior of the state tie , the state is split into two parts: J/P= $2” + #?‘, one (J/i”) for which the coefficients of the expansion
244
FONDA
AND
GHIRARDI
(2.5) satisfy ml + m2 + . . . + mN 5 E, ; the other for which the coefficients satisfy ml + m2 + . . . + ma > E, . From Eq. (2.18) of ref. (I), it follows that, when consideration is given to the relative coordinate rl of any two chosen particles, the wave function (a ( #?‘) goes to zero exponentially as rl goes to infinity. Coming to the consideration of $l”, it is better to take into account first its projection $2:’ on a two-particle state. Calling r the relative coordinate of the two particles, we have
k@(r)= & Integrating
1 d3peiP%2(p).
(3.1)
once by parts over angles we obtain dpp[eiP'a2(p)
-
e-ipra2(
+
-p)]
i(2?r)-3’2 d3pek”’ 1 h(p) s r i i!GFi
(3.2) ’
where cos e = p. r/pr; in the first integral by kp we mean &p(r/r). It is clear that the second term goes down asymptotically at least like r-‘, so that attention will now be paid to the first term at the right-hand side of Eq. (3.2). Use of Eq. (2.7) together with time reversal invariance shows that a~( -p) = a**(p) [see, for example, footnote 3 of ref. (I)]. We get, therefore, -l/2 ,j,;;‘(r)
According
=
j(2T;
m l
dpp
[eiPr
(q2$)f;h) e
_
c.c.
1
+ OF2).
(3.3)
to Eq. (2.10), at the point p = p, we have f(p)
~ P(cpz(P), Hr 9.) pyp, (Pe E, - E
P2+’
(3.4)
f(p) does, of course, depend on the chosen fixed direction r/r. Since (p2(p), HI+,) is not singular in the range of integration there follows that the function f(p) is singular, if it is at all, only at the point p = p, . f(p) can be expressed as f(P)
= (Pe - P)-“*+‘wP)
with h(p) regular at the point p, . By expanding we get f(p)
h(p)
= fo(pe - p)-l’z+e + fl(pe - p)““+’
in the neighborhood + ** * .
of p, (3.5)
f(P) - fo(Pe - P)-1’2+’ is a function with first derivative absolutely integrable in an interval including +p, . If the first derivative of f(p) is well behaved at infinity (6), the asymptotic behavior of the Fourier transform of f(p) is evalu-
BOUND
STATES
ON
THE
ated by using the standard theory of Fourier asymptotic spatial behavior of #z:’ (r) : G’(r)
;s6,
215
CONTINUUM
analysis,4
which
furnishes
[c exp (@. r> + c.c.1 + o(y-2) , )“3/2+t
the
(3.6)
where c is an unimportant constant. The oscillatory term in Eq. (3.6) is respon sible of the fact that the Fourier transform of J/j;‘(r) is divergent for p = p,, as indicated by Eq. (3.4), if 6 < 95. The three- or more particle projected portions of $~b<’ are analyzed as follows. Let $$ be the projection of #i<’ on a certain N-particle state, and let rl be the relative distance of the two chosen particles. From Eq. (2.5) we have s &P* gp2 . . . d3p~-1 exp (ip1 rl)a,(pl ‘(PN-l(p2
We now make a substitution P1 1P2 , . . . , PN-1 + E, pl , @a
similar ) a2
, . . . )
, pz , . . . , p3,
, PN-1)
(3.7)
. . ’ , PN-1).
to that used for the inequality a3N-7 ; we get
s
d3pl
exp
ipl
’ rl
(2.8) :
bN ( pl ) ,
where h,(pd
= ~;+m,+,,,+mN
dEj
da1 da2 . . . dw,-7
a~(-&
Note that bN(pl) is a vector in the Hilbert space. For Eq. (3.8) we use now the very same procedure we obtain
hIfiI2)
= -
i(27r-1’2 1’1
s0
OD dpl phxp(ipl
PI,
a).
a)pN-l(E,
(3.9)
used for Eq. (3.1),
and
dbN(pl)
(3.10)
- exp (-ipl
?*l)bN(-pl)]
+
o(rl’),
where fpl = fpl(r,/r,). It is immediately seen from (3.10) that the representative of the vector bN( fpl) in the coordinate r2 , r3 , . . . , rTpl representation is not infinite for any value of pl for fixed (rl/rl), r2 , r3 , . . . , rN-I , since the singular behavior of ox at the point E = E, is washed out by the integration on E’, and for E # E, the matrix element ((oN( E, pl , (.y), HI+,) is never divergent. It follows that the representative of the vector plbx(pl) has limited total fluctuation in the range 0 S pl < + 00 for fixed (rl/rl), r2 , r3 , . . . , rxV1 . Therefore by the use of the Riemann-Lebesgue lemma (3), we obtain from (3.10) that 4 See ref.
9, p. 52, Theorem
19; and p. 43, Table
I.
246
FONDA
&Y
AND
e.0
GHIRARDI
, TN-1
inthelimitrl+cc,,forri,r3,... , rN-1 clearly independent of the choice of the particle state. Comparing the asymptotic behaviors e < x in Eq. (3.6), #k;’ determines the
)
is
O(ryZ)
(3.11)
kept fixed. The result now obtained is pair of particles in the considered Nof #62’, #kf’ and I/:‘) we see that, asymptotic spatial behavior of #e .
if
ACKNOWLEDGMENT
We thank Dr. A. Rimini RECEIVED
1. 2. 3.
for discussions.
June 26, 1963
REFERENCES L. FONDA, Ann. Phys. (N.Y.) 22, 123 (1963). M. J. LIGHTHILL, “Introduction to Fourier Analysis and Generalized Functions,” p. 49, Definition 20. Cambridge Univ. Press, London and New York, 1959. E. T. WHITTAKER AND G. N. WATSON, “A Course of Modern Analysis,” p. 172. Cambridge Univ. Press, London and New York, 1958.
OF
Properties
PHYSICS:
26, 246-246
of the
(1964)
Bound
States
Embedded
LUCIANO Istituto
di Fisica
dell’Universitic, Gruppo
in
the
Continuum
FONDA
Parma; Istituto Nazionale di Parma, Italy
di Fisica
Nucleare,
AND GIAN Istituto
di Fisica
CARLO
GHIRARDI
dell’Universitd,
Trieste,
Italy
The energy dependence of the matrix element connecting the state vector describing the generic hound state embedded in the continuum with the unperturbed states is investigated in detail. The energy behavior is such that the scattering matrix turns out to be not infinite at the energy of the bound state embedded in the continuum. The asymptotic spatial behavior of the state vectors relative to these kinds of eigenvalues is analyzed.There results that the two-particle projected part of these vectors determines in most cases the asymptotic behavior of such states. I. INTRODUCTION
As it has been recently shown (1) , when discrete eigenvalues happen to be embedded in the continuum part of the spectrum of the Hamilton operator, the matrix element connecting the corresponding normalizable eigenstates with the unperturbed state pertaining to the same energy is zero. This circumstance allows the definition of the scattering matrix, together with the realization that this matrix does not exhibit a polar singularity at the energies of these bound states. The main purpose of this paper is to investigate the behavior of the aforementioned matrix element when the energy E of the unperturbed state is made to approach E, , the energy of the bound state embedded in the continuum. As will be shown in Section II, this matrix element behaves like (E, - JT)~“+‘, with c positive. This result makes the scattering matrix not infinite at E = E, . In Section III consideration has been given to the asymptotic spatial behavior of the wave functions of the bound states embedded in the continuum. It is found that the two-particle projected portion of these states behaves as r-312--r (t > 0) times an oscillatory term of wave number pe , relative momentum for the two particles of center of mass total energy E, . If E < 35, this oscillatory 240
BOUND
STATES
ON
THE
241
CONTINUUM
term makes the wave function infinite in the momentum representation just at the point p, . On the contrary, if the energetic threshold of t’he considered twoparticle state is higher t’hen E, , the asymptotic spatial behavior is exponentially decreasing. The three- or more particle projected portions of these bound states have a different asymptotic behavior in the relative coordinates of the various particles. In fact they vanish at least as Y2 if their energetic thresholds are lower or equal to E, , and they vanish exponentially if their energetic thresholds occur at energies higher than E, . II.
EXERGk-
DEPEKDEKCE
We will assume that some of the discrete eigenvalues of the total Hamiltonian, H, fall onto its continuous spectrum, which is supposed to stretch from the mass, ~2, of the lightest particle to E = + 00. For the eigenstates belonging to these discrete eigenvalues of H we have
The total Hamiltonian is split into two parts, Ho and HI , which are the free and the interaction Hamiltonian, respectively.’ In ref. (1) , by making use of the Fock free-particle coordinate representation, consideration has been given to the spatial behavior of the states #/, ; from the normalization condition it has been established both that (~a(&),
H&!
(2.2)
= 0
and that +
.
(3.3)
is the eigenstate of Ho belonging to the eigenvalue E, and @‘(E,)
G@“(E,)
is given
(2.4)
= 6 E 1 H , e 0
where CPmeans that the Cauchy principal value has to be taken for the integral over the continuous spectrum of Ho understood at the right-hand side of (2.4 ) . In order to study the behavior of the element (p(E), HItie) for E --+ I!‘, , we make use of the Fock free-particle momentum representation. In what follows we shall restrict ourselves for simplicity to spinless particles. The introduction of spin and other degrees of freedom would make our notations very cumbersome, and it is anyway straightforward. 1 The continuous spectra of Ho and H are suitably h = c = 1 will be used.
chosen
to be the same.
Natural
units
242
FONDA
AND
GHIRARDI
We expand $, as follows: A = al PI + NT2 /- d3pl d3p, . . . d$i,-I U&PI , pz , . . - , PM)
-
.(pN(Pi
, P2,
. * * , PN-11,
N free particles where (ON(PI , pz , . . * , pN--I) is an eigenstate of Ho containing with linear relative momenta pl , p2 , . - - , pN-1, constructed in the usual way.3 The (ON’s satisfy h&l
>P2,
. . ’ PN-11,
+%U(pi,
pi,
. . ’ , &f-l) =
&fNs(pl
> -
&)@2
-
pz’>
-‘*
and (ol is normalized. The elimination of the center of mass motion and the zero eigenvalue for the total linear momentum is assumed Evaluating the norm of the state vector #, and requiring that normalized, one obtains the following condition on the meEicients sion (2.5) :
6(pN-1
-
&-1)
is understood, for simplicity. this state be of the expan-
* In Eq. (2.5) a sum over all possible kinds of one-particle (many-particle) states is understood. An accidental coincidence of the energies of some of these free one-particle states with some of the discrete eigenvslues of H will not be considered here. a One of the possible waps of constructing this system of relative momenta is to start with a set of relative coordinates chosen as follows : = the distance between any two chosen particles to be called 1 and 2, rl = the distance between particle 3 and the center of mass of the system (1 + 2), r2 r,%J-1 = the distance between particle N and the center of mass of the system ... + N - l), (1+2+ = the distance of the center of mass of the total system (1 + 2 + .f. + N) from the rN origin of the chosen frame of reference. The generating function 8’2 for the canonical transformation connecting the Cartesian coordinates ri’ and the new coordinates ri (i = 1, 2, ... , N) is given by Fz = 5 i=l
pk(pk’) are the momenta canonically is simply constructed from the recipe then
5 Ah; ri’.pk
.
k=l
conjugate to the coordinates given above; its determinant
rk(rk’). The is equal to +l.
matrix A We have
N
rk = f!?‘f = c Akdri’; pi’ = a; = gl Aki I aPk i=l The connection of the system)
between the relative and the old momenta
pk.
momenta (note that pN is the total linear momentum pk’ (k = 1, 2, . . . , N) is therefore given by pi = $l(A-l)aipk’
BOUND
STATES
ON
d3p2 ... s&I1 r?p,-1)
THE
UN(P1,
243
CONTINUUM
p2,
. . . ,
PN-1)
I2 5
0.6)
1.
By using Eq. (2.3) one obtains for the unr’s: dPl,PZ,
*.*
, PN-1)
=
((PN(PI
, P2,
*. * , PN-d,
=
(cpnr(pl
, PZ , . . . , ~~-11,
ik) HI
A)
E, - E
(2.7)
>
where E is the relativistic energy for the system of N free particles with relative momenta p1 , p2 , . . * , pN-l , and total linear momentum equal to zero. We have dropped the principal value symbol 6 since Eq. (2.2) holds. Substitution of Eq. (2.7) into (2.6) yields
s
d3pl d3p2
1 (ndp1,
. . . d3p,-l
~2,
. . . , ~~-11,
HI
tie)
(E, - E)3
I2 <
= I.
If we make the substitution pl , p2 , . . . pN-i + E’, (Y~, (~2 , . . . , a3N-4 , where the CX’Sare the eigenvalues of a set of variables which together with the energy operator H form a complete set of commuting observables, the above inequality transforms into the following: &3N-4 ’
<
L;+m2+...+mN
(j-j c
“E,,2
j
da1
da2
J
Ep’
’ ’ ’
P2 al(Y2
* ’ ’ PN--1 "'
(Y3N-4
>
(2.8)
. I (pN(E, a), HI $4 I2 I 1. J( Pl;LZ’PN;&-4) is the Jacobian of the transformation from the pi’s to the E, a's variables and is a function which never vanishes. ~11 , mz , . . . , )nN are the masses of the particles in the considered N-particle state. As the integral in E must converge, the integral in the ar’smust behave like (E:, - E)‘+’ with E > 0, if E, 2 ml + ~1~+ 3. . -I- ?)lN. Due to the fact that the integrand in Eq. (2.8) has a definite sign, it follows: 1 w+r From Eq. (2.7) we have also that
f&&E, a) z'ye (E, - E)-I'"+'.
(2.10)
Obviously, Eqs. (2.9) and (2.10) hold only for those states (PN(E,a) which are present at the energy E, , i.e., for those satisfying ml + na2 -f + . . -/- mN $ E, . It follows from Eqs. (2.9) and (3.13) of ref. (1) that the scattering amplitude is not infinite at E = h', . III.
SPATIAL
BEHAVIOR
For the study of the spatial behavior of the state tie , the state is split into two parts: J/P= $2” + #?‘, one (J/i”) for which the coefficients of the expansion
244
FONDA
AND
GHIRARDI
(2.5) satisfy ml + m2 + . . . + mN 5 E, ; the other for which the coefficients satisfy ml + m2 + . . . + ma > E, . From Eq. (2.18) of ref. (I), it follows that, when consideration is given to the relative coordinate rl of any two chosen particles, the wave function (a ( #?‘) goes to zero exponentially as rl goes to infinity. Coming to the consideration of $l”, it is better to take into account first its projection $2:’ on a two-particle state. Calling r the relative coordinate of the two particles, we have
k@(r)= & Integrating
1 d3peiP%2(p).
(3.1)
once by parts over angles we obtain dpp[eiP'a2(p)
-
e-ipra2(
+
-p)]
i(2?r)-3’2 d3pek”’ 1 h(p) s r i i!GFi
(3.2) ’
where cos e = p. r/pr; in the first integral by kp we mean &p(r/r). It is clear that the second term goes down asymptotically at least like r-‘, so that attention will now be paid to the first term at the right-hand side of Eq. (3.2). Use of Eq. (2.7) together with time reversal invariance shows that a~( -p) = a**(p) [see, for example, footnote 3 of ref. (I)]. We get, therefore, -l/2 ,j,;;‘(r)
According
=
j(2T;
m l
dpp
[eiPr
(q2$)f;h) e
_
c.c.
1
+ OF2).
(3.3)
to Eq. (2.10), at the point p = p, we have f(p)
~ P(cpz(P), Hr 9.) pyp, (Pe E, - E
P2+’
(3.4)
f(p) does, of course, depend on the chosen fixed direction r/r. Since (p2(p), HI+,) is not singular in the range of integration there follows that the function f(p) is singular, if it is at all, only at the point p = p, . f(p) can be expressed as f(P)
= (Pe - P)-“*+‘wP)
with h(p) regular at the point p, . By expanding we get f(p)
h(p)
= fo(pe - p)-l’z+e + fl(pe - p)““+’
in the neighborhood + ** * .
of p, (3.5)
f(P) - fo(Pe - P)-1’2+’ is a function with first derivative absolutely integrable in an interval including +p, . If the first derivative of f(p) is well behaved at infinity (6), the asymptotic behavior of the Fourier transform of f(p) is evalu-
BOUND
STATES
ON
THE
ated by using the standard theory of Fourier asymptotic spatial behavior of #z:’ (r) : G’(r)
;s6,
215
CONTINUUM
analysis,4
which
furnishes
[c exp (@. r> + c.c.1 + o(y-2) , )“3/2+t
the
(3.6)
where c is an unimportant constant. The oscillatory term in Eq. (3.6) is respon sible of the fact that the Fourier transform of J/j;‘(r) is divergent for p = p,, as indicated by Eq. (3.4), if 6 < 95. The three- or more particle projected portions of $~b<’ are analyzed as follows. Let $$ be the projection of #i<’ on a certain N-particle state, and let rl be the relative distance of the two chosen particles. From Eq. (2.5) we have s &P* gp2 . . . d3p~-1 exp (ip1 rl)a,(pl ‘(PN-l(p2
We now make a substitution P1 1P2 , . . . , PN-1 + E, pl , @a
similar ) a2
, . . . )
, pz , . . . , p3,
, PN-1)
(3.7)
. . ’ , PN-1).
to that used for the inequality a3N-7 ; we get
s
d3pl
exp
ipl
’ rl
(2.8) :
bN ( pl ) ,
where h,(pd
= ~;+m,+,,,+mN
dEj
da1 da2 . . . dw,-7
a~(-&
Note that bN(pl) is a vector in the Hilbert space. For Eq. (3.8) we use now the very same procedure we obtain
hIfiI2)
= -
i(27r-1’2 1’1
s0
OD dpl phxp(ipl
PI,
a).
a)pN-l(E,
(3.9)
used for Eq. (3.1),
and
dbN(pl)
(3.10)
- exp (-ipl
?*l)bN(-pl)]
+
o(rl’),
where fpl = fpl(r,/r,). It is immediately seen from (3.10) that the representative of the vector bN( fpl) in the coordinate r2 , r3 , . . . , rTpl representation is not infinite for any value of pl for fixed (rl/rl), r2 , r3 , . . . , rN-I , since the singular behavior of ox at the point E = E, is washed out by the integration on E’, and for E # E, the matrix element ((oN( E, pl , (.y), HI+,) is never divergent. It follows that the representative of the vector plbx(pl) has limited total fluctuation in the range 0 S pl < + 00 for fixed (rl/rl), r2 , r3 , . . . , rxV1 . Therefore by the use of the Riemann-Lebesgue lemma (3), we obtain from (3.10) that 4 See ref.
9, p. 52, Theorem
19; and p. 43, Table
I.
246
FONDA
&Y
AND
e.0
GHIRARDI
, TN-1
inthelimitrl+cc,,forri,r3,... , rN-1 clearly independent of the choice of the particle state. Comparing the asymptotic behaviors e < x in Eq. (3.6), #k;’ determines the
)
is
O(ryZ)
(3.11)
kept fixed. The result now obtained is pair of particles in the considered Nof #62’, #kf’ and I/:‘) we see that, asymptotic spatial behavior of #e .
if
ACKNOWLEDGMENT
We thank Dr. A. Rimini RECEIVED
1. 2. 3.
for discussions.
June 26, 1963
REFERENCES L. FONDA, Ann. Phys. (N.Y.) 22, 123 (1963). M. J. LIGHTHILL, “Introduction to Fourier Analysis and Generalized Functions,” p. 49, Definition 20. Cambridge Univ. Press, London and New York, 1959. E. T. WHITTAKER AND G. N. WATSON, “A Course of Modern Analysis,” p. 172. Cambridge Univ. Press, London and New York, 1958.