a mathematical approach to the derivation of the …...centro di matematica e di pisica teorica del...

IL NUOV0 CIMENTO VoL. 2 A, N. 3 1 Aprile 1971

A Mathematical Approach to the Derivation of the Optical Potential.

]V[. BERTERO and G. t)ASSATORE

Isti tuto di ~cienze Fisiche dell' Unive~'sitd - Genova Istituto ~Yazionale di _Fisica Nucleate - Sezione di Genova

Centro di Matematica e di Pisica Teorica del C .N .R . - Genova

(ricevuto il 4 Agosto 1970)

'Iheoretical

S u m m a r y . - - A rigorous analysis of the foundations and of the properties of the theoretical optical potential is attempted. It is shown that, under suitable conditions on the two-body potentials, the optical potential exists both in a time-dependent and in a time-independent theory and is a bounded operator. This property does not require the boundedness of the two-body potentials. The analytic and asymptotic properties of the optical potential in the energy plane are obtained. The connection between the tirne-independent and time-dependent ~ormulation is discussed in a rigorous way.

1 . - I n t r o d u c t i o n .

The optical potent ia l has been derived from the Schr6dinger equation for

many-part icle systems in several ways.

The derivation has been made both in the framework of the time-depen-

dent (1) and of the t ime-independent scattering theory (2.3). However in both

eases a rigorous analysis of the foundations and of the propert ies of the optical

potent ial was lacking. Such an analysis can now be a t t empted since a rigorous

t r ea tmen t of the SchrSdinger equation and of scattering theory for many-

particle systems has been achieved in recent years.

(1) F. COESTER and It. K~MMEL: Nucl. Phys . , 9, 225 (1958). (2) H. FESn~ACH: A n n . o] Phys. , 5, 357 (1958); 19, 287 (1962). (3) K.M. WATSON: Phys. Rev., 89, 575 (1953); 105, 1388 (1957).

579

~ 8 0 M. BERTERO ~nd G. PASSATORE

This approach can give definite answers to many questions which remain completely open in a formal treatment. First of all there is the problem of the existence of the optical potential itself. Next, there are its general properties, such as boundedness, nonlocality, analyticity and asymptotic behaviour for large values of the energy. Then a clear connection among the different formal derivations, in particular between the time-dependent and the time- independent formalisms should be shown.

Such results may be employed to suggest models having the correct mathematical properties and containing a minimal number of parameters directly connected with the two-body potentials. These models wou]d be more signi- ficant than the empirical local optical potential and even the fictitious nonlocal potentials introduced in the last years (4).

In this paper we give an answer only to a part of the previous questions. We think it useful to summarize some concepts and results of scattering

theory which are needed in the following. We consider a Hamiltonian operator for the N-particle system of the type

± (1.1) 11'= - - ~. A i ~- ~ Vij(ri--rj)

i-12mi i<;=1

The centre-of-mass motion can be separated out and the ttamiltonian operator for the relative motion (in 3 N - - 3 dimensions) is

(1.2)

where

(1.3)

and

(1.4)

H = 11o + ~ v,,(xi) + ~ v i , ( x , - ~,1, i=1 i < ~ = 1

X i = r i - - r N

~V--1

1to = - - ~ ai~ V,. V~ , i , j = l

{a.} being a positive-definite symmetric matrix with real and constant elements. The operator H0 is self-adjoint in L*(R ~v-3) with domain

{ f( } (1.5) 9 0 = /eL~(R3X-8): l + ~p~) ](~,~/)(p,, ...,p~_l)l~dp,...dp~< co , 1=1

where ~'] denotes the Fourier transform of 1. Besides, if the two-body potentials satisfy the condition

(1.6) Vt~e L~(R ~) + L~(R3),

(4) F. PEREY and B. :BucK: Nucl. Phys., 32, 353 (1962).

A M A T H E M A T I C A L A P P R O A C t I TO T I l E D E R I V A T I O N ETC. ~ l

then the operator H, eq. (1.2), with domain ~o, is self-adjoint and is bounded from below (6). This implies tha t H has a complete sys tem of eigenfunctions

and tha t its spect rum is bounded from below, i.e. t ha t the N-part icle system has a ground state .

A channel g of the N-part icle sys tem is defined by specifying a par t i t ion of the 2V particles in f ragments F1, . . . , / ~ and a bound s ta te for each fragment . A state in the channel ~ is an e lement J~ ~ L~(/~ 8~-3) of the form

(1.7) 1~ =/(y~, ..., y¢_~) f l ~v;(x~), J=l

where the vectors y~ are the relat ive co-ordinates of the eentres of mass of the fragments , J(y~, ..., y~_~) is an a rb i t ra ry square-integrable funct ion of the variables y~ and ~0/.0~j) is the bound-s ta te wave function in te rms of the in ternal co-ordinates of the f ragment Fj . We denote by P~ the project ion operator on the subspacc formed by elements of the form (1.7) (e-channel subspace).

The channel Hami~tonian H~ is the full Hamil tonian H minus all interactions between the fragments. We call V~ ihe sum of all the interactions between the fragment~, so tha t

(1.8)

Clearly H~ commutes with P~

(1.9)

and

(1.10)

H = H ~ + V .

~CI=(Ho,~+~)~ / , le~o,

where Ho." is a differential opera tor of the type (1A) (describing the free motion of the centres of mass of the fragments) and s~ is the sum of the internal energies of the fragments, e~ is the lower bound of the spectrum of H~ and we call it the energy-threshold o/ the channel oc. I f we assume tha t the number of channels is finite, t hen we c~u order the channels with increasing thresholds. As proved by HUNZlKEI~ (6), if the two-body potentials satisfy eq. (1.6) in such a way tha t the component in L ' ( /~ 3) has an arbi t rar i ly small Z ' - n o r m ,

then the in te rva l [el, + cxD) belongs to the spectrum of H and the par t of the spect rum of H externa l to this in terval contains only isolated eigenvalues of finite mult ipl ici ty which accumulate to &.

The main results of the t ime-dependent scat ter ing theory are contained in the following theorem proved by HACK (7): if the two-body potent ials satisfy

(s) T. KATO: Trans. Am. Math. Soc., 70, 195 (1951). (o) W. HUNZIKER: Helv. Phys. Acta, 39, 451 (1966). (v) M. N. HACK: NUOVO Cimento, 13, 231 (1959).

5 8 2 ~ . B E R T E R O and o . P A S S A T O R E

the condit ion (*)

(1.11) v~j ~ L~(R ~) + .L'(R ~) , 2 4 p < 3 ,

then, for any channel a, the wave operators U~.~ exist and are defined on the whole L2(R 3~-a) as the s trong limits

(1.12) U~.~ ] = s-lira exp [itH] exp [-- itH~] P~ 1. t--->=i= co

U~.e are isometric operators on the a-channel subspace and satisfy the inter- twining relat ions

(1.13) HU~+ ~_ U~.±H~P~,

(1.14) P(~) ~L,~ = u~.~P'='(~) ,

where P(2) and P(~)(2) are the resolutions of the iden t i ty associated to H and

H~ P~ respectively. F rom this theorem it follows tha t there exists a unique solution of the problem

(1.15)

( 1 . 1 6 )

(1.17)

d i ~ /=( t ) = H/=(t),

H L ( t ) - e x p [ - i tH , ]L_H + 0

L._+ ~0 , P . L . - = £ . - , II1=.-II = 1 .

(t - + - - c ~ ) ,

The funct ion /~(t) is a time-dependent scattering solution originating in channel o~ and it is given by

(1.1s) i=(t) = exp [-itH] U=_L._ •

]~(t) is a s trongly differentiable funct ion with strongly continuous derivative. In this paper we use the previous general results of scattering theory in

order to prove that , if the two-body potent ials satisfy condition (1.11), the

t ime-dependent optical potent ia l exists for any channel ~ and is a bounded operator on L~(R 3~-3) (Sect. 2). The t ime- independent optical potent ia l is in t roduced as a (~ Fourier t rans form ~) of the t ime-dependent optical potent ia l

and is defined at least for the values of the energy which do not belong to the spec t rum of the operator ( 1 - - P ~ ) H ( 1 - - P ~ ) . In the case of the channel with the lowest threshold (i.e. a = 1) this restr ic t ion excludes the region, ve ry im- po r t an t physically, where inelastic scat ter ing is present . Such a l imitation

(') Condition (1.11) implies condition (1.6) and is satisfied by the potentials which are locally square integrable and decrease more rapidly than the Coulomb potential at infinity.

A M A T t t E M A T I C A L A P P R O A C H TO TI=IE D E R I V A T I O N E T C . 383

may probably be removed, but this requires detailed knowledge of the resol- ven t of the operator (1- -P~)H(1--P~) . The t ime-independent optical potent ia l we obtain coincides with the potent ial derived by :FEsttBACtt (2) and WATSO~ (3).

I t turns out to be a bounded operator and a holomorphic function of the energy in the energy plane from which the spectrum of the operator (1--P~). • H(1--P~) has been excluded; it also satisfies a dispersion-type relation (Sect. 3).

Finally the connection between the t ime-dependent and the time-independent (~ SchrSdinger equation ~) for the theoretical optical wave function is shown (Sect. 4).

2. - The t ime-dependent optical potential.

The derivation of the t ime-dependent optical potent ial has been outlined by COES~ER and K(~MMEL (1) aS follows. The t ime-dependent SchrSdinger equation (eq. {1.15)) is t ransformed into a set of coupled equations for the components P~ ]~(t) and (1 -- P~) ]~(t) of the t ime-dependent scattering solution originating in channel a (eqs. (1.16), (1.17)); then the component (1--P~)l~(t) is eliminated in order to obtain a t ime-dependent (( Sehr6dinger equation ,> for the component P~,]~(t) alone, i.e. for the component in the g-channel subspace.

In (1) only a channel with two fragments is considered ((~ nucleon-nucleus ~> scattering), bu t i t is not difficult to consider an arbi t rary number of fragments. In this way to every channel ~ corresponds a t ime-dependent optical potential. We introduce the notations

(2.1)

and

(2.2)

uAt) =P~tAt), vat) = (1 -P~) /A t )

~( t ) = exp [ite~]%(t);

we shall call V~(t) a time-dependent optical scattering solution originating in channel o~. We prove that , i] the two-body potentials satis]y condition (1.11), then F~(t), eq. (2.2), is a solution o] the equation

t

d t" (2.3) i ~ y~(t) = (Ho.~+ V(~av')y&(t) +_1 U~(t--s) V~(s) ds

--co

with the asymptotic behaviour

(2.4) II ~v~,(t) - exp [ - itHo.,] ]~,.-II ~ 0 (t -~ -- co),

584 M. B]~RTERO and G. PASSATORE

where Ho.~, is defined in eq. (1.10); V(~ "~ is a multiplicative operator o/ the type

(2.5) V (a')'' ~ V c'v, ( - ) , c` (YJ = r ~ , ~ Yah a < b - 1

V ( & V ) / ~',.~'btYab) being, in terms of the relative position Yah o/ the eentres o/ mass o/ the fragments ~'~, ~b, the sum of the interactions between the two fragments averaged over the fragments' wave functions; finally U~(t) is defined by

(2.6) U~(t) : --iPc` V~,(1--Pc`) exp [ - - i tYfa](1--Pc, ) Vc̀ Pc`,

(2.7) ,ff~, = ( 1 - P ~ , ) H ( 1 - P ~ , ) - ~ , .

The average potentials V (~ satisfy condition (1.11) and are bounded and contin- • p a , p b

uous functions of Yah so that they define bounded operators on L2(t~3~v-s). Uc`(t) is also a bounded operator and there exists a constant C~ independent o/t , so that(*)

(2.8) [ U~,(t)[ < C~,

/or any t e ( - - c ~ , +oo). Besides,/or any/eL2(R3~-3), U~(t)/ is a strongly continuous /unction o/ t.

Proof. F r o m the SchrSdinger equa t ion (1.15)~ b y means of eqs. (2.1), {1.9), (1.10)~ (1.16), i t follows t h a t

d (2.9a) i-~tuc`(t) = (Ho.~+ ec̀ +P~V~P~)uc`(t) +-P~Vc`(1--P~)vc`(t),

d (2.95) i-~tvc`(t ) = (1--Pc`)V~Pc`u~(t) + (1--_P~)H(1--P~)vc`(t) ,

(2.9c) ]Juc`(t) - - e x p [--itH~]/~._ll -->0 ( t -+- - (x)) ,

(2.9d) II vat)11 -~ o (t - ~ - ~ ) .

In o rder to e l imina te vc`(t) we p u t

(2.10) v ~(t) = exp [-- it(1 - - P~) H (1 - - P~) ]z ,(t) .

zc`(t) is s t rong ly d i f ferent iable wi th s t rong ly cont inuous der iva t ive , as ]c`(t) is. Prec i se ly ,

d (2.11a) i~zc`( t ) ---- exp[it(1--Pc`)H(1--P~)](1--P~)Vc`Pc`uc`(t),

(2.11b) ll~(t) ll ~ o (t - ~ - ~ ) .

(*) In this paper we denote by ]A] the norm of a bounded operator A.

A M A T H E M A T I C A L A P P R O A C H TO T H E D E R I V A T I O N E T C . 5 ~

Both sides are integrable on any finite interval, so that , by means of the defi- nitions (2.2) and (2.7),

(2.12) z~(t) = z~( T) -- ifexp [is~/f~](1 -- P~) V~ P~ yJ~(s) ds .

T

This equation holds for any T. Since, by eq. (2.11b), z~(T) tends to zero for T-+ c% the integral in eq. (2.12) has a limit, and we have

(2.13)

t

z~(t) ---- -- i f exp [is~f~](1 -- P~) V¢, P~ ~f~(s) ds ,

the integral in this equation being precisely the limit in norm of the integral in eq. (2.12).

From eqs. (2.13), (2.10), (2.9a) it follows tha t

(2.14) d

- - iP~V~(1 --P~) exp [ - - i t ( ~ + s )]fexp - P~)v~ P~ ~f~,(s) ds . --co

In Appendix A (Lemma A.2) it is shown tha t the operator P~ V~( I - -P } can be extended, in a unique way, to a bounded operator on L2(R 3~-3) if the potentials V~ satisfy condition (1.11) (it must be noted tha t the operator V~ is not in general a bounded operator). Then in eq. (2.14a) the operator P~V~(1--_P~) can be brought under the integral. Moreover in Appendix A (Lemma A.3) it is shown tha t the operator P~ V~_P~ is of the form V~v)P~, where V~ ~) is defined as in eq. (2.5) and has the properties stated above. By subst i tut ing in eq. (2.14) ~o(t) (defined in eq. (2.2)), eq. (2.3) is obtained.

The asymptot ic condition (2.4) follows from the condition (2.9c) and from the eqs. (2.2) and (1.10). Equat ion (2.8) follows from the fact tha t the oper- ,~tors P V~(1- -P) and (1--P~)V~P~ can be extended in a unique way to bounded operators on L2(Ra~-3). Finally, the strong continuity of U~(t)]~ JEL~(R3~-3), follows from the strong continuity of

exp [-- i t ~ ] g, g 6 L~(R 3~-3) ,

where g = (1 - -P~)V~P~]. We conclude this Section with a few remarks. In the derivation of eq. (2.3)

(the t ime-dependent (~ SehrSdinger equation ~) for the optical scattering solution) we have followed the procedure of COESTER and K/2~x~EL, but considering

586 M. BERTERO and G. PASSATORE

a channel with an arbi t rary number of fragments. Besides, differing from these authors, who define the time dependent optical potential by means of a functional equation, we have given an explicit expression for the operator U~(t) (eqs. (2.6) and (2.7)). Such an expression, as we discuss in the following Section, allows the introduction of the Feshbaeh t ime-independent optical potential. Fur thermore we give sufficient conditions on the potentials V~j for the existence of the optical potential operator. Such conditions do not include potentials which are strongly singular at the origin nor the Coulomb potential. As it concerns the behaviour at infinity, however, it is sufficient tha t the two- body potentials decrease faster than the Coulomb potential .

3. - The time-independent optical potential.

The t ime-independent optical potential is often defined as the Fourier t ransfo lm of the t ime-dependent optical potential (~.s.9). However a correct s ta tement of this definition requires some care.

Let ~(Jt(~) denote the resolvent set of the operator ,,%f~ (eq. (2.7)); then, for any E ~ ~(j/o ) and real (*) we can define on L~(R 3~-~) the following operator:

(3.1) 0

U~(t) being defined in eq. (2.6). indeed, since the operator Jt~ is self-adjoint, for any ]~L~(R 8~-a) the following relation holds (see (~0), p. 241):

+co

(3 .2) ~o~(Z)f = ( Z - - ~ ) i f = _ifexp[i t(z_~to)]/dt , I m z > 0 ,

o

so tha t it clearly results tha t the operator SZ~(E) exists, is bounded and is given by

(3.3) ~ ( E ) ! = V'2")_VJ + ~ 7~(1 -- P~)~(E) (1 --P~) Y~P~I.

Equation (3.3) shows tha t the operator ~ ( E ) coincides with the time-independent optical potent ial introduced by FESttBACI-I (2).

(s) R. LIPPERHEIDE: Yuc1. Phys., 89, 97 (1966). (9) J . M . CORNWALL and M. RUDERMAN: Phys. Rev., 128, 1474 (1962). (*) As will be clear in the next Section, the parameter E is the total energy of the N-particle system minus the energy threshold of the channel ~, e~. (lO) K. YOS~DA: Functional Analysis (Berlin, 1965).

.4. MATItEMATICAL APPROACH TO THE DERIVATION ETC. ~ 7

The restr ict ion E ~ ~(5tF~) can perhaps be weakened. Indeed eq. (3.3) makes it clear tha t ~ ( E ) is not defined for the values of E which belong to the point spectrum of ~ (i.e. t h e eigenvalues of 5~ ) , bu t it is not excluded tha t ~ ( E ) can be defined for values of E which belong to the continuous spec t rum o f ~ . However, in this case, we need a knowledge of the proper t ies of the resolveut of Y['~, which is beyond the scope of this work.

In order to point out the phy~ical meaning of the condition E ~ ~(~f~), it is impor tan t to know the s t ructure of the spectrum of the operator YF~. In a subsequent paper (11) a three-part icle system is considered and this result is proved: if the two-body potent ia ls satisfy the condition (1.6) in such a way tha t the eomponen/~s in L ~ can be chos~,n arbi t rar i ly small, then the spectrum of ( 1 - - P ~ ) H ( 1 - - P ~ ) contains the in terval [s, ~- c~), where s =- s~ for ~.-~ 1 {channel with the lowest-energy threshold) and ~- -S l for the other chan-

nels. The par t of the spect rum of (] - -P ~ )H (1 - -P~) externa l to [s, d- c~) contains only isolated eigenvalues of finite mult iplici ty which accumulate at most to ~. Besides, the lowest eigenvalne of ( 1 - - P ~ ) H ( 1 - - P f l cannot be below the lowest eigenvalue of H.

We assume tha t these results, which have been proved only for a three- particle system, hold in the general case. I t follows tha t only for the channel a = l does eq. (3.3) define the optical potent ia l for positive values of the channel energy E ( 0 < E < e 2 - - e l ) , This in terval corresponds to purely elastic scat- ter ing (usually this channel is a two-fragment channel) and for almost all values of E in this in terval the operator ~ ( E ) exists and is a bounded and self-adjoint operator. As regards the isolated eigen-¢alues of ~ on this interval, t hey are poles of ~ ( E ) of order one.

In the case of an a rb i t ra ry channel ~, eq. (3.1) defines the operator $/:(E) only for almost all the values of E < Q - - ~ < O.

As a consequence of the theorem proved by I-IuNzIKER (6) and mentioned in the Introduct ion, only bound states of the iv-particle system are possible for these values of the energy.

The connection between the bound states of the IV-particle system and the (( eigenvectors ~ of the optical <~ Hamil tonian )~ Ho. ~ -~- :~ (E) is the following:

I f ~fli i8 a sol~t ion o] the p r o b l e m

(3.4) ~2 i G 9 o ~ H~fli --- ~ i (2 i< el),

then ~p~.~ = P~Fi sat is] ies the equat ion

(3.5)

(11) M. BEaTERO" IYuovo Cimento, 2, 605 (1971).

~ M. BE, R T E R 0 a n d G. PASSATORE,

Conversely, i/ y~:.i ~ ~o is a solution of eq. (3.5), then

,~, = ~=., + ~AE=.,)(~ --P=) v=~=.,

is a solution o/ the problem (3.4) with 2~- E:,~ + s:. The proof is s traightforward if one t ransforms eq. (3.4) into a set of equations

for the vectors P~y~ and (1--P~)~ and if one observes tha t E / ~ _ ~(~f ) (apart f rom an accidental coincidence with an eigenvalue of g f : ) (*). For the channel a = 1 the correspondence between the eigenvectors of H and of rio. ~ ÷ ~ ( E ) can be ex tended to the region 0<E~<~2--e~. Indeed bound states embed- ded in the cont inuum cannot be excluded for a many-body Hamil tonian (like H) as well as for an effective two-body t tamil toniun with a nonlocal potent ial

(like Ho.1 -~-$/1(E)). We prove now some general proper t ies of the operator ~ f l E ) : if E E ~(.~f:),

then:

i) $/':(E) is a bounded operator on L2(R 3~-3) and $/:(E)* -- ~ ( L ' ) ;

ii) $F(E) is a holomorphic function of E in ~(gf~);

iii) ] ~ ( E ) - - V<:~)-P:I -+ O, IEI -> 0% largE] > 5 > 0;

iv) /or any ] ~ L~(R 3z+-~) the following relation holds:

(3.6)

s-lim s-lim A ~ ( E ' ) f dE ' , ÷ ~/~:+o+ , : + o + E ' - - E

a.-+-} -¢° ~a.O-- ~

where the numbers #~.i are the eigenvalues of gf~ bdow #~.o (#~.o - Q -- ~ ~ 0 if ~ :/: 1, #1.0----s2--~ ~ 0); ~ is the projection operator on the eigenspace o/ gg~ corresponding to eigenvalue #~/, and

(3.7) A, ?f~(E) -~ ~'=(E ~- i~) -- ~ ( E - - i~) .

/1~$/~(E) is a dissipative operator, i.e.

(3.S)

for any ] c L~(t~3~-a).

Im (A ~ (E) f, l) < 0

(*) The state P ~ is physically important to the extent in which the N-particle bound state can be approximately described as made up of the fragments corresponding to the ~-channel.

A M A T H E M A T I C A L A P P R O A C H TO T H E D E R I V A T I O N E T C . ~

Pro@ In eq. (3.1) the operator ~ ( E ) is defined only for real values of E. However, by means of eq. (3.3), it can be easily ex tended to all w l u e s of E e ~(~=).

Proper t ies i) and ii) follow immedia te ly from similar propert ies of the resolvent of a self-adjoint operator (we recall once more tha t the operators P~,V~,(1--P~,) and (1--P~)V~P~ are bounded- - see Appendix A).

The p rope r ty iii) follows from the inequalities

1 (3.9a) I~=(E)I < ii m_E~ , Im E ¢ 0

(which holds for any resolvent of a self-adjoint operator) and

1 (3.9b) I~=(E)I < , Re E < #=.~

IE--

(which holds for any resolvent of a self-adjoint operator whose spectrum has a lower bound).

In order to prove p rope r ty iv) we introduce the operator

= ~,(E + i ~ ) - - : ~ ( E - - i~]) , r t > O, (3.10) A,~ , (E ~) ~ ' ,

and we note that , if / n l E : ~ 0 , then (E'--E)-~A,~,(E ') is ~ bounded and continuous operator-valued function of E ' ; therefore it is integrable over any finite in terval [~, a]. Moreover, if ~(~)(2) is the resolution of the ident i ty associated with ~%f, the following relat ion holds (*):

(3.11)

0 O--0

s-lim,._~+ ..r E l - - E ~ I ,~(E ' , / dE' : 27df E 1 ~ d~<~'( ,~) ]_ + q 0+0

ni 7d + ~ [~<~'(e) -- ~(~>(~ -- 0 ) ] / + ~ _ ~ [~<~>(~) -- ~'~>(~-- 0)]] .

In the limit ~ - ~ - - c ~ , ~-+ + c ~ it follows, recalling the general propert ies of

a resolution of the ident i ty and the spectral representa t ion of the resolvent of a self-adjoint operator:

(3.12) f 1 2 ~ i ~ ( E ) ] = s - l i m s-lim ~ An~,(E')]dE'. O-+--a~ ~--)-0 + a-->+eo O

(') This relation call easily be proved by means of the spectral resolution of the resolvent of a self-adjoint operator - - see, for instance 0o), p. 324.

~ 9 0 M . B E R T E R O and G. P A S S A T O R E

As the operator P~ V~(I--P~) is bounded, from eqs. (3.12) and (3.3) it follows that

(3.13) $/=(E)/= v<~)P ~ + ~ . s-lim s l im A , ~ ( E ' ) / d E ' a:-++¢o

(Ira E =/- 0).

Only the values of E' in the spectrum of g / contribute to this integral because otherwise IAnY/=(E')]-+0, as follows from the resolvent equation (see 0°), p. 211)

(3.14) I r / ~ o • ~ A E ) - - ~ ( E ) = (E - - E ) ~ A E ) ~ = ( E ) , E, ,E ' ee (~ , )

If we assume, in the general case, the results proved in the subsequent paper (~) for a three-particle system, then the integral in eq. (3.]3) is an integral over the interval [v~:.+,-F oo) plus contributions from the isolated eigenvahms #~.~ in the domain E < #~.o. Now, in a neighbourhood [#~.~-- d, #~.j q- ~] of #~.~ with 5 sufficiently small in order to exclude other eigenvalues, we can write

l (3.15) ~ : (E ' ± iT) -- E ' - - #:,j -[: i T

(~) ~ j + 5P=.gE'± iv) ,

E! where ,9o( ± iT) is a continuous function of E' and ~]>0. Therefore,

(3.16) s~lim~_+0+ f ~ / ~ 1 AnY/(E,)]dE,=

~ a . j + ~

=-P~,V~,(I--P~,)s lim f 1 2i T ±T2~,,(l_p~,)V~,p~,/dE, =

2~i - - a~.j

From eqs. (3.13), (3.16) and from the previous remarks, eq. (3.6) follows for complex values of E. Then, by a limiting procedure eq. (3.6) follows for any E e e(=~,,).

We have still to preve eq. (3.8); frGm eq. (3.]4) we get

(3.]7) I m (z],7 ~ ( E ) / , l) = ]m (zJn~c,(E)(1--P~,) V~,Pc, I, (1 --Pc,)Ve, Pc,/) =

= Im {--2iv(~=,(E--iT)~(E + iT)(1 --P~,) V~,P~,I, (1 --P~)V~,P~,I)} =

= --2T ]-{e (RgE + iV)(] - - ~ ) V= e~t, ~ ( E + iT)(1 --P~) V~ L I ) =

A M A T H E M A T I C A L A P P l Z O A C I I T O T I I E D E R I V A T I O N E T C . 591

We conclude this Section by observing tha t in ref. (~) and ref. (3) the operator Yf~(E) is defined through a functional equation. This equat ion can be wri t ten as follows:

(3.~8) ~ ( E ) = 1 + G ( E + c=)(~ --P~) V. ~AE) ,

where R~(2) = (2 - - H~) -~. I t easy to see that , if I m E ~ - 0 , then the operator ~ ( E ) defined in

eq. (3.3) is a solution of eqs. (3.18). Anyway, apar t from the specific interest of this formulation, such as, for example, mult iple-scat tering approach, eqs. (3.18) are less convenient than the explicit form (3.3), because they are less suitable for the analysis of the propert ies of Yf~(E) derived in this Section.

4 . - Connection between the time-dependent and the time-independent for- mulations.

In this Section, from the t ime-dependent equat ion proved in Sect. 2 (eq. (2.3)) we derive a t ime- independent equat ion which involves the potent ia l ~ ( E ) .

I t is convenient to wri te eq. (2.3) in a modified form which, on the other hand, turns out to be more general than eq. (2.3) since it allows also bound-state solutions. The limit for T-~--cxD of the t-integral which appears in eq. (2.3) exists so tha t it can be replaced by the Abelian limit:

(4.1)

t

s-lim f G,(t--s)y,~,(s)ds= s-lim f ,--~0+ exp [-- ~(t - - s)] y&(s) ds .

Equa t ion (2.3) can then be wri t ten as follows:

t

• d c,v, f (4.2) ~ y & ( t ) = ( H o . ~ + V, )y&( t )+s - l im U~(t--s) exp[--~](t--s)]yJ~(s)ds. --co

In order to derive the t ime- independent equat ion we need a representat ion of the t ime-dependen t optical-scattering solution F~(t) in terms of (~eigenfunctions ~ with given energy. Such a representa t ion is obtained as follows

592 M. ~E~T~aO and G. PASSATORE

(see eqs. (1.18), (2.1) and (2.2)):

(4.3) ~p:(t) = exp [ite:] P : exp [ - - i tH] U : _ ] : _ = a

= exp [item] P~ s-lim fexp [-- it2] dP(~) r)~_ J~._ = a----~+ ~ d

E~

-- s-lim ~exp [-- irE] dE(P~,P(E + s~,) V~,._]~,._) = " ~ + ~ J ~-~ ~ J s-lim fexp [-- itE]dq~=(E) 0 D

where E = ~ - - s : (energy in the channel a), P().) is the resolution of the iden- t i ty associated with H and

(4.4) + : (E) = P : P ( E + ~:) U : . _ L _ •

We note tha t the (( state ,) ~ ( E ) would be the projection on the ~-chan~el subspace of a superposition of the usual t ime-independent outgoing scattering solutions with energy less than E + E~, The integrals in eq. (4.3) are defined as strong limits of l~ieman_a sums; the lower l imit of the +~-integral in eq. (4.3) is s~ owing to eq. (1.14).

In this Section we consider only wave packets, i.e. opticM-scattering solutions ~f=(t) formed by stutes with energy in an interval [Eo--5, E0 + 5]; they have the form

Eo+6

(4.5) ~0=(t) = / e x p [-- itE] d ~ ( E ) (Eo-- 5 > 0, 5 > O)

Eo--O

and they correspond to an ]~_ which satisfies the condition

(4.6) ]=._ = [PC~(E o + e~ + 5) -- P<~)(E o + s~-- 5)]1=._,

where P(~>(2) is again the resolution of the ident i ty associated with H~P~. Equat ion (4.5) follows from eqs. (4.3) and (4.6) by means of eq. (14).

h~ow the following results can be proved:

i) Equation (4.2) has stationary solutions o] the type

(4.7) exp [--itE~.~]~.~, Y~.~ ~ ~o ,

with E~.~ ~O(gf~) i/ and only i] ~=.~ is a solution o] the equation

(.t.s) (Ho.= + W~(E=.,)) ~,=., = E=.,W=.,.

A MATHEMATICAL APPROACH TO THE DERIVATION ETC. 59~

The proof follows immediately from eqs. (4.2) and (3.2). These solutions ~re related to the bound stutes of the/V-part icle system (see eqs. (3.4) and (3.5)).

ii) I] the time-dependent optical-scattering solution ~f~(t) is a wave packet o] the type (4.5), then

go+6

(4.9) s-lim [{H o : ÷ ~ ' : (E + i~) - - E} dF:(E) = 0 . ~--~0+ ,J

Eo--c~

Besides in the channel with the lowest threshold, i/

[Eo-- ~, Eo + 5] c ~)(~'~)

(purely elastic scattering), then

(4Ao) ~o+~

({Ho.1 + W I ( E ) - E}d~0~(E) = 0 .

Eo--6

In order to prove these results, the following lemmu is needed. I] yJ~(t) is a wave packet as in eq. (4.5), then

t

<4.H) exp [ - -~( t - - s ) ]~ , ( s )ds = - - oo

Eo+~ =fexp [ - i t E ] P~ V ~ ( 1 - P~,):~,(E & i~])(1 --P=) V~ P~ d~=(E), Eo--(~

where the integral at the r.h.s, is the strong limit o] the Riemann sums

( 4 . 1 2 ) ~ e x p [ - - item] -~a Vc,(1 - - Po~)~a(E I. -~/~])(] - - Pa ) V~ r ~ [(~c~(Ej+1) - - ~ a ( E ~ ) ] J

! with Eo-- 5 =~E~<E2< . . .< E,~+I ++Eo + 5, Ej e (E~, Ej+~] when max IEj+~--Ejl -~ O.

Proof. By substi tut ing the representution (4.5) into the 1.h.s. of eq. (4.11) and reculling that the integral in eq. (4.5) is the limit of Riemann sums, we have

(4.13)

t

f u A t - - s ) exp [ - - ~ ( t - - s)] ~f=(s) ds = --¢o

t

--co

38 - II Nuovo Cimento A.

5 9 4 M. B E R T E R O a n d G. P A S S A T O R E

In order to prove eq. (4.11) it suffices to show tha t in the r.h.s, of eq. (4.13) the limit and the integration can be interchanged (then, by eqs. (2.6) and (3.2), the r.h.s, of eq. (4.13) becomes the limit of Riemann sums of the type (4.12)). As this is possible if the integration interval is finite, such interchanging in eq. (4.13) is assured if the Riemalm sums are uniformly integrable. Now, the Riemann sums in eq. (4.13) are uniformly bounded for t e (--c~, -+ ~ ) . Indeed, from the definition (4.4), from the properties of P(~) and of the wave packet (4.6), we have

(4.14) I] ~ exp [--itEm] [~0~(E~+l) -- ~(E~)] I12 ---- J

= I]P~ ~ exp [--itEm] [P(E~+~ + e~) - -P(Ej + e~)] U~_J~._[I2< J

U 2 ~ < ll~ exp [-- itEm] [P(E~+I + s~) -- P(Ej + s~)] ~.-]~.-[1 J

= ~ II[P(E~+~ + s ~ ) - - P ( E ~ + s~)] U~_]~_[] 2-- i

= U 2 U 2 II[P(Eo + s~, + (~)--P(Eo + e:-- d)] ~,.-/~,.-II -: I] ~,.-/~,.-II = 1 .

F rom eq. (2.8) it follows that , if d' is an arbi trary positive number, then a T < 0 can be found, with IT[ large enough, such that , for any t fixed, the following relation holds:

(4.15)

T

f ) U~(t-- s) exp [-- ~(t-- s)] exp [-- ~sEi] [ ~ z ( E ] . [ _ I ) - - q~(Ej)] ds < --co

T

< C=fexp [-- ~](t -- s)] ~ e x p [--isE;] [<~,(Ej+~)--<p:(E,)] d s<

T

< C~fexp [-- ~(t -- s)] ds = C_~ exp [~(T--t)] < d' .

Such a bound is independent of the particular Riemann sum, thus the lemma is proved.

I t is now straightforward to prove eq. (4.9). From eqs. (4.2), (4.5) and (4.11) it follows tha t

(4.16) f E exp [-- itE] dq~(E) = f (Ho.~ ÷ V<2V)) exp [-- itE] dq~(E) +

Eo+5

+ s-lim (P~ V~(1 -- P~)~(E + iV)(1 -- P~) V~ P~ exp [-- itE] d?~(E). 71----~0+ J

go--(~

A MATHEMATICAL APPROACH TO TIIE DERIVATION ETC. 595

If we consider this equat ion at the t ime t = O, remember ing eq. (3.3), we obtain eq. (4.9).

In order to prove eq. (4.10) we first observe tha t , if

(4.17) Ec[Eo--b , Eo+ 5] c ~)( ~f~),

t hen the operator ~ ( E + i~), V > 0 has a l imit for ~ ~ 0 and the limit is bounded operator on L~(R 3N-3) (see Sect. 3). As a consequence of a resul t proved in Appendix B, the integral

(4.18)

Eo+6

f G ( E ) d%(E)

Eo--6

exists as a limit of Riemann sums. i ndeed the hypotheses of Lemma B.2 are satisfied because ~11(E) fulfils the conditions (B.3e) and (B.3d) (as follows from eq. (3.14)) and %(E) is ~ continuous function of E as follows from

(4.19) ~ ( E ) = / )1 U~.-P('~( E + s~)/~,-

(see eqs. (1.14) and (4.4)), if we recall tha t HxPs has a purely continuous spectrum. We are then reduced to showing tha t

(4.20) Eo+~ Eo+6

s-limn~o+ fY~(E + ~)d(]91(J~ff ) =f~ l ( J~) d~Pl(E) . Eo--6 Eo--6

From eq. (3.14) and from the boundedness of the operator P1VI(1--P~) it follows tha t

(4.21)

Eo+6

f [G(E +

E0--5

i~) - - G ( E ) ] d~,(E) =

Eo+6

E o --~

i,t]) ~I(E)(1 - - _P~)V 1 _Pl d%(E) .

As the in tegrand satisfies the conditions (B.3c) and (B.3d) with a constant N independent of ~, from eq. (B.15) we have

~e+6

(4.22) f[ (E + i,]) -- ~I(E)] d%(E) < ~N(45 + 1) -~ 0 (~ -~ 0). E0--3

Some concluding remarks are worth-while mentioning.

~ 9 6 M. BERTERO and G. PASSATORE

Firs t ly we point out tha t in eq. (4.9) the operator ~ ( E + iT) with ~ > 0 appears. This follows from the fact t ha t we have considered t ime-dependent scattering solutions originating in channel ~ at t = - c~ (i.e. superpositions of t ime-independent outgoing scattering solutions originating in channel a) and from the causal character of eq. (2.3). Indeed the source of the factor ~ > 0 is eq. (4.1).

Therefore eq. (4.9) can be the s tar t ing point in order to just i fy that , in the energy region where inelastic scat ter ing is present , the optical potent ia l is defined as the l imiting value of the holomorphic function $':(E) on the upper r im of the cut in the energy plane. Of course the existence of the limiting value cannot be deduced from the results of this paper and requires u separate analysis. General ly speaking, in order to prove the existence of the limiting value, the class of the two-body potent ials must be strongly restr icted.

Besides eq. (4.11) shows tha t the wave packet v~(t), eq. (4.5), with [Eo-- 5, Eo + 5] c ~(,Zf~), satisfies approximate ly the equation

d (4.23) i ~ yJl(t) = (Ho,~ + ~(Eo)) ~p~(t),

if 5 is ve ry small. Indeed from eq. (4.11) in the l imit ~ = 0 it follows tha t

(4.24)

t

V~V'y~l(t ) + s-lim,~o + f u,(t - s) - - c o

exp [-- ~/(t - - s)] Vii(s) ds - - 3~(Eo)y~l(t) =

Ea+6

=- fexp [--itE]{Y/'~(E)--3~(Eo)} d?l(E), Eo--/t

and evaluat ing the integral by means of eq. (3.14) and of the bound (B.15), we get the result tha t the norms of both sides of eq. (4.24) are of the order of 5.

Finally, f rom eq. (4.23), it follows tha t the wave packet ~vl(0) satisfies ap- p rox imate ly the equat ion

(4.25) (5/o., + ¢~(Eo))~,(0) = E o ~ ( 0 ) ,

where again the neglected terms have a norm of the order of 5.

We wish to t hank Prof. J. P. CECCONI for helpful discussions.

A M A T H E M A T I C A L A P P R O A C H T O T H E D E R I V A T I O N E T C . 597

A P P E N D I X A

I n this Append ix we prove some proper t ies of the operators P~V~P~, P~V~(1--P~) and (1--P~)V~P~ which have been essent ial in der iving the results conta ined in this paper .

We quote first a lenima proved by KAT0 ('~).

Lemma A.1. - Define /or every /~L2 (R 3N-3) the operator

(A.~) (A~/)(xi) = { f l/(x)12dx~.., dx~-~dX~+l.., dx,v-~} ~ •

I / ] c ~ o , then the /unctions (A~/)(x~) ( i = = a , . . . , N - - 1 ) are bounded and continuous and satis/y the relation

(A.2) o < (A J)(x~) < a' II Ho] IIz~(R3N-3, ÷ b' I i /I1. ,~-~,,

where a' and b' are independent o/ / and a' is arbitrarily small. We do not need this L e m m a in its complete form bu t only the result t h a t

for any ]~ ~o there exists a const~mt Cj > 0, such t ha t

(Ai / ) (x~) < C¢ (i = 1, . . . , N - - i ) .

We prove now the following

Lemma A.2. - I / the two-body potentials Vi~ satis/y the condition (1.11) with p ~ 2, then, /or any channel ~, the operators P~V~(1--P~) and (1--P~)V~P~ have a unique bounded extension on L2(R~V-3).

Proo/. - Of course it suffices to p rove such result for V~P~ only because its adjoin t P~i~ would then be ~ bounded operator . We consider for sim- pl ic i ty a channel with only two f ragments F1 = (1, 2, .. . , j) and F2 ~ (j ÷ 1 , ..., N). We shall show at the end of the proof t ha t such rest r ic t ion is easily renmved.

The po ten t ia l V~ is a sum of two-body potent ia ls of the t ype

( A . 3 ) ~ - - V ( J ~ l , F 2 ) = ~ Vik, i 6 F 1 , }6F 2

I t suffices then to show t h a t every opera tor V~P~ has a. unique bounded extension. We in t roduce the var iables

(A.4)

mlrl ~- ... -~ mjr~ mj+lr~+i + ... + m~rz~ ~ 1 2 = - - ,

ml + ... + mj m~+i ~- . . . -~ m~v

X ~ 11 r 1 - - r j , x ~ 2) = = r i + 1 - - r ~ ,

(1) X{~ 2) 1 X j - - 1 = r j - 1 - - r j , _ - = r v _ 1 - - r ~ .

5 9 8 M. BERTERO a n d G. PASSATORE

The variable Y~2 gives the relat ive position of the centres of mass of the two fragments, while _m and (2) ~,, x,~ are internal co-ordinates of the two fragments respectively• The potent ia l V~ depends only on the variable

(A.5) i-i //-i-i

r i r ~ = y12-~- ~ o~i.x(n 1' ~ n (2) - - - - p ~ m X m = y12 ~ - X i - - X ~ ,

where O~in and f l~ are constants and X~ and X~ denote the first and the second sum in the r.h.s, of eq. (A.5).

I f V~ satisfies condition (1.11), it is cer ta inly defined on the whole ~o (5) (al though it may be not defined on the whole L2(R3~-3)). So for any ] e 2o, we can wri te

(A.6) ( V ~ k P ~ , / ) ( y , z ; x(1), • . • , X$_l , (1) . x ? ) , o.• , 3£(292) 1) =

= v , ~ ( y ~ + x , - x~) ~ ( x ; ~,, .., (~" " ,2, ,2, • Xj--1) ~ 2 ( X l ' " ' " , X~r--j--1) "

f (~>" ' '~" x~" : ~ " x(~ w, ' " " x(~ 2, ' , x~-)'~_~) dx ' . • ~o~(x(~)'~ . . . ~ x ~ - ~ ) ~ 2 [ x x ~ . . . , - ~ - ~ ) l [ Y ~ 2 ; . . . ~ x ~ _ x , ...,

By the Schwartz inequal i ty we obtain ([]W~II~,,~-,, = llw211-,~ .... ~ ) = 1)

(A.7) <fe,k(y~)ll(yx~; x i ' , - , . x (z ) , x ( ~ _ x ) 1 2 d y 1 2 d x [ ~) a (2, . . . ~ x i - 1 ; . . . ~ . . . o-Xar-j- 1 ,

=I I v,~(y~2 + x , - X~)l 2. ~i~(Y12) • l w , ( x T , . . . , ' ~ " ,2, , ,2_, (1) dx2_)_l x¢-~JI21w2(x? ,, . . . , x~- ; - , ) l ~x~ . . . .

As Vik v(2) (p) v(2) v(~) - ,k + V~ with ~L2(R 3) and _ ~ ~L~(R3), p~>2 (eq. (1.11)), ---- ~ i k we have correspondingly .(2~ and (r) ~ik ~i~, and, by eq. (A.7),

(A.S) ~ik(Y12) (2) (, , 2~)ik (y12) < 2 Q i k (Y12) -~-

After introducing the variable Xi in place of X~ 1), by the H61der inequal i ty with p r = ½p and q ' = p / ( p - - 2 ) , (1 /p '+ 1/q '= 1), we have

+ x , - x ~ ) ] Iw, (x , , x,_~,, ( A . 9 ) ~(')/'" ~ ( O ~ i l I J[ ik ~ 1 2 X(21), " • ' ' (1)~[2

• 1 ~ ( x 2 ', , ~ - s - ~ , J = . . . . . . ~-~N--j--l l

[V,k (y~z+X,--Xk) l (A,w~)(Xi)2lwu(xl 2), • . . , .(2)~N_j_I], ' ]2 c]X~ i~' l t ' ] ~(2) " '" dX~y--,--ll 2'" /

(for p = 2 the equal i ty holds). F ro m L e m m a (A.2) it follows tha t the function (Ari~fl~)(Xi) is bounded,

so t ha t

(A.10) (~)

x x U /

A M A T H E M A T I C A L A P P R O A C H T O T H E D E R I V A T I O N ~ T C . 599

where C is a constant . in place of X~, we have

(Aal)

Finally, int roducing the var iable x~k = y~2+ X~--X~

ik ~.~v12] ~ - Co~i,1 I] W i k [ ] L ' ( R a ' °

As this re lat ion holds for p > 2 , f rom eq. (A.3) it follows t h a t ~k (y l~ )<N ~, where N is a constant . Equa t ion (A.7) then implies

V/~ 2o.

Therefore V~kP~ is a bounded opera tor on 2o and, by closure, i t can be ex- t ended in a unique way on the whole L~(R3~-3). The extension is given b y the eq. (A.6) itself. I f the num ber of f ragments is larger t h a n two, the poten- t ial V~ is a sum of t e rms like V(E, , Fb) (see eq. (A.3)), one for each pair of f ragments . Therefore a given two-body potent ia l V~k contained in V(F~, Eb) depends only on the co-ordinates of the corresponding f ragments . I t follows t h a t in an inequal i ty like (A.7), the quan t i ty ~ does not contain the wave functions of the other f ragments , because these are e l iminated b y in tegra t ion on the corresponding variables .

We prove now the following

Lemma A.3. - I] the channel ~ contains the ]ragments Fx, ..., Eq, the operator P~V~P~ has the ]orm

(A.12) P~,V~, P~ = V',,"P --o¢ ~g¢ ,

where V(2 ") is a multiplicative operator o/ the type

(A.13) V(~"(y) ~ V ̀ ,,) (- ' Fa,aVb~fab ~ , a < b = l

Y~b being the relative position o] the centres o/ mass of the ]ragments F, and Fb. Moreover, i] the two-body potentials V, satis]y eq. (1.11) with p ~ 2 , the ]unc-

• (av) t~ons VF~.~b(Y~) are bounded and continuous, and V (~) ~L2(R 3) + L~(R3). Fa,F b Proof. - As in L e m m a A.2 i t is sufficient to consider a channel with two

f ragments because the extension to the general case is s traightforward. B y means of the co-ordinates (A.4) it is easy to show t h a t

(A.14)

( P a W i k P a / ) ( y l e ; x(11), (1) . x(12), ~(~) • . . , X i - - 1 , . . - , .a~g--j--1] =

(.v)(y)(p~[)(,e . x ~ l ) (1) . x ( 2 ) , _ ( ~ ) = X j - - I , ~I¢--j--l)

V ( a v ) [ ~ . ik ~¢Vlg] =

f v i k ( y l 2 ~ _ X i _ _ X k ) ] ~ l ( ( 1 ) X (1) ~]2]a/, / X (2) ~ { 2 ) ~ [ 2 A ~ ( 1 ) ~[,~(2)

Then, by the Schwartz inequa l i ty we have

(3..15) V (~') ~ ~<

6 0 0 ~ . B E R T E R O a n d G . P A S S A T O R ~

and f rom the inequali t ies (A.8) and (A.11) it follows tha t the function V (~')¢-, ~k ~J12! is bounded. I t is also continuous because, b y proceeding as in the deriva- t ion of eq. (A.11), we obtain

(A.16) ~) [Vik (Y12-~ h) - ~ikT/'~l*'~.Y12~[~ < t --3 (2) < c ~.~{ll v,~ (. + h) _ _~,v,~. )]1 ~,,~., + Ii-~v'~'~', + h) - - -,~v'~. )/I- ,~,} ~ o (]hi ~>0)

(indeed, the t rans la t ion group is s t rongly continuous in the L ~ norm). Finally, we prove t h a t --~kTZ(~') E L2(R 3) + L~(Ra). I f we introduce the vari-

able X~ in place of x~ 1) and Xk in place of x(~ 2), we can write ( ] . g means the convolut ion product)

(A.17)

where the functions (A'iY/1)(X~) and (A'ky/~)(X~) are defined as in eq. (A.1). As Vik = V(2) + v(v) v(~) --~k --~k, we have a corresponding decomposi t ion of --~k :

Since

(A.18)

(av) V,~ (y~) = u",~)(y~) + v('.~)(yl~).

I l ! 2 ! 2 = ~111-,~,~-., = (~i.~) ~, ]I(A,~) ]l~',,', [1 I ! 2 /

b y a r epea ted use of the Young inequal i ty ( 1 2 ) w e get

ll(Ak~v=) L,,R.,II v(2, , 2 (A.19) [I u`''k, Hz,,R., <-< (a,,lflk.1) -a ' , 2 --,k * (A~

--3 ! r 2 t I 2 (2)

and s imi lar ly

i,k (~) (A.20) ]Iv II-,~., < It vik I]LV(Rtt, "

The L e m m a is complete ly proved.

A P P E N D I X B

I n this Appendix we consider integrals of the t y p e int roduced in Sect. 4:

b

(B.1) f T(,~) dl(~), a

where T($) is an opera tor -valued funct ion defined on the in terva l [a, b] whose values are bounded operators on L2(R~g-~), while ](2) is a vec tor -va lued fune-

(12) G. H. HARDY, J. E. LITTLEWOOD and G. POLYA: Inequalities (Cambridge, 1934).

A M A T H E M A T I C A L A P P R O A C I I TO T H E D ~ I ~ I V A T I O N E T C . ~ O I

t ion on [a, b] wi th values in L2(R3~-3). Such integrals have been first considered b y GowuRI~ ~ (~a) in the more general case of a Banach space ~ . GowuRI~ also obta ined a necessary and sufficient condition on f(A) for all the functions T(2), continuous in tile norm of the bounded operators on ~ , to be integrable wi th respect, t,o /(2), the integrals being l imits of R i emann sums. The condition is

i

for any par t i t ion of the in te rva l [a, b] and for any choice of the operators Ti wi th IT i l<l (the norm is t ha t of the bounded operators on ~ ) . In the case of the integrals in t roduced in Sect. 4 this condition is not satisfied because f(2) is of the form f (~)= PP(2)] , where P is a project ion and P(2) is a resolution of the ident i ty.

However , since in our case P(~)f is s t rongly continuous and T(/~) is differentiable, the existence of the integrM (B.1), as e% strong l imit of tCiemann sums, can still be shown. Indeed the following l e m m a holds:

Lemma B.1. - I f the function f(~) is strongly continuous and there exist two constants M and N such that, for any ~ [ a , b ] and for any partition of the interval [a, b],

(B.3a) ilf(~) II~ < M, (B.3b) ~ fir(A/+,) - - f(2i)]l 2< M 2 '

i

(B.3c) IT(;t)l < N ,

(B.3d) IT(2)-- T(~)I <NI~ - ~1 (where the norm in eqs. (B.3c) and (B.3d) is that of the bounded operators on ~ ) , then the integral (B.1) exists as a strongly limit of the Riemann sums

(B.4) y, T(2;)[/(2~+1) --/(M], i

where a -- As< 22< . . .< An+l =I b, 2~ e (Ai, ~+~], when m a x IAj+~-- At] -~0. More- over the following inequality holds:

b

(B.5) T(~) df(~) <;712M(b-- a) + Ill(b)-- f(a)!1~]. a

(B.6)

Proof. - For the R i e m a n n sum (B.4) the following iden t i ty holds:

T (~ ) [ I ( ,L+~) - ]('L)] = ~ [T(,L+~)f(,L+~)- T ( i L ) ] ( ' L ) ] - J i

- - ~ [T(A~+~) - - T(Aj)]f(A~) + ~ [T(2/) - - T(~j+l)][/(~i+l) - - / ( ~ j ) ] = i j

= T ( b ) / ( b ) - T ( a ) / ( a ) - ~ [ T ( I ~ + I ) - T(~)]/(A~) -~- i

+ ~ [T(,~;) - - T(,L+~)][f(,L+~) - - / ( , L ) ] • i

(13) M. GOWURIN: Fund. Math., 27, 255 (1936).

602 M. BERTERO ~nd O. PASSATORE

The th i rd t e r m on t he last side of this equa t ion vanishes in n o r m for (~ = m~x [~+~--~s[--~0. I n d e e d f r o m eqs. (B.3b) and (B.3d) we huve

(B.7) II ~ IT(A;) -- T(As+I)] [I(4s+i) --/(As)] I]~ < S

< N ~ Ix; - 4s+,l IIl(x,+~) - I(~)J[~ < N ~: ( x , . - 4,)II/(x,+l) - 1(~,1 [1~, <

W e m u s t t h e n show t h a t t h e second t e r m on t h e last side in eq. (B.6) has a l imi t for $ - > 0. As /(4) is u n i f o r m l y con t inuous on [a, b], for fixed e the re exis ts (~ > 0 such t h a t

(B .8) 1[/(~) - - / ( ~ ) I[~ < ~, i f 14 - - ~l < (~.

Consider now two pa r t i t ions of the in te rva l [a, b]:

a n d

w i t h

a = 21< 2z< . . . < 2~+1---- b

a = ~ul </~2 < . . . </t~+l = b ,

m~x IAs+, -- As] < ~ and max ]/~J+l --/~J[ < ~"

I f a = v l < v2 < . . . < v~ = b is t he superpos i t ion of these two par t i t ions (p < m + n), t h e n we have

(B.9)

w i t h

(B.IO)

S k

= ~ [~'(~,+1) - T(~,)] a,! i

F r o m cond i t ion (B.3d) i t follows t h a t

(B J1) iI~ ET(Xs+1)-- T(L)]/(L) -- ~ [/'(/~+i) -- T(/~)]/(~)II~ < J

< 2 N 8 ~ i~i+l--~di] : 2N(b--a)e --+0 i

(~ -> 0).

As t he G-space is comple te , i t is p r o v e d t h a t t he R i e m a n n sums (B.4) have a l imi t for (~ = m a x 14~+i - - 4~ I --> 0.

As regards t he inequa l i ty (B.5), f rom the i d e n t i t y (B.6) we get

(B J2) T(2~)[ / (2 j+l ) - / (2s) ] = [ T ( b ) - T(a)]/(b) + T(a)L/(b ) - / (a)] - - J

- ~ ET(~.I)- r(~,'l]/(~S+l)- ~ [T(~;)- T(~,)]/(~s), S S

A M A T H E M A T I C A L A P P R O A C H T O T H E D E R I V A T I O N E T C .

so that , by means of eqs. (B.3a), (B.3c) and (B.3d), we have

(B.13)

603

i] X T ( ~ ) [ ] ( ~ . ~ ) -- ] (~ ) ] II~ < 2VM(b - - a ) + N [It(b; - - ] ( a )J [~ + i

+ ~ NM(~j+, -- ~t;) + ~ 5TM(~I;- ~¢) = zV[2M(b - - a) + ]!](b) - - ](a)I1~] i j

and inequali ty (B.5) is proved. F rom lemma (B.1) we obtain

Lemma B.2. - Let T(~) be a ]unction on [a, b] whose values are bounded operators on the Hilbert space )F. I f T(~) satis]ies conditions (B.3e) and (B.3d) and i] P(~) is a resolution o] the identity so that ]or ]~..Yf and ~ [ a , b ] , P(~)] is strongly continuous, thcn the integral

b

(B.14) f T ( 2 ) dP(~)]

a

exists as a limit o] Riemann sums. Moreover the ]ollowing inequality holds:

b

(B.~5) fr(Z)dP(a)/ <N[2(b--a)ll/:l~+l'~(P(b)--P(a))]!l~]. a

Proo]. - I t must only be shown tha t ] ( ~ ) = P(~)] satisfies the conditions (B.3a) and (B.3b). This is straightforward, since

IlP(;~)l[J~< [l/][~ (B.16)

and

(B.17) Z II (P(~J+, ) - P(~,)) ] I'} = [! (P(b) - -P(a) ) ! I!~ < I1! II ~.

Thus, from inequali ty (B.5), inequali ty (B.15) then follows.

• R I A S S U N T O

In questo lavoro si effettua un tentativo di analisi rigorosa de] fondamenti e delle propriet£ de1 potcnziale ottico. Si dimostra che, se i potenziali a due corpi soddisfano ad opportune condizioni, allora il potcnziale ottico esiste ed g u n opera]ore limitato sia nella teoria dipendente dal tempo che in quella indipendente dal tempo. Questa propriet~ non richiede che i potenziali a duc corpi siano lira]tat]. Si ottengono inoltre le propriet£ di analiticit~ e i l comportamento asintotico del potcnziale ottico nel piano dell'energia. Infine si discute in modo rigoroso la connessione tra la formulazione dipendente dal tempo e quella indipendente dal tempo.

604 M. BERTERO &rid G. PASSATORE

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