Spectral properties of the Kirkwood-Salsburg operator

Physica 87A (1977) 331-343 (~ North-Holland Publishing Co.

SPECTRAL PROPERTIES OF THE KIRKWOOD-SALSBURG OPERATOR

H. M O R A A L

lnstitut fiir theoretische Physik der Universitiit zu KiJln, D-5000 KiSln 41, West Germany

Received 29 October 1976

A mathemat ical ly precise definition of the " inf ini te-volume" Ki rkwood-Sa lsburg operator as a bounded linear operator in a Banach space is given. It is shown that this operator has a bounded inverse for a bounded, stable and regular pair potential. These facts are exploited to establish the connect ion be tween the Ki rkwood-Sa l sburg and the Mayer-Montro l l equat ions and to give a classification of the spectra and resolvents of the Ki rkwood-Sa l sburg operator and of its inverse. The theorems proved in this article const i tute a f ramework for the derivation of any more precise resul ts for special potentials.

I. Introduction

In the statistical mechanics of classical cont inuous systems, a large role is played by those sys tems of linear integral equations which have as solutions the n-particle distribution functions p ( z ; x l . . . . . xn) in the grand canonical ensemble. These distribution functions depend parametr ical ly on the fugacity z and give the probabil i ty of finding n different particles at the prescr ibed posit ions xl . . . . . xn. Two such sets of equations, the Ki rkwood-Sa l sburg 1-5) and the Mayer -Mont ro l l 1,6.7) equations, have been used extensively to obtain bounds on the radius of convergence of the virial expansion. Since these equations can be formulated for infinite systems, it would seem that more information, for example on phase transitions, can be extracted f rom them.

The present article is a first step in this direction. After defining the Ki rkwood-Sa l sburg opera tor in a mathemat ical ly rigorous way, we prove a number of theorems relating the solvability of the Ki rkwood-Sa l sburg equations to the spectral propert ies of the opera tor and of its inverse. We also establish rigorously the connect ion with the Mayer -Mont ro l l equations. Al- though results of this nature cannot immediately be applied to a specific physical problem, they at least consti tute a rigorous f r amework for fur ther research. More precise results can presumably only be obtained by specifying the pair potential in a much more detailed way.

331

332 H. MORAAL

2. The Kirkwood-Salsburg operator and its inverse

We restr ict ourse lves in this article to the s tudy of the spectral proper t ies of the K i r k w o o d - S a l s b u r g (KS) opera to r for an interact ion consis t ing of an external potential &(x) and a pair potential &2(x ,y )= &2(Ix y[) in d-dimen- sional Eucl idean space R a. The potential energy of a set of n part icles at x~, x2 . . . . . x, = X, is then given as

u(x . ) = ~, 6(xi) + ~ ~(~,, xp. i=1 I ~ i < j ~ n

(2.1)

As usual, we will have to assume that the potential is stable/),

U(X,,) >i -nB, (2.2)

with B a posi t ive constant . This implies in par t icular that &(x) and &2(x, y) are bounded f rom below. H o w e v e r , we will not assume that the integral

f exp [-/3~b(x)] dx (2.3)

exists as was done in ref. 2 in order to discuss the spec t rum of the KS opera to r in a special space. This necess i ta tes the requi rement that the pair potential be regularS'Z), i.e.

f lexp [-t~4,~(Ixl)] - 1[ dx : C(/3) < ~. (2.4)

Finally, we assume the pair potential to be bounded f rom above, so that it is absolute ly bounded .

The KS equat ions are now given as '-3)

p(x)=zt(x)r=~ K(x, Yr) p(Yr)dYr+zt(x),

p(xU Xp)= zt(x)s(x, Xp) ~ K(x, Yr)p(Xp U Yr) dYr (p~> l). (2.5)

Here o(X,) is the n-part icle distr ibution funct ion, z the fugaci ty , t ( x ) = e x p - / 3 & ( x ) and the kernels K(Xp, X,) are given by

K(Xp, Yr) = 1"I [ s (Xp , y ) - 1] (2.6) v C Y r

with

s(Xp, Xq) = [exp - 13U(X~, U X, ) l [ e xp / 3U(Xp) ] [ exp /3U(X, ) ] . (2.7)

SPECTRAL PROPERTIES OF KIRKWOOD-SALSBURG OPERATOR 333

Note that the boundedness of the pair potential is necessary to write this form. Since in eqs. (2.5) t(x) is multiplied by z and since t(x) is bounded, we may assume ess sup t(x)= 1 f rom now on.

We now define the KS opera tor formally by the equations

(Kf)(x) = ,=, ~ K(x, Yr)t(Yr)f(Yr) d Y ,

r = 0

(2.8)

with t (Yr)=IIyeyt (y) . The KS equation is then formally defined as

f = zKf + zot, (2.8)

In these equations f is an infinite vector with components f(Xp) and a is given by a(x) = 1, a(Xp) = 0 for p ~> 2. We will consider the KS equation for all complex values of the fugacity z. It is clear that if eq. (2.8) has a solution vector f with components f(Xp), then eqs. (2.5) have the solution p(Xp)= t(Xp)f(Xp).

The definitions in the preceding paragraph are formal in the sense that the space of vectors f in which K operates has as yet not been defined. We will, however , pos tpone this definition and first show that K has, at least formally, an inverse K -~. This opera tor may be defined by the equation

(g-lf)(Xp): ~ (-1) ' r K(x, Yt) t=0 t! J s(x, XpU Y,) t ( Y ' ) f ( x U X p U Y')dY' ' (2.9)

That this opera tor is formally the left- inverse of K may be seen by inspection; the necessary interchanges of sums and integrals are allowed in the spaces E~ defined below.

We now give the definitions of the Banach spaces E~ introduced by Ruelle 1) and of two of their subspaces.

Definition ! Ee is the Banach space of all vectors f with, as components , symmetr ic , essentially bounded, Lebesgue measurable functions f(Xp) satisfying ]lf]l = supp ess SUpxp rf(xp)lU p < ~.

Definition 2 Se is the set of all f E E e such that Kf has again symmetr ic vectors as components .

Definition 3 Te is the set of all f E E~ such that K-I f does not depend on the redundant

334 H. MORAAL

variable x in eq. (2.9). The l inearity of the subspaces S t and T~ is a trivial c o n s e q u e n c e of the l inearity of K and of K '.

Theorem 1 S t and T~ are c losed linear subspaces of E~ sat isfying the relations* (i) S t = s p a ~ ) K S ~ ; (ii) S~CT~; and (iii) if f E T e but I ~ S t then exists a non -cons t an t func t ion g(x) such that f = got + h, h C S e

Proof (a) KS~ C S e Let f E S t, then K f has symmet r i c componen t s . The com-

ponen t s of K2f are then, fo r p I> I, given by

(KZf)(x,, xz U Xp) = s(x,, xz)s(xl u xz, Xp)

× g(XlUX2, Yt)t(Yt)f(Xp U Yt )dY . t=O

(2.10)

as is shown by an easy calculat ion. This is clearly a symmet r i c func t ion of x, and x2, but also of the set of indices x2 U Xp since f E S e There fo re , K2f has symmet r i c c o m p o n e n t s for three or more variables. A separate p roof for two variables is easily obtained, namely ,

(K2f)(xl, x2) = s(x,, x2) ~ K(Xl U x2, Yt)t(Yt)f(Yt) dY,.

For later use we note the general fo rmula

(2.11)

(Kqf)(Xq U X p ) = [ e x p - ~U(Xq)ls(Xq, Xp)

x ~ l f K(X~'Y~)t(Y~)f(X~U . (2.12)

which may be p roved by induction. To comple te the p roof of this par t we must show that K f E E e Since K f is

symmet r ic , we can copy Ruel le ' s p roo f of the boundednes s of K wi thout in t roducing an " index- juggl ing" opera tor :

ess sup [(Kf)(x)I<~ ~_ ~ ~, ]K(x, Yr)[ dYe']Ill] x r=l r ' d

= {exp [srC(/3)] - 1}Ill]I, (2.13)

ess sup ](Kf)(x U Xp)] ~< s(x, Xp)~ p exp [~:C(/3)] ~< s r" exp [~:C(/3) + 2/3B], x,Xp

*spf is the closed linear subspace consisting of all scalar multiples of f.


where the last inequality results from the fact that, by stability,

I-I s(x, Xp) = exp [-2/3U(Xp+,)] <~ exp [/3(2p + 2)BI, xEXp+I

so that we can always arrange the indices in such a way that s(x, Xp)<~ exp2/3B without changing the value of KS. From eqs. (2.13) we then have immediately

IIK.fll ~< ¢- ' exp [~C(/3) + 2flBll l f l l , (2.14)

thus showing that Kf @ E e (b) K 'T~C T e The proof that K-2f does not depend on a redundant

variable if K- ' . f does not, is again the result of a simple calculation. In fact, the following general formula is easily derived:

~-, (- 1)' f K(X~,Y,) t (Y , ) f (XqUXpUY, ) . (K-"D(XP) = 7"o t! s(X,, X~ u Y,)

(2.15)

Since this is symmetric in the set of indices Xq, it cannot depend on these indices if it does not depend on one of them.

To prove that K I r E E~, consider eq. (2.9) for x so far from the Xp that s(x, Xp)~ 1; this is possible since K-~f does not depend on x and since the regularity of the pair potential implies &2(x,y)-*O for I x - y ] ~ o o . We then easily see that

I esssup l ( K ,f)(Xp)l<~ ~ 1 K(x, Y~) dr~o+,+,ll#. x~ ,=o s(x, YD

(2.16)

Since the integral

f lK(x, y)/s(x, Y)r dy = C'(/3) (2.17)

exists by virtue of the regularity and boundedness of the pair potential, it follows that

IlK 'Ill ~ [exp ¢C'(t3)]C:llfll, (2.18)

and K-If ~ E~. (c) K 'T~ = S e Clearly, KS~C T~ and K-'(KS~)= S~. It is further easily

seen by inspection that if f E T~, then K - ' f E S~, since KK 'l differs from f at most by a vector of the form g(x)ot.

(d) S~ = sp et 0 KS~. From (b) and (c) we have S¢ C T e Now let f C S and

336 H. MORAA1,

K I f = g. Clear ly , g E S¢ and if we wr i te f = / x + Kg, then tt @ Se. Then KK-~Ia = 0, but this di f fers f rom p on ly by a v e c t o r g(x)a, and thus ~ = g(x)a. The v e c t o r K # has as s e c o n d c o m p o n e n t

(K t t ) (x l , x~) = s(xl, X 2 ) g ( X 2 ) ,

and this can be s y m m e t r i c on ly if g(x) is a cons t an t , which p r o v e s the a s se r t ion . The same a r g u m e n t shows that if f E T¢ but f ~ S e then there is a g @ S¢ and a n o n c o n s t a n t func t ion g(x) such that f - g(x)a + Kg.

(e) S~ and T~ are c losed . W e on ly have to p r o v e this for S~ on ly by the a b o v e c o n s i d e r a t i o n s . Le t f , ~ f , f,, E S e f E E e By the b o u n d e d n e s s of K, the s e q u e n c e Kf, is a C a u c h y s e q u e n c e in E e It t h e r e f o r e has a l imit in E~ which can be no th ing else but Kf. But K f E E~ impl ies f E S e Q .E .D.

Corollary 1 E v e r y v e c t o r f E S~ can be wr i t t en as

f = ~ a i ( f ) K H a + K"f,, f . = K "f, i=1

(2.19)

for all n. If f has only a finite n u m b e r n of n o n v a n i s h i n g c o m p o n e n t s ,

f = ~ ai(f)K i la. (2.20) i=1

The c~i are b o u n d e d l inear func t iona l s on Se with no rm

I1~,11 = 0 1 - KK-1)K '÷111. (2.21)

Proof Since S~ = s p a (~) KS e t he re ex i s t s a n u m b e r a l ( f ) such that

f = oq(f)a + Kf,, fl = K if. (2.22)

R e p e a t i n g this a r g u m e n t n - 1 t imes g ives eq. (2.19). If f has on ly n non- van i sh ing c o m p o n e n t s , then c l ea r ly K "f = O. This y ie lds eq. (2.20). W e fu r the r have f rom eq. (2.22) tha t a l ( f ) is g iven b y

Ogl(f)tflg = (1 - KK-' ) f , (2.23)

or, gene ra l ly ,

ai(f)a = (1 - KK-l)K-i+lf. (2.24)

S ince Hall = ~: ', eq. (2.21) fo l lows i m m e d i a t e l y . Q .E .D.


An obvious choice for the common domains of K and K -1 is now the closed linear subspace W e of S e defined as the closed linear span of the vectors K~-~a, i = 1,2 . . . . . This space is mapped into itself by K and K -~ and satisfies all requirements derived above for S e Although the identity of W~ and Se cannot be asserted, it is clear that the vectors of S J W e are in a sense impossible to find since they cannot be approximated by finite vectors from S e It should be noted that the vectors of W e are all scalars with respect to the rotation group of R a, so that we cannot expect to find solutions of the K S equation corresponding to a particularly oriented solid phase4). In the next section we can still take the whole of S e as domain.

3. Kirkwood-Salsburg and Mayer-Montroll equations

From the definition of the KS equation we can immediately derive the following theorem.

Theorem 2 If f E S~ satisfies the KS equation (2.8), then K-~ f = z f and a~(f)= z( If f satisfies f = zKf , then K - I f = z f as well, but now a~(f) = 0 for all i. If, on the other hand, K - ~ f = zf, then either a~(f)=/~z ~ and i z - i z f satisfies the KS equation or a i ( f ) = 0 and f is an eigenvector of K with eigenvalue z 1.

Proo f From eq. (2.26) it follows that a , ( K g ) = 0 for all g ~ S e and a i ( K g ) = ai-~(g) for i/> 2. Application of the functionals ot~ to eq. (2.8) therefore gives the system of equations

al( f ) = z; ai(f) = zai l(f), for i I> 2, (3.1)

with the solution ai(f) = z i. That an f satisfying eq. (2.8) is an eigenvector of K -~ with eigenvalue z is trivial as is the same statement i f f satisfies f = zK$. That a/(f) = 0 in this latter case follows immediately from a~(f)= 0 and a i ( f )= zai-l(f), i >I 2.

Conversely, let K - I f = zf. From eq. (2.26) we get ai(K-~g) = ai+~(g) for all i and g ~ Se, and therefore ai+l(f) = zai(f). Now if oq(f) = / x z # 0, it follows that ai(f) =/xz i. Furthermore, f can be written as f = t~zot + K K - ~ f and this inserted into K-~ f = z f gives K - i f = z K K - ~ f + i~za and hence g - ~ K - l f = zf]ix satisfies the KS equation. Otherwise, a l ( f ) = 0 implies a~(f)= 0 for all i and f = K K - I f so that K - i f = z K K - ~ f or f = zKf. Q.E.D.

Before considering the derivation of the Mayer-Montrol l (MM) equations, we state two explicit formulae for a~(f). The next equation is easily proved using the explicit forms of K - q f and K f given in eqs. (2.15) and (2.7), respectively:

338 H. MORAAI,

%q)exp[-~U(Xp)l : 2 (-1) ' fK(Xp, Y,)t(Y,)f(X~ U Y,) dY,. ,~o~T-J siX,,, Y,)

(3.2)

Using the fact that f (gp L3 Yt) = ( KoK Pf)(Sp L3 Y,) for t /> 1, which fol lows f rom the p roof [of part (c)] of T he o re m 1 by induct ion, this can also be wri t ten as

% ( f ) = f ( X p ) [ e x p f l U ( X p ) l - k = ~ . K(Xf,, Y D t ( Y D ( K ~'f)(YDdYk. (3.3)

Theorem 3 If f E S¢ satisfies the KS equat ion, it satisfies the MM equation~'6):

f = ~ ( z ) + M ( z ) f ; ¢ l ( z ; X p ) = z P e x p - C l U ( X p ) ;

~ f ~ 1 [ M (z)f](Xp) = zO[exp - ~3U ( Xp)] k~_fl~ ~T. J K (Xp, YDt( Yk)f( Yk) d Y~. (3.4)

If f @ Se satisfies z K f = f, then also f = M(z)f .

Proof Let f = z K f + z . It fol lows f rom T h e o r e m 2 that K P f = z ~ f and a p ( f ) = z p. Inser t ing this into eq. (3.3) yields eq. (3.4). If f = zKf. then K of = z~,f and % ( f ) = 0; eq. (3.3) gives f = M(z) f . Q.E.D.

The MM equat ion appears to be more amenable to t rea tment than the KS equat ion, since it can be defined on all of Ee wi thout difficulty. The fol lowing theorem ex tends slightly some results of P e n r o s e 7) and shows that the MM opera to r is not too useful (not even for posi t ive potentials), since it does not map E¢ into itself for large values of Izl.

Theorem 4 For a posi t ive potential and for t (x )= 1, the norm of M(z) on Ee is given by

[IM(z)ll = Izl~-'[exp {~C(B)}- l] < 1, if Izl ~ ~: exp - ~C(/3). (3.5)

For Izl > ~ e x p - ~ C ( / 3 ) , M(z) does not map E~ into itself. If M(z) maps Ee into itself, the MM equat ion has a unique solution in Ee.

Proof For any f E E~ we easily obtain the est imate

ess sup ]f(Xp)l ~< Izl ~ ~ [p~C(~)lkllfll = Izlp{exp [p!~C(fl)] - 1}llfll Xp k = 1

(3.6)


for a positive potential since it is easy to prove the inequality

f lK(Xp, YDI dYk ~< [pC(fl)] k, (3.7)

where the right-hand side is the value of the left-hand side for widely separated Xp, in this case. However , taking the special vector h E Ee, given by h(Xp) = (-~)P so that Ilhll-- 1, yields

(Mh)(Xp) = z p exp [- /3U(Xp)] ~. ¢k f [K(Xp, Yk)I dYk, (3.8) k=lk!

SO that we get for h:

ess sup I(Mh)(X~)I = Iz l °{exp [p~:C(fl)] - 1}. xp

(3.9)

Therefore , Mh E E~ if and only if

sup (fzl/~)P{exp [p#C(/3)I - 1} p

exists and this is precisely the case if I z l~<exp-£C( /3) . The supremum alluded to must then by eqs. (3.6) and (3.9) be the norm of M(z) and is attained for p = 1, yielding eq. (3.5). Since clearly IlM(z)ll < 1 if it exists, eq. (3.4) then has a unique solution given by the L iouv i l l e -Neumann series 8) for the inverse of l - M :

f (z) = ~ [M(z)]"lJ(z). (3.10) m = 0

Q.E.D.

4. S p e c t r a a n d reso ivent s

Due to the fact that K - l is a bounded linear operator in the Banach space W~, every point z of the complex plane belongs to one of the following four mutually exclusive sets8):

(1) R(K-1), the resolvent set of K -l, is the (open) set of all z such that K -l - z has a bounded everywhere defined inverse;

(2) P(K l), the point spectrum, is the set of all z such that K-if = zf has a nonzero solution;

340 H. MORAAL

(3) CO(K-‘1. the continuous spectrum, is the set of all z for which K-’ - z is one-to-one, but the range of this operator. 8!{K-’ - z), satisfies %(K- - z) f W,, B(K-‘- z)== W,; here the superscript “a” indicates the closure of a set; and

(4) RS(K-‘), the residual spectrum, is the set of all z for which K-’ - z is one-to-one, 92(1(-’ - 2)” # W,.

Since K is also a bounded linear operator in W, z-“ belongs to one of the four sets R(K) (again an open set), P(K), CO(K) and RS(K) defined as above but with K-r - z replaced by I - zK, everywhere. This gives a classification of all points of the complex plane in sixteen different classes. We will now prove a number of lemmas concerning the ranges of K-’ -z and 1- ZK which reduce the number of classes to six and also show the connections between the solvability of the KS equation and these ranges.

Lemma I %(K-’ - z) always contains all finite vectors from WC; 91(K-’ _ z)” = W, for all z; %(K-’ - z) = sp fx @ K%!(K-’ - z).

Proof We observe that the vectors KPa! are always in the range of K--r -2, since they are the images of the finite vectors

6c,(z) = _ f: z-P+i-lKi-la~

i=l

Therefore, %(K-” - z) contains a11 finite vectors and the second assertion of the lemma follows from the definition of W,. Since this is the case, %(K-‘- z) = sp 1~ @ KA for some set A. But since

A = K-‘%(K--’ - z) = K-‘(K-l - 2) W, = (K-’ - z)K-‘W; = (K-l- z>Wf = W(P - z),

the lemma is proved. Q.E.D.

Lemma 2

C%!(R-’ - z) = sp (Y @ %(I - zK); !%!(I - zly) = .%?(K-’ - z) if and only if the KS equation has a solution.

Proof

%(K-’ - z) = (K-I - z) W, = (K-’ - z)[sp LY @ KWzl

=splu0(1-zK)w~=spcuO~(l-zT()‘

Since ZQ! E .B(K-‘- z) by Lemma I, the equality of the ranges implies the existence of an f E WC with (1 - zK)f = zcr. If, conversely, such an f exists,

SPECTRAL PROPERTIES OF KIRKWOOD--SALSBURG OPERATOR 341

then sp a C ~ ( 1 - zK) and the first pa r t of the l e m m a impl ies the e q u a l i t y of the ranges . Q .E .D.

Lernma 3 If the K S equa t ion has no so lu t ion , t he re ex i s t s a l inear f unc t i ona l ;t~(f) on ~ ( K - l - z) such tha t

°~(1 - zK) = {Az(f)a + Kf; f @ (K- ' - z)}, (4.1)

~ ( 1 - - z K ) " : ~ W e if and on ly if A~(f) is b o u n d e d .

Proof The fo rm (4.1) fo l lows i m m e d i a t e l y f rom K - ~ ( 1 - z K ) = ~ ( K -~ - z) and the a s s u m e d unso lvab i l i t y of the K S equa t ion . I f A~(f) is b o u n d e d on ~ ( K -I - z), it m a y be e x t e n d e d to the who le of W~ s ince this r ange is dense . This impl ies

°,~(1 -- zK) a = {Az(f)a + Kf; f E W$} ¢ W$. (4.2)

If At(f) is u n b o u n d e d , the re is a s e q u e n c e {fn} C ~ ( K - 1 - -7) such tha t f , ~ 0 , hz(fn)--* 1 and, t h e r e f o r e , ~ ( 1 - z K ) a c o n t a i n s a . I t is e a s y to see tha t this impl ies ~ ( 1 - zK) a= W~. Q.E.D.

W e r e m a r k tha t the K S equa t ion is nea r ly so lvab l e in the sense tha t there ex i s t s a s e q u e n c e {f,} such tha t (1 - zK)fn ~ zet if and on ly if ~ ( 1 - zK) a = W¢. Since this p r o p e r t y is c lo se ly c o n n e c t e d wi th the na tu re o f hz(f), we give s o m e m o r e p r o p e r t i e s o f this f unc t i ona l in the nex t l emma.

Lemma 4 On a finite v e c t o r of length m, the func t i ona l )tz(f) is g iven by

Az(f) = - ~ z-iai(f). (4.3) i - 1

If K 1 - z has an inver se , it is g iven by

h~ ( f ) a = (1 - K K - I ) ( K 1_ z)-lf.

F o r all f E ~ ( K - ~ - z) we have

(4.4)

Xz(Kf) = z- 'Az( f ) ; A z ( K - ' f ) = zAz(f) + c~(f). (4.5, 4.6)

I f A~(f) is b o u n d e d , ( K - l - z)g, ~ f impl ies oti(g,)---> Az(K i+~f).

Proof F r o m (1 - zK)g = )t~(f) + K f f o l l ows a l ( g ) = )t~(f) and ( K -~ - z )g = f. N o w if f is a finite vec to r , g is the sum of a finite v e c t o r g ' , wh ich m a y be f o u n d as

342 H. MORAAL

ind ica t ed in the p r o o f of L e m m a I, and an e i g e n v e c t o r go of K ' with e igenva lue z, which , by T h e o r e m 2 and the a s s u m e d inso lvab i l i t y of the K S equa t ion mus t have ~(g0) = 0 for all i. T h e r e f o r e , Az(f) = a~(g') and this may be seen to have the fo rm (4.3) by m e a n s of an ea sy ca lcu la t ion . If K - ~ - z has an inverse , we i m m e d i a t e l y find

A:(f) = (1 - z K ) g - K f = [(1 - z K ) ( K - ' - z) ' - K ) I L

f rom which eq. (4.4) fo l lows by app ly ing 1 - K K ~ to bo th s ides . F r o m the a b o v e we have ( 1 - z K ) K g = K [ a : ( J ' ) o e + K f ] and a : [ a = ( f ) a +

Kf] = 0. Eq. (4.5) fo l lows then f rom a = ( ~ ) = - z ~, which is a spec ia l ca se of eq. (4.3). F r o m eq. (4.5) we get A : ( K K ' f ) = z ~az(K ~f). C o m b i n i n g this with K K If = f - a~( f )a yie lds eq. (4.6).

To p r o v e the last a s se r t i on , we o b s e r v e that by the def in i t ion of W~, (K ~ - z ) g , - ~ f impl ies that there ex is t s for all e > 0 a s e q u e n c e of finite v e c t o r s {hn} sa t i s fy ing

tlgn h. l l < ~/n+ ( K ' - z ) h . = L - ~ I

N o w since h, and f , are finite, an expl ic i t ca lcu la t ion shows o t i ( h n ) =

A,(K ,+if,) and this impl ies ~i(h,)--+az(K i<f) by the b o u n d e d n e s s of az. The a s se r t i on now fo l lows f rom the inequa l i ty

la , (g . ) - ~ , (h . ) l ~ II~,llLlg. - h . I I ~ II~, l l~/n. Q.E.D.

Theorem 5 E v e r y c o m p l e x n u m b e r z is a m e m b e r of on ly one of the fo l lowing six c lasses . (1) z E P ( K ~), z I E R ( K ) ; the K S equa t ion has a unique so lu t ion given by

fCz) = (1 - z K ) 'zoe. (2) z ~ P ( K ~), z ' E P ( K ) ; the K S o p e r a t o r has at leas t one e i g e n v e c t o r wi th

e igenva lue z ~; the K S equa t ion has e i the r no so lu t ion or more than one so lu t ion .

(3) z E P(K-L) , z ~ C O ( K ) ; the K S equa t ion has a un ique solut ion . (4) z G C O ( K 1), z ~ E C O ( K ) ; the K S equa t ion has no so lu t ion , the K S

o p e r a t o r has no e i g e n v e c t o r wi th e igenva lue z ~; &.(f) is u n b o u n d e d . (5) z E C O ( K ~), z ~E R S ( K ) ; as for case (4) but Az0') is b o u n d e d . (6) z E R ( K '), z I E R S ( K ) ; as for case (5) but K ~ - z has a b o u n d e d

inverse .

P r o o f Let z E P ( K ~); by T h e o r e m 2 the K S o p e r a t o r can then have an e i g e n v e c t o r wi th e igenva lue z i this is case (2). If l - z K has an inve r se and this is b o u n d e d , the range of this o p e r a t o r is c lo sed ; this impl ies e i ther (i) z -~ E R ( K )


or (ii) z -1 C R S ( K ) . The s e c o n d pos s ib i l i t y is, h o w e v e r , ru led ou t b y L e m m a s 1 and 2. This , then , g ives ca se (1). If 1 - zK has an u n b o u n d e d inve r se the s a m e a r g u m e n t s h o w s tha t z - l E C O ( K ) , case (3). L e t z E C O ( K - I ) ; then the K S equa t ion has no so lu t ion and the K S o p e r a t o r has no e i g e n v e c t o r , so tha t z 1 mus t be long to e i the r C O ( K ) or R S ( K ) . The d i s t inc t ion b e t w e e n these two c a s e s (4) and (5), is p r o v i d e d by L e m m a 3.

I f z~R(K-I) , the a r g u m e n t of L e m m a 3 s h o w s tha t ~ ( 1 - z K ) = {Az(f) + Kf; f E We} and Az(f) is b o u n d e d s ince ( K -I - z) -I is b o u n d e d . The re - fo re , z - l E R S ( K ) , case (6). The on ly r e m a i n i n g cases w o u l d c o r r e s p o n d to z E R S ( K I) and these are ru led ou t by I- ,emma 1. Q .E .D.

5. Conclusion

Since all po in t s of (6) mus t have a finite d i s t a n c e f rom (I) and s ince (1) and (6) are open se ts , the un ion U~ of the se ts (2), (3), (4) and (5) is no t e m p t y and c lo sed and even the i n t e r s ec t i on of U~ wi th the pos i t i ve rea l axis is no t e m p t y . This s h o w s tha t the p o s s i b i l i t y of some t y p e of p h a s e t r ans i t i on ex i s t s in this t heo ry . M o r e def in i te i n f o r m a t i o n d e p e n d s on the b e h a v i o u r of the se ts (1)- (6) as a func t ion o f ~; the fo l lowing c o n j e c t u r e can be g u e s s e d to hold fo r su i tab le pa i r po ten t i a l s .

Conjecture The sets U e sa t i s fy U e,D U~ if ~:'>~:. The spec t r a l r ad ius of K i, i.e. the n u m b e r sup~cu~ Izl, a p p r o a c h e s infini ty fo r ~ ~ ~. This c o n j e c t u r e , if p r o v e d to be t rue , impl ies the e x i s t e n c e of at leas t one p h a s e t r ans i t ion due to the i nc rea s ing p r o p e r t y of the U e It f u r t he r impl ies tha t t he re is no f u n d a m e n t a l r e a s o n w h y the K S e q u a t i o n shou ld no t have a so lu t ion in s o m e W~ for all z.

References

1) D. Ruelle, Statistical Mechanics, Rigorous Results (Benjamin, New York/Amsterdam, 1%9). 2) H. Moraal, Physica 81A (1975) 469. 3) J.G. Kirkwood and Z.W. Salsburg, Discuss. Faraday Soc. 15 (1953) 28. 4) W. Klein, J. Math. Phys. 16 (1975) 1482. 5) L.A. Pastur, Teor. Mater. Fiz. 18 (1974) 233. 6) J.E. Mayer and E. Montroll, J. Chem. Phys. 9 (1941) 2. 7) O. Penrose, J. Math. Phys. 4 (1%3) 1312. 8) K. Yosida, Functional Analysis (Springer, Berlin/Heidelberg/New York, 1968).

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Spectral properties of the Kirkwood-Salsburg operator