Ergodic properties of groups of the Bogoliubov transformations of CAR C∗-algebras

ANNALS

OF PHYSICS

175. X-266

(1987)

Ergodic Properties of Groups of the Bogoliubov Transformations of CAR C*-Algebras A. G.

SHUHOV

AND

M.

Yu.

Received

Various formulated

SUHOV

October

*

IO. 1985

conditions on a group of Bogoliubov’s transformations of a CAR which guarantee the convergence to a stationary quasi-free state.

C*-algebra

are

1 19x7 Acsdemlc

Preaa. Inc

0. INTRODUCTION

0.1. Let Y. and .X’ be complex separable Hilbert spaces, 91 be the CAR operators the fermion creationannihilation C*-algebra generated by acting in Y (we suppose that a + (,f) depends on ,f linearly, (~+m~(g),.Lg~~‘) and a(,/‘) antilinearly). In the algebraic formulation of quantum mechanics (see [l-4]) the CAR C*-algebras describe fermion systems (with finitely or infinitely many degrees of freedom). Assume an operator matrix T=(

‘,:I:

;I’)

(0.1)

acting in Y - 0 Y - is given where T’.‘, T’.’ are linear and T’.‘, T’.’ antilinear ded operators in $ - which satisfy the conditions T’.’ = 73.2 = T” 1 Tl.2 = T2.1 = T”’ T’z)*T(l)+ T’“‘*T”‘+

* Permanent Sciences, GSP-4

address: Moscow

T’1’*T’?l= T”‘*T”‘-

Institute 101447,

-

for Problems USSR.

T”‘T”‘*+ T”‘T”‘*

boun-

T”‘T’1’*=0, +

T’“‘T’2’*T’?‘*

of Information

Transmission.

(0.2) (0.3 1

=

E

(0.4) USSR

Academy

of

‘31 OOO3-4916/87

$7.50

CopyrIght ((‘8 1987 by Academc Press, Inc. All rights of reproducuon in any form reserved

232

SHUHOV

AND

SUHOV

By setting

we define the * -automorphism J: VI-+ 9I, called the Bogoliubov transfornzu~ion of the C*-algebra ‘LI (see, e.g., [2, Theorem 5.2.51). The Bogoliubov transformations are, in a sense, the simplest examples of *-automorphisms of 91. The problem of constructing and studying “natural” *automorphisms (as well as groups of * -automorphisms) of U is of great importance for physical applications. In particula., r this is related to the problem of constructing time dynamics generated by a given Hamiltonian interaction. For some classes of systems (spin systems) this problem is solved in [S-7] under general assumptions on the interaction (see also [2, Chap. 6.2 and the references therein]). However, for such a realistic system as that of interacting particles in Euclidean space R”, this is still an open question (which probably has no positive solution for general interaction potentials). For this type of systems the Bogoliubov transformations form a unique known class of examples of dynamics (corresponding to quadratic Hamiltonians, w.r.t. creation and annihilation operators). A particular example of such a dynamics is the free motion in R”. For the rigorous formulation of problems and results we introduce one more object which plays an important role in the sequel. This is a v-parameter group { U,, n E Z’ i or ( U,, x E R” i of unitaries in Y (in the second case it is assumed to be strongly continuous in s). Physically, the group f U,,; or { tJ,l represents the space shift translations: the discrete case corresponds to lattice systems and the continuous one to continuous ones, v being the dimension of the system. The “natural” character of a given Bogoliubov transformation may be noted by its properties w.r.t. the group ([Jr,} or (U.,). In the present paper we consider Bogoliubov transformations which commute with the corresponding group of space translations. We assume that the group { CJ,,) or { CJ,.) has the simple Lebesgue spectrum. 0.2. Let a group ( T,, t E R’ ) of operator matrices (0.1) be given which satisfy conditions (0.2)-(0.4). The infinitesimal generator D for jT,) is given by D= The operators

B and C in the spectral (Fourier)

(0.5) representation

(0.6a)

aw)=ww(~), Q(H) = c(Qf(

have the form

-8).

(0.6b)

Here H is a point of the torus [ --7~, 7~)~for lattice system and of the Euclidean space R’ for a continuous one, b is a real, and c an odd function either on [ --71, R)\’ or on R’, SE L,( [ - 7c,7~)~) or L,( R“), respectively.

ERGODIC

233

PROPERTIES

The (continuous) model of the free motion corresponds to h = i ll0li’, c = 0, 0~ R”. In the spectral representation the operators T:,‘, Tf,’ are the multiplication operators by the function

(

elrhz and Tj*‘, T,,’ are the product operator by the function

COS(

ih tlr’) + --! sin( [iv) H’ 1

of the operator

j:j’~,f.(

-

) and the multiplication

Here

h,(H)=~(h(tr)+h(-O)),

(0.9a)

hz(n)=~(h(e)-h(--O)),

w(e)=(b;(fl)+

jc(r))l’)’

l.

(0.9b)

An important role is played in the sequel by the non-degeneracy mulated in Sect. 0.6 below) imposed upon the functions

conditions

(11* = h, f \I‘.

mations

(0.10)

0.3. The group {T,, t E R’ 1 generates the group of the Bogoliubov (35, TV R’ ). For a given state Q of the C*-algebra 91 we set .F;Q(A)=

Q(S

,A),

(for-

A EYI.

transfor-

(0.11)

We are interested in the question on convergence of states .YFQ, t E R’, to a limit when r--t :tx. It is natural to expect that the limiting state P will be stationary (g:P = P)I. In connection with this we consider the class of quasi-free states (see

[IlAl,. A state jD is called qzta.+f~ee if for any ,f’, ,..., j;,,, R, ,..., g,, E Y ‘,

,?I p(,=I n u+c.t;, I’r 4Rk)1 A=I = 1 (-l)d”fiL i E“‘Vf.g) x ,,,,,fi:‘: IP(4h, ) 4h,)) - p(ah, )I R4b))l x Here

Z’)(f,

g)

denotes

n”‘P(li(h)). r/,,tr the

set

(0.12) of partitions

of the

sequence

of symbols

234

SHUHOV

AND

SUHOV

(fi,..., f,, g, ,..., g,) each element of which contains not more than two ones (speaking on an element containing two symbols we always write them down in the order of the sequence). The product n(” is over all elements with two symbols, and over all elements with one symbol. Depending on the symbol h, h(h) reads as rs” a+(h) or a(h). Finally, deg 5 is defined as follows. We order the elements of the partition w.r.t. the order of their “first” symbols. Then deg { is the parity of the (m+n)-permutation which transforms the whole set of symbols into the initial

sequence.f, ,...,fm, g, ,...,g:,. It is clear that a quasi-free the form

state is determined

by its value on elements of ‘9l of

For a given state Q (not necessarily quasifree), it is convenient functionals (which are mutually connected in the obvious way) K;,“‘(f)

= ecu+(f)),

K~~a'cf,,f,)=Q(~+(.f,)~+(.fi)), K;W; g)=Q(a+(f,4g,,>

qy

‘( g) = ads)),

to introduce

the

(0.13)

Kh""'(,,, gJ=Q(a(g,) u(gz)h (0.14) (0.15) L g,f,, giE I‘, i= 1,2.

We call them the “lower” (moment) functionals corresponding to the state Q (in an analogous way one can introduce “higher” functionals). These functionals are linear and/or antilinear and with a norm < 1. Every (consistent) set of lower moment functionals determines uniquely a quasi-free state (see [2, Sect. 5.2.31). It is not hard to check that a quasi-free state P is stationary (invariant) w.r.t. a group j,q, tER’) iff the lower functionals Kpl.‘:?). E, + E? ~2, have the corresponding invariance property. 0.4. Let a state Q of 9I and a pair of orthogonal be given. We set

subspaces

Y ^‘I ‘, $ -“’ c Y.

ERGODIC

23’5

PROPERTIES

Here the sum is taken over all possible sets f’l” )...) f$,\,, g\” ,...) g),fAE %“I), j-y’,..., f&, with norms

< 1. Furthermore, plp(

g’l” ).‘., g$ E I “?‘?

we set

y -(‘I, $ ““) = max p# ,,,“‘,,,“‘).(,,‘?‘.,,‘?‘1](Y ‘(II 3 .j .(?I 17

where the max is taken over all representations integers 111= ??I” ) + r7P),

(0.17)

of m and II as sums of non-negative ,, = 11”’ + )+2’

with (m”

+ n”‘)(m”’

+ d2’) > 0.

coefficient” The quantity pg.“‘( Y (I’, Y -“‘) may be considered as a “correlation between the subspaces $ -‘I’ and Y “?’ (more precisely, between C*-subalgebras ‘?l(Y““) and ‘u(Y^“‘) generated by I+, a(g) with f; ge Y“” and 1; gE Y““, respectively). In the sequel we shall deal with the subspace of the form or L,(R’\I(Jl,r+s)), $,“I’ = [,(I( y, r) n Z’) or L,(Z( J, r)). Y ‘“‘=I,(Z’\/(~,r+s)) where I( J’, r) = )(y= ’ [y, - r, ~1,+ r] is the cube in R” centered at J’ = (r, ,..., y,) with the edge length 2r. 0.5. The main results of this paper are related to the convergence of states S,?Q, t E R’, to a stationary quasi-free state when t -+ +;c under some nondegeneracy conditions upon the functions w f (see (0.10)) and conditions of decay of space correlations in the initial state Q. Chronologically, this fact has been proven first by Lanford and Robinson [8 J for the free motion in R’ and translationally invariant, “square integrable” initial state Q (the method from [S] may be extended to a general class of the Bogoliubov transformations provided the translational invariance and square integrability of Q are assumed).The slightly later paper [9] of Haag, Kadison, and Kastler should be especially quoted. The result of [9] relates to the case of free motion as well (the authors deal with R’ but their arguments go in any dimension v), but does not need the assumption of the translational invariance of Q. However, the method developed in [9] (as it is described therein) essentially uses some particular features of free motion (it seemsthat, after technical modifications, this method might be applied to some other models). In addition, the condition of the decay of space correlations from [9] is too restrictive: for v 3 2 it lacks justification even for Gibbs (KMS) states with small activity (within the l-phase region). In the present paper we deal with general groups of the Bogoliubov transformations satisfying the non-degeneracy conditions mentioned before (notice that the free motion is a “perfect” non-degenerate case in our classification (seebelow)).

236

SHUHOVANDSUHOV

Such an approach allows us to treat some models of interacting particle systems which may be reduced to Bogoliubov transformations: one-dimensional KY-model, lattice, and continuous hard rod model (this is the subject of separate papers, see, e.g., [l&12]). The method we use is close, in some basic points, to that of [9] and goes back to well-known ideas of probability theory (seethe next section). In the course of proving the results we use many ideas and results from [13]. We mention as well the papers [ 14, 151, where some elements of our approach have been used earlier. Moreover, unlike [S]. we deal with not necessarily translationally invariant states, and, unlike [9], our conditions of decay of space correlations in the initial state Q may be verified by means of the Gibbs states theory. We refer the reader to the papers [ 16, 171 for the details of this verification. Considering non-translationally invariant states allows us to treat the problem of the so-called hydrodynamical limit for groups of the Bogoliubov transformations (this is again the subject of a forthcoming paper). 0.6. In this section we formulate our main result in the one-dimensional case (v = 1 ): this requires minimal preliminaries. We suppose that the functions h, CE C’ (see (0.6a), (0.6b)). Furthermore, for some ~,:=2, 3,... each function o,,, E= f, is a Pi + ‘-function, and 11, +’ (0.18 B(o,:, PI:+ 1) = n B,(w) = Bzr, /=2 where (0.19 the number pr is chosen to be the minimal one for which (0.18) holds. For p, = 2 this condition has been used in [ 131 (see [ 13, Sect. 2, condition IV]). Now we formulate conditions on initial state Q. Let YE, & c R’ or Z’, denote the subspaceof ^I. spanned by the vectorsf with supp,f c 0. We set (0.20) (in the lattice case R’ is replaced by Z’ and (s,, s2) by (sr , s2) n Z’). Suppose that for some d = d(m, n) > 0 lim s”pg.“‘(s) = 0. , - -x THEOREM

(0.21 )

1. Let operutors B, C, and a state Q satisj~ the uhove conditions. Then

the state Q, = S:Q

ERGODIC PROPERTIES converges, as t + +a, to a quasbfiee state P W’ the lobrer ftinctionals C, + c2 < 2, converge to the corresponding functionals K$‘.“z’. Remark.

The same statement

237 K$.‘?‘,

holds if t + +YJ is replaced by t + -K.

The question about convergence of the lower functionals will be studied separately (see Sect. 2.5-2.8, 3.3-3.5 ). In the multidimensional case the non-degeneracy conditions are formulated in terms of the structure of singularities of the functions UI i (see Chaps. 4, 5 ). 0.7. The paper is (unfortunately) quite long and contains. besides the introduction, six sections divided into subsections. Section 1 contains an auxiliary result proven in [ 161 which is used in the sequel. In Section 2 we prove Theorem I for the case of a discrete group [ Ii,,, tz E Z’ ) (i.e., for one-dimensional quantum lattice fermions). Section 3 is devoted to the one-dimensional continuous case (Z’ being replaced by R’ ). In Sections 4 and 5 we consider the multidimensional lattice and the continuous case, respectively.

1. AK AUXILIARY TOOL: THE "NON-COMMUTATIVE" BERNSTEINMETHOD Consider the following condition (A) on Q and (T, ] : (A) For any t E R’ there exists a finite or denumerable orthogonal decomposition Y = @ ,I~ o Y‘I,” such that ( I ) for all m, II E Zl, with n1+ II > 2,

where

(2)

for every vector h from a set 7 dense in Y‘, lim r--r fl

where

sup d,( h, I) = 0. /a0

238

SHUHOVANDSUHOV

and 7tf III stands for the orthoprojector the sequel ), (3)

forallm,nEZ’+

9’ + 9 “5” (the latter notation will be used in

withm+n>3andany,f‘,,...,

lim

fm,g ,,..., g,,EY’.,

2 e,(f,, .... .L,;gi, .... g,,;O=O,

I--+ k-r ,a,

where

the max is over all sequences &jr I,..., ECI, e’,” ,...,~jl’ = 1, 2 and f- is the dense set from (2).

The following result has been proven in [ 161: PROPOSITION 1.1. Let a stute Q and a group of’ operator matrices jT,J he given which obey (A). Then the states Y:Q, t E R’, converge as t -+ fm to u quasi-free state P ifs lower ,functionals K!2;$‘, E~+ c2= 1, 2, converge weakly to the ’ corresponding,functionals K$I.QI.

Proposition 1.1 reduces, under condition (A), the question of convergence for states to the simpler question about convergence of lower functionals. However, condition (A) is imposed upon the pair (Q, {T, 1). But from the point of view of physical applications it is more convenient to formulate “separate” conditions on Q and {T,j. (more precisely, on the generator D). In particular, this will allow us to compare ergodic properties groups of the Bogoliubov transformations. We do this in Theorem 1 and Theorems 2-6 below.

FIGURE I

239

ERGODIC PROPERTIES

The “physical” meaning of condition (A) is as follows: (1) indicates that every C*-subalgebra 9l( # ‘I”), 13 1, becomes “asymptotically independent” on the family of C*-subalg,ebras (2l( +“j!‘), I’ # 0,1) in the state Q. The subspace $‘g’ plays the role of a “corridor” separating “rooms” %-!I), I> 1. In what follows we set %‘l‘j’)to be equal either to I,(Oj’)) or to L,(Oi”), where 0;‘. O’,t’,... will be a finite partition either of Z“ or of R’ (see Fig. 1). Conditions (2) and (3) mean that every vector Ti;.,% is “spread out” sufficiently fast when t -+ XI. The original formulation of this condition is due to S. N. Bernstein; it is intensively used in probability theory for proving various limit theorems for weakly dependent random variables (seereferences in [16]).

2. LINEAR MODELSON ONE-DIMENSIONALLATTICE 2.1. The proof of Theorem 1 for the lattice case is divided into several stages. At first, we prove that

,?, lim PI n fi dg,)> I- I I ( ,=Ia+(.f;) k=l -Q

(fi cF’i’h+(,f,) ,=I

fi

-/;‘%(,,))I

=O,

(2.1)

k=l

where ~~“‘a+(f)=u+(n”~‘T’:‘,f) q’s(g)

x(“) is the orthoprojector I, = [-it,

= a+(n”i’pg)

+a(711’tIp2,f‘),

(2.2)

+ a(n”“p.‘g);

(2.31

Y + ^I;, and At]

n

Z’,

I.>

max Iw’,(H)I. Ot[ -n.n,

(2.4)

For proving (2.1) it is sufficient to show that (2.5) This follows from bounds contained in [ 131 (see [ 13, Proposition A.2). 2.2. In what follows we give a detailed analysis of the cut-off quantity

240

SHUHOV

which is based on vanishing

AND

SUHOV

(as t -+ m) coefficients

( Pl.rBJ)k = ( TE.‘.,“‘e,, r,+k),

1, k E Z’,

(2.6)

where {e,, n EZ’ ] is the orthonormal basis in ‘I for which UP,,= r,, + , . These coeflkients are written in the form

-i =-

n

2n s n

& p

_ -de) sm(rltje)). w(e)

r/h,(O)

(2.7b

In assertion which follows some positive constants appear that do not depend on ZER’, 8, etc. Their concrete values may change from one statement to another. We shall denote them by r, r’, etc. LEMMA

2.1. The following hound bolls: max sup I(Y,,‘:),I 2,-l.?=I,? k

where p=max{p+, ProojI

d~(l + Iti)

’ il’+‘),

(2.8)

p.. ).

We restrict ourselves to the proof of the bound (2.9)

all other integrals in (2.8) as well as the case t < 0 are considered in the same way.’ The proof proceeds along the same scheme as that of Proposition A.3 of [ 131, where the case p + = 2 is treated. Let us denote S,,,(e) = -kB + to + (0) and write fl,(c~+ ) = I[IO,..., (I,,) (from (0.18) it follows that the sets p,(o+) are finite). We surround the points (I,,..., [I,, by neighborhoods I’(@,),..., V(0,,) of the length 2t “, where 6 > 0 will be chosen later. For t large enough

HEC .,b$,v,e,,

‘wic(0)’

>oTjl,, , ,

‘wI;(e,*

rr”)l >

nl+

2(p+ -l)!

If’ - I ,d f

’

(2.10)

’ Notice one difference which will appear in considering other integrals in (2.8 1. This is connected with the presence of the functions b,/,c and c/w. For studying these integrals it is suflicient to divide L - x, n) into the intervals of monotonicity and constancy of the sign of the above-mentioned functions and then to apply the second mean-value theorem.

ERGODIC

241

PROPERTIES

where (2.11 ) Hence, IS;‘,(H)( >

G+

fl Itz+ - I)!

and according to Van der Korput’s Lemma 13.4.1]),

f/l+ I )d

lemma

&I cos S&J()) [ X.7rl u;’ ,,CliJ,)

HE[-I.n)\ij

,,=,I

V(U,),

(see [ 19, Chap. 2, Lemma 1; [20,

< r’f- I,~+ll/r+

1) 2)“.

(2.12)

Since the integral over V(U,) is < 22 -~(‘, by choosing S = l/(/l + + 1 ) we get (2.9). Lemma 2.1 is proven. In general, the bound (2.8) cannot be improved: it may be reached for k=c~‘+(O,)t, O,E/?~(~C)+) (see [21, Chap. III, Sect. 1, Theorem 1.5). It is possible to show that for all k E Z’ which satisfy

the following

bound holds: max I( F1,,iz),I 6 Y’( 1 + /tl ) ’ ‘, ,,,i.:= 1.7

where r’ depends on r. More detailed information is given by LEMMA

heE Z’

2.2. Let ~=max(~+,~ satisfj~ing the inqralit~

). Fis x~((,r+l)

‘, 1). Th~,fbr

anal PER’,

(2.13) Proyf:

We restrict ourselves to the proof that for t > 0,

242

SHUHOV

AND

SUHOV

implies that x dQcos(-kB+ro+(@)


+I))“+++

-1));2~-.

(2.14)

All other integrals in (2.13) as well as the case t > 0 can be proved in an analogous way (see the footnote in the preceding lemma). Again we surround the points 0 cl,..., 6, E p2(w k ) by neighborhoods of the length 2t -‘, E = /.L+ ‘( 1 - a). The bound (2.12) implies that

dBcos(-kB+to+(O)) C-n. nl\U:=o P’(O,’


“+z(f’+

Now we estimate the integral over V(Oj). To be definite derivative UJ~I + “(8,) > 0 and consider the integral

“““‘+.

we suppose

that the

o,+r8d%cos S,,,( 0 ). sfl,

(2.15)

It convenient to consider is three cases: (a)k/t‘+(Oi)-f2‘, (b) k/t>w’+(O,+ t-“), (c) w’+(O,) + t’- ’
o’+ (d,,,) = k/t. From the inequality

(c) it follows

(2.16)

that

P ’ < w’, (e,,,) - w’, (0,) < 2~‘f’++“w(o~, + P+!

Ai

-d ){‘+. i

We surround U,,, by a neighborhood Vk., of the (2~+))‘( 1 + a(~+ - 1)). The integral (2.15) is represented as

length

(2.17) 2t I’.

6=

(2.18) The second term in (2.18) is <2tP(‘+“P+ ~ ““lr-. The first and the last ones are similar and we consider only one of them, say, the first. Given 8 E [0,, 0,,, - tP”1, the following inequality holds

(2.19)

ERGODIC

243

PROPERTIES

For
>

1 w+

- l)!

O~++“(ej)(ek,,-t-6-e,)~+-‘.

(2.20)

(2.19), and (2.20)

This inequality (2.18) is

and the second mean-value

For 6=(2~~+))‘(1 proven.

theorem

imply that the first term in

+M(P+ - 1)) this gives the estimate

required.

Lemma 2.2 is

Remarks 1. For h, c E C” the assertion of Lemma 2.1 may be obtained by using a result of Vinogradov (see [22, Chap. 2, Lemma 41) and Hua (see [23]). See also [24, Sect. 4, Example 4.1.2; and 26, Chap. II, Sect. 6, Theorem 81. Another way to obtain this assertion is to use results of [27, and 281. 2. For C’-functions Theorem 3 of [26].

h, c and p = 2, the assertion

of Lemma 2.2 follows

from

From Lernma 2.2 and the bound (2.8) we deduce LEMMA

2.3.

For any tE RI and 6 E (0, 2(p + 1) ‘) there exists a set D”,‘)c

Z’

such that (1 )

lirn, _ % xkE Dll,lliI ( Pi.,Cz)k12 = 0,

(2)

I(‘F:.;L)kI


+ ltl)~‘i2+“,

(2.21) (2.22)

k $ D”. ‘).

a constant r being independent on t and 6. Proof.

Consider

the recurrent

2 To=--p+l

6 2’

sequence tO, z, ,... given by 7, =

1

l(p+Tj-

P

1)

-2i+”

6

j2

1.

Let us denoste

Dl.1. u [to;(e)-t’~, tw’+(e)+t”]nZ’. I 6I- 0Epz(oJ* 1 595475

2-3

(2.23)

244

SHUHOV

AND

SUHOV

First, we estimate the sum of the same type as in (2.2 1) but with k E II:‘,“‘. This sum may be represented as

According

to Lemmas 2.1 and 2.2, we have, respectively,

we get that for large enough j 1 +q(P-l)>~-6 3’ holds. Hence, according following bound is valid

2

to Lemma 2.2, for large enough j and /it Z’\D;.‘.“)

I(pl.,EZ)&l
+ Itl)--I’+r,(ll~l)!l/l)
+ IfI)

the

‘#?+I$.

For completing the proof of Lemma 2.3 one can take, as the set D”. “, the set DC’, / hl for large enough j. 2.3. Now we are able to finish the proof of Theorem 1. Fix ~2, n = 0, I,... with m + n b 2. Choose 6 > 0 and a pair /I, y of functions which are monotonically increasing to + cc and satisfy the conditions (a) (b) (c)

lim,,, (p(r)-2j,(t))= lim,, x P(t) Pp’=O, lim,, x (1 -wBuw=0,lim,,,= t~(t)~~‘p~~“)(~(t)-2y(t))=O.

cc,

Such functions and a number 6 exist due to (0.21). For every t > 0 we take the set D (I. 6’2’ indicated in Lemma 2.3. Choose a value r. > maxHEC+n.n, Iw’,(Q)l. We introduce the finite partition Og), O!‘),... of the lattice Z’ as follows. At first we set arr’=([va(t)-~(t),yB(t)+y(r)]n[-r~t,r,t]nZ’)\D”.“/‘“‘,

.VEZ’,

ERGODIC

245

PROPERTIES

and

ob”=z’ u al!‘. \ ,,FZ’ The number of points y for which a.;!’ # fa is finite and <2r,tj?(t) ‘. We label them by the naturals in an arbitrary order. These will be the elements of our partition Ojr’, toy’,... . We set I“:”

= 12(op,,

k 30.

Now use Proposition 1.1 from Section 1. We have to check the conditions ( I)-( 3) of this proposition. The role of the dense set T’ will be played by span je,? ). (1) follows immediately from (c). For simplicity we check (2) for the case k = e,,. In this case

(in the general “shifted” set of If l#O, then is estimated by

situation one must consider the sum of the same type but with a summation indices). we have D(r.iii2’nOO)r)= 0 and hence, according to (2.22), this sum the bound

and tends to 0 in view of (a). For l==O (the contributions

,.c,,,=

from corridors)

= h t 7’ =[ml. r,,rl + kE ,]lr.o2,

+

this sum may be represented

k t co/;‘”

[

1 r,j,. roll)

n”.,‘:’

as

.

The first term in the RHS of this equality tends to 0 according to Proposition A.2 from [ 131. The second term tends to 0 due to (2.21). The third term is estimated by

and tends to 0 in view of (b). The “worst” case in (3) corresponds

to the minimal value I?Z+ fz = 3. In this case

1 eLfI ,..., f,,; g, ,..., g,; 1) I>, I < const

tb( t)

max ’ y( t)3’L max max />,I ,EOj”i,.EZ=I.Z

< const . b(t) -‘l’(r)(y(r)

1( pJ.,f?), I3

f ~’ +36)“2.

The RHS tends to 0 in view of (a). Theorem

1 for the lattice case is proven.

246

SHUHOV

AND

SUHOV

2.4. We now pass to the question on the convergence of functional K$l,&‘), E, + .s2 6 2. First, consider the case where the initial functionals K$l’E’) are invariant w.r.t. U,. In this case K$.“=O, Kp ‘)=O. The functionals K$I’~~), E, +E? =2, are written in the form K~,“‘(.f,

3fi) = U-i 3MI:: fi >,

Kg%,,

.d=

oqyg,,

g,h

(2.25)

Kg. “(A g) = (.L M’$ .- g) ’ where IV’,~~ is a linear, Ml?) and M$) are antilinear d 1 commuting with U,. In addition, we set Mr,f?,‘= E-

(2.24)

operators

$‘ + $“ with norm

(M’,fj)*;

(2.26)

then

!2(4g)a+(f))=wq% s>. The quadruple

of operators

IVJ:;~ is written

(2.27)

as the matrix

Ml”‘=

(2.28)

In the same way one introduces the operator matrix M’“l’ corresponding state Q,. It is easy to check that McQ’ and MtQf’ are related by M’Qt’=T”I,M’Q’T_ The stationarity

condition

,.

(2.29)

reads (iD)*M’Q’+M’Q’(iD)=O,

where D is the infinitesimal matrix. Now we pass to the spectral representation.

(2.30)

The matrix 0 is written

as the sum

~,+O*,

(2.31)

where fro is the diagonal matrix which contains the multiplication operators functions h, (see (0.9a)), 0, = fi - oo. The operators &lj,o;2 read Afi~f(e)=~~ @ym=

,.,(e)f(e (1 -m,.Ae))f(e,,

A:f:f(e) the following

=m,,,(Qf(

condition

eG c-n,

n),

(2.32b)

n),

(2.33a)

OE [ -71, 7r),

(2.33b)

66 C-n, -fl),

by the (2.32a)

ee c--71, xl*

~II?Ifce)=m,,,(e)S(-e,,

Consider

to the

for the functions

h, c:

247

ERGODIC PROPERTIES

(B) F’or some p,, = 1, 2,... the function up(see(0.9b)) is of class CPoand the set np=, p,(w) (see (0.19)) is empty. PROPOSITION 2.1. The matrix MtQo is weakly converging when t + iaz stationar-v matrix which, in the spectral representation, reads

~fi’Q’+$ Proof:

to the

(2.34)

(iO,)*A’Q)(&,).

Equality (2.29) may be written in the form

(

K4(Qfi= cos(tw)e+ (ifi,)* x

sin( -;w) ht’ >

&I’Q) cos(tw) E + sin(;fM.) (io,j) (

(2.35)

and it is an easy exercise to check that this matrix is weakly converging to (2.34). 2.5. The arguments of the preceding section arc easily extended to the case where the initial functionals K$l.‘:?), E, + e2d 2, have a “periodicity” property w.r.t. the translation operator U, . This means the existence of a number s E Z\ such that K’*:I. “z’(h) = K’Q. a)(~ I1’ ). Q K$l.=)(;h,, h,) =K;‘,“?‘(L$h,,

U,h,).

E,+C?= I,

(2.36a)

F, +z2=2.

(2.36b)

As before KC’.” and K$‘, ‘) must vanish in this case. As to the functionals K’$‘, cz), E, + E?= 2, iFis convenient to write them again in the form (2.24) (2.25). The periodicity is expressed in terms of the operators Mj:),, i,, i, = 1, 2, as follows. We set

Wfj2(j,,j2)=

(W$2e,,,

e,,),j,,h~Z'.

(2.37)

Then, for every m EZ’, Mlfj2(j,,

j2) = Mf.l$i2(jl + MS,j, + ms).

(2.38)

In the sequel we take the minimal s for which (2.38) holds. As before, the matrix M’Qo has the form (2.29). It is convenient to pass again to the spectral representation. The operators M(,ei and ML!,’ are now written as

248

SHUHOV

AND

SUHOV

and M\Q/ and M$fi as

where yr!:! ,2, I = 0 ,..., s - 1, i, , i, = 1, 2, are functions on [ - 7r, 71). Let S,, u E R’, be the shift operators (on [ --7c, n)): $Jw

=f(d

UE R’.

+ u),

(2.41)

Then IWQJ = c

IWQ. “slnlir

(2.42)

/=O

where the matrix I@(Q,‘) consists

of operators

commuting

with

0, and

0 S,=( 0s, 3,; 1 Notice that &IcQ.” corresponds to the matrix averaging the initial periodic matrix M(Q):

Taking

all this into account, we write \-

M’Q,”

the formula

which

is obtained

by

(2.29) as

I

&I(Ql) = c Tt,A(QJ’$Zrr,,bT ,,

(2.43)

/=O

where f-*

-I

fi(Q.Os

sin( - tw) zn,,,T-l=

e,rhz

COS(tM')e+(is,,*

>

w

( x m(Q.

0s

2nll.s

e -rrhJ

(cf. (2.35)). Further analysis is based on properties consider subsequently the cases below:

cos(tw)

e

of functions

I. For any I = O,..., s - 1 the function

0,i e+y>-o-(e)

+

w f (see (0.10)). We shall

ERGODIC

PROPERTIES

249

satisfies condition (B) (see Sect. 2.4). Further specifications are related to the periodicity of o + . It is clear that there exist values IF = l,.... s with the properties (1)

1: are divisors of s,

(2)

w,(e+27df/s)~0+(0),

BE(-7L,

71).

We assume in the sequel that 1: are minimal numbers obeying (1 ), (2). The condition is that for all I* = l,..., s - 1 which are not multiples of If the functions

satisfy again condition II.

(B ).

For some I, = l,..., s - 1,

(3+( N+? 1=w(O),BE[-7cn,7r). As before, it is convenient to choose the minimal I, satisfying this equality. In this case we require that for all I = I,..., s- 1 which are not multiples of I, the function

satisfies condition (B). Further conditions in case II, as in case I, are connected with the periodicity of functions tit+. It is clear that there exists a value I, = l,..., s with the property w+(8 + 27c/,/s) = o),(Q), 8~ [ -7c, n). In the sequel we deal with the minimal I, satisfying this condition. Clearly, I, > lo. We require that for all I= l,..., s - 1 which are not multiples of I, the function o + (0 + 2711/s)- w + (0) satisfies condition (B) as well. II’. Function o + coincide: w + z o ~. This corresponds to the value I, = 0 in (2.44). In this case \V= 0. Further conditions are connected with the periodicity of the function hz. It is clear that there exists a value I, = l,..., s with the property h,(0 + Znl,/s) = L(Q), 0 E [ -7~ 7~). As above, we choose the minimal such I,. The condition is that for all I not multiples of I, the function h,(B + 2x1/s)-h,(8) satisfies condition (B). Now we write down the limiting matrices for all the three cases (cf. [ 13, Theorem 4.21). PROPOSITION 2.2. Under the conditions stated above the matrix M’“” is converging to a limit when t --f &CO. The limiting matrix in the spectral representation reads as follows:

250

SHUHOV

AND

SUHOV

Case I.

where ST = s/l:,

s; = sll;.

Case II.

where s, = s/l,. Case II’. .A, - I 1

iQ(Q.“~‘sZn,,,,J,

/=O

where si = s/I,. The cases where the entries of the limiting matrix commute with special mention: Il.

U, are worthy

of

Let I: = 1; = s (Case I). The limiting matrix in the spectral representation

is

11’1. Let I, = s (Case II’). The limiting coincides with $l(Q*‘).

matrix

in the spectral representation

2.6. In this section we consider initial functionals Kg”‘, Q), E, + E?= 2, which are “almost periodic” w.r.t. the operator U,. For the precise formulation we again pass to the spectral representation.

ERGODIC PROPERTIES

251

We assumethat the operator matrix klCQ)is of the form a(Q)

=

x y(da) s -A

61(Q.“)~6,

(2.45)

where p is a finite Bore1 measure on [ -71, rr) and {i%‘Q,u), (TE [ -rr, rr)} is a family of matrices which consists of operators commuting with 6,. The integral (2.45) can be understood in the weak sense. The formula (2.45) gives an obvious generalization of (2.42) where the role of p is played by the uniform distribution on the finite set {27-~l/s,1= O,..., s - 1 1. We suppose that the matrices i$lCQ,C’ are uniformly bounded in norm

and that the measure ~1has a positive atom at the origin:

As for the functions o + , we assumethat for any GE C--71,n)\{O) difference

and s, s’= Ycthe

o,(O + 0) - %,(W satisfies condition (B) (see above). PROPOSITION 2.3. Under the conditions stated before, the matrix M’Qi” converges to a limit when t + f cc and the limiting matrix in the spectral representation is

cP,Oj+L (i~,,)*jQ(Q.“)(if),) w*

(2.46)

As it is easy to check, the matrix elements of (2.46) commute with ii,, which corresponds to the translation invariance. Proof.

The convergence to (2.46) is based on the formula

which is a generalization of (2.43). By using the Lebesgue dominant convergence theorem we reduce the problem to proving that for any non-zero c E [ - rr, XL)the matrix 9??,AcQ~0)S,T ~, tends weakly to 0. This is verified exactly in the same way as in the preceding sections. 2.7. One can consider the case where the initial functionals K$,Ez’,

252

SHUHOVANDSUHOV

.sI $ c2= 2, have different “asymptotics” on k 8% (cf. [ 13, Theorem 4.11). In the precise form this means that ul-lim U k Mf.,VjzIi, = M!Q. I,,,, ~ ’’ k + ~ IL,

i, , i, = 1, 2,

(2.47)

w-lim k-tr

i,, iz = 1, 2,

(2.48 )

U~)J4~~pk=M’Q~+’ I,.I? ’ ._

where Mj$z* ) commute with U,. As in [ 131 we impose one more condition,

We suppose now that b,/w and c/u, are C’-functions, and w * are of class C”+ ’ and satisfy (B) (see Sect. 2.4); the same condition is supposed for the difference Co, -CL = 2w. To understand when the matrix M’“!’ converges to a limit it is sufficient to study the particular case where

(see [13, pp. 139P141]). PROPOSITION 2.4. Under the conditions qf‘ this section the matrix MfQ” converges to a limit when t + co and the limiting matrix in the spectral representations is

In the opposite limit t + -a~ the limiting matrix changesthe sigfr. Proof The proof is along the same line as that for Theorem 4.1 of [ 131. We omit the details for the sake of brevity.

3. LINEAR MODELS ON THE LINE 3.1. The proof of Theorem 1 for the continuous case proceeds as above in several steps. As in the lattice case, first we prove the relation (2.1 ) where I, is now the interval [ -q(t)t, ?I(t) t] and q(t) is a monotonic function R’+ -+ R’+ which increasesto cc when t + cc. It is again sufficient to prove the equality (2.5) (replacing Z’ by R’). The action of the operators T;l.“’ is written as

ERGODK-

253

PROPERTIES

(3.la)

(3.lb) where the functions h, , h2, and )V are defined in (0.9). Note that it is sufficient to prove (2.5). not for all he Y ‘* but for a dense set of vectors. We choose the set .f of functions whose Fourier transforms are C ’ functions with compact support (C; -functions). For definiteness, assume that E, = E,. The square of the norm l(n’@ ‘i’T;l.‘V!l ‘, 12E .P, is written as

cos(rrc(k))+-

ih,(k) sm(rn,(k)) M.(k)

where i; E C;;’ with supp h c [a, h], (I, h E R’, u < h. Now the exponents under the integral may be separated into the real and imaginary parts and, after squaring, we obtain a sum of several terms of the same type. Every term is treated in the same way and we restrict ourselves to the proof of dkcos(m+(k)-kx)I;(k) Integrating

‘=O.

(3.3)

by parts in dk we arrive at the expression (3.4)

which

is h

j Furthermore,

L’(k) dk

< ”

(I tw’+(k)-x

(3.5)

(3.5) does not exceed r

1 sup kE[‘,.hl Ifo;(fi)--d

(3.6)

254

SHUHOVANDSUHOV

Taking into account that x > tq( t), we get that for large enough f (3.6) is br’x-‘. Hence, for large t, h Ii a

dkcos(to+(k)-kx)h(k)


+x))’

(see also [21, Chap. III, Corollary 3.1 I). By squaring and integrating we obtain that (3.3) is GrAV(t))

-’ x

0.

In the next stage of the proof of Theorem 1 we estimate the expression

ih,(k) cos(rw(k)) +- u,(k) sln(tw(k)) where XE [-[q(t). Lemma 2.1)

[q(t)]

and h E C;.

The following

~~(t,.u)~~r(l+~t~)~“~~+‘),

hk),

(3.7)

bound is fulfilled

~=mmax{~+,~~

).

(see

(3.8)

As before, we need more detailed information about the decrease of cp(t, X) (cf. Lemma 2.2). LEMMA

3.1. For cm)’ t, x E R’ satisjjing the inequality k;171(<,~~n”FL,Prh b-O~‘,(k)fl

M’ithC(E((l(+l)~‘,l),~Lmax(C1+,~l}, Iq(t,x)l
’ Ifl”

thefollowinghoundholds +ltl)

“++

“12P.

(3.9)

For the proof it is sufficient to divide supp fi into the monotonicity intervals of h and apply the second mean-value theorem together with Lemma 2.2. As in Sect. 2.2, Lemma 3.1 implies LEMMA 3.2. For any t ER’ D”, ” c [-q(t), rr](t)] such that

(I) (2)

and

6 E (0, 2/p + 1)

there

exists

a

set

lim,,,S,,,.n,d~I~(~,-u)12=0, (cp(t, .x)1
where r does not depend on t and 6. The proof of this lemma repeats that of Lemma 2.3. Fix m, n = 0, l,... with

255

ERGODIC PROPERTIES

+ n 3 2. Choose 6 > 0 and a triple of monotone functions p(t), y(r), q(t) which tend to co and satisfy the conditions

m

(a)

lim,,

x v(l)

P(t)

f3”-

’ = 0, lim,,

lb)

lim,,

T rl(f)(

1-

&(fYB(t))

(cl

lim,,

, v(t)

G(f)

‘,gT

,*, (P(t)

- 2y(t))

= co,

t” = 0,

vdr)-21’(f))=0.

Such functions and a number 6 exist in view of (0.21). For every ,f> 0 take the set D(‘.sizi figuring in Lemma 3.2. We construct the finite partition Ol;‘, O(,“,... of R’ as follows. We set a:l’=(ryp(t)-j’(t),

yJ(t)+y(t)]

n [ - t,/(r), q(t)] ,\D”, +?‘, op=

.VEZ’,

RI\: u qr1. i, E.zl

The number of points J’ for which d\‘) # Qr is finite and < 2tr7(I) b( 1)~~‘. As above, we number them in an arbitrary order. These will be the elements Oil’, I= I, 2,... . Now set

y -:“, = L,( oy,,

13 0.

The role of the dense set Y-’ will be played by .f. Further arguments do not differ from those in the proof of Theorem 1 for the lattice case. Theorem 1 is proven. 3.2. Now consider the question on convergence of the functionals Kg;. Q), E’ + cl < 2. First of all, assumethat the initial functionals Kg” Q) are U.,-invariant. In this case, as before, Kg. ” = 0, Kg Ii = 0. The functionals Kg” r2), 8, + c2 = 2, arc written in the form (2.24)-(2.27), where the operators M],f:,, i,, i, = 1, 2, commute with I/,, .YE R’. We again use the matrix representation (i.28). The matrix M’“o which corresponds to the state Q, is connected with M(p) by (2.29). The stationarity condition has the form (2.30). PROPOSITION the

matrix

61’QC’

3.1. Let the ,fimction is weakly

converging,

w satisfj as t +

condtion CC, to the

(B) (see Sect. 2.4). Then (2.34).

mutrirv

3.3. The above arguments may be extended to the case where the initial functionals K(;;I,‘,~‘, F, + E?6 2, are “periodic” w.r.t. the group (U, ). This property means that for some s E RI, , K$I.“‘(+

K$-‘(hl,

K$.““(UJl),

h,) = Kg’-‘(

U,7hl, u,,h,),

E’+EZ= 1,

(3.10)

E’+Ez=2.

(3.11)

In this case again K,$, O1= 0, Kp I ) = 0. Th e functionals Kgl”.z), cl + E, = 2, are writ-

256

SHUHOV

AND

SUHOV

ten in the forms (2.24), (2.25). The periodicity is written in terms of M:.Qj?, i,, iz = 1, 2, in the obvious way. Fix the minimal s for which (3.11) holds. The operator matrix M’Qo is again given by (2.29). As before, it is convenient to pass to the spectral representation of { U,). The operators kI!Qi and A4kQ/ in this representation have the form

I@Q/ and #Qj

are given by

where rni:! iz, 1E 2’. i, , i2 = 1, 2, are bounded functions. We shall assume the following condition is fulfilled: 1 p(” < cc’, /t/l

(3.12)

where

This condition means a “rapid’ in the periodic functions

decrease in the contribution

.XYH (Mf~/,U,h,,

lJ,.h,),

Let s,,, u E R’, denote, as before, the shift operators the spectral representation is written as

of “higher”

harmonics

i,. iz= 1, 2. on R’. Then the matrix M’Q’ in

where the matrix fiCQ.‘i, f~ Z’, consists of operators commuting with : fi, 1. As earlier, the matrix fi’Q,O’ corresponds to the matrix MIQ,O’ obtained by averaging the periodic matrix A4 (Q):

For &l(QCJ we obtain the formula (3.13)

ERGODIC

251

PROPERTIES

Further analysis does not differ from the corresponding steps in the investigation of &IcQlr in the lattice case. We can consider the Cases I, II, and II’ which are analogous to the corresponding cases of Section 2.5 (replacing the values I = 0 ,..., s - 1 by I E Z’ ). The differences

(11 +

- (‘1+ (x-1,

etc., should satisfy condition (B). Condition (3.12) is used here for passing to the limit in (3.13). For the sake of brevity we do not formulate here the corresponding assertion.

A’“’

3.4. As in Section 2.6, one can consider an “almost in the spectral representation is written as

periodic”

case where

where p is a. Bore1 measure on RI and fi’“, O’, GE R’, is a family of matrices commuting with the group ( 0,). We shall assume that the integral

and the measure p has a non-zero atom at the origin. By imposing condition (B) upon the differences

o,,( k + a) - o,,(k), it is possible to get that fiIcQr) converge, Proposition 2.3).

E, El = *,

as t + vj, to the limit matrix

3.5. Finally, we discuss very briefly the case Kg’, Q), E, + c? = 2, have different “asymptotics” on + {U,> (see Sect. 2.7). The arguments used in the lattice to the continuous one. As a result, the limiting matrix the details for the sake of brevity.

(2.46) (see

where the initial functionals lx, and - lx: w.r.t. the group case may be extended easily (2.49) arises. We again omit

258

SHUHOVANDSUHOV

4. LINEAR MODELS ON MULTIDIMENSIONAL

CUBIC LATTICE

4.1. In this section we assume that each of the functions 6, c (see (0.6)) is of class CT.’ In what follows, the essential role is played by the behavior of the functions 0 + in neighborhood of the sets P2(~i)=iHt[-n,n)‘:det~

(fl)=O }

(4.1)

IZ We shall call the sets p,(o, for any XE R” the sets

) the manifolds of degeneracy of IX+. We assume that {0: grad w,(0) =x1,

are finite. Furthermore,

E= f,

(4.2)

we assume that for any 0, E fi?(oJ the function 0~

-00’ grad CO,:(~)+ w,(8),

&= +,

(4.3 1

has at 8 = o. a simple sirlgulurity (in Arnold’s terminology (see [23, 281 and the books [25-261) this is a singularity of one of the following types: A, (k 3 1 ), D, (k > 4), Eh, E,, IT,). Let F(Q)i f = sup dg, fi,), (4.4) (l(,E/iJ(“’f )

where E(CJ,e,) is the singularity exponent of the function (4.3) at 8= il,,, (T= a(B,, ok ) is the singularity type at this point (see [24, 26, 301). For simple singularities the exponent I takes the following values (see [24, 26-28, 301):

1 E(0)

1

--2 k+l

1

1

5 12

--~ 2 2(&l)

4 5

7 -5

We impose the condition v>6c(l

-2~)

‘,

(4.5)

where s=max{E(a+),s(0~

)i.

(4.6)

Now we formulate conditions for the initial state Q. In analogy with the Introduction we denote VI, Ic Z’, the subspace of 9 spanned by ek, k E I. We set p:". ')(r3 3) = SUP ~2 ytZ' ‘This integrals

"Y y jcy, r,, ‘/.Z'.,,(5. r+ y,),

condition is introduced to apply some well-known (see below). In some cases it may be weakened.

facts

about

decrease

(4.7) of the oscillating

ERGODIC PROPERTIES

259

where I(q, U) = X;= , qi - U, q, + u is the cube in Z” centered at q = (q, ,..., q) with the edge length 2~ (cf. (0.20)). We assume that for any a>O, lirn

t6”p~.n)(fl

6~:v,

cctl

~6’(1+“:3)/‘)=0.

(4.8)

I + -r

THEOREM 2. Assume that B, C, and Q satisiy the above conditions. Then the assertion of Theorem 1 holds, i.e., convergence qf the state Q, is equivalent to the convergence sf the lower ,functionals K$;, “2), E, + E?d 2.

4.2. The proof of Theorem 2 is along the same line as that of Theorem 1. As the first step of the proof we establish the equality (2.12) where i>

I,=[-l~,E~t]“nZ’,

max lgrado+(B)I. OE[ =.n)’

(4.9)

As a dense set ? c %- we choose the family .a of vectors with C: Fourier transforms. For the definiteness consider the case E, = s2. The norm square /lx ‘z” ‘“T”!,,“‘hll ‘, /zE9, is written as

(4.10) The expression in brackets may be rewritten as a sum of the exponents. After obvious computations the problem is reduced to estimating the sum of the integrals of the form 2 C-n,n) ,,-ir.f~+;~~uy(H)~(,) , (4.11 ) where I? is a P-function. The set Z”\l, is separated from the set of values taken by t grad w+(0), 8 E C-n, rr)‘, and hence, we can use the well-known bounds for

)1[-ddee-

rr.n+;n”~lHIfi((j)

(4.12)

(see, e.g., Corollary 3.1 from [21, Chap. III]). We obtain that the sum (4.11) vanishes as t + co. As the second step of the proof we need to estimate the quantity dt?e rr’o+irh~lH) x cos(tw(B)) + z ( 595/175;2-4

sin(tw(8))

>

h(0)

1

(4.13)

260

SHUHOV

AND

SUHOV

uniformly in XE I,. Here h is the Fourier transform of the vector h E .Y. Again writing cos( tuj 0)) and sin( tw( 0)) as the sum of the exponents we reduce the problem to estimating the integral (4.14) with HE C”. The integral (4.14) is a dconst. t ViZ+‘. (see Proposition 4.2.1 from [28] or Theorem 4 and Remark 2.4 from [27], as well as Chap. II of [2.6]). Now we finish the proof of Theorem 2. Fix nz, II = 0, l,... with LIZ+ n 3 2. Choose a pair of monotone functions /l(t), y(t) which increase to x with t and satisfy the conditions (a)

lim,,,

p(t)t~‘+61”‘=0,

lim,,,

(P(t)-2y(t))=‘x,

(b) (c)

lim,, x P( 1 - (2y(t)/B( 1))‘) = 0, lim,,, (tfi(t)P’)‘p~~“‘(2y(t),/l(t)-2g(t))=0.

Such functions exist in view of (4.8). As in Section 2.3, we introduce partition Ot’, Oi”,... of Z’ by setting

the finite

The number of points .V for which d-i0 # @ is finite and d (2ti$(t) I)‘. We label them in an arbitrary order by naturals; this gives us the elements O’,“, Oy’,... . Next we set Y ‘y’ = I,( OL”,,

k 30,

and choose .a as the dense set YY c ‘I -. The final step of the proof of Theorem 2 (i.e., using Proposition 1.1 formulated in Sect. 1) is an obvious modification of the arguments from Section 2.3. Theorem 2 is proven. 4.3. For even initial states Q conditions be weakened. We assume that

(4.5) and (4.8) in Theorem 2 may

V>41(1 -2&)-l,

(4.15)

where E = max (E(O + ), E(O _ ) } and for any CI> 0, lim

,-l-I

141:p~.

nl([l

4C/~,

~1’

j”:~~

‘“)

=

0.

(4.16)

THEOREM 3. Let he v > 4&( 1 - 2~) ‘. If operators B, C satisf:1. the conditionsfkm Section 4.1 and Q is an even state satis~~~ing (4.16), then again the assertion qf Theorem 1 holds.

ERGODIC

261

PROPERTIES

The proof proceeds in the same way as above. Instead of Proposition here Theorem 2 of [ 161.

1.1 one uses

Remark. The conditions of Theorems 2 and 3 allow us to treat singularities which are not “too strong” among those which may occur. in a generic situation, for a given v. For instance, the singularity of the type A, may appear for v 3 2. However. Theorems 2 and 3 deal with A 3 only, for v > 4 and 1’3 3, respectively. 4.4. Theorems 2 and 3 are from the category of “rough” assertions in which one uses only uniform bounds on matrix elements (F,‘. ‘z)~, h-g Z”, of operators T:I,“~, unlike the “line” assertions in Theorem 1 where. besides uniform bounds, one analyses the behavior of the matrix elements in a neighborhood of a “manifold” of critical directions (caustics). In a multidimensional case a similar analysis may be done by using the methods and results of [27] (see [27. Theorems 3 and 41). However, here one encounters difficulties connected with a complicated structure of the manifolds b(tu, ) (more precisely, of their submanifolds corresponding to singularities of a given co-dimensionality). In this paper we restrict ourselves to the use of estimations of [27] in a neighborhood of the strongest singularity. This trick already allows us to treat stronger singularities for a given 11.For simplicity we consider the case v < 3. Furthermore, assume that the function 6,H+~.++tu*(H).

.ucR’,

OE [-7~. n)‘,

(4.17)

is a minimal versa1 deformation (see [2.5]) of a simple singularity of the type A,. kbvt 1. Assume that Q is an even state and for some n satisfying the bounds O
‘;

(4.18)

the relation (4.16) holds with (-:= 2,. + d v = 2, 3, where 2: = +, d, = A. Notice that the second inequality in (4.18) is equivalent to (4.15) with I: = C, + (I. THEOREM

4.

ditions forrmluted t E RI, holds true.

Let v he d 3 und the oprrutors B, C mui u state Q sutisf~j~ the muin this section. Thtw the assertion c?f‘ Tlworew 1 ,fbr tk stutes Q,,

For I’= 2 the conditions of this section describe a generic situation for Renwk. the functions w, (in class C’ ), see [24, 251. The proof of Theorem 4 is given by a combination of arguments from the proofs of Theorems 1-3. Instead of Lemma 2.2 one usesTheorem 4 from [27]. The details are omitted. 4.3. Finally, we discuss briefly the question on convergence of the lower functional Kg;.‘:‘, e, + E: < 2. We notice here that the results of Sections 2.4 ~~2.6 may be easily exlended to the multidimensional case, where instead of the circumference

262

SHUHOVANDSUHOV

[ -7c, rr) the corresponding torus [ -rc, rc)’ arises. Condition (B) upon MI may be replaced, e.g., by the following condition: (C)

WE C”, the set {& grad w(0) =0} is finite and

In dealing with oscillating integrals one usesestimates of [28, Sect. 1.31. The details are omitted.

5. LINEAR MODELS IN SPACE

As in the preceding Section, we assumethat 6, c E C’ and for any x ER’ the set (4.2) is finite. The manifolds /Iz(o+) are given by (4.1) ([ -71, rc)’ replaced by R”). To begin with, we assumethat /12(w*) = 0, i.e., for any kE R’, det 3 /I

(k) # 0. I?

(5.1)

This case is worthwhile only for continuous models. It is useful in view of its simplicity. Further on we shall formulate a more general result (see Theorem 6). One imposes the following condition on Q: For any cx> 0, lim p h”,“‘(1, cd) = 0. ,+ %

(5.2)

This condition is obtained from (4.8) by setting E= 0. THEOREM 5. Assume that operators B, C, and a state Q satisfy the above conditions. Then for the state Qr the assertion of Theorem 1 holds.

Remark.

In the one-dimensional case condition (5.1) takes the form co’; #O

(5.3)

and instead of (5.2) one can use the relation lim pg. “j(s) = 0, J- K

(5.4)

where p-(m,n) is introduced in (0.20). The proof of Theorem 5 is simple. It is based on a combination of results used in the proof of Theorem 1 and 2. In the first stage we introduce a monotone function v increasing to co and prove the equality (2.1), where

I,= C-a(t)t, q(t)tl”.

ERGODIC

263

PROPERTIES

For the dense set 3” we choose, as in Section 3, the family .a of functions with Cc Fourier transforms. The norm /Ire‘RV”“Trl,EZhlj, 12~9, is written as in (3.2) (with the obvious changes). As in Section 4 (see(h.lO)-(4.12)) the problem is reduced to estimating the integrals of the form dk e lkY+,l
2,

R’

(5.5)

where fi is a C;--function. For sufficiently large I the set R“\I, is separated from the set of values of the functions t grad W* on supp I? (since q(t) + cx and supp fi is a compact set independent of t). By using the bounds of Corollary 3.1, Chap. 111 of [21], one concludes that the integral (5.5) vanishes as t + KCI. In the second stage of the proof we need to estimate the integral (3.7) (R’ is replaced by R’) uniformly in x E I,. As in Section 4 (see (4.13))(4.14)), we reduce the problem to the integral

where H is a C” -function.

It is convenient

to introduce

the constants (5.7)

Notice that for .KE I,\Z(O, r$ t) the integral (5.6) is estimated in the same. way as for xeR”\Z, (see above) and for all N= 1, 2,... is 2. Choose a triple 8, y, q of monotone functions increasing to CT,with t such that (a)

hi, _ .r q(f)P(t)t

lb)

h,+,,

(c)

~‘=O,lim,-,,,

(fl(t)-2j(t))=O,

$t)‘(l - (2y(t)/b(t))‘)=O, ‘)“p~,“‘(2y(t),P(t)-21’(t))=o. linl,+ Tm(iq(t),R(t)

We introduce We set

the finite partition

Ol;‘, Oi”,... following

the same principle

Again the number of points JJ for which aif) # 0 is finite and < (2q( t) P(r) label them by 1, 2 ,... and get the elements O’,‘), Oy) ,... . Finally, we set

as before:

’ )“. We

264

SHUHoV

AND

SUHoV

and choose the set .a as the dense set $? c ^I‘ figuring in condition (A). The end of the proof of Theorem 5 is an obvious modification of arguments of Section 2.3. Theorem 5 is proven. 5.2. As in Section 4.1, we assume now that for any 8,, E fi2( w + ) the function (4.3) has a simple singularity at 0 = do. Let us assume, as well, that the condition (4.5) is fulfilled. As to the initial state Q, it is supposed to satisfy (4.8). THEOREM 6. Under the conditions qf’this section the assertion of Theorem I holds for the state Qt.

The proof of Theorem 6 coincides in the first stage with that of Theorem 5. The second stage is a combination of arguments used in the proofs of Theorems 4 and 5. Concluding this section we notice that in the same way as in the lattice case, one can immediately extend the arguments of Sections 3.2-3.4 on the multidimensional situation (condition (B) must be, of course, replaced by (C) of Sect. 4.4.)

6. FINAL REMARKS The paper contains some results which connect non-degeneracy properties of the infinitesimal operator matrix D which generates the group of the Bogoliubov trans[.c7J) of the CAR C*-algebra ‘II and properties decay of the spatial formations correlations of an initial state Q with the convergence of the time-evolved states STQ, as t + fm, to a stationary quasi-free state P. We formulate here some open questions related of these results (several questions have been mentioned already above ). 6.1. It is interesting to weaken the non-degeneracy conditions, especially in the multidimensional case. At present, this is connected with the modern situation in the problem of estimating oscillatory integrals dependent on singularity types of the integrand exponent. 6.2. Another possible generalization is connected with some stronger conditions for the decay of spatial correlations of the initial state Q. For example. one can require so-called cluster bounds for “cumulants” of the initial state (see, e.g., [31]). Apparently, such an approach will help to weaken the conditions on the operator matrix D and thereby to answer, at least partially, the question 6.1. 6.3. Similar results may be established subject of forthcoming papers. 6.4. Erratum. In our article [32], are announced there are several incorrect

for boson systems as well. This is a

where some of the results of this paper statements. Theorem 3 on page 89 of

ERGoDlC PROPERTIES

265

[32] is proven only for p = 0 (this case occurs for continuous systems (I’” = L,(R))). In Theorem 4 on page 90 one needs to assumethat the F’ have simple singularities (thereby the parabolic singularities P,, X9, J,,, are excluded). In addition, condition (17) on page 89 of [32] should be replaced by condition (4.8) of this paper. 6.5. Addendum. The referencesin [ 10, 12, 16, 17, 321 should be completed by the paper [9] which, unfortunately, were not brought to our attention.

ACKNOWLEDGMENTS One of the authors (Yu. S.) is deeply grateful to Professors M. Fannes and A. Verbeure hospitality at the Instituut voor Theoretische Fysica. Katholieke Universiteit Leuven. express their deep gratitude to the referee for the comments and remarks. We are particularly pointing out the significance of [9].

for the warm The authors grateful for

No/r udder1 111proof: After sending this paper in print it was dlscovered by the authors that the main results (Theorems I-6) may be essentially strengthened without making the proof more complicated (on the contrary, with a simpler proof). Namely, (i) in the one-dimensional case (iatticc as well as continuous) one can assume that the mixing cocficient p simply vanishes at r (the value ((!~ ) v f12(tu i ). For such a vector /I the integrals (3.7) and (4.13) may be estimated uniformly by const. I “. This gives some answers to Remarks 6.1 and 6.2. Furthermore, the statements mentioned in Remark 6.6 become a fortiori correct.

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2. 3. 4. 5. 6. 7. 8. 9.

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XY-Model: Ergodic Properties and 10. A. G. SHUHOV ANV YU. M. SUHOV. “One-Dimensional Hydrodynamic Limit,” J. Stat. Phys. 45 (1986), 669-694. 11. A. G. SHUHOV AND Yu. M. SUHOV, “Convergence to a Stationary State for the One-Dimensional Quantum Model of Hard Rods,” Preprint, IPT-85-16, Universitk Catholique de Louvain. 12. Yu. M. SUHOV. Convergence to equilibrium state for the one-dimensional quantum systems of hard rods, Ix. Akad. Nauk SSSR Math., 46, No. 6 ( 1982), 1274-1315. 13. C. BOLDRIGHINI, A. PELLEGRINOTTI, AND L. TRIOLO. Convergence to stationary states for infinite harmonic systems. J. Slat. Phys. 30, No. 1 (1983), 123-155. 14. V. A. MALYSHEV AND Yu. A. TERLEZKII. A limit theorem for non-commutative fields, Vesm. Mask. (iniu. Ser. Maf. Mekh. No. 3 (1978), 47-51. [Russian] 15. E. V. GUSEV, Limit states of the planar Heisenberg dynamics with a transverse magnetic field, Usp. Math. Nauk. 1981. v. 36, No. 5. p.177-178. [Russian] 16. Yu. M. SUHOV. Linear boson models of time evolution in quantum statistical mechanics, Izv. AN, SSSR, ser. Math., 1984. V. 48, No. 1. p. 155-191. [Russian] 17. Yu. M. SUHOV, Convergence to equilibrium for the free Fermi-gas, Theor. Math. fhys. 55. No, 2 (1983), 282-290. [Russian] 18. D. D. BOTVICH AND V. A. MALYSHEV, Unitary equivalence of temperature dynamics for ideal and locally perturbed Fermi-gas, Commun. Math. Phys. 91, No. 4 ( 1983), 301-312. 19. L. HUA, “Method of Trigonometric Sums and its Applications in the Number Theory,” Mir, Moscow. 1964. [Russian] 20. T. KAWATA. “Fourier Analysis in Probability Theory,” Academic Press, New York, 1972. 21. M. V. FEUORIUK. “Method of the Saddle Point,” Nauka. Moscow, 1977. [Russian] 22. 1. M. VINOGRAVOV, “Method of Trigonometric Sums in the Number Theory,” Nauka, Moscow. 1971. [Russian] 23. L. HUA. On the number of solutions of Tarry’s problems. .4rla Sci. Sin. 1 (1953) l-76. 24. V. 1. ARNOLD, Remarks on the stationary phase method and Coxeter numbers, C/.yp. Mat. Nauk 28, No. 5 (1973). 1744. [Russian] 25. V. 1. ARNOLI), A. N. VARCHENKO. AND S. M. GUSSEIN-ZADEH, “Singularities of Differentiable Mapings,” Vol. 1, Nauka. Moscow, 1982. [Russian] 26. V. 1. ARNOLV. A. N. VARTHENKO, AND S. M. GUSSEIN-ZADEH. “Singularities of Differentiable Mappings,” Vol. 2 Nauka. Moscow, 1984. [Russian] 27. Y. COLIN DE VERDIERE. Nombre de points entiers dans une famille homothktique de domaines de R”, .4nn. Sci. EC. Norm Sup., S&r. 4 10 (1977). 559-576. 28. J. T. DUISTERMAAT. Oscillatory integrals, Lagrange immersions and unfolding of singularities, Comm. Pure Appi. Math. 21. No. 2 ( 1974), 207-28 I 29. V. I. AKNOLV, Normal forms of functions closely to degenerated critical points, the Weyl groups A,, D,. E,, and Lagrange singularities, Funk;. Anal. Pril. 6, No. 4 (1972), 3-25. [Russian] 30. V. I. ARNOLD, Integrals of rapidly oscillating Functions and Singularities of Projections of Lagrange Manifolds (Russian). Funk:. Anal. Pril. 6. No. 3 (1972). 61-62. 31. V. A. MALYSHEV, Cluster expansions in lattice models of statistical physics and quantum field theory, I/sp. Math. Nauk 35, No. 2 (1980). 3-53. [Russian] 32. A. G. SHUHOV AND Yu. M. SUHOV. Linear and related models of time evolution in quantum statistical mechanics, in “Statistical Physics and Dynamical Systems. Rigorous Results,” pp. 83-104, Birkhaiiser, Boston/Basel/Stuttgart, 1985.