LIMITING DENSITY OF STATES FOR QUANTUM ROTATOR MODEL I. A. Koshovets The concept of the limiting density of states is introduced for operators associated with a dynamical system, and some smoothness properties of the density are studied. The limiting density of states for the evolution operator of a quantum rotator is found explicitly. i.

Introduction

We consider a model of a quantum rotator with evolution described by the SchrSdinger equation +~

i~

a o~ o~ "~-=~-~ O~2+i~7--+•176

2 6(t--n)%

where ~ is a function of the space L 2 on the circle S of unit circumference with Lebesgue measure d% This model was introduced in [i]. We define the evolution operator U by

It has the form

(1)

U=QV, where Q = exp

2r~ OqF"+

~

,

V=exp(-i•

The operator Q is diagonal in the Fourier representation, and Qek=exp (2~i~) 9e~, where

(2 )

~h----~k2+~k,eh=exp(2~iq)).

The operator U belongs to the family of operators introduced by Sinai [2]. It can be described as follows. We consider the space X with o algebra ~ and normalized measure on it. Let T be an automorphism of the space X, g be a measurable (with respect to the o algebra ~ ) function on X, and h be a real-valued function on S. We shall say that U belongs to the family of operators associated with the dynamical system (X, v, T) and the functions g and h if U can be represented in the form of the product (i), where V=exp(ixh), is a real parameter, and Q is an operator diagonal in the Fourier representation and for which (2) holds with

~h=g(Tkx), xEX.

(3)

We shall denote the family by ~q(X, v, T, g, h), or, for brevity, ~({~h}, h). For the operators of this family, we can introduce the concept of the limiting density of states. We shall follow the standard construction (see, for example, [3,4]), in accordance with which the limiting density of states of an operator is defined as the weak limit of the point measures concentrated on the spectra of its finite-dimensional approximations. In Sec. 2, we shall give a precise definition of the limiting density of states. We shall also obtain an explicit expression for the Fourier coefficients of the limiting density of states (Theorem i). This will enable us to investigate the density for some types of operators in the family 9)~({~h},h). In Sec. 3, we consider the case of a sequence ~k corresponding to the evolution operator of a quantum rotator: Sk = ak2 + ~k. We show that in this case the limiting density of states is a uniform density on the interval [0, i] if at least one of the numbers a or ~ is irrational. In the same section, we investigate the case when gk form a sequence of independent equally distributed random variables with distribution given by an analytic density. In this case, the problem of estimating the Fourier coefficients of Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 85, No. 2, pp. 193-204, November, 1990. Original article submitted December 30, 1989.

1146

0040-5779/90/8502-1146512.50 9 1991 Plenum Publishing Corporation

the limiting density of states reduces to the problem of estimating the mean values of a certain functional on the trajectories of a symmetric random walk, and this makes it possible to prove the existence of an analytic density also for the limiting density of states of the operator U for sufficiently small values of the parameter • (Theorem 2). 2.

Definition of the Limiting Density of States

Let U~(X, v, T, g, h). It is natural to define the finite-dimensional approximation U N of the operator U as follows. Consider the space L~ of sequences f(j), j = --N . . . . . N, with scalar product N

(/"f~)= 2N+1,=_ f~(J)f~(])" It

is readily

verified

f o r m s an o r t h o n o r m a l as follows:

that basis

the set i n L~.

of vectors

{gk}~=-s ..... s,

L e t VN be an o p e r a t o r

where ~(])=exp

of multiplication

2hi 2 N ~ -

'

on L~ t h a t

acts

exp i• and QN be an o p e r a t o r

diagonal

in the basis

{~h}~=-~ . . . . . x:

We d e f i n e UN a s t h e p r o d u c t QNVN. By c o n s t r u c t i o n , UN i s a u n i t a r y v a l u e s h a v e t h e f o r m e x p ( 2 ~ i ~ ) , ] = N , . . . , N , ~G[0, t]. L e t

operator.

Its

eigen-

1

We d e f i n e t h e l i m i t i n g d e n s i t y o f s t a t e s ~ o f t h e o p e r a t o r U a s t h e weak l i m i t ( i f i t e x i s t s ) a s N ~ ~ o f t h e s e q u e n c e o f m e a s u r e s ~N (~ may d e p e n d on x , s i n c e U = U ( x ) ) . THEOREM 1. L e t U ~ ( X , % T,g, h), h~C~(S). Then f o r a l m o s t a l l ( i n t h e s e n s e o f t h e m e a s u r e ~) p o i n t s x~X t h e l i m i t i n g d e n s i t y o f s t a t e s ~ o f t h e o p e r a t o r U e x i s t s . If the a u t o m o r p h i s m T i s e r g o d i c , t h e n t h e m e a s u r e ~ d o e s n o t d e p e n d on x and f o r a l l s~Z+ i

j exp (2~is~) d~ (~) =< (U~eo, eo) >. 0

(Here, < > denotes averaging over the space X.) Proof.

~y virtue of the definition of the measure ~N, 1

i ~ e x p (2nisT) drt~ (~) = 2 - ~ t r

U~ ~.

o

Thus, to prove the existence of the limiting density of states it is sufficient to show that i for almost all x and for any 8~Z+ there exists the limit l i r a - - t r U ~ ~ L e t {~}k=--N . . . . . ~ b e t h e b a s i s i n LN d e s c r i b e d a b o v e . o p e r a t o r VN h a s t h e f o r m (v~(k'--k))k,~=_~v ..... ~, w h e r e

In this

basis,

the matrix

of the

2W

v~-(k)= 2 N + I ~=0

2N+t

\2N+t /

I t i s o b v i o u s t h a t V N ( . ) i s a p e r i o d i c f u n c t i o n w i t h p e r i o d 2N + 1. In addition, for any k we have VN(k) ~ v(k) as N § ~, where v(k'-k)=(Ve~,, e~), e~=exp(2nikcp). Using an Abel transformation, we can readily show that Iv~(k) I ~ -k2+l C particular, IvN(k) l _-< C for all k and N. operator U N

for all k, N such that k _-
In

Further, by virtue of the definition of the

1147

(Uu~G, G) ~=

~

/ 4

vN 01

-

k) ~,~ (i~

-

-

i : ) . . , v~ (k

-

-

-

i~_:) •

exp (2~i (g (T~x) + . . . -4- g (T:~-:x))). extension of the function exp(2w• We d e n o t e b y d N ( k ) t h e p e r i o d i c k ~ [ - N , N] t o ~. L e t i ~ = k + ] ~ , . . . , i ~ _ ~ = k + ] ~ + . . . + L - , Then the expression rewritten in the form

( U ~ G , e~) =

~,

for

from the interval ( U ~ G , g~) c a n be

vN 0 1 ) . . .

--k--N~_jt~--k+N ~F< --(jr+..,-~js_2)--N~-~js_l~--(j,+...-~js_2)+N

~ 0,-1) ~,N (-- (]1 + . - . + L-:)) d:~ (k)'... dN (k + (A + - . . + ]8-:))Since the f u n c t i o n v~(j~)...v,~(-(j~+...+j~-~))d~(k)...d~(k+(jt+ .+7~-~)) is p e r i o d i c w i t h p e r i o d 2N + 1 in all the v a r i a b l e s ]~,, s'=i .... ,s--i, we can take as r a n g e of v a r i a t i o n of any v a r i a b l e ]~, in the s u m m a t i o n the i n t e r v a l [--N, N]. Thus

( u ~ G , G) ~

Y~ vt; (Jl). 9 9 vtr (-- (j~ + . . . + j~_:)) dN ( k ) . . . dN (~ + 01 + . . . + J~-:))Ij,I<~N lJs-tI~N

We i n t r o d u c e

the n o t a t i o n

2

/~"~" (x) =

~ ( / 3 . . - v~ (-- 01 + . . . + ]~-:)) &v ( a ) . . . dN (k + (Jl + . . . . § ]~-1)),

lid--
.

~

}Js-l.l~L I~ 'w(x)-~-

)'j Y(jl).-- y(-(jl Ij,i~
~- " ' " ~- is-t)) X

IJs-ll~ L exp (2ai (g (T~x) § . .. + g (T ~+(i'§

1) x))),

v ( ] 1 ) ' " v ( - - (il -6 . . . + L-l)) exp (2at (g (T~x) + . . .

.f~'= (x) ---- Z

_ g (T ~(i~§

31,---, Js-1

In this

notation,

We f i x

trU~ s=

~ > 0.

Z

N,N

/~ Ikl~
(x).

F o r L _-< N

l/#,N (x)-- ]~'N(x) l----I ~ ,

,N ( ] i ) . . . d~ (k + (]i + . . . + ]~-~)) i < C% -I,

N~/'~>/L N>~ljs_1>~L where

1 ]2+i ,

rL=

rL-~0 as L ~ ~.

We c h o o s e L 0 so that

C~r~-I Lo


l]~~

"N (x) I <

IZ~>L

e/4 for all x, k and all N => L 0. Let

N>~sL~, IkI<~N--sLo, I]~I<~no

Similarly,

]/~,=

for Z = 1 . . . . .

(x)-/~ ,~

s -

i,

(x)[
for all x and k.

Then for all x

]v~(L) . . . v ~ C - ( L + . .. + L - J ) G ( k ) . . . d ~ C k + C L + . . . Jr],_,))-. - - v ( L ) ... v ( - - ( j ~ + . . . +L_,) ) e x p ( 2 ~ i ( g ( T k x ) + . . . w h e r e e N d e p e n d s o n l y on N and eN ~ 0 as N § ~. that [k] < N -- sL 0

+ g ( T ~§

Therefore,

-+J'-')x)))[
for all N > sL 0 and all k such

[ ]~o,N (X ) --/~[. . . (x) . . l <. L o 1eN u n i f o r m l y w i t h r e s p e c t to x. W e c h o o s e N O such that N O > sL o and for all N > N O

1148

I s
all

such t h a t

" -

(z) -s<~': (~) I'< ~4

and a l l

X.

.e

estimate

the s

difference:

t

2N+ i

ik[

1

2N+ l

Z

2N+l"

L~

--

Ilk

Lo,=:

(x)-/~.

(x).

I~

s + 2sLo ~-i ~ 2--2--~-~20r..

N--sLo<~]hI'~N

2sLo

We choose N z such that N l > N O and

2N~+-----~20r~-'< s-4 " Then for all N > N z and all x

2--k-iq ~

,hi<.-

i

i

w

' :~<.,

~0,.

i

/:0.:

A"

(x)

2 N + I Ikl<~-

(~)l

i,~ ~ (x)-- 2N+t

1

ThHs

i

2N+] as N + ~ uniformly Further,

the

with

respect

function

-trU~- ~ ( x ) -

-

9ff+i,

(x)

-~0

t o x.

] o '~ c a n b e r e p r e s e n t e d

/0= ' ~ ( x ) =

/~'

E

in the

form

v(h)...v(-(h+...+.L-J)exp(2~i(g(x)+...

Jf,...,.is-I

g (Ts'+'+j'-'x) ) ) =

E

v ( i J . . . v ( - % _ J exp (2hi (g (x) + ... + g (T'~-'x)) ) = ( U ~(x) eo, eo).

Thus, for all x [f~,~ (x)i~i , and therefore ]o ,~ 6L~(X,v). In addition /~'~(x)=/0~(T~x). the Birkhoff-Khinchin ergodic theorem, there exists for almost all x the limit

lim-:w~ 2 N~ +[t

L ]hl~N

By

/o~'~ (T%).

If the automorphism T is ergodic, this limit does not depend on x and is

/:o x

(W(X)eo, eo d (x) X

The theorem is proved. Note that in the definition of the limiting density of states of the operator U we can consider as finite-dimensional approximations the operators UN, which act on L2(S). Following basically the scheme of proof of the spectral theorem [5], we can show that there exists a bounded self-adjoint operator H such that a)

exp(iH) = U,

b) the operation of shift along the principal diagonal conserves the finite-dimensional distributions of the elements of the matrix of the operator H in the Fourier basis {e~}~6g. Then we define U N as exp(iPNHPN), where PN is the projector onto the linear hull of the vectors ek, k = --N, ..., N. One can show that the limiting density of states constructed by means of such finite-dimensional approximations is exactly equal to the limiting density of states introduced at the beginning of this section. 3.

Applications

of Theorem 1

We now consider some examples. i. Let U6~({~k}, h), where~h==k2+~k, h is an arbitrary function of the class C2(S). The sequence ~k can be constructed as a sequence of the type (3) if we consider

1149

X = T o r ~,

T(x,, x : ) =

(x~+2a(mod 1),

x~+x~(mod t ) ) ,

g(x,, xz)=x=.

Then ~k = g ( T k ( a + ~ ' 0 ) ) . Simple calculations

show t h a t

for any set
(ii,...,i,-l) +~_,) ) >=0.

I f a i s an i r r a t i o n a l number, the automorphism T defined above is strictly M o d i f y i n g somewhat t h e p r o o f o f T h e o r e m 1 f o r t h i s c a s e , we c a n show t h a t l i m i t i n g d e n s i t y o f s t a t e s ~ o f t h e o p e r a t o r U e x i s t s and f o r a n y i n t e g e r

~exp(2ais~)dp(~)=O.

T h i s means t h a t

for almost

all

~ the density

~ is

ergodic [6]. for all ~ the s we h a v e

the Lebesgue measure

o

on the interval

[0, 1].

We can similarly investigate the case when a is a rational number, irrational. It is necessary to consider

X:zgXS ~, T(n,x)=(n+t(modq),x+~(modi)), Then ~k = g ( T k ( 0 , 0 ) ) , and i n t h i s e a s e t h e i s a l s o t h e L e b e s g u e m e a s u r e on [ 0 , 1 ] .

limiting

~ = p/q, and ~ is

g(n,x)=~n~+x.

density

of states

of the operator

U

Finally, s u p p o s e b o t h n u m b e r s ~ and ~ a r e r a t i o n a l . It is readily seen that in this case the limiting density of states of the operator U exists but is not the Lebesgue measure on [0, i] (at least, for small values of the parameter ~). 2. We now consider the case U~({%h}, h), where gk form a sequence of independent equally distributed random variables with density P0. Without loss of generality, it can be assumed that ~6[0, i] and, thus, p0(~) = 0 for T~[0, I]. Suppose that for TE[0,1] p0(T)= I

p0(exp(2~iT)), where the function ~0(z) is analytic in a certain ring: --~Iz[~Ro, R~

fl0>l.

This means that the Fourier coefficients of the function P0 decrease at exponential rate, i.e., C [ [< - - , m~O, (4) ~0 m for some C > 0. THEOREM 2. Under the assumptions made above, there exists Xcr > 0 such that with probability i for all x in the interval (--Ucr, • the limiting density of states of the operator U is given by the density p~ At the same time p~(x)=p(exp(2~i~), ~), where the i function p(z, x) is analytic with respect to the variables z and u in the region [ul
is a piecewise smooth decreasing function such that R(O) = R 0, R(•

<

= i.

In R~ For C > l, the value of Zcr can be estimated as follows*:

Zcr~--k,

=

where k depends

C

only on the function h.

In addition if the Fourier coefficients of the function h decrease ]n R0

at exponential rate, then we also have the estimate Mcr ~

i+in~k',

where k' depends only

on the function h. Proof. We denote b~(z)=<(U~eo, eo)>=<((Qexp(i• e0)>, where H is the operator of multiplication by the function h. It is obvious that for all s6Z+ b~(• is an analytic function of the argument ~ in the complete complex plane. From this fact there follows a simple but important lemma, which is given below without proof. LEMMA I.

Let r~ and r 2 be non-negative numbers such that

x=O

m!

*From this estimate it does not follow that ncr = ~ when R0 = ~, since the constant C in (4) actually depends on R 0.

1150

Then the series Eb,(u)z'

converges in the region

]z[<~r~,]•

to a function that is

s~0

analytic

in t h i s

domain w i t h r e s p e c t

t o z and • b~]~=o. We write

We estimate the value of

(5) l~Jl,...,Jm~s m

w h e r e k~=k~(j~,...,jm)

and E

k~----s. F u r t h e r

l=0

<(Q~HQ~,... HQ~eo, eo) > -

E

hoi,h~,~. . . . h~m_~o (exp(2ni(ko~o+... + k~-,~m-~+km~o) ) >,

zh..-~?m-t

where

l

h~,5,= ~ h (~) exp (--2~i (],--j~) ~) dq~=a (j,-j~). o

Since h is a real-valued function, it follows that a(-n) = a--(-n-~. We denote h(z)=:

I a (])] z i. We consider a homogeneous symmetric random walk o(t), t~Z+, ~(0)=0, with probabilities of the steps

:~{~o(t+ ~ ) = , ~ + , ~ [ o ( t ) = n ~ } =

la(,*)l

la(])l

=

I.(n)(

~(t)

In what follows, we shall denote by 9~ the probability associated with this random walk. It is readily seen that

<(Q~~

r~'... HQr%eo, eo)> j < (h(l)) "*

Z

~ (0, i I . . . . . i.,-1, O) .~ (0, i 1. . . . . 6~-i, 0),

(6)

(0, i~,..., ;~m-l'O)

where 9~(0, h,...,im-~,0) is the probability of the trajectory (0, i~,...~i~_~,0), ~(0, i~....,i~)= I(exp(2~g(k0~0+...+k~))>l , and the summation is over all paths that end at the origin. LEMMA 2.

Let the function ~(x) be analytic in the ring

t ~7-<[zlI.

Then for

~r

k~=s

any set k0, .... km such that

and any real number g ~ l =

~-,I

n

,i

we have the

estimate .....

c~'

o)

.....

(0, i~..... irn_l , O)

o) <

\ h(~_) /

where ~ = in R0/in C~ and M is a coefficient that depends only on y. Proof.

It follows from (4) that for C >_- 1

~ ( 0 , i~....

( C~(Od ... ,~-,~)

i~_,, O) <~ min t,

where p(0, i,.....i~_~) is the number of different indices in the set (0, i1,...,i~_~).Therefore .E

~ (0, i 1. . . . .

~,n-1,

O) ~ (0, ~1. . . . , tin-l, O) <

(0, i~..., ira_ I , o)

~{maxj~<,,_11~176

+~

-j~<.~-lmaxI o, (j) I.~. L o~7,

(7)

1151

where [ ] denotes the integral part of the enclosed number. walk,

By the symmetry of the random

{max I ~ (i) 1i> ~', ~o(m) = O}~< 29~{o)(m) = 2k}. y~m--

1

This last probability can be represented

in the form

i \~'~/

~{~

z~r

-dz,

Sy

ra

,]

,u,

that

'

zESy

we obtain

DI I %1 y2};. {m~x I(o (S) l ~ k, ,,, (m) = O} < ! J-<.,-~ x ~(t) 1

(8)

Then

(vC)2~. For any g~I we have yC ~ I.

Therefore

[a~]

2C

(9)

~p ,

(yC)2~--~ 2C

(v~CD ~ y~C~ - - t _

, _

(10)

<~ (yC)c~. M y

2

where M I depends only on y.

Further,

by virtue of (8) we have

~/ (~(~) / ~ [~ 4 < \ -(~(~) F g~'M v

~{max{(o(]){~E~-s],.~<~.n_l o ) ( m ) = O , < 2 \ ~ - - - ~ - / where M 2 depends only on y. (9), and (i0).

The lemma follows from the last inequality and also from (7),

The lemma together with (5) and (6) lead to the estimate

bsl•

\ on/

"

for

I

a l l yeI.

Let r I and r 2 be two non-negative numbers. We verify for them the fulfillment of the condition of Lemma i. By virtue of the last estimate, we have

r2~~.
bs I~=o ~ i'D,>

7)%, s~O

"--0 oo

oo

M Z e x p (sh (g) q) gC*~r2~~- M Z e x p s (h (g) r 1 + a In g + In r2). S~O

S~O

To prove the convergence of the last series,

it is sufficient to establish the existence

of a yeI such t h a t h(y)r,+a In y+ln r2
~'={(r~, ra): r~>/O, r2>O, Xy6I, h(y)r~+~Iny+lnr2
If

(Q, r2)e.g-, then by

virtue of Lemma I the series 2 b ~ ( x ) z ~ converges to a function that is analytic with respect s=O

to z and t

• in the region Izl~
Now suppose

(r~, r2)6~o={(r~, ~:2)EJ', r2>/t}.

Then in the

region --~
*Of course,

1152

~ depends on the function h.

respect to z and •

(11) s=O

s=!

By Theorem i, for any • rl] the limiting density of states of the operator U has with probability 1 the density 9~, which is given by the formula p~(T)=p(exp(2ai~),z),

~6[0, i].

(12)

Using the uniformly continuous dependence of U on the parameter • we can readily show that with probability i formula (12) gives the limiting density of states of the operator U at once for all values of the parameter • in the interval [--rl, rl]. We now turn to the description of the set ~0. It is easy to see that ~ 0 is part of the plane bounded by the two lines r I = 0, r 2 = i and the graph of the function F defined by

F (rl) = sup {r~ > O : :~y E I, h (y) r 1 + a ln y + lnr2~0}----sup {r 2 ~ 0 :,~IgE I, r 2 ~ exp (-- h(y) r 1 -- ~ In g)} sup exp (-- h(g) r 1 -- a In g). yEf

It is obvious that F is a decreasing, c a n be d e f i n e d a s f o l l o w s :

piecewise

smooth function.

The i n v e r s e

function

F -1 (r~) ~ sup {r 1 ~ 0 : ~ y E I, h(y) rl § a In y + In r 2 ~ 0 } ~ sup{r~>0:5[g6I,

rl<

- ~ l n g - -}l n r ~=

~-sup

h(u)

--alny--lnr~

~ex

h(u)

U s i n g t h e e x p r e s s i o n s o b t a i n e d f o r F and F - z , we c a n r e a d i l y c o n s t r u c t t h e s e t ~-0. I t i s shown i n F i g . 1 ( t h e b r o k e n c u r v e i s t h e g r a p h o f t h e f u n c t i o n F, and t h e s e t 9~0 i s shown by t h e v e r t i c a l hatching). We e s t i m a t e

the values

of F(0)

and r c r :

F(0) = sup exp ( - a In y) = min (R0l~ R'/ln /~ R0), yGI

and t h e r e f o r e

1 < F(0)

_-< R 0.

Further,

d(C)=--sup I n C uex The f u n c t i o n

d(C) decreases

k0/in C, where

leo----- max

rcr

= F-l(0)

lz(g)

monotonically,

'

= in R0"d(C),

I=

,1

d(0+) = 1/fi(1),

where

~Q \ C ' and f o r C > R' we h a v e d ( C ) =

--Iny . Thus, we can choose a value of k '

such that for all

R" k' Cd(C)>~ l+ln------C T h u s ,

in No r c r ~ i+ln------~k', w h e r e k '

d e p e n d s o n l y on t h e f u n c t i o n

h.

We now c o n s i d e r t h e c a s e when t h e f u n c t i o n ~ i s n o t a n a l y t i c in the neighborhood of the unit circle. To i n v e s t i g a t e t h i s c a s e , we m u s t r e p e a t a l l t h e a r g u m e n t s made e a r l i e r , e x c e p t t h a t i n p l a c e o f t h e e s t i m a t e o f Lemma 2 we u s e t h e f o l l o w i n g o b v i o u s e s t i m a t e (we a s s u m e t h a t C > 1 ) :

r(0)

rer Fig.

1

1153

2

Cm

~ ( 0 , i~. . . . . ,i~_~, 0 ) ~ (0, i~ . . . . .

i~_,, O) ~< Ro'

'

( 0,1I,..-,/m-i,0)

Then we arrive at the following result.

Let

F'(r,)=Roexp(-~h(t)Cr~),

9~'o'={(r~, ri): r,>/O, i<~ri
If (rl,ri)E~-~', then the function p(z, u) defined by Eq. (Ii) is analytic with respect to z i and x in the region --<~Izl<~ri, l• and with probability i formula (12) determines for r~

all •

r,] the limiting density of states of the operator U.

the horizontai Further, analytic in a It is obvious completes the

hatching

in F i g .

1.

The v a l u e of r c r

is

The set ~0' is shown by

lnR o

i

C

h(1)

we set R------max(F',F), • = max(rcr, rcr) in the case when the function ~ is certain neighborhood of the unit circle; otherwise we set R~f', Xcr = rcr. that R is a piecewise smooth decreasing function, R(0)=R0, R(Ucr) = I. This proof of the theorem.

In conclusion, we give an example of the density P0 for which the limiting density of states can be found explicitly. We consider p0(T)= 1_2kcos(2a~)+% 2, ~[0,1],

%
In this case = ~k and the coefficients <(USeo, eo)> can be calculated exactly:

<(U~eo, eo) > = ~? (V~eo, eo) = %~j exp (i•

((p)) dcp.

o

It follows from Theorem 1 that for all real • the limiting density of states of the operator U is given with probability i by the density

p~(~)=p(exp(2~i~), • where I

~(z,• It is obvious that the function

i -- -~ exp (i• p(z, •

+

(~))

i exp ~_~• ~ ~

d~-1.

i is analytic with respect to z in the ring % < I z l < - -

for all real values of the parameter x. Although in the considered case Theorem 2 establishes the existence of a limiting density of states only for values of ~ in the interval (-• • t does guarantee the possibility of analytic continuation of the function p(z, • f r o m this interval to the disk l•215 I thank Ya. G. Sinai for suggesting the problem and for helpful discussions. LITERATURE CITED i. 2. 3. 4. 5. 6.

1154

G. Casati, B. V. Chirikov, J. Ford, and F. M. !zraelev, Lecture Notes in Phys., 92, 334 (1977). Ya. G. Sinai, Physica (Utrecht) A, 164, No. I (1990). A . L . Figotin and L. A. Pastur, Selecta Math. Soc., ~, 69 (1983/84). M . M . Benderskii and L. A. Pastur, Mat. Sb., 82, 245 (1970). M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. i, Academic Press, New York (1972). I . P . Kornfel'd, Ya. G. Sinai, and S. V. Fomin, Ergodic Theory [in Russian], Nauka, Moscow (1980).